-CategoriesfortheWorkingMathematician ... · ∞-CategoriesfortheWorkingMathematician EmilyRiehlandDominicVerity DepartmentofMathematics,JohnsHopkinsUniversity,Baltimore,MD21218,USA

infin-Categories for the Working Mathematician

Emily Riehl and Dominic Verity

Department of Mathematics Johns Hopkins University Baltimore MD 21218 USAEmail address eriehlmathjhuedu

Centre of Australian Category Theory Macquarie University NSW 2109 AustraliaEmail address dominicveritymqeduau

This text is a rapidly-evolving work in progress mdash use at your own risk The most recent version canalways be found here

wwwmathjhuedusimeriehlICWMpdfWe would be delighted to hear about any comments corrections or confusions readers might havePlease send to

eriehlmathjhuedu

Commenced on January 14 2018

Last modified on June 15 2018

Contents

Preface vii

Part I Basicinfin-category theory 1

Chapter 1 infin-Cosmoi and their homotopy 2-categories 311 Quasi-categories 312 infin-Cosmoi 1213 Cosmological functors 2214 The homotopy 2-category 24

Chapter 2 Adjunctions limits and colimits I 3121 Adjunctions and equivalences 3122 Initial and terminal elements 3723 Limits and colimits 3924 Preservation of limits and colimits 45

Chapter 3 Weak 2-limits in the homotopy 2-category 4931 Smothering functors 5032 infin-categories of arrows 5333 The comma construction 5834 Representable commainfin-categories 6435 Sliced homotopy 2-categories and fibered equivalences 73

Chapter 4 Adjunctions limits and colimits II 8141 The universal property of adjunctions 8142 infin-categories of cones 8443 The universal property of limits and colimits 8744 Loops and suspension in pointedinfin-categories 97

Chapter 5 Fibrations and Yonedarsquos lemma 10351 The 2-category theory of cartesian fibrations 10352 Cocartesian fibrations and bifibrations 12353 The quasi-category theory of cartesian fibrations 12554 Discrete cartesian fibrations 12955 The external Yoneda lemma 135

Part II Homotopy coherent category theory 145

Chapter 6 Simplicial computads and homotopy coherence 14761 Simplicial computads 148

62 Free resolutions and homotopy coherent simplices 15363 Homotopy coherent realization and the homotopy coherent nerve 157

Chapter 7 Weighted limits ininfin-cosmoi 16571 Weighted limits and colimits 16572 Flexible weighted limits and the collage construction 17173 Homotopical properties of flexible weighed limits 17674 Moreinfin-cosmoi 18375 Weak 2-limits revisited 193

Chapter 8 Homotopy coherent adjunctions and monads 19781 The free homotopy coherent adjunction 19882 Homotopy coherent adjunction data 20983 Building homotopy coherent adjunctions 21684 Homotopical uniqueness of homotopy coherent adjunctions 219

Chapter 9 The formal theory of homotopy coherent monads 22791 Homotopy coherent monads 22792 Homotopy coherent algebras and the monadic adjunction 23093 Limits and colimits in theinfin-category of algebras 23694 The monadicity theorem 24095 Monadic descent 24996 Homotopy coherent monad maps 256

Part III The calculus of modules 263

Chapter 10 Two-sided fibrations 265101 Four equivalent definitions of two-sided fibrations 265102 Theinfin-cosmos of two-sided fibrations 272103 Representable two-sided fibrations and the Yoneda lemma 275104 Modules as discrete two-sided fibrations 280

Chapter 11 The calculus of modules 285111 The double category of two-sided fibrations 286112 The virtual equipment of modules 293113 Composition of modules 297114 Representable modules 303

Chapter 12 Formal category theory in a virtual equipment 311121 Exact squares 311122 Pointwise right and left extensions 316123 Formal category theory in a virtual equipment 321124 Limits and colimits in cartesian closedinfin-cosmoi 323

Part IV Change of model and model independence 329

Chapter 13 Cosmological biequivalences 331131 Cosmological functors 331

132 Biequivalences ofinfin-cosmoi as change-of-model functors 336133 Properties of change-of-model functors 340134 Ad hoc model invariance 344

Chapter 14 Proof of model independence 347141 A biequivalence of virtual equipments 350

Appendix of Abstract Nonsense 353

Appendix A Basic concepts of enriched category theory 355A1 Cartesian closed categories 356A2 Enriched categories 359A3 Enriched natural transformations and the enriched Yoneda lemma 362A4 Tensors and cotensors 367A5 Conical limits 369A6 Change of base 372

Appendix B Introduction to 2-category theory 377B1 2-categories and the calculus of pasting diagrams 377B2 The 3-category of 2-categories 382B3 Adjunctions and mates 383B4 Right adjoint right inverse adjunctions 384B5 A bestiary of 2-categorical lemmas 384

Appendix C Abstract homotopy theory 385C1 Lifting properties weak factorization systems and Leibniz closure 385

Appendix of Semantic Considerations 387

Appendix D The combinatorics of simplicial sets 389D1 Simplicial sets and markings 389

Appendix E Examples ofinfin-cosmoi 391

Appendix F Compatibility with the analytic theory of quasi-categories 393

Bibliography 395

Preface

In this text we develop the theory of infin-categories from first principles in a model-independentfashion using a common axiomatic framework that is satisfied by a variety of models In contrast withprior ldquoanalyticrdquo treatments of the theory of infin-categories mdash in which the central categorical notionsare defined in reference to the combinatorics of a particular model mdash our approach is ldquosyntheticrdquoproceeding from definitions that can be interpreted simultaneously in many models to which ourproofs then apply While synthetic our work is not schematic or hand-wavy with the details of howto make things fully precise left to ldquothe expertsrdquo and turtles all the way downsup1 Rather we prove ourtheorems starting from a short list of clearly-enumerated axioms and our conclusions are valid in anymodel ofinfin-categories satisfying these axioms

The synthetic theory is developed in any infin-cosmos which axiomatizes the universe in whichinfin-categories live as objects So that our theorem statements suggest their natural interpretation werecast infin-category as a technical term to mean an object in some (typically fixed) infin-cosmos Severalmodels of (infin 1)-categoriessup2 areinfin-categories in this sense but ourinfin-categories also include certainmodels of (infin 119899)-categoriessup3 as well as fibered versions of all of the above This usage is meant to in-terpolate between the classical one which refers to any variety of weak infinite-dimensional categoryand the common one which is often taken to mean quasi-categories or complete Segal spaces

Much of the development of the theory ofinfin-categories takes place not in the fullinfin-cosmos butin a quotient that we call the homotopy 2-category the name chosen because an infin-cosmos is some-thing like a category of fibrant objects in an enriched model category and the homotopy 2-categoryis then a categorification of its homotopy category The homotopy 2-category is a strict 2-category mdashlike the 2-category of categories functors and natural transformations mdash and in this way the foun-dational proofs in the theory of infin-categories closely resemble the classical foundations of ordinarycategory theory except that the universal properties that characterize eg when a functor betweeninfin-categories defines a cartesian fibration are slightly weaker than in the familiar case In Part Iwe define and develop the notions of equivalence and adjunction between infin-categories limits andcolimits in infin-categories cartesian and cocartesian fibrations and their discrete variants and provean external version of the Yoneda lemma all from the comfort of the homotopy 2-category In PartII we turn our attention to homotopy coherent structures present in the full infin-cosmos to define

sup1A less rigorous ldquomodel-independentrdquo presentation ofinfin-category theory might confront a problem of infinite regresssince infinite-dimensional categories are themselves the objects of an ambient infinite-dimensional category and in de-veloping the theory of the former one is tempted to use the theory of the latter We avoid this problem by using a veryconcrete model for the ambient (infin 2)-category of infin-categories that arises frequently in practice and is designed to fa-cilitate relatively simple proofs While the theory of (infin 2)-categories remains in its infancy we are content to cut theGordian knot in this way

sup2 (naturally marked quasi-categories) all define theinfin-categories in aninfin-cosmossup3Θ119899-spaces iterated complete Segal spaces and 119899-complicial sets also define theinfin-categories in aninfin-cosmos as do

(nee weak) complicial sets a model for (infininfin)-categories We hope to add other models of (infin 119899)-categories to this list

and study homotopy coherent adjunctions and monads borne by infin-categories as a mechanism foruniversal algebra

Whatrsquos missing from this basic account of the category theory of infin-categories is a satisfactorytreatment of the ldquohomrdquo bifunctor associated to an infin-category which is the prototypical example ofwhat we call a module In Part III we develop the calculus of modules betweeninfin-categories and applythis to define and study pointwise Kan extensions This will give us an opportunity to repackageuniversal properties proven in Part I as parts of the ldquoformal category theoryrdquo ofinfin-categories

This work is all ldquomodel-agnosticrdquo in the sense of being blind to details about the specificationsof any particular infin-cosmos In Part IV we prove that the category theory of infin-categories is alsoldquomodel-independentrdquo in a precise sense all categorical notions are preserved reflected and createdby any ldquochange-of-modelrdquo functor that defines what we call a biequivalence This model-independencetheorem is stronger than our axiomatic framework might initially suggest in that it also allows usto transfer theorems proven using ldquoanalyticrdquo techniques to all biequivalent infin-cosmoi For instancethe four infin-cosmoi whose objects model (infin 1)-categories are all biequivalent It follows that theanalytically-proven theorems about quasi-categories⁴ from [36] transfer to complete Segal spaces andvice versa

⁴Importantly the synthetic theory developed in theinfin-cosmos of quasi-categories is fully compatible with the analytictheory developed by Joyal Lurie and many others This is the subject of Appendix F

Part I

Basicinfin-category theory

CHAPTER 1

infin-Cosmoi and their homotopy 2-categories

11 Quasi-categories

Before introducing an axiomatic framework that will allow us to develop infin-category theory ingeneral we first consider one model in particular namely quasi-categories which were first analyzedby Joyal in [24] and [25] and in several unpublished draft book manuscripts

111 Notation (the simplex category) Let 120491 denote the simplex category of finite non-empty ordi-nals [119899] = 0 lt 1 lt ⋯ lt 119899 and order-preserving maps These include in particular the

elementary face operators [119899 minus 1] [119899] 0 le 119894 le 119899 and the

elementary degeneracy operators [119899 + 1] [119899] 0 le 119894 le 119899

120575119894

120590119894

whose images respectively omit and double up on the element 119894 isin [119899] Every morphism in 120491 fac-tors uniquely as an epimorphism followed by a monomorphism these epimorphisms the degeneracyoperators decompose as composites of elementary degeneracy operators while the monomorphismsthe face operators decompose as composites of elementary face operators

The category of simplicial sets is the category 119982119982119890119905 ≔ 119982119890119905120491op

of presheaves on the simplex cat-egory We write Δ[119899] for the standard 119899-simplex the simplicial set represented by [119899] isin 120491 andΛ119896[119899] sub 120597Δ[119899] sub Δ[119899] for its 119896-horn and boundary sphere respectively

Given a simplicial set 119883 it is conventional to write 119883119899 for the set of 119899-simplices defined byevaluating at [119899] isin 120491 By the Yoneda lemma each119899-simplex 119909 isin 119883119899 corresponds to amap of simplicialsets 119909∶ Δ[119899] rarr 119883 Accordingly we write 119909 sdot 120575119894 for the 119894th face of the 119899-simplex an (119899 minus 1)-simplexclassified by the composite map

Δ[119899 minus 1] Δ[119899] 119883120575119894 119909

Geometrically 119909 sdot 120575119894 is the ldquoface opposite the vertex 119894rdquo in the 119899-simplex 119909Since the morphisms of120491 are generated by the elementary face and degeneracy operators the data

of a simplicial setsup1 119883 is often presented by a diagram

⋯ 1198833 1198832 1198831 1198830

1205750

1205751

1205752

1205753

1205901

1205900

1205902

1205751

1205752

12057501205900

1205901 1205751

12057501205900

identifying the set of 119899-simplices for each [119899] isin 120491 as well as the (contravariant) actions of the elemen-tary operators conventionally denoted using subscripts

sup1This presentation is also used for more general simplicial objects valued in any category

112 Definition (quasi-category) A quasi-category is a simplicial set119860 in which any inner horn canbe extended to a simplex solving the displayed lifting problem

Λ119896[119899] 119860 119899 ge 2 0 lt 119896 lt 119899

Δ[119899]

Quasi-categories were first introduced by Boardman and Vogt [8] under the name ldquoweak Kancomplexesrdquo a Kan complex being a simplicial set admitting extensions as in (113) along all horninclusions 119899 ge 1 0 le 119896 le 119899 Since any topological space can be encoded as a Kan complexsup2 in thisway spaces provide examples of quasi-categories

Categories also provide examples of quasi-categories via the nerve construction

114 Definition (nerve) The category 119966119886119905 of 1-categories embeds fully faithfully into the categoryof simplicial sets via the nerve functor An 119899-simplex in the nerve of a 1-category 119862 is a sequence of 119899composable arrows in 119862 or equally a functor [119899] rarr 119862 from the ordinal category 120159 + 1 ≔ [119899] withobjects 0hellip 119899 and a unique arrow 119894 rarr 119895 just when 119894 le 119895

115 Remark The nerve of a category 119862 is 2-coskeletal as a simplicial set meaning that every sphere120597Δ[119899] rarr 119862 with 119899 ge 3 is filled uniquely by an 119899-simplex in 119862 (see Definition D11) This is becausethe simplices in dimension 3 and above witness the associativity of the composition of the path ofcomposable arrows found along their spine the 1-skeletal simplicial subset formed by the edges con-necting adjacent vertices In fact as suggested by the proof of the following proposition any simplicialset in which inner horns admit unique fillers is isomorphic to the nerve of a 1-category see Exercise11iii

We decline to introduce explicit notation for the nerve functor preferring instead to identify1-categories with their nerves As we shall discover the theory of 1-categories extends toinfin-categoriesmodeled as quasi-categories in such a way that the restriction of each infin-categorical concept alongthe nerve embedding recovers the corresponding 1-categorical concept For instance the standardsimplex Δ[119899] is the nerve of the ordinal category 120159 + 1 and we frequently adopt the latter notationmdash writing 120793 ≔ Δ[0] 120794 ≔ Δ[1] 120795 ≔ Δ[2] and so on mdash to suggest the correct categorical intuition

To begin down this path we must first verify the assertion that has implicitly just been made

116 Proposition (nerves are quasi-categories) Nerves of categories are quasi-categories

Proof Via the isomorphism 119862 cong cosk2 119862 and the adjunction sk2 ⊣ cosk2 of D11 the requiredlifting problem displayed below-left transposes to the one displayed below-right

Λ119896[119899] 119862 cong cosk2 119862 sk2Λ119896[119899] 119862

Δ[119899] sk2 Δ[119899]

sup2The total singular complex construction defines a functor from topological spaces to simplicial sets that is an equiv-alence on their respective homotopy categories mdash weak homotopy types of spaces correspond to homotopy equivalenceclasses of Kan complexes

For 119899 ge 4 the inclusion sk2Λ119896[119899] sk2 Δ[119899] is an isomorphism in which case the lifting problemson the right admit (unique) solutions So it remains only to solve the lifting problems on the left inthe cases 119899 = 2 and 119899 = 3

To that end consider

Λ1[2] 119862 Λ1[3] 119862 Λ2[3] 119862

Δ[2] Δ[3] Δ[3]

An inner hornΛ1[2] rarr 119862 defines a composable pair of arrows in119862 an extension to a 2-simplex existsprecisely because any composable pair of arrows admits a (unique) composite

An inner hornΛ1[3] rarr 119862 specifies the data of three composable arrows in 119862 as displayed in thediagram below together with the composites 119892119891 ℎ119892 and (ℎ119892)119891

1198881

1198880 1198883

1198882

ℎ119892119891

119892119891

(ℎ119892)119891

ℎ119892

Because composition is associative the arrow (ℎ119892)119891 is also the composite of 119892119891 followed by ℎ whichproves that the 2-simplex opposite the vertex 1198881 is present in119862 by 2-coskeletality the 3-simplex fillingthis boundary sphere is also present in119862 The filler for a hornΛ2[3] rarr 119862 is constructed similarly

117 Definition (homotopy relation on 1-simplices) A parallel pair of 1-simplices 119891 119892 in a simplicialset 119883 are homotopic if there exists a 2-simplex of either of the following forms

1199091 1199090

1199090 1199091 1199090 1199091

119891119891

119892 119892

or if 119891 and 119892 are in the same equivalence class generated by this relation

In a quasi-category the relation witnessed by any of the types of 2-simplex on display in (118) isan equivalence relation and these equivalence relations coincide

119 Lemma (homotopic 1-simplices in a quasi-category) Parallel 1-simplices 119891 and 119892 in a quasi-categoryare homotopic if and only if there exists a 2-simplex of any or equivalently all of the forms displayed in (118)

Proof Exercise 11i

1110 Definition (the homotopy category) By 1-truncating any simplicial set 119883 has an underlyingreflexive directed graph

1198831 11988301205751

1205750

1205900

the 0-simplices of 119883 defining the ldquoobjectsrdquo and the 1-simplices defining the ldquoarrowsrdquo by conventionpointing from their 0th vertex (the face opposite 1) to their 1st vertex (the face opposite 0) The free

category on this reflexive directed graph has1198830 as its object set degenerate 1-simplices serving as iden-tity morphisms and non-identity morphisms defined to be finite directed paths of non-degenerate1-simplices The homotopy category h119883 of 119883 is the quotient of the free category on its underlyingreflexive directed graph by the congruencesup3 generated by imposing a composition relation ℎ = 119892 ∘ 119891witnessed by 2-simplices

1199091

1199090 1199092

119892119891

ℎThis implies in particular that homotopic 1-simplices represent the same arrow in the homotopy cat-egory

1111 Proposition The nerve embedding admits a left adjoint namely the functor which sends a simplicialset to its homotopy category

119966119886119905 119982119982119890119905perph

Proof Using the description of h119883 as a quotient of the free category on the underlying reflexivedirected graph of 119883 we argue that the data of a functor h119883 rarr 119862 can be extended uniquely to asimplicial map 119883 rarr 119862 Presented as a quotient in this way the functor h119883 rarr 119862 defines a mapfrom the 1-skeleton of119883 into119862 and since every 2-simplex in119883 witnesses a composite in h119883 this mapextends to the 2-skeleton Now119862 is 2-coskeletal so via the adjunction sk2 ⊣ cosk2 of Definition D11this map from the 2-truncation of 119883 into 119862 extends uniquely to a simplicial map 119883 rarr 119862

The homotopy category of a quasi-category admits a simplified description

1112 Lemma (the homotopy category of a quasi-category) If 119860 is a quasi-category then its homotopycategory h119860 hasbull the set of 0-simplices 1198600 as its objectsbull the set of homotopy classes of 1-simplices 1198601 as its arrowsbull the identity arrow at 119886 isin 1198600 represented by the degenerate 1-simplex 119886 sdot 1205900 isin 1198601bull a composition relation ℎ = 119892 ∘ 119891 in h119860 if and only if for any choices of 1-simplices representing these

arrows there exists a 2-simplex with boundary

1198861

1198860 1198862

119892119891

Proof Exercise 11ii

1113 Definition (isomorphisms in a quasi-category) A 1-simplex in a quasi-category is an isomor-phism just when it represents an isomorphism in the homotopy category By Lemma 1112 this meansthat 119891∶ 119886 rarr 119887 is an isomorphism if and only if there exist a 1-simplex 119891minus1 ∶ 119887 rarr 119886 together with apair of 2-simplices

119887 119886

119886 119886 119887 119887

119891minus1 119891119891 119891minus1

sup3A relation on parallel pairs of arrows of a 1-category is a congruence if it is an equivalence relation that is closedunder pre- and post-composition if 119891 sim 119892 then ℎ119891119896 sim ℎ119892119896

The properties of the isomorphisms in a quasi-category are most easily proved by arguing in aslightly different category where simplicial sets have the additional structure of a ldquomarkingrdquo on a spec-ified subset of the 1-simplices subject to the condition that all degenerate 1-simplices are markedmaps of these so-called marked simplicial sets must then preserve the markings Because these objectswill seldom appear outside of the proofs of certain combinatorial lemmas about the isomorphisms inquasi-categories we save the details for Appendix D

Let us now motivate the first of several results proven using marked techniques Quasi-categoriesare defined to have extensions along all inner horns But if in an outer hornΛ0[2] rarr 119860 orΛ2[2] rarr 119860the initial or final edges respectively are isomorphisms then intuitively a filler should exist

1198861 1198861

1198860 1198862 1198860 1198862

ℎ119891minus1

119892sim119891

119892minus1ℎ

and similarly for the higher-dimensional outer horns

1114 Proposition (special outer horn lifting)(i) Let 119860 be a quasi-category Then any outer horns

Λ0[119899] 119860 Λ119899[119899] 119860

Δ[119899] Δ[119899]

119892 ℎ

in which the edges 119892|01 and ℎ|119899minus1119899 are isomorphisms admit fillers(ii) Let 119860 and 119861 be quasi-categories and 119891∶ 119860 rarr 119861 a map that lifts against the inner horn inclusions

Then any outer horns

Λ0[119899] 119860 Λ119899[119899] 119860

Δ[119899] 119861 Δ[119899] 119861

119892

119891

in which the edges 119892|01 and ℎ|119899minus1119899 are isomorphisms admit fillers

The proof of Proposition 1114 requires clever combinatorics due to Joyal and is deferred toAppendix D⁴ Here we enjoy its myriad consequences Immediately

1115 Corollary A quasi-category is a Kan complex if and only if its homotopy category is a groupoid

Proof If the homotopy category of a quasi-category is a groupoid then all of its 1-simplices areisomorphisms and Proposition 1114 then implies that all inner and outer horns have fillers Thusthe quasi-category is a Kan complex Conversely in a Kan complex all outer horns can be filled andin particular fillers for the horns Λ0[2] and Λ2[2] can be used to construct left and right inverses forany 1-simplex of the form displayed in Definition 1113⁵

⁴The second statement subsumes the first but the first is typically used to prove the second⁵In a quasi-category any left and right inverses to a common 1-simplex are homotopic but as Corollary 1116 proves

any isomorphism in fact has a single two-sided inverse

A quasi-category contains a canonical maximal sub Kan complex the simplicial subset spannedby those 1-simplices that are isomorphisms Just as the arrows in a quasi-category 119860 are representedby simplicial maps 120794 rarr 119860 whose domain is the nerve of the free-living arrow the isomorphisms in aquasi-category are represented by diagrams 120128 rarr 119860 whose domain is the free-living isomorphism

1116 Corollary An arrow 119891 in a quasi-category119860 is an isomorphism if and only if it extends to a homo-topy coherent isomorphism

120794 119860

120128

119891

Proof If 119891 is an isomorphism the map 119891∶ 120794 rarr 119860 lands in the maximal sub Kan complex con-tained in 119860 The postulated extension also lands in this maximal sub Kan complex because the inclu-sion 120794 120128 can be expressed as a sequential composite of outer horn inclusions see Exercise 11iv

The category of simplicial sets like any category of presheaves is cartesian closed By the Yonedalemma and the defining adjunction an 119899-simplex in the exponential 119884119883 corresponds to a simplicialmap 119883 times Δ[119899] rarr 119884 and its faces and degeneracies are computed by precomposing in the simplexvariable Our aim is now to show that the quasi-categories define an exponential ideal in the simpli-cially enriched category of simplicial sets if 119883 is a simplicial set and119860 is a quasi-category then119860119883 isa quasi-category We will deduce this as a corollary of the ldquorelativerdquo version of this result involving aclass of maps called isofibrations that we now introduce

1117 Definition (isofibrations between quasi-categories) A simplicial map 119891∶ 119860 rarr 119861 is a isofibra-tion if it lifts against the inner horn inclusions as displayed below left and also against the inclusionof either vertex into the free-standing isomorphism 120128

Λ119896[119899] 119860 120793 119860

Δ[119899] 119861 120128 119861

119891 119891

To notationally distinguish the isofibrations we depict them as arrows ldquo↠rdquo with two heads

1118 Observation(i) For any simplicial set 119883 the unique map 119883 rarr lowast whose codomain is the terminal simplicial

set is an isofibration if and only if 119883 is a quasi-category(ii) Any class of maps characterized by a right lifting property is automatically closed under com-

position product pullback retract and limits of towers see Lemma C11(iii) Combining (i) and (ii) if 119860 ↠ 119861 is an isofibration and 119861 is a quasi-category then so is 119860(iv) The isofibrations generalize the eponymous categorical notion The nerve of any functor

119891∶ 119860 rarr 119861 between categories defines a map of simplicial sets that lifts against the innerhorn inclusions This map then defines an isofibration if and only if given any isomorphism in119861 and specified object in119860 lifting either its domain or codomain there exists an isomorphismin 119860 with that domain or codomain lifting the isomorphism in 119861

We typically only deploy the term ldquoisofibrationrdquo for a map between quasi-categories because our usageof this class of maps intentionally parallels the classical categorical case

Much harder to establish is the stability of the class of isofibrations under forming ldquoLeibniz ex-ponentialsrdquo as displayed in (1120) The proof of this result is given in Proposition in AppendixD

1119 Proposition If 119894 ∶ 119883 119884 is a monomorphism and 119891∶ 119860 ↠ 119861 is an isofibration then the inducedLeibniz exponential map

119860119884

bull 119860119883

119861119884 119861119883

1198941114023⋔119891

119860119894

119891119884

119891119883

119861119894

(1120)

is again an isofibration⁶

1121 Corollary If119883 is a simplicial set and119860 is a quasi-category then119860119883 is a quasi-category Moreovera 1-simplex in 119860119883 is an isomorphism if and only if its components at each vertex of 119883 are isomorphisms in 119860

Proof The first statement is a special case of Proposition 1119 see Exercise 11vi The secondstatement is proven similarly by arguing with marked simplicial sets See Lemma

1122 Definition (equivalences of quasi-categories) A map 119891∶ 119860 rarr 119861 between quasi-categories isan equivalence if it extends to the data of a ldquohomotopy equivalencerdquo with the free-living isomorphism120128 serving as the interval that is if there exist maps 119892∶ 119861 rarr 119860 and

119860 119861

119860 119860120128 119861 119861120128

119860 119861119892119891

120572

120573

119891119892 ev0

We write ldquo⥲rdquo to decorate equivalences and119860 ≃ 119861 to indicate the presence of an equivalence119860 ⥲ 119861

1123 Definition A map 119891∶ 119883 rarr 119884 between simplicial sets is a trivial fibration if it admits liftsagainst the boundary inclusions for all simplices

120597Δ[119899] 119883 119899 ge 0

Δ[119899] 119884

119891 (1124)

We write ldquo⥲rarrrdquo to decorate trivial fibrations

1125 Remark The simplex boundary inclusions 120597Δ[119899] Δ[119899] ldquocellularly generaterdquo the monomor-phisms of simplicial sets mdash see Definition C12 and Lemma D12 Hence the dual of Lemma C11

⁶Degenerate cases of this result taking 119883 = empty or 119861 = 1 imply that the other six maps in this diagram are alsoisofibrations see Exercise 11vi

implies that trivial fibrations lift against any monomorphism between simplicial sets In particularapplying this to the map empty rarr 119884 it follows that any trivial fibration 119883 ⥲rarr 119884 is a split epimorphism

The notation ldquo⥲rarrrdquo is suggestive the trivial fibrations between quasi-categories are exactly thosemaps that are both isofibrations and equivalences This can be proven by a relatively standard althoughrather technical argument in simplicial homotopy theory given as Proposition in Appendix D

1126 Proposition For a map 119891∶ 119860 rarr 119861 between quasi-categories the following are equivalent(i) 119891 is at trivial fibration(ii) 119891 is both an isofibration and an equivalence(iii) 119891 is a split fiber homotopy equivalence an isofibration admitting a section 119904 that is also an equivalence

inverse via a homotopy from 119904119891 to 1119860 that composes with 119891 to the constant homotopy from 119891 to 119891

As a class characterized by a right lifting property the trivial fibrations are also closed undercomposition product pullback limits of towers and contain the isomorphisms The stability ofthese maps under Leibniz exponentiation will be verified along with Proposition 1119 in Proposition

1127 Proposition If 119894 ∶ 119883 rarr 119884 is a monomorphism and 119891∶ 119860 rarr 119861 is an isofibration then if either 119891 isa trivial fibration or if 119894 is in the class cellularly generated⁷ by the inner horn inclusions and the map 120793 120128then the induced Leibniz exponential map

119860119884 119861119884 times119861119883 1198601198831198941114023⋔119891

a trivial fibration

1128 Digression (the Joyal model structure) The category of simplicial set bears a model structure(see Appendix D) whose fibrant objects are exactly the quasi-categories all objects are cofibrant Thefibrations weak equivalences and trivial fibrations between fibrant objects are precisely the classesof isofibrations equivalences and trivial fibrations defined above Proposition 1126 proves that thetrivial fibrations are the intersection of the classes of fibrations and weak equivalences Propositions1119 and 1127 reflect the fact that the Joyal model structure is a closed monoidal model category withrespect to the cartesian closed structure on the category of simplicial sets

We have declined to elaborate on the Joyal model structure for quasi-categories alluded to in Di-gression 1128 because the only aspects of it that we will need are those described above The resultsproven here suffice to show that the category of quasi-categories defines an infin-cosmos a concept towhich we now turn

Exercises

11i Exercise Consider the set of 1-simplices in a quasi-category with initial vertex 1198860 and finalvertex 1198861

(i) Prove that the relation defined by 119891 sim 119892 if and only if there exists a 2-simplex with boundary1198861

1198860 1198861

119891

119892

is an equivalence relation

⁷See Definition C12

(ii) Prove that the relation defined by 119891 sim 119892 if and only if there exists a 2-simplex with boundary1198860

1198860 1198861

119891

119892

(iii) Prove that the equivalence relations defined by (i) and (ii) are the sameThis proves Lemma 119

11ii Exercise Consider the free category on the reflexive directed graph

1198601 11986001205751

1205750

1205900

underlying a quasi-category 119860(i) Consider the relation that identifies a pair of sequences of composable 1-simplices with com-

mon source and common target whenever there exists a simplex of119860 in which the sequences of1-simplices define two paths from its initial vertex to its final vertex Prove that this relation isstable under pre- and post-composition with 1-simplices and conclude that its transitive clo-sure is a congruence an equivalence relation that is closed under pre- and post-composition⁸

(ii) Consider the congruence relation generated by imposing a composition relation ℎ = 119892 ∘ 119891witnessed by 2-simplices

1198861

1198860 1198862

119892119891

ℎand prove that this coincides with the relation considered in (i)

(iii) In the congruence relations of (i) and (ii) prove that every sequence of composable 1-simplicesin 119860 is equivalent to a single 1-simplex Conclude that every morphism in the quotient of thefree category by this congruence relation is represented by a 1-simplex in 119860

(iv) Prove that for any triple of 1-simplices 119891 119892 ℎ in 119860 ℎ = 119892 ∘ 119891 in the quotient category if andonly if there exists a 2-simplex with boundary

1198861

1198860 1198862

119892119891

This proves Lemma 1112

11iii Exercise Show that any quasi-category inwhich inner horns admit unique fillers is isomorphicto the nerve of its homotopy category

11iv Exercise(i) Prove that 120128 contains exactly two non-degenerate simplices in each dimension

⁸Given a congruence relation on the hom-sets of a 1-category the quotient category can be formed by quotientingeach hom-set see [39 sectII8]

(ii) Inductively build 120128 from 120794 by expressing the inclusion 120794 120128 as a sequential composite ofpushouts of outer horn inclusions⁹ Λ0[119899] Δ[119899] one in each dimension starting with 119899 =2sup1⁰

11v Exercise Prove the relative version of Corollary 1116 for any isofibration 119901∶ 119860 ↠ 119861 betweenquasi-categories and any isomorphism119891∶ 120794 rarr 119860 any homotopy coherent isomorphism in119861 extending119901119891 lifts to a homotopy coherent isomorphism in 119860 extending 119891

120794 119860

120128 119861

119891

119901

11vi Exercise Specialize Proposition 1119 to prove the following(i) If 119860 is a quasi-category and 119883 is a simplicial set then 119860119883 is a quasi-category(ii) If 119860 is a quasi-category and 119883 119884 is a monomorphism then 119860119884 ↠ 119860119883 is an isofibration(iii) If 119860 ↠ 119861 is an isofibration and 119883 is a simplicial set then 119860119883 ↠ 119861119883 is an isofibration

11vii Exercise(i) Prove that the equivalences defined in Definition 1122 are closed under retracts(ii) Prove that the equivalences defined in Definition 1122 satisfy the 2-of-3 property

12 infin-Cosmoi

In sect11 we presented ldquoanalyticrdquo proofs of a few of the basic facts about quasi-categories Thecategory theory of quasi-categories can be developed in a similar style but we aim instead to developthe ldquosyntheticrdquo theory of infinite-dimensional categories so that our results will apply to many modelsat once To achieve this our strategy is not to axiomatize what these infinite-dimensional categoriesare but rather axiomatize the ldquouniverserdquo in which they live

The following definition abstracts the properties of the quasi-categories and the classes of isofibra-tions equivalences and trivial fibrations introduced in sect11 Firstly the category of quasi-categoriesand simplicial maps is enriched over the category of simplicial sets mdash the set of morphisms from 119860to 119861 coincides with the set of vertices of the simplicial set 119861119860 mdash and moreover these hom-spaces areall quasi-categories Secondly a number of limit constructions that can be defined in the underlying1-category of quasi-categories and simplicial maps satisfy universal properties relative to this simpli-cial enrichment And finally the classes of isofibrations equivalences and trivial fibrations satisfyproperties that are familiar from abstract homotopy theory In particular the use of isofibrations indiagrams guarantees that their strict limits are equivalence invariant so we can take advantage of up-to-isomorphism universal properties and strict functoriality of these constructions while still workingldquohomotopicallyrdquo

As will be explained in Digression 129 there are a variety of models of infinite-dimensional cat-egories for which the category of ldquoinfin-categoriesrdquo as we will call them and ldquofunctorsrdquo between them is

⁹By duality mdash the opposite of a simplicial set 119883 is the simplicial set obtained by reindexing along the involution(minus)op ∶ 120491 rarr 120491 that reverses the ordering in each ordinal mdash the outer horn inclusions Λ119899[119899] Δ[119899] can be used instead

sup1⁰This decomposition of the inclusion 120794 120128 reveals which data can always be extended to a homotopy coherentisomorphism for instance the 1- and 2-simplices of Definition 1113 together with a single 3-simplex that has these as itsouter faces with its inner faces degenerate

enriched over quasi-categories and admits classes of isofibrations equivalences and trivial fibrationssatisfying analogous properties This motivates the following axiomatization

121 Definition (infin-cosmoi) Aninfin-cosmos119974 is a category enriched over quasi-categoriessup1sup1 mean-ing that it hasbull objects 119860119861 that we callinfin-categories andbull its morphisms define the vertices of functor-spaces Fun(119860 119861) which are quasi-categories

that is also equipped with a specified class of maps that we call isofibrations and denote by ldquo↠rdquoFrom these classes we define a map 119891∶ 119860 rarr 119861 to be an equivalence if and only the induced map119891lowast ∶ Fun(119883119860) ⥲ Fun(119883 119861) on functor-spaces is an equivalence of quasi-categories for all 119883 isin 119974and we define 119891 to be a trivial fibration just when 119891 is both an isofibration and an equivalence theseclasses are denoted by ldquo⥲rdquo and ldquo⥲rarrrdquo respectively These classes must satisfy the following two axioms

(i) (completeness) The simplicial category 119974 possesses a terminal object small products pull-backs of isofibrations limits of countable towers of isofibrations and cotensors with all sim-plicial sets each of these limit notions satisfying a universal property that is enriched oversimplicial setssup1sup2

(ii) (isofibrations) The class of isofibrations contains all isomorphisms and any map whose codo-main is the terminal object is closed under composition product pullback forming inverselimits of towers and Leibniz cotensors with monomorphisms of simplicial sets and has theproperty that if 119891∶ 119860 ↠ 119861 is an isofibration and119883 is any object then Fun(119883119860) ↠ Fun(119883 119861)is an isofibration of quasi-categories

122 Digression (simplicial categories) A simplicial category 119964 is given by categories 119964119899 with acommon set of objects and whose arrows are called 119899-arrows that assemble into a diagram120491op rarr 119966119886119905of identity-on-objects functors

⋯ 1199643 1199642 1199641 1199640 ≕ 119964

1205750

1205751

1205752

1205753

1205901

1205900

1205902

1205751

1205752

12057501205900

1205901 1205751

12057501205900 (123)

The data of a simplicial category can equivalently be encoded by a simplicially enriched categorywith a set of objects and a simplicial set 119964(119909 119910) of morphisms between each ordered pair of ob-jects an 119899-arrow in119964119899 from 119909 to 119910 corresponds to an 119899-simplex in119964(119909 119910) (see Exercise 12i) Eachendo-hom-space contains a distinguished identity 0-arrow (the degenerate images of which define thecorresponding identity 119899-arrows) and composition is required to define a simplicial map

119964(119910 119911) times 119964(119909 119910) 119964(119909 119911)∘

the single map encoding the compositions in each of the categories 119964119899 and also the functoriality ofthe diagram (123) The composition is required to be associative and unital in a sense expressed by

sup1sup1This is to say119974 is a simplicially enriched category whose hom-spaces are quasi-categories this will be unpacked in122

sup1sup2This will be elaborated upon in 124

the commutative diagrams

119964(119910 119911) times 119964(119909 119910) times 119964(119908 119909) 119964(119909 119911) times 119964(119908 119909)

119964(119910 119911) times 119964(119908 119910) 119964(119908 119911)1times∘

∘times1

119964(119909 119910) 119964(119910 119910) times 119964(119909 119910)

119964(119909 119910) times 119964(119909 119909) 119964(119909 119910)

id119910 times1

1timesid119909 ∘

the latter making use of natural isomorphisms 119964(119909 119910) times 1 cong 119964(119909 119910) cong 1 times 119964(119909 119910) in the domainvertex

On account of the equivalence between these two presentations the terms ldquosimplicial categoryrdquoand ldquosimplicially-enriched categoryrdquo are generally taken to be synonymssup1sup3 The category1199640 of 0-arrowsis the underlying category of the simplicial category119964 which forgets the higher dimensional simpli-cial structure

In particular the underlying category of an infin-cosmos 119974 is the category whose objects are theinfin-categories in119974 and whose morphisms are the 0-arrows in the functor spaces In all of the examplesto appear below this recovers the expected category ofinfin-categories in a particularmodel and functorsbetween them

124 Digression (simplicially enriched limits) Let 119964 be a simplicial category The cotensor of anobject 119860 isin 119964 by a simplicial set 119880 is characterized by an isomorphism of simplicial sets

119964(119883119860119880) cong 119964(119883119860)119880 (125)

natural in 119883 isin 119964 Assuming such objects exist the simplicial cotensor defines a bifunctor

119982119982119890119905op times119964 119964

(119880119860) 119860119880

in a unique way making the isomorphism (125) natural in 119880 and 119860 as wellThe other simplicial limit notions postulated by axiom 121(i) are conical which is the term used

for ordinary 1-categorical limit shapes that satisfy an enriched analog of the usual universal propertysee Definition 7114 When these limits exist they correspond to the usual limits in the underlying cat-egory but the usual universal property is strengthened Applying the covariant representable functor119964(119883 minus) ∶ 1199640 rarr 119982119982119890119905 to a limit cone (lim119895isin119869119860119895 rarr 119860119895)119895isin119869 in1199640 there is natural comparison map

119964(119883 lim119895isin119869119860119895) rarr lim

119895isin119869119964(119883119860119895) (126)

and we say that lim119895isin119869119860119895 defines a simplicially enriched limit if and only if (126) is an isomorphismof simplicial sets for all 119883 isin 119964

Considerably more details on the general theory of enriched categories can be found in [31] andin Appendix A Enriched limits are the subject of sectA

127 Remark (flexible weighted limits ininfin-cosmoi) The axiom 121(i) implies that anyinfin-cosmos119974 admits all flexible limits (see Corollary 733) a much larger class of simplicially enriched ldquoweightedrdquolimits that will be introduced in sect72

sup1sup3The phrase ldquosimplicial object in119966119886119905rdquo is reserved for themore general yet less commonnotion of a diagram120491op rarr 119966119886119905that is not necessarily comprised of identity-on-objects functors

Using the results of Joyal discussed in sect11 we can easily verify

128 Proposition The full subcategory 119980119966119886119905 sub 119982119982119890119905 of quasi-categories defines aninfin-cosmos with theisofibrations equivalences and trivial fibrations of Definitions 1117 1122 and 1123

Proof The subcategory 119980119966119886119905 sub 119982119982119890119905 inherits its simplicial enrichment from the cartesianclosed category of simplicial sets note that for quasi-categories 119860 and 119861 Fun(119860 119861) ≔ 119861119860 is again aquasi-category

The limits postulated in 121(i) exist in the ambient category of simplicial setssup1⁴ The defininguniversal property of the simplicial cotensor is satisfied by the exponentials of simplicial sets We nowargue that the full subcategory of quasi-categories inherits all these limit notions

Since the quasi-categories are characterized by a right lifting property it is clear that they areclosed under small products Similarly since the class of isofibrations is characterized by a right liftingproperty Lemma C11 implies that the fibrations are closed under all of the limit constructions of121(ii) except for the last two Leibniz closure and closure under exponentiation (minus)119883 These lastclosure properties are established in Proposition 1119 This completes the proof of 121(i) and 121(ii)

Remark 1125 proves that trivial fibrations split once we verify that the classes of trivial fibrationsand of equivalences coincide with those defined by 1123 and 1122 By Proposition 1126 the formercoincidence follows from the latter so it remains only to show that the equivalences of 1122 coincidewith the representably-defined equivalences those maps of quasi-categories 119891∶ 119860 rarr 119861 for which119860119883 rarr 119861119883 is an equivalence of quasi-categories in the sense of 1122 Taking 119883 = Δ[0] we seeimmediately that representably-defined equivalences are equivalences and the converse holds sincethe exponential (minus)119883 preserves the data defining a simplicial homotopy

We mention a common source of infin-cosmoi found in nature at the outside to help ground theintuition for readers familiar with Quillenrsquos model categories a popular framework for ldquoabstract ho-motopy theoryrdquo but reassure others that model categories are not needed outside of Appendix E

129 Digression (a source of infin-cosmoi in nature) As explained in Appendix E certain easily de-scribed properties of a model category imply that the full subcategory of fibrant objects defines aninfin-cosmos whose isofibrations equivalences and trivial fibrations are the fibrations weak equiva-lences and trivial fibrations between fibrant objects Namely any model category that is enriched assuch over the Joyal model structure on simplicial sets and with the property that all fibrant objects arecofibrant has this property This compatible enrichment in the Joyal model structure can be definedwhen the model category is cartesian closed and equipped with a right Quillen adjoint to the Joyalmodel structure on simplicial sets whose left adjoint preserves finite products In this case the rightadjoint becomes the underlying quasi-category functor (see Proposition 133(ii)) and the infin-cosmoiso-produced will then be cartesian closed (see Definition 1218) The infin-cosmoi listed in Example1219 all arise in this way

The following results are consequences of the axioms of Definition 121 The first of these resultstells us that the trivial fibrations enjoy all of the same stability properties satisfied by the isofibrations

1210 Lemma (stability of trivial fibrations) The trivial fibrations in aninfin-cosmos define a subcategorycontaining the isomorphisms are stable under product pullback forming inverse limits of towers the Leibnizcotensors of any trivial fibration with a monomorphism of simplicial sets is a trivial fibration as is the Leibniz

sup1⁴Any category of presheaves is cartesian closed complete and cocomplete mdash a ldquocosmosrdquo in the sense of Beacutenabou Ourinfin-cosmoi are more similar to the fibrational cosmoi due to Street [57]

cotensor of an isofibration with a map in the class cellularly generated by the inner horn inclusions and the map120793 120128 and if 119864 ⥲rarr 119861 is a trivial fibration then so is Fun(119883 119864) ⥲rarr Fun(119883 119861)

Proof We prove these statements in the reverse order By axiom 121(ii) and the definition of thetrivial fibrations in aninfin-cosmos we know that if 119864 ⥲rarr 119861 is a trivial fibration then is Fun(119883 119864) ⥲rarrFun(119883 119861) is both an isofibration and an equivalence and hence by Proposition 1126 a trivial fibra-tion For stability under the remaining constructions we know in each case that the maps in ques-tion are isofibrations in the infin-cosmos it remains to show only that the maps are also equivalencesThe equivalences in an infin-cosmos are defined to be the maps that Fun(119883 minus) carries to equivalencesof quasi-categories so it suffices to verify that trivial fibrations of quasi-categories satisfy the corre-sponding stability properties This is established in Proposition 1127 and the fact that that class ischaracterized by a right lifting property

Additionally every trivial fibration is ldquosplitrdquo by a section

1211 Lemma (trivial fibrations split) Every trivial fibration admits a section

119864

119861 119861

Proof If 119901∶ 119864 ⥲rarr 119861 is a trivial fibration then by the final stability property of Lemma 1210 sois 119901lowast ∶ Fun(119861 119864) ↠ 119865119906119899(119861 119861) By Definition 1123 we may solve the lifting problem

empty = 120597Δ[0] Fun(119861 119864)

Δ[0] Fun(119861 119861)

sim 119901lowast119904

id119861

to find a map 119904 ∶ 119861 rarr 119864 so that 119901119904 = id119861

A classical construction in abstract homotopy theory proves the following

1212 Lemma (Brown factorization lemma) Any functor 119891∶ 119860 rarr 119861 in an infin-cosmos may be factoredas an equivalence followed by an isofibration where this equivalence is constructed as a section of a trivialfibration

119875119891

119860 119861

119901

sim119902

119891

sim119904

(1213)

Proof The displayed factorization is constructed by the pullback of an isofibration formed bythe simplicial cotensor of the inclusion 120793 + 120793 120128 into theinfin-category 119861

119860120128

119860 119875119891 119861120128

119860 times 119861 119861 times 119861

119891120128

sim119904

(119860119891)(119902119901)

(ev0ev1)

119891times119861

Note the map 119902 is a pullback of the trivial fibration ev0 ∶ 119861120128 ⥲rarr 119861 and is hence a trivial fibration Itssection 119904 constructed by applying the universal property of the pullback to the displayed cone withsummit 119860 is thus an equivalence

1214 Remark (equivalences satisfy the 2-of-6 property) Exercise 11vii proves that the equivalencesbetween quasi-categories mdash and hence also the equivalences in any infin-cosmos mdash are closed underretracts and have the 2-of-3 property Using this latter fact we see that in the case where 119891∶ 119860 rarr 119861 isan equivalence the map 119901 of (1213) is also a trivial fibration and in particular has a section by Lemma1211 Combining these facts a result of Blumberg and Mandell [7 64] reproduced in Appendix Capplies to prove that the equivalences in any infin-cosmos satisfy the stronger 2-of-6 property for anycomposable triple of morphisms

119861

119860 119863

119862

simℎ119892119891

sim119892119891

ℎ119892119891

119892ℎ

if 119892119891 and ℎ119892 are equivalences then 119891 119892 ℎ and ℎ119892119891 are too

By a Yoneda-style argument the ldquohomotopy equivalencerdquo characterization of the equivalences intheinfin-cosmos of quasi-categories extends to an analogous characterization of the equivalences in anyinfin-cosmos

1215 Lemma (equivalences are homotopy equivalences) A map 119891∶ 119860 rarr 119861 between infin-categories inan infin-cosmos 119974 is an equivalence if and only if it extends to the data of a ldquohomotopy equivalencerdquo with thefree-living isomorphism 120128 serving as the interval that is if there exist maps 119892∶ 119861 rarr 119860 and

119860 119861

119860 119860120128 119861 119861120128

119860 119861119892119891

120572

120573

119891119892 ev0

(1216)

in theinfin-cosmos

Proof By hypothesis if 119891∶ 119860 rarr 119861 defines an equivalence in theinfin-cosmos119974 then the inducedmap on post-composition 119891lowast ∶ Fun(119861119860) ⥲ Fun(119861 119861) is an equivalence of quasi-categories Evalu-ating the equivalence inverse ∶ Fun(119861 119861) ⥲ Fun(119861119860) and homotopy ∶ Fun(119861 119861) rarr Fun(119861 119861)120128at the 0-arrow 1119861 isin Fun(119861 119861) we obtain a 0-arrow 119892∶ 119861 rarr 119860 together with an isomorphism120128 rarr Fun(119861 119861) from the composite 119891119892 to 1119861 By the defining universal property of the cotensorthis isomorphism internalizes to define the map 120573∶ 119861 rarr 119861120128 in119974 displayed on the right of (1216)

Now the hypothesis that 119891 is an equivalence also provides an equivalence of quasi-categories119891lowast ∶ Fun(119860119860) ⥲ Fun(119860 119861) and the map 120573119891∶ 119860 rarr 119861120128 represents an isomorphism in Fun(119860 119861)from 119891119892119891 to 119891 Since 119891lowast is an equivalence we can conclude that 1119860 and 119892119891 are isomorphic in thequasi-category Fun(119860119860) such an isomorphism may be defined by applying the inverse equivalenceℎ ∶ Fun(119860 119861) rarr Fun(119860119860) and composing with the components at 1119860 119892119891 isin Fun(119860119860) of the iso-morphism ∶ Fun(119860119860) rarr Fun(119860119860)120128 from 1Fun(119860119860) to ℎ119891lowast Now by Corollary 1116 this isomor-phism is represented by a map 120128 rarr Fun(119860119860) from 1119860 to 119892119891 which internalizes to a map120572∶ 119860 rarr 119860120128in119974 displayed on the left of (1216)

The converse is easy the simplicial cotensor construction commutes with Fun(119883 minus) so homotopyequivalences are preserved and by Definition 1122 homotopy equivalences of quasi-categories defineequivalences of quasi-categories

One of the key advantages of the infin-cosmological approaches to abstract category theory is thatthere are a myriad varieties of ldquofiberedrdquo infin-cosmoi that can be built from a given infin-cosmos whichmeans that any theorem proven in this axiomatic framework specializes and generalizes to those con-texts The most basic of these derived infin-cosmos is the infin-cosmos of isofibrations over a fixed basewhich we introduce now Other examples of infin-cosmoi will be introduced in sect74 once we have de-veloped a greater facility with the simplicial limits of axiom 121(i)

1217 Proposition (slicedinfin-cosmoi) For anyinfin-cosmos119974 and any object 119861 isin 119974 there is aninfin-cosmos119974119861 of isofibrations over 119861 whose

(i) objects are isofibrations 119901∶ 119864 ↠ 119861 with codomain 119861(ii) functor-spaces say from 119901∶ 119864 ↠ 119861 to 119902 ∶ 119865 ↠ 119861 are defined by pullback

Fun119861(119901 ∶ 119864 ↠ 119861 119902 ∶ 119865 ↠ 119861) Fun(119864 119865)

120793 Fun(119864 119861)

119902lowast

119901

and abbreviated to Fun119861(119864 119865) when the specified isofibrations are clear from context(iii) isofibrations are commutative triangles of isofibrations over 119861

119864 119865

119861

119903

119901 119902

(iv) terminal object is 1∶ 119861 ↠ 119861 and products are defined by the pullback along the diagonal

times119861119894 119864119894 prod119894 119864119894

119861 prod119894 119861

(v) pullbacks and limits of towers of isofibrations are created by the forgetful functor119974119861 rarr119974(vi) simplicial cotensors 119901∶ 119864 ↠ 119861 by 119880 isin 119982119982119890119905 are denoted 119880 ⋔119861 119901 and constructed by the pullback

119880 ⋔119861 119901 119864119880

119861 119861119880

119901119880

(vii) and in which a map

119864 119865

119861

119891

119901 119902

over 119861 is an equivalence in theinfin-cosmos119974119861 if and only if 119891 is an equivalence in119974

Proof Note first that the functor spaces are quasi-categories since axiom 121(ii) asserts that forany isofibration 119902 ∶ 119865 ↠ 119861 in 119974 the map 119902lowast ∶ Fun(119864 119865) ↠ Fun(119864 119861) is an isofibration of quasi-categories Other parts of this axiom imply that that each of the limit constructions define isofibra-tions over 119861 The closure properties of the isofibrations in 119974119861 follow from the corresponding onesin 119974 The most complicated of these is the Leibniz cotensor stability of the isofibrations in 119974119861which follows from the corresponding property in 119974 since for a monomorphism of simplicial sets119894 ∶ 119883 119884 and an isofibration 119903 over 119861 as above the map 119894 1114024⋔119861 119903 is constructed by pulling back 119894 1114023⋔ 119903along Δ∶ 119861 rarr 119861119884

The fact that the above constructions define simplicially enriched limits in a simplicially enrichedslice category are standard from enriched category theory It remains only to verify that the equiva-lences in theinfin-cosmos of isofibrations are created by the forgetful functor119974119861 rarr119974 First note thatif 119891∶ 119864 rarr 119865 defines an equivalence in119974 then for any isofibration 119904 ∶ 119860 ↠ 119861 the induced equivalenceon functor-spaces in119974 pulls back to define an equivalence on corresponding functor spaces in119974119861

Fun119861(119860 119864) Fun(119860 119864)

Fun119861(119860 119865) Fun(119860 119865)

120793 Fun(119860 119861)

119891lowast119901lowast sim

119891lowast

119902lowast

119903

This can be verified either by appealing to Lemmas 1210 and 1212 and using standard techniquesfrom simplicial homotopy theorysup1⁵ or by appealing to Lemma 1215 and using the fact that pullbackalong 119903 defines a simplicial functor

For the converse implication we appeal to Lemma 1215 If 119891∶ 119864 rarr 119865 is an equivalence in 119974119861then it admits a homotopy inverse in119974119861 The inverse equivalence 119892∶ 119865 rarr 119864 also defines an inverseequivalence in119974 and the required simplicial homotopies in119974 are defined by composing

119864 120128 ⋔119861 119864 119864120128 119865 120128 ⋔119861 119865 rarr 119865120128120572 120573

with the top horizontal leg of the pullback defining the cotensor in119974119861

1218 Definition (cartesian closed infin-cosmoi) An infin-cosmos 119974 is cartesian closed if the productbifunctor minus times minus∶ 119974 times119974 rarr 119974 extends to a simplicially enriched two-variable adjunction

Fun(119860 times 119861119862) cong Fun(119860 119862119861) cong Fun(119861 119862119860)in which the right adjoint (minus)119860 preserve the class of isofibrations

For instance theinfin-cosmos of quasi-categories is cartesian closed with the exponentials definedas (special cases of) simplicial cotensors This is one of the reasons that we use the same notation forcotensor and for exponentialsup1⁶

1219 Example (infin-cosmoi of (infin 1)-categories) The following models of (infin 1)-categories definecartesian closedinfin-cosmoi

(i) Rezkrsquos complete Segal spaces define the objects of aninfin-cosmos 119966119982119982 in which the isofibra-tions equivalences and trivial fibrations are the corresponding classes of the model structureof [45]sup1⁷

(ii) The Segal categories defined byDwyer Kan and Smith [18] and developed byHirschowitz andSimpson [23] define the objects of aninfin-cosmos119982119890119892119886119897 in which the isofibrations equivalencesand trivial fibrations are the corresponding classes of the model structure of [43] and [3]sup1⁸

(iii) The 1-complicial sets of [61] equivalently the ldquonaturally marked quasi-categoriesrdquo of [36] de-fine the objects of an infin-cosmos 1-119966119900119898119901 in which the isofibrations equivalences and trivialfibrations are the corresponding classes of the model structure from either of these sources

Proofs of these facts can be found in Appendix E

Appendix E also proves that certain models of (infin 119899)-categories or even (infininfin)-categories defineinfin-cosmoi

sup1⁵In more detail any functor between the 1-categories underlying infin-cosmoi that preserves trivial fibrations alsopreserves equivalences

sup1⁶Another reason for this convenient notational conflation will be explained in sect23sup1⁷Warning the model category of complete Segal spaces is enriched over simplicial sets in two distinct ldquodirectionsrdquo

mdash one enrichment makes the simplicial set of maps between two complete Segal spaces into a Kan complex that probesthe ldquospacialrdquo structure while another enrichment makes the simplicial set of maps into a quasi-category that probes theldquocategoricalrdquo structure [26] It is this latter enrichment that we want

sup1⁸Here we reserve the term ldquoSegal categoryrdquo for those simplicial objects with a discrete set of objects that are Reedyfibrant and satisfy the Segal condition The traditional definition does not include the Reedy fibrancy condition becauseit is not satisfied by the simplicial object defined as the nerve of a Kan complex enriched category Since Kan complexenriched categories are not among our preferred models of (infin 1)-categories this does not bother us

1220 Example (119966119886119905 as an infin-cosmos) The category 119966119886119905 of 1-categories defines a cartesian closedinfin-cosmos inheriting its structure as a full subcategory 119966119886119905 119980119966119886119905 of the infin-cosmos of quasi-categories via the nerve embedding which preserves all limits and also exponentials the nerve of thefunctor category 119861119860 is the exponential of the nerves

In theinfin-cosmos of categories the isofibrations are the isofibrations functors satisfying the dis-played right lifting property

120793 119860

120128 119861

119891

The equivalences are the equivalences of categories and the trivial fibrations are surjective equiva-lences equivalences of categories that are also surjective on objects

1221 Definition (discrete infin-categories) An object 119864 in an infin-cosmos 119974 is discrete just when forall 119883 isin 119974 the functor-space Fun(119883 119864) is a Kan complex

In theinfin-cosmos of quasi-categories the discrete objects are exactly the Kan complexes by Propo-sition the Kan complexes also define an exponential ideal in the category of simplicial sets Similarlyin theinfin-cosmoi of Example 1219 whoseinfin-categories are (infin 1)-categories in some model the dis-crete objects are theinfin-groupoids

1222 Proposition (infin-cosmos of discrete objects) The full subcategory 119967119894119904119888(119974) 119974 spanned bythe discrete objects in anyinfin-cosmos form aninfin-cosmos

Proof We first establish this result for the infin-cosmos of quasi-categories By Proposition 1114an isofibration between Kan complexes is a Kan fibration a map with the right lifting property withrespect to all horn inclusions Conversely all Kan fibrations define isofibrations Since Kan com-plexes are closed under simplicial cotensor (which coincides with exponentiation) It follows that thefull subcategory 119974119886119899 119980119966119886119905 is closed under all of the limit constructions of axiom 121(i) Theremaining axiom 121(ii) is inherited from the analogous properties established for quasi-categoriesin Proposition 128

In a generic infin-cosmos 119974 we need only show that the discrete objects are closed in 119974 underthe limit constructions of 121(i) The definition natural isomorphism (126) characterizing thesesimplicial limits expresses the functor-space Fun(119883 lim119895isin119869119860119895) as an analogous limit of functor spaceFun(119883119860119895) If each 119860119895 is discrete then these objects are Kan complexes and the previous paragraphthen establishes that the limit is a Kan complex as well This holds for all objects 119883 isin 119974 so it followsthat lim119895isin119869119860119895 is discrete as required

Exercises

12i Exercise Prove that the following are equivalent(i) a simplicial category as in 122(ii) a category enriched over simplicial sets

12ii Exercise Elaborate on the proof of Proposition 128 by proving that the simplicially enrichedcategory 119980119966119886119905 admits conical products satisfying the universal property of Digression 124 That is

(i) For quasi-categories 119860119861119883 form the cartesian product 119860times 119861 and prove that the projectionmaps 120587119860 ∶ 119860 times 119861 rarr 119860 and 120587119861 ∶ 119860 times 119861 rarr 119861 induce an isomorphism of quasi-categories

(119860 times 119861)119883 119860119883 times 119861119883cong

(ii) Explain how this relates to the universal property of Digression 124(iii) Express the usual 1-categorical universal property of the product119860times119861 as the ldquo0-dimensional

aspectrdquo of the universal property of (i)

12iii Exercise Prove that any object in aninfin-cosmos has a path object

119861120128

119861 119861 times 119861

(ev0ev1)

constructed by cotensoring with the free-living isomorphism

12iv Exercise(i) Use Exercise 11iv and results from Appendix D to prove that a quasi-category 119876 is a Kan

complex if and only if the map119876120128 ↠ 119876120794 induced by the inclusion 120794 120128 is a trivial fibration(ii) Conclude that aninfin-category 119860 is discrete if and only if 119860120128 ⥲rarr 119860120794 is a trivial fibration

13 Cosmological functors

Certain ldquoright adjoint typerdquo constructions define maps between infin-cosmoi that preserve all ofthe structures axiomatized in Definition 121 The simple observation that such constructions definecosmological functors betweeninfin-cosmoi will streamline many proofs

131 Definition (cosmological functor) A cosmological functor is a simplicial functor 119865∶ 119974 rarr ℒthat preserves the specified classes of isofibrations and all of the simplicial limits enumerated in 121(i)

132 Lemma Any cosmological functor also preserves the equivalences and the trivial fibrations

Proof By Lemma 1215 the equivalences in an infin-cosmos coincide with the homotopy equiva-lences defined relative to cotensoring with the free-living isomorphism Since a cosmological functorpreserves simplicial cotensors it preserves the data displayed in (1216) and hence carries equivalencesto equivalences The statement about trivial fibrations follows

In general cosmological functors preserve any infin-categorical notion that can be characterizedinternally to theinfin-cosmos mdash for instance as a property of a certain map mdash as opposed to externallymdash for instance in a statement that involves a universal quantifier From Definition 1221 it is notclear whether cosmological functors preserve discrete objects but using the internal characterizationof Exercise 12iv mdash aninfin-category 119860 is discrete if and only if 119860120128 ⥲rarr 119860120794 is at trivial fibration mdash thisfollows easily cosmological functors preserve simplicial cotensors and trivial fibrations

133 Proposition(i) For any object 119860 in aninfin-cosmos119974 Fun(119860 minus) ∶ 119974 rarr 119980119966119886119905 defines a cosmological functor(ii) Specializing eachinfin-cosmos has an underlying quasi-category functor

(minus)0 ≔ Fun(1 minus) ∶ 119974 rarr 119980119966119886119905

(iii) For anyinfin-cosmos 119974 and any simplicial set 119880 the simplicial cotensor defines a cosmological functor(minus)119880 ∶ 119974 rarr 119974

(iv) For any object 119860 in a cartesian closed infin-cosmos 119974 exponentiation defines a cosmological functor(minus)119860 ∶ 119974 rarr 119974

(v) For any map 119891∶ 119860 rarr 119861 in aninfin-cosmos119974 pullback defines a cosmological functor 119891lowast ∶ 119974119861 rarr119974119860(vi) For any cosmological functor119865∶ 119974 rarr ℒ and any119860 isin 119974 the inducedmap on slices119865∶ 119974119860 rarr ℒ119865119860

defines a cosmological functor

Proof The first four of these statements are nearly immediate the preservation of isofibrationsbeing asserted explicitly as a hypothesis in each case and the preservation of limits following fromstandard categorical arguments

For (v) pullback in aninfin-cosmos119974 is a simplicially enriched limit construction one consequenceof this is that 119891lowast ∶ 119974119861 rarr119974119860 defines a simplicial functor The action of the functor 119891lowast on a 0-arrow119892 in119974119861 is also defined by a pullback square since the front and back squares in the displayed diagramare pullbacks the top square is as well

119891lowast119864 119864

119891lowast119865 119865

119860 119861

119891lowast(119892)

119901

119892

119902

119891

Since isofibrations are stable under pullback it follows that 119891lowast ∶ 119974119861 rarr 119974119860 preserves isofibrationsIt remains to prove that this functor preserves the simplicial limits constructed in Proposition 1217In the case of connected limits which are created by the forgetful functors to 119974 this is clear Forproducts and simplicial cotensors this follows from the commutative cubes

times119860119894 119891lowast119864119894 prod119894 119891

lowast119864119894 119880 ⋔119860 119891lowast(119901) (119891lowast119864)119880

times119861119894 119864119894 prod119894 119864119894 119880 ⋔119861 119901 119864119880

119860 prod119894119860 119860 119860119880

119861 prod119894 119861 119861 119861119880

(119891lowast119901)119880

119891 prod119894 119891

119891 119891119880

119901119880

Since the front back and right faces are pullbacks the left is as well which is what we wanted toshow

The final statement (vi) is left as Exercise 13i

134 Non-Example The forgetful functor119974119861 rarr119974 is simplicial and preserves the class of isofibra-tions but does not define a cosmological functor failing to preserve cotensors and products Howeverby Proposition 133(v) minus times 119861∶ 119974 rarr 119974119861 does define a cosmological functor

135 Definition (biequivalences) A cosmological functor defines a biequivalence 119865∶ 119974 ⥲ ℒ if ad-ditionally it

(i) is essentially surjective on objects up to equivalence for all 119862 isin ℒ there exists119860 isin 119974 so that119865119860 ≃ 119862 and

(ii) it defines a local equivalence for all 119860119861 isin 119974 the action of 119865 on functor quasi-categoriesdefines an equivalence

Fun(119860 119861) Fun(119865119860 119865119861)sim

136 Remark Cosmological biequivalences will be studied more systematically in Chapter 13 wherewe think of them as ldquochange-of-modelrdquo functors A basic fact is that any biequivalence of infin-cosmoinot only preserves equivalences but also creates them a pair of objects in aninfin-cosmos are equivalentif and only if their images in any biequivalentinfin-cosmos are equivalent (Exercise 13ii) It follows thatthe cosmological biequivalences satisfy the 2-of-3 property

137 Example (biequivalences betweeninfin-cosmoi of (infin 1)-categories)(i) The underlying quasi-category functors defined on the infin-cosmoi of complete Segal spaces

Segal categories and 1-complicial sets

119966119982119982 119980119966119886119905 119982119890119892119886119897 119980119966119886119905 1-119966119900119898119901 119980119966119886119905sim(minus)0 sim(minus)0 sim(minus)0

are all biequivalences In the first two cases these are defined by ldquoevaluating at the 0th rowrdquoand in the last case this is defined by ldquoforgetting the markingsrdquo

(ii) There is also a cosmological biequivalence 119980119966119886119905 ⥲ 119966119982119982 defined by Joyal and Tierney [26](iii) The functor 119966119982119982 ⥲ 119982119890119892119886119897 defined by Bergner [5] that ldquodiscretizesrdquo a complete Segal spaces

also defines a cosmological biequivalence(iv) There is a further cosmological biequivalence (minus) ∶ 119980119966119886119905 rarr 1-119966119900119898119901 that gives each quasi-

category its ldquonatural markingrdquo with all invertible 1-simplices and all simplices in dimensiongreater than 1 marked

Exercises

13i Exercise Prove that for any cosmological functor 119865∶ 119974 rarr ℒ and any 119860 isin 119974 the inducedmap 119865∶ 119974119860 rarr ℒ119865119860 defines a cosmological functor

13ii Exercise Let 119865∶ 119974 ⥲ ℒ be a cosmological biequivalence and let 119860119861 isin 119974 Sketch a proofthat if 119865119860 ≃ 119865119861 inℒ then 119860 ≃ 119861 in119974 (and see Exercise 14i)

14 The homotopy 2-category

Small 1-categories define the objects of a strict 2-categorysup1⁹119966119886119905 of categories functors and naturaltransformations Many basic categorical notions mdash those defined in terms of categories functorsand natural transformations and their various composition operations mdash can be defined internallyto the 2-category 119966119886119905 This suggests a natural avenue for generalization reinterpreting these same

sup1⁹A comprehensive introduction to strict 2-categories appears as Appendix B Succinctly in parallel with Digression122 2-categories can be understood equally asbull ldquotwo-dimensionalrdquo categories with objects 0-arrows (typically called 1-cells) and 1-arrows (typically called 2-cells)bull or as categories enriched over 119966119886119905

definitions in a generic 2-category using its objects in place of small categories its 1-cells in place offunctors and its 2-cells in place of natural transformations

In Chapter 2 we will develop a non-trivial portion of the theory of infin-categories in any fixedinfin-cosmos following exactly this outline working internally to a strict 2-category that we refer to asthe homotopy 2-category that we associate to anyinfin-cosmos The homotopy 2-category of aninfin-cosmosis a quotient of the fullinfin-cosmos replacing each quasi-categorical functor-space by its homotopy cat-egory Surprisingly this rather destructive quotienting operation preserves quite a lot of informationIndeed essentially all of the work in Part I will take place in the homotopy 2-category of aninfin-cosmosThis said we caution the reader against becoming overly seduced by homotopy 2-categories for thatstructure is more of a technical convenience for reducing the complexity of our arguments than afundamental notion ofinfin-category theory

The homotopy 2-category for the infin-cosmos of quasi-categories was first introduced by Joyal inhis work on the foundations of quasi-category theory

141 Definition (homotopy 2-category) Let 119974 be an infin-cosmos Its homotopy 2-category is thestrict 2-category 120101119974 whosebull objects are theinfin-categories ie the objects 119860119861 of119974bull 1-cells 119891∶ 119860 rarr 119861 are the 0-arrows in the functor space Fun(119860 119861) of119974 ie theinfin-functors and

bull 2-cells 119860 119861119891

119892dArr120572 are homotopy classes of 1-simplices in Fun(119860 119861) which we call infin-natural

transformationsPut another way 120101119974 is the 2-category with the same objects as119974 and with hom-categories defined by

hFun(119860 119861) ≔ h(Fun(119860 119861))that is as the homotopy category of the quasi-category Fun(119860 119861)

The underlying category of a 2-category is defined by simply forgetting its 2-cells Note that aninfin-cosmos 119974 and its homotopy 2-category 120101119974 share the same underlying category of infin-categoriesandinfin-functors in119974

142 Digression The homotopy category functor h ∶ 119982119982119890119905 rarr 119966119886119905 preserves finite products as ofcourse does its right adjoint It follows that the adjunction of Proposition 1111 induces a change-of-base adjunction

2-119966119886119905 119982119982119890119905-119966119886119905perp

hlowast

whose left and right adjoints change the enrichment by applying the homotopy category functor orthe nerve functor to the hom objects of the enriched category Here 2-119966119886119905 and 119982119982119890119905-119966119886119905 can each beunderstood as 2-categories mdash of enriched categories enriched functors and enriched natural trans-formations mdash and both change of base constructions define 2-functors [10 643]

143 Observation (functors representing (invertible) 2-cells) By definition 2-cells in the homotopycategory 120101119974 are represented by maps 120794 rarr Fun(119860 119861) valued in the appropriate functor space andtwo such maps represent the same 2-cell if and only if their images are homotopic as 1-simplices inFun(119860 119861) in the sense defined by Lemma 119

Now a 2-cell in a 2-category is invertible if and only if it defines an isomorphism in the appropriatehom-category hFun(119860 119861) By Definition 1113 and Corollary 1116 it follows that each invertible2-cell in 120101119974 is represented by a map 120128 rarr Fun(119860 119861)

144 Lemma Any simplicial functor 119865∶ 119974 rarr ℒ between infin-cosmoi induces a 2-functor 119865∶ 120101119974 rarr 120101ℒbetween their homotopy 2-categories

Proof This follows immediately from the remarks on change of base in Digression 142 but wecan also argue directly The action of the induced 2-functor 119865∶ 120101119974 rarr 120101ℒ on objects and 1-cells isgiven by the corresponding action of 119865∶ 119974 rarr ℒ recall an infin-cosmos and its homotopy 2-categoryhave the same underlying 1-category Each 2-cell in 120101119974 is represented by a 1-simplex in Fun(119860 119861)which is mapped via

Fun(119860 119861) Fun(119865119860 119865119861)119865

119860 119861 119865119860 119865119861119891

119892dArr120572

119865119891

119865119892

dArr119865120572

to a 1-simplex representing a 2-cell in 120101ℒ Since the action 119865∶ Fun(119860 119861) rarr Fun(119865119860 119865119861) on functorspaces defines a morphism of simplicial sets it preserves faces and degeneracies In particular homo-topic 1-simplices in Fun(119860 119861) are carried to homotopic 1-simplices in Fun(119865119860 119865119861) so the action on2-cells just described is well-defined The 2-functoriality of these mappings follows from the simplicialfunctoriality of the original mapping

We now begin to relate the simplicially enriched structures of an infin-cosmos to the 2-categoricalstructures in its homotopy 2-category The first result proves that homotopy 2-categories inheritproducts from their infin-cosmoi which satisfy a 2-categorical universal property To illustrate recallthat the terminal infin-category 1 isin 119974 has the universal property Fun(119883 1) cong 120793 for all 119883 isin 119974Applying the homotopy category functor we see that 1 isin 120101119974 has the universal property hFun(119883 1) cong120793 for all119883 isin 120101119974 This 2-categorical universal property has both a 1-dimensional and a 2-dimensionalaspect Since hFun(119883 1) cong 120793 is a category with a single object there exists a unique morphism119883 rarr 1in 119974 And since hFun(119883 1) cong 120793 has only a single identity morphism we see that the only 2-cells in120101119974 with codomain 1 are identities

145 Proposition (cartesian (closure))(i) The homotopy 2-category of anyinfin-cosmos has 2-categorical products(ii) The homotopy 2-category of a cartesian closedinfin-cosmos is cartesian closed as a 2-category

Proof While the functor h ∶ 119982119982119890119905 rarr 119966119886119905 only preserves finite products the restricted functorh ∶ 119980119966119886119905 rarr 119966119886119905 preserves all products on account of the simplified description of the homotopycategory of a quasi-category given in Lemma 1112 Thus for any set 119868 and family of infin-categories(119860119894)119894isin119868 in 119974 the homotopy category functor carries the isomorphism of quasi-categories displayedbelow left to an isomorphisms of hom-categories displayed below right

Fun(119883prod119894isin119868119860119894) prod119894isin119868 Fun(119883119860119894) hFun(119883prod119894isin119868119860119894) prod

119894isin119868 hFun(119883119860119894)cong h cong

This proves that the homotopy 2-category 120101119974 has products whose universal properties have both a 1-and 2-dimensional component as described for terminal objects above

If 119974 is a cartesian closed infin-cosmos then for any triple of infin-categories 119860119861 119862 isin 119974 there existexponential objects 119862119860 119862119861 isin 119974 characterized by natural isomorphisms

Fun(119860 times 119861119862) cong Fun(119860 119862119861) cong Fun(119861 119862119860)Passing to homotopy categories we have natural isomorphisms

hFun(119860 times 119861119862) cong hFun(119860 119862119861) cong hFun(119861 119862119860)which demonstrates that 120101119974 is cartesian closed as a 1-category functors 119860 times 119861 rarr 119862 transpose todefine functors 119860 rarr 119862119861 and 119861 rarr 119862119860 and 2-cells transpose similarly

There is a standard definition of isomorphism between two objects in any 1-category Similarlythere is a standard definition of equivalence between two objects in any 2-category

146 Definition (equivalence) An equivalence in a 2-category is given bybull a pair of objects 119860 and 119861bull a pair of 1-cells 119891∶ 119860 rarr 119861 and 119892∶ 119861 rarr 119860bull a pair of invertible 2-cells

119860 119860 119861 119861119892119891

congdArr120572

119891119892

congdArr120573

When119860 and119861 are equivalent we write119860 ≃ 119861 and refer to the 1-cells 119891 and 119892 as equivalences denotedby ldquo⥲rdquo

In the case of the homotopy 2-category of aninfin-cosmos we have a competing definition of equiva-lence from 121 namely a 1-cell 119891∶ 119860 ⥲ 119861 that induces an equivalence 119891lowast ∶ Fun(119883119860) ⥲ Fun(119883 119861)on functor-spaces mdash or equivalently by Lemma 1215 a homotopy equivalence defined relative to theinterval 120128 Crucially these two notions of equivalence coincide

147 Theorem (equivalences are equivalences) A functor 119891∶ 119860 rarr 119861 between infin-categories defines anequivalence in theinfin-cosmos119974 if and only if it defines an equivalence in its homotopy 2-category 120101119974

Proof Given an equivalence 119891∶ 119860 ⥲ 119861 in the infin-cosmos 119974 Lemma 1215 provides an inverseequivalence 119892∶ 119861 ⥲ 119860 and homotopies 120572∶ 119860 rarr 119860120128 and 120573∶ 119861 rarr 119861120128 in119974 By Observation 143 thisdata represents an equivalence in the homotopy 2-category 120101119974

For the converse first note that if parallel 1-cells ℎ 119896 ∶ 119860 119861 in the homotopy 2-category areconnected by an invertible 2-cell as displayed below left then ℎ is an equivalence in theinfin-cosmos119974if and only if 119896 is

119860 119861ℎ

119896

congdArr120574119860

119861 119861120128 119861

120574ℎ 119896

simev0

simev1

By Observation 143 the invertible 2-cell can be represented by a map 120574∶ 119860 rarr 119861120128 in119974 as displayedabove right now apply the 2-of-3 property for the equivalences in119974

Now it follows immediately from the 2-of-6 property of the equivalences in anyinfin-cosmos estab-lished in Remark 1214 and the fact that the equivalences contain the identities that any 2-categorical

equivalence defines an equivalence in the infin-cosmos since 119892119891 and 119891119892 are isomorphic to identitiesthey must be equivalences in119974 and hence so must 119891 and 119892

148 Digression (on the importance of Theorem 147) It is hard to overstate the importance ofTheorem 147 to the work that follows The categorical constructions that we will introduce forinfin-categories infin-functors and infin-natural transformations are invariant under 2-categorical equiva-lence in the homotopy 2-category and the universal properties we develop similarly characterize a2-categorical equivalence class of infin-categories Theorem 147 then asserts that such constructionsare ldquohomotopically correctrdquo both invariant under equivalence in theinfin-cosmos and precisely identi-fying equivalence classes of objects

The equivalence invariance of the functor space in the codomain variable is axiomatic but equiv-alence invariance in the domain variable is notsup2⁰ But using 2-categorical techniques there is now ashort proof

149 Corollary Equivalences of infin-categories 119860prime ⥲ 119860 and 119861 ⥲ 119861prime induce an equivalence of functorspaces Fun(119860 119861) ⥲ Fun(119860prime 119861prime)

Proof The simplicial functors Fun(119860 minus) ∶ 119974 rarr 119980119966119886119905 and Fun(minus 119861) ∶ 119974op rarr 119980119966119886119905 induce2-functors Fun(119860 minus) ∶ 120101119974 rarr 120101119980119966119886119905 and Fun(minus 119861) ∶ 120101119974op rarr 120101119980119966119886119905 which preserve the 2-categor-ical equivalences of Definition 146 By Theorem 147 this is what we wanted to show

Similarly there is a standard 2-categorical notion of an isofibration defined in the statementof Proposition 1410 and any isofibration in an infin-cosmos defines an isofibration in its homotopy2-categorysup2sup1

1410 Proposition (isofibrations define isofibrations) Any isofibration 119901∶ 119864 ↠ 119861 in an infin-cosmos119974 also defines an isofibration in the homotopy 2-category 120101119974 given any invertible 2-cell as displayed belowleft abutting to 119861 with a specified lift of one of its boundary 1-cells through 119901 there exists an invertible 2-cellabutting to 119864 with this boundary 1-cell as displayed below right that whiskers with 119901 to the original 2-cell

119883 119864 119883 119864

119861 119861

119890

119887

119901congdArr120573 =

119890

congdArr120574

119901

Proof Put another way the universal property of the statement says that the functor

119901lowast ∶ hFun(119883 119864) ↠ hFun(119883 119861)is an isofibration of categories in the sense defined in Example 1220 By axiom 121(ii) since 119901∶ 119864 ↠119861 is an isofibration in 119974 the induced map 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861) is an isofibration of quasi-categories So it suffices to show that the functor h ∶ 119980119966119886119905 rarr 119966119886119905 carries isofibrations of quasi-categories to isofibrations of categoriessup2sup2

sup2⁰Lemma 132 does not apply since Fun(minus 119861) is not cosmologicalsup2sup1In this case the converse does not hold nor is it the case that a representably-defined isofibration of quasi-categories

is necessarily an isofibration in theinfin-cosmos consider the case of slicedinfin-cosmoi for instancesup2sup2Alternately argue directly using Observation 143

So let us now consider an isofibration 119901∶ 119864 ↠ 119861 between quasi-categories By Corollary 1116 ev-ery isomorphism 120573 in the homotopy category h119861 of the quasi-category 119861 is represented by a simplicialmap 120573∶ 120128 rarr 119861 By Definition 1117 the lifting problem

120793 119864

120128 119861

119890

119901120574

120573

can be solved and the map 120574∶ 120128 rarr 119864 so-produced represents a lift of the isomorphism from h119861 to anisomorphism in h119864 with domain 119890

1411 Convention (on ldquoisofibrationsrdquo in homotopy 2-categories) Since the converse to Proposition1410 does not hold there is a potential ambiguity when using the term ldquoisofibrationrdquo to refer to a mapin the homotopy 2-category of an infin-cosmos We adopt the convention that when we declare that amap in 120101119974 is an isofibration we always mean this is the stronger sense of defining an isofibration in119974 This stronger condition gives us access to the 2-categorical lifting property of Proposition 1410but also to the many homotopical properties axiomatized in Definition 121 which guarantee that thestrictly defined limits of 121(i) are automatically equivalence invariant constructions

The 1- and 2-cells in the homotopy 2-category from the terminal infin-category 1 isin 119974 to a genericinfin-category 119860 isin 119974 define the objects and morphisms in the homotopy category of 119860

1412 Definition (homotopy category of aninfin-category) The homotopy category of aninfin-category119860 in aninfin-cosmos119974 is defined to be the homotopy category of its underlying quasi-category that is

h119860 ≔ hFun(1 119860) ≔ h(119865119906119899(1119860))

As we shall discover homotopy categories generally bear ldquoderivedrdquo analogues of structures presentat the level ofinfin-categories See the remark after the statement Proposition 217 for an early exampleof this

Exercises

14i Exercise Let 119865∶ 119974 ⥲ ℒ be a cosmological biequivalence and let 119860119861 isin 119974 Prove that if119865119860 ≃ 119865119861 in ℒ then 119860 ≃ 119861 in 119974 and ruminate on why this exercise is considerably easier thanExercise 13ii)

14ii Exercise(i) What is the homotopy 2-category of theinfin-cosmos 119966119886119905 of 1-categories(ii) Prove that the nerve defines a 2-functor 119966119886119905 120101119980119966119886119905 that is locally fully faithful

14iii Exercise Let 119861 be an infin-category in the infin-cosmos 119974 and let 120101119974119861 denote the 2-categorywhosebull objects are isofibrations 119864 ↠ 119861 in119974 with codomain 119861bull 1-cells are 1-cells in 120101119974 over 119861

119864 119865

119861

bull 2-cells are 2-cells in 120101119974 over 119861

119864 119865

119861119901

119891

119892

dArr120572

119902

in the sense that 119902120572 = id119901Argue that the homotopy 2-category 120101(119974119861) of the slicedinfin-cosmos has the same underlying 1-categorybut different 2-cells How do these compare with the 2-cells of 120101119974119861sup2sup3

sup2sup3A more systematic comparison will be given in Proposition 353

CHAPTER 2

Adjunctions limits and colimits I

Heuristicallyinfin-categories generalize ordinary 1-categories by adding in higher dimensional mor-phisms and weakening the composition law The dream is that proofs establishing the theory of1-categories similarly generalize to give proofs for infin-categories just by adding a prefix ldquoinfin-rdquo every-where In this chapter we make this dream a reality mdash at least for a library of basic propositionsconcerning equivalences adjunctions limits and colimits and the relationships between these no-tions

Recall that categories functors and natural transformations assemble into a 2-category119966119886119905 Sim-ilarly theinfin-categoriesinfin-functors andinfin-natural transformations in anyinfin-cosmos assemble intoa 2-category namely the homotopy 2-category of the infin-cosmos introduced in sect14 By Exercise 14ii119966119886119905 can be regarded as a special case of a homotopy 2-category In this chapter we will use strict2-categorical techniques to define adjunctions betweeninfin-categories and limits and colimits of diagramsvalued in aninfin-category and prove that these notions interact in the expected ways In the homotopy2-category of categories these recover the classical results from 1-category theory As these proofs areequally valid in any homotopy 2-category our arguments also establish the desired generalizations bysimply appending the prefix ldquoinfin-rdquo

21 Adjunctions and equivalences

In sect14 we encountered the definition of an equivalence between a pair of objects in a 2-categoryIn the case where the ambient 2-category is the homotopy 2-category of aninfin-cosmos we observed inTheorem 147 that the 2-categorical notion of equivalence precisely recaptures the notion of equiva-lence introduced in Definition 121 betweeninfin-categories in the fullinfin-cosmos In each of the exam-ples ofinfin-cosmoi we have considered the representably-defined equivalences in theinfin-cosmos coin-cide with the standard notion of equivalences between infin-categories as presented in that particularmodelsup1 Thus the 2-categorical notion of equivalence is the ldquocorrectrdquo notion of equivalence betweeninfin-categories

Similarly there is a standard definition of an adjunction between a pair of objects in a 2-categorywhich when interpreted in the homotopy 2-category ofinfin-categories functors and natural transfor-mations in aninfin-cosmos will define the correct notion of adjunction betweeninfin-categories

211 Definition (adjunction) An adjunction betweeninfin-categories is comprised ofbull a pair ofinfin-categories 119860 and 119861bull a pair of functors 119906∶ 119860 rarr 119861 and 119891∶ 119861 rarr 119860bull and a pair of natural transformations 120578∶ 1119861 rArr 119906119891 and 120598 ∶ 119891119906 rArr 1119860 called the unit and counit

respectively

sup1For instance as outlined in Digression 129 the equivalences in theinfin-cosmoi of Example 1219 recapture the weakequivalences between fibrant-cofibrant objects in the usual model structure

so that the triangle equalities holdsup2

119861 119861 119861 119861 119861 119861

119860 119860 119860 119860 119860 119860dArr120598 119891 dArr120578 = =

119891dArr120578 dArr120598

119891 = = 119891119891119906119906

119906 119906 119906

The functor 119891 is called the left adjoint and 119906 is called the right adjoint a relationship that is denotedsymbolically in text by writing 119891 ⊣ 119906 or in a displayed diagram such assup3

119860 119861119906perp119891

212 Digression (why this is the right definition) For readers who find Definition 211 implausiblemdash perhaps too simple to be trusted mdash we offer a few words of justification Firstly the correct notionof adjunction between quasi-categories is well established though the definition appearing in [36 sect52]takes a quite different form In Appendix F we prove that in the infin-cosmos of quasi-categories ourdefinition of adjunction precisely recovers Luriersquos As explained in Part IV each of the models of(infin 1)-categories described in Example 1219 ldquohas the same category theoryrdquo so Definition 211 agreeswith the community consensus notion of adjunction between (infin 1)-categories

But what about those infin-cosmoi whose objects model (infin 119899)- or (infininfin)-categories For in-stance in the infin-cosmos of complicial sets the adjunctions defined in the homotopy 2-category arethe ldquopseudo-stylerdquo adjunctions While these are not the most general adjunctions that might be con-sidered mdash for instance one could have (op)lax units and counits mdash they are an important class ofadjunctions One reason for the relevance of Definition 211 in allinfin-cosmoi is its formal propertiesvis-a-vis the related notion of equivalence which Theorem 147 has established is morally ldquocorrectrdquoand with the notions of limits and colimits to be introduced

Finally a reasonable objection is that Definition 211 appears too ldquolow dimensionalrdquo comprisedof data found entirely in the homotopy 2-category and ignoring the higher dimensional morphismsin aninfin-cosmos This deficiency will be addressed in Chapter 8 when we prove that any adjunctionbetween infin-categories extends to a homotopy coherent adjunction and moreover such extensions arehomotopically unique

The definition of an adjunction given in Definition 211 is ldquoequationalrdquo in character stated interms of the objects 1-cells and 2-cells of a 2-category and their composites Immediately

213 Lemma Adjunctions in a 2-category are preserved by any 2-functor

Lemma 213 provides an easy source of examples of adjunctions between quasi-categories The2-functors underlying the cosmological functors of Example 137 then transfer adjunctions defined inone model of (infin 1)-categories to adjunctions defined in each of the other models

214 Example (adjunctions between 1-categories) Via the nerve embedding 119966119886119905 120101119980119966119886119905 any ad-junction between 1-categories induces an adjunction between their nerves regarded as quasi-categories

sup2The left-hand equality of pasting diagrams asserts the composition relation 119906120598 sdot 120578119906 = id119906 in the hom-categoryhFun(119860 119861) while the right-hand equality asserts that 120598119891 sdot 119891120578 = id119891 in hFun(119861119860)

sup3Some authors contort adjunction diagrams so that the left adjoint is always on the left we instead use the turnstilesymbol ldquoperprdquo to indicate which adjoint is the left adjoint

215 Example (adjunctions between topological categories) The homotopy coherent nerve defines a2-functor 120081∶ 119974119886119899-119966119886119905 rarr 120101119980119966119886119905 from the 2-category of Kan complex enriched categories simpli-cially enriched functors and simplicial natural transformations to the homotopy 2-category 120101119980119966119886119905In this way topologically enriched adjunctions define adjunctions between quasi-categories

216 Remark Topologically enriched adjunctions are relatively rare More prevalent are ldquoup-to-homotopyrdquo topologically enriched adjunctions such as those given by Quillen adjunctions betweensimplicial model categories These also define adjunctions between quasi-categories though the proofwill have to wait until Part II

The preservation of adjunctions by 2-functors proves

217 Proposition Given any adjunction 119860 119861119906perp119891

betweeninfin-categories then

(i) for anyinfin-category 119883

Fun(119883119860) Fun(119883 119861)

119906lowast

119891lowast

defines an adjunction between quasi-categories(ii) for anyinfin-category 119883

hFun(119883119860) hFun(119883 119861)

119906lowast

119891lowast

defines an adjunction between categories(iii) for any simplicial set 119880

119860119880 119861119880

119906119880

perp119891119880

defines an adjunction betweeninfin-categories(iv) and if the ambientinfin-cosmos is cartesian closed then for anyinfin-category 119862

119860119862 119861119862

119906119862

perp119891119862

defines an adjunction betweeninfin-categories

For instance taking 119883 = 1 in (ii) yields a ldquoderivedrdquo adjunction between the homotopy categoriesof theinfin-categories 119860 and 119861

Proof Any adjunction 119891 ⊣ 119906 in the homotopy 2-category 120101119974 is preserved by the 2-functorsFun(119883 minus) ∶ 120101119974 rarr 120101119980119966119886119905 hFun(119883 minus) ∶ 120101119974 rarr 119966119886119905 (minus)119880 ∶ 120101119974 rarr 120101119974 and (minus)119862 ∶ 120101119974 rarr 120101119974

218 Remark There are contravariant versions of each of the adjunction-preservation results ofProposition 217 the first of which we explain in detail Fixing the codomain variable of the functor-space at anyinfin-category 119862 isin 119974 defines a 2-functor

Fun(minus 119862) ∶ 120101119974op rarr 120101119980119966119886119905that is contravariant on 1-cells and covariant on 2-cells⁴ Similarly the cotensor or exponential 119862(minus) iscontravariant on 1-cells and covariant on 2-cells⁵ Such 2-functors preserve adjunctions but exchangeleft and right adjoints for instance given 119891 ⊣ 119906 in119974 we obtain an adjunction

Fun(119860 119862) Fun(119861 119862)

119891lowast

119906lowast

between the functor- spaces

219 Proposition Adjunctions compose given adjoint functors

119862 119861 119860 119862 119860119891prime

perp119891

perp119906prime 119906

119891119891prime

perp119906prime119906

the composite functors are adjoint

Proof Writing 120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860 120578prime ∶ idc rArr 119906prime119891prime and 120598prime ∶ 119891prime119906prime rArr id119861 for therespective units and counits the pasting diagrams

119862 119862 119862

119861 119861 119861 119861

119860 119860 119860

119891prime dArr120578prime dArr120598prime119891prime

119891dArr120578

119906prime119906prime

dArr120598119891

119906119906

define the unit and counit of 119891119891prime ⊣ 119906prime119906 so that the triangle equalities

119862 119862 119862 119862 119862 119862

119861 119861 119861 119861 119861 119861

119860 119860 119860 119860 119860 119860

119891119891prime119891119891prime

=dArr120598prime 119891

primedArr120578prime

119891dArr120578

119906prime

dArr120598119891

119906prime

dArr120598 119891 dArr120578

119906prime

=119906 119906

119906

119906prime119906 119906prime119906=

⁴On a strict 2-category the superscript ldquooprdquo is used to signal that the 1-cells should be reversed but not the 2-cells thesuperscript ldquocordquo is used to signal that the 2-cells should be reversed but not the 1-cells and the superscript ldquocooprdquo is usedto signal that both the 1- and 2-cells should be reversed see Chapter B

⁵In the case of the simplicial cotensor the domain can safely be restricted to the homotopy 2-category of quasi-categories or can be regarded as an analogously-defined homotopy 2-category of simplicial sets

An adjoint to a given functor is unique up to natural isomorphism

2110 Proposition (uniqueness of adjoints)(i) If 119891 ⊣ 119906 and 119891prime ⊣ 119906 then 119891 cong 119891prime(ii) Conversely if 119891 ⊣ 119906 and 119891 cong 119891prime then 119891prime ⊣ 119906

Proof Writing 120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860 120578prime ∶ idc rArr 119906119891prime and 120598prime ∶ 119891prime119906 rArr id119861 for therespective units and counits the pasting diagrams

119861 119861 119861 119861

119860 119860 119860 119860119891prime

dArr120578prime119891

119891dArr120578

119891prime119906 dArr120598 119906 dArr120598prime

define 2-cells 119891 rArr 119891prime and 119891prime rArr 119891 The composites 119891 rArr 119891prime rArr 119891 and 119891prime rArr 119891 rArr 119891 are computed bypasting these diagrams together horizontally on one side or the other Applying the triangle equalitiesfor the adjunctions 119891 ⊣ 119906 and 119891prime ⊣ 119906 both composites are easily seen to be identities Hence 119891 cong 119891primeas functors from 119861 to 119860

Part (ii) is left as Exercise 21i

We will make repeated use of the following standard 2-categorical result which says that anyequivalence in a 2-category can be promoted to an equivalence that also defines an adjunction

2111 Proposition (adjoint equivalences) Any equivalence can be promoted to an adjoint equivalence bymodifying one of the 2-cells That is the invertible 2-cells in an equivalence can be chosen so as to satisfy thetriangle equalities Hence if 119891 and 119892 are inverse equivalences then 119891 ⊣ 119892 and 119892 ⊣ 119891

Proof Consider an equivalence comprised of functors 119891∶ 119860 rarr 119861 and 119892∶ 119861 rarr 119860 and invertible2-cells

119860 119860 119861 119861119892119891

congdArr120572

119891119892

congdArr120573

We will construct an adjunction 119891 ⊣ 119892 with unit 120578 ≔ 120572 by modifying 120573 The ldquotriangle identitycompositerdquo

120601 ≔ 119891 119891119892119891 119891119891120572 120573119891

is an isomorphism though likely not an identity Define

120598 ≔ 119891119892 119891119892 id119861 ≔ 119891119892 119891119892119891119892 119891119892 id119861120601minus1119892 120573 120573minus1119891119892 119891120572minus1119892 120573

This ldquocorrectsrdquo the counit so that now the composite 120598119891 sdot 119891120578 displayed on the top of the diagram

119891119892119891

119891 119891119892119891 119891

119891

120601minus1119892119891120598119891119891120572

120601minus1

120573119891

119891120572120601

which agrees with the bottom composite by ldquonaturality of whiskeringrdquo is the identity id119891Now by another diagram chase the other triangle composite 119892120598 sdot 120578119892 is an idempotent

119892 119892119891119892 119892

119892119891119892 119892119891119892119891119892 119892119891119892

119892119891119892 119892

120578119892

120578119892 120578119892119891119892

119892120598

120578119892119892119891120578119892 119892119891119892120598

119892120598119891119892 119892120598

119892120598

By cancelation any idempotent isomorphism is the identity proving that 119892120598 sdot 120578119892 = id119892

One use of Proposition 2110 is to show that adjunctions are equivalence invariant

2112 Proposition (equivalence-invariance of adjunctions) A functor119906∶ 119860 rarr 119861 betweeninfin-categoriesadmits a left adjoint if and only if for any pair of equivalentinfin-categories119860prime ≃ 119860 and 119861prime ≃ 119861 the equivalentfunctor 119906∶ 119860prime rarr 119861prime admits a right adjoint

Proof Exercise 21ii

As we will discover all ofinfin-category theory is equivalence invariant in this way

2113 Lemma For anyinfin-category 119860 the ldquocompositionrdquo functor

119860120794 times119860119860120794 119860120794∘perpperp

(minusiddom(minus))

(idcod(minus)minus)

(2114)

admits left and right adjoints which respectively ldquoextend an arrow into a composable pairrdquo by pairing it withthe identities at its domain or its codomain

Proof There is a dual adjunction in119966119886119905whose functors we describe using notation for simplicialoperators introduced in 111 the full subcategory of 119966119886119905 spanned by the finite non-empty ordinals isisomorphic to 120491

120795 120794 119860120795 1198601207941199040⊤

1199041⊤1198891 1198601198891

1198601199040

1198601199041perp

Any infin-category 119860 in an infin-cosmos 119974 defines a 2-functor 119860(minus) ∶ 119966119886119905op rarr 120101119974 carrying the adjointtriple displayed above-left to the one displayed above-right

Now we claim there is a trivial fibration 119860120795 ⥲rarr 119860120794 times119860 119860120794 constructed as follows The pushoutdiagram of simplicial sets displayed below-left is carried by the simplicial cotensor119860(minus) ∶ 119982119982119890119905op rarr119974to a pullback diagram displayed below-right since the legs of the pushout square are monomorphisms

the legs of the pullback square are isofibrations

Λ1[2] 120794 119860Λ1[2] 119860120794

120794 120793 119860120794 119860

1198890

1198891

Lemma 1210 tells us that the cotensor of the inner horn inclusion Λ1[2] 120795 with the infin-category119860 defines a trivial fibration119860120795 ⥲rarr 119860Λ1[2] and the pullback square above left recognizes its codomainas the desiredinfin-category of ldquocomposable pairsrdquo Any section 119904 to 119902 ∶ 119860120795 ⥲rarr 119860120794 times119860 119860120794 can be madeinto an equivalence inverse By Proposition 2111 these functors are both left and right adjointsComposing the adjunction 119902 ⊣ 119904 ⊣ 119902 with the adjunction constructed above defines the desiredadjunction

Note that the adjoint functors of (2114) commute with the ldquoendpoint evaluationrdquo functors to119860 times 119860 In fact the units and counits can similarly be fibered over 119860 times 119860 see Example 3512

Exercises

21i Exercise Prove Proposition 2110(ii)

21ii Exercise Prove Proposition 2112 given an adjunction 119860 119861119906perp119891

and equivalences119860 ≃ 119860prime

and 119861 ≃ 119861prime construct an adjunction between 119860prime and 119861prime

22 Initial and terminal elements

Employing the tactic used to define the homotopy category of 119860 in Definition 1412 we use theterminal infin-category 1 to probe inside the infin-category 119860 The objects 119886 isin h119860 of the homotopycategory of 119860 were defined to be maps of infin-categories 119886 ∶ 1 rarr 119860 but to avoid the proliferation ofthe term ldquoobjectsrdquo we refer to maps 119886 ∶ 1 rarr 119860 as elements of theinfin-category 119860 instead

Before introducing limits and colimits of general diagram shapes we warm up by defining initialand terminal elements in aninfin-category 119860

221 Definition (initialterminal element) An initial element in an infin-category 119860 is a left adjointto the unique functor ∶ 119860 rarr 1 as displayed below left while a terminal element in an infin-category119860 is a right adjoint as displayed below right

1 119860 1 119860119894

perp 119905

Let us unpack the definition of an initial element dual remarks apply to terminal elements

222 Lemma (the minimal data required to present an initial element) To define an initial element in119860 it suffices to specifybull an element 119894 ∶ 1 rarr 119860 and

bull a natural transformation1

119860 119860119894

dArr120598

so that the component 120598119894 ∶ 119894 rArr 119894 is the identity in h119860

Proof Proposition 145 demonstrates that the infin-category 1 isin 119974 is terminal in the homotopy2-category 120101119974 The 1-dimensional aspect of this universal property implies that 119894 defines a section ofthe unique map 119860 rarr 1 and from the 2-dimensional aspects we see that there exist no non-identity2-cells with codomain 1 In particular the unit of the adjunction 119894 ⊣ is necessarily an identity andone of the triangle equalities comes for free The data enumerated above is what remains of Definition211 in this setting

Put more concisely an initial element 119894 defines a left adjoint right inverse to the functor ∶ 119860 rarr 1Such adjunctions are studied more systematically in sectB4 In fact it suffices to assume that the counitcomponent 120598119894 is an isomorphism not necessarily the identity see Lemma B41

223 Remark Applying the 2-functor Fun(119883 minus) ∶ 120101119974 rarr 120101119980119966119886119905 to an initial or terminal elementof aninfin-category 119860 isin 119974 yields adjunctions

120793 cong Fun(119883 1) Fun(119883119860) 120793 cong Fun(119883 1) Fun(119883119860)

119894lowast

119905lowast

Via the isomorphisms Fun(119883 1) cong 120793 that express the universal property of the terminalinfin-category1 we see that initial or terminal elements of119860 define initial or terminal elements of the functor-spaceFun(119883119860) namely the composite functors

119883 1 119860 or 119883 1 119860 119894 119905

In particular initial or terminal elements are representably initial or terminal at the level of theinfin-cosmos

This representable universal property is also captured at the level of the homotopy 2-category Thenext lemma shows that the initial element 119894 ∶ 1 rarr 119860 is initial among all generalized elements 119891∶ 119883 rarr 119860in the following precise sense

224 Lemma An element 119894 ∶ 1 rarr 119860 is initial if and only if for all 119891∶ 119883 rarr 119860 there exists a unique 2-cellwith boundary

119883 119860119894

dArrexist

119891

Proof If 119894 ∶ 1 rarr 119860 is initial then the adjunction of Definition 221 is preserved by the 2-functorhFun(119883 minus) ∶ 120101119974 rarr 119966119886119905 defining an adjunction

120793 cong hFun(119883 1) hFun(119883119860)

119894lowast

Via the isomorphism hFun(119883 1) cong 120793 this adjunction proves that the element 119894 ∶ 119883 rarr 119860 is initial inhFun(119883119860) and thus has the universal property of the statement

Conversely if 119894 ∶ 1 rarr 119860 satisfies the universal property of the statement applying this to thegeneric element of 119860 (the identity map id119860 ∶ 119860 rarr 119860) easily produces the data of Lemma 222

225 Remark Lemma 224 says that initial elements are representably initial in the homotopy 2-cat-egory Specializing the generalized elements to ordinary elements we see that initial and terminalelements in 119860 respectively define initial and terminal elements in the homotopy category h119860

226 Lemma If119860 has an initial element and119860 ≃ 119860prime then119860prime has an initial element and these elements arepreserved up to isomorphism by the equivalences

Proof By Proposition 2111 the equivalence119860 ≃ 119860prime can be promoted to an adjoint equivalencewhich can immediately be composed with the adjunction characterizing an initial element 119894 of 119860

1 119860 119860prime119894

simperpsim

The composite adjunction provided by Proposition 219 proves that the image of 119894 defines an initialelement of 119860prime which by construction is preserved by the equivalence 119860 ⥲ 119860prime

To see that the equivalence 119860prime ⥲ 119860 also preserves initial elements we can use the invertible2-cells of the equivalence to see that 119894 is isomorphic to the image of the image of 119894 in 119860prime In case theinitial objects in mind are not the ones being considered here we can appeal to the uniqueness ofinitial elements proven in Exercise 22ii

Exercises

22i Exercise Prove that initial elements are preserved by left adjoints and terminal elements arepreserved by right adjoints

22ii Exercise Prove that any two initial elements in aninfin-category 119860 are isomorphic in h119860

23 Limits and colimits

Our aim is now to introduce limits and colimits of diagram valued inside aninfin-category119860 in someinfin-cosmos We will consider two varieties of diagramsbull In a genericinfin-cosmos119974 we shall consider diagrams indexed by a simplicial set 119869 and valued in

aninfin-category 119860

bull In a cartesian closedinfin-cosmos119974 we shall also consider diagram indexed by aninfin-category 119869 andvalued in aninfin-category 119860⁶

231 Definition (diagraminfin-categories) For a simplicial set 119869mdash or possibly in the case of a cartesianclosed infin-cosmos an infin-category 119869 mdash and an infin-category 119860 we refer to 119860119869 as the infin-category of119869-shaped diagrams in 119860 Both constructions define bifunctors

119982119982119890119905op times119974 119974 119974op times119974 119974

(119869 119860) 119860119869 (119869 119860) 119860119869

In either indexing context there is a terminal object 1 with the property that 1198601 cong 119860 for anyinfin-category 119860 Restriction along the unique map ∶ 119869 rarr 1 induces the constant diagram functorΔ∶ 119860 rarr 119860119869

We are deliberately conflating the notation for infin-categories of diagrams indexed by a simplicialset or by another infin-category because all of the results we will prove in Part I about the former casewill also apply to the latter For economy of language we refer only to simplicial set indexed diagramsfor the remainder of this section

232 Definition An infin-category 119860 admits all colimits of shape 119869 if the constant diagram functorΔ∶ 119860 rarr 119860119869 admits a left adjoint while119860 admits all limits of shape 119869 if the constant diagram functoradmits a right adjoint

119860119869 119860

colimperp

limperpΔ

233 Warning Limits or colimits of set-indexed diagrams mdash the case where the indexing shape is acoproduct of the terminal object 1 indexed by a set 119869mdash are called products or coproducts respectivelyIn this case theinfin-category of diagrams itself decomposes as a product 119860119869 cong prod119869119860 As the functor

120101119974 119966119886119905

119860 h119860

hFun(1minus)

that carries aninfin-category to its homotopy category preserves products when 119869 is a set there is a chainof isomorphisms

h(119860119869) cong h(prod119869119860) cong prod119869 h119860 cong (h119860)119869

Thus in this special case the adjunctions of Definition 232 that define products or coproducts in aninfin-category descend to the adjunctions that define products or coproducts in its homotopy category

However this argument does not extend to more general limit or colimit notions and suchinfin-cat-egorical limits or colimits are generally not limits or colimits in the homotopy category⁷ In sect32 we

⁶In Proposition 1334 proven in Part IV we shall discover that in the case of the infin-cosmoi of (infin 1)-categoriesthere is no essential difference between these notions in 119980119966119886119905 they are tautologically the same and in all biequivalentinfin-cosmsoi theinfin-category of diagrams indexed by aninfin-category119860 is equivalent to theinfin-category of diagrams indexedby its underlying quasi-category regarded as a simplicial set

⁷This sort of behavior is expected in abstract homotopy theory homotopy limits and colimits are not generally limitsor colimits in the homotopy category

shall see that the homotopy category construction fails to preserve more complicated cotensors evenin the relatively simple case of 119869 = 120794

The problem with Definition 232 is that it is insufficiently general manyinfin-categories will havecertain but not all limits of diagrams of a particular indexing shape So it would be desirable tore-express Definition 232 in a form that allows us to define the limit of a single diagram 119889∶ 1 rarr 119860119869or of a family of diagrams To achieve this we make use of the following 2-categorical notion thatop-dualizes the more familiar absolute extension diagrams

234 Definition (absolute lifting diagrams) Given a cospan 119862 119860 119861119892 119891

in a 2-categoryan absolute left lifting of 119892 through 119891 is given by a 1-cell and 2-cell as displayed below-left

119861 119883 119861 119883 119861

119862 119860 119862 119860 119862 119860uArr120582

119891 uArr120594

119887

119888 119891 =existuArr120577

uArr120582

119887

119888 119891

119892

so that any 2-cell as displayed above-center factors uniquely through (ℓ 120582) as displayed above-rightDually an absolute right lifting of 119892 through 119891 is given by a 1-cell and 2-cell as displayed below-left

119861 119883 119861 119883 119861

119862 119860 119862 119860 119862 119860dArr120588

119891 dArr120594

119887

119888 119891 =existdArr120577

dArr120588

119887

119888 119891

119892

119903

119892

119903

119892

so that any 2-cell as displayed above-center factors uniquely through (119903 120588) as displayed above-right

The adjectives ldquoleftrdquo and ldquorightrdquo refer to the handedness of the adjointness of these construc-tions left and right liftings respectively define left and right adjoints to the composition functor119891lowast ∶ hFun(119862 119861) rarr hFun(119862119860) with the 2-cells defining the components of the unit and counit ofthese adjunctions respectively at the object 119892 The adjective ldquoabsoluterdquo refers to the following stabilityproperty

235 Lemma Absolute left or right lifting diagrams are stable under restriction of their domain object if(ℓ 120582) defines an absolute left lifting of 119892 through 119891 then for any 119888 ∶ 119883 rarr 119862 the restricted diagram (ℓ119888 120582119888)defines an absolute left lifting of 119892119888 through 119891

119861

119883 119862 119860uArr120582

119891

119888 119892

Proof Exercise 23i

Units and counits of adjunctions provide important examples of absolute left and right liftingdiagrams respectively

236 Lemma A 2-cell 120578∶ id119861 rArr 119906119891 defines the unit of an adjunction 119891 ⊣ 119906 if and only if (119891 120578) definesan absolute left lifting diagram displayed below-left

119860 119861

119861 119861 119860 119860uArr120578

119906dArr120598

119891119891 119906

Dually a 2-cell 120598 ∶ 119891119906 rArr id119860 defines the counit of an adjunction if and only if (119906 120598) defines an absolute rightlifting diagram displayed above-right

Proof We prove the universal property of the counit Given a 2-cell 120572∶ 119891119887 rArr 119886 as displayedbelow left

119883 119861 119883 119861

119860 119860 119860 119860

119887

119886 dArr120572 119891 =

119887

119886dArr120573

dArr120598119891119906

there exists a unique transpose 120573∶ 119887 rArr 119906119886 as displayed above-right across the induced adjunction

hFun(119883 119861) hFun(119883119860)

119891lowast

119906lowast

between the hom-categories of the homotopy 2-category see Proposition 217(ii) From right to lefttransposes are composed by pasting with the counit hence the left-hand side above equals the right-hand side The converse is left as Exercise 23ii

In particular the unit of the adjunction colim ⊣ Δ of Definition 232 defines an absolute leftlifting diagram

119860

119860119869 119860119869uArr120578

Δcolim

By Lemma 235 this universal property is retained upon restricting to any subobject of theinfin-categoryof diagrams This motivates the following definitions

237 Definition A colimit of a family of diagrams 119889∶ 119863 rarr 119860119869 indexed by 119869 in aninfin-category 119860 isgiven by an absolute left lifting diagram

119860

119863 119860119869uArr120578

Δcolim

119889

comprised of a colimit functor colim ∶ 119863 rarr 119860 and a colimit cone 120578∶ 119889 rArr Δ colim

Dually a limit of a family of diagrams 119889∶ 119863 rarr 119860119869 indexed by 119869 in aninfin-category119860 is given by anabsolute right lifting diagram

119860

119863 119860119869dArr120598

119889comprised of a limit functor lim ∶ 119863 rarr 119860 and a limit cone 120598 ∶ Δ limrArr 119889

238 Remark If 119860 has all limits of shape 119869 then Lemma 235 implies that any family of diagrams119889∶ 119863 rarr 119860119869 has a limit defined by evaluating the limit functor lim ∶ 119860119869 rarr 119860 at 119889 ie by restrictinglim along 119889 In certain infin-cosmoi such as 119980119966119886119905 if every diagram 119889∶ 1 rarr 119860119869 has a limit then 119860has all 119869-indexed limits because the quasi-category 1 generates theinfin-cosmos of quasi-categories in asuitable sense but this result is not true for allinfin-cosmoi

For example a 2-categorical lemma enables general proof of a classical result from homotopy the-ory that computes geometric realizations of ldquosplitrdquo simplicial objects Before proving this we introducethe indexing shapes involved

239 Definition (split augmented (co)simplicial objects) Recall 120491 is the simplex category of finitenon-empty ordinals and order-preserving maps introduced in 111 It defines a full subcategory of acategory 120491+ which freely appends the empty ordinal ldquo[minus1]rdquo as an initial object This in turn definesa wide subcategory of a category 120491perp which adds an ldquoextrardquo degeneracy 120590minus1 ∶ [119899 + 1] ↠ [119899] betweeneach pair of consecutive ordinals including 120590minus1 ∶ [0] ↠ [minus1]

Diagrams indexed by 120491 sub 120491+ sub 120491perp are respectively called cosimplicial objects coaugmentedcosimplicial objects and split coaugmented cosimplicial objects if they are covariant and simplicialobjects augmented simplicial objects and split augmented simplicial objects if they are contravariant

A simplicial object 119889∶ 1 rarr 119860120491opin an infin-category 119860 admits an augmentation or admits a split-

ting if it lifts along the restriction functors

119860120491perpop

119860120491op+

1 119860120491op

119889

The family of simplicial objects admitting an augmentation and splitting is then represented by thegeneric element 119860120491perp

op↠ 119860120491op

The following proposition proves that for any simplicial objectadmitting a splitting the augmentation defines the colimit cone dual results apply to colimits of splitcosimplicial objects The limit and colimit cones are defined by cotensoring with the unique naturaltransformation

120491 120491+

120793 [minus1]

uArr120584 (2310)

that exists because [minus1] ∶ 120793 rarr 120491+ is initial see Lemma 22443

2311 Proposition (geometric realizations) Let 119860 be any infin-category For every cosimplicial object in119860 that admits a coaugmentation and a splitting the coaugmentation defines a limit cone Dually for everysimplicial object in 119860 that admits an augmentation and a splitting the augmentation defines a colimit coneThat is there exist absolute right and left lifting diagrams

119860 119860

119860120491perp 119860120491+ 119860120491 119860120491perpop

119860120491op+ 119860120491op

dArr119860120584Δ

uArr119860120584op Δ

ev[minus1]

res res res

ev[minus1]

in which the 2-cells are obtained as restrictions of the cotensor of the 2-cell (2310) with 119860 Moreover suchlimits and colimits are absolute preserved by any functor 119891∶ 119860 rarr 119861 ofinfin-categories

Proof By Example B52 the inclusion 120491 120491perp admits a right adjoint which can automaticallybe regarded as an adjunction ldquooverrdquo 120793 since 120793 is 2-terminal The initial element [minus1] isin 120491+ sub 120491perpdefines a left adjoint to the constant functor

120491 120491+ 120491perp

120793

[minus1] perp

with the counit of this adjunction (2310) defining the colimit cone under the constant functor atthe initial element These adjunctions are preserved by the 2-functor 119860(minus) ∶ 119966119886119905op rarr 120101119974 yielding adiagram

119860

119860120491perp 119860120491+ 119860120491dArr119860120584

ev[minus1]

res⊤

By Lemma B51 these adjunctions witness the fact that evaluation at [minus1] and the 2-cell from (2310)define an absolute right lifting of the canonical restriction functor 119860120491perp rarr 119860120491 through the constantdiagram functor as claimed The colimit case is proven similarly by applying the composite 2-functor

119966119886119905coop 119966119886119905op 120101119974(minus)op 119860(minus)

Finally by the 2-functoriality of the simplicial cotensor any 119891∶ 119860 rarr 119861 commutes with the 2-cellsdefined by cotensoring with 120584 or its opposite

119860 119861 119861

119860120491perp 119860120491+ 119860120491 119861120491 119860120491perp 119861120491perp 119861120491+ 119861120491dArr119860120584

119891

Δ =dArr119860120584

ev[minus1]

res 119891120491 119891120491perp res

ev[minus1]

Since the right-hand composite is an absolute right lifting diagram so is the left-hand compositewhich says that 119891∶ 119860 rarr 119861 preserves the totalization of any split coaugmented cosimplicial object in119860

Exercises

23i Exercise Prove Lemma 235

23ii Exercise Re-prove the forwards implication of Lemma 236 by following your nose through apasting diagram calculation and prove the converse similarly

24 Preservation of limits and colimits

Famously right adjoint functors preserve limits and left adjoints preserve colimits Our aim inthis section is to prove this in the infin-categorical context and exhibit the first examples of initial andfinal functors in the sense introduced in Definition 246 below

The commutativity of right adjoints and limits is very easily established in the case where theinfin-categories in question admit all limits of a given shape under these hypotheses the limit functoris right adjoint to the constant diagram functor which commutes with all functors between the baseinfin-categories Since the left adjoints commute the uniqueness of adjoints (Proposition 2110) impliesthat the right adjoints do as well This outline gives a hint for Exercise 24i

A slightly more delicate argument is needed in the general case involving say the preservation ofa single limit diagram without a priori assuming that any other limits exist This follows easily froma general lemma about composition and cancelation of absolute lifting diagrams

241 Lemma (composition and cancelation of absolute lifting diagrams) Suppose (119903 120588) defines an ab-solute right lifting of ℎ through 119891

119862

119861

119863 119860

dArr120590119892

dArr120588119891

119903

119904

Then (119904 120590) defines an absolute right lifting of 119903 through 119892 if and only if (119904 120588 sdot 119891120590) defines an absolute rightlifting of ℎ through 119891119892

Proof Exercise 24ii

242 Theorem (RAPLLAPC) Right adjoints preserve limits and left adjoints preserve colimits

The usual argument that right adjoints preserve limits proceeds like this a cone over a 119869-shapeddiagram in the image of 119906 transposes across the adjunction 119891119869 ⊣ 119906119869 to a cone over the original diagramwhich factors through the designated limit cone This factorization transposes across the adjunction119891 ⊣ 119906 to define the sought-for unique factorization through the image of the limit cone The use ofabsolute lifting diagrams to express the universal properties of limits and colimits (Definition 237)and adjoint transposition (Lemma 236) allows us to economize on the usual proof by suppressingconsideration of a generic test cone that must be shown to uniquely factor through the limit cone

Proof We prove that right adjoints preserve limits By taking ldquocordquo duals the same argumentdemonstrates that left adjoints preserve colimits

Suppose 119906∶ 119860 rarr 119861 admits a left adjoint 119891∶ 119861 rarr 119860 with unit 120578∶ id119861 rArr 119906119891 and counit 120598 ∶ 119891119906 rArrid119860 Our aim is to show that any absolute right lifting diagram as displayed below-left is carried toan absolute right lifting diagram as displayed below-right

119860 119860 119861

119863 119860119869 119863 119860119869 119861119869dArr120588

ΔdArr120588

119906

119889

119889 119906119869

The cotensor (minus)119869 ∶ 120101119974 rarr 120101119974 carries the adjunction 119891 ⊣ 119906 to an adjunction 119891119869 ⊣ 119906119869 with unit 120578119869and counit 120598119869 In particular by Lemma 236 (119906119869 120598119869) defines an absolute right lifting of the identitythrough 119891119869 which is then preserved by restriction along the functor 119889 Thus by Lemma 241 thediagram on the right of (243) is an absolute right lifting diagram if and only if the pasted compositedisplayed below-left

119860 119861 119861 119861

119863 119860119869 119861119869 119860 119860 119860

119860119869 119863 119860119869 119860119869 119863 119860119869

dArr120588Δ

119906

ΔdArr120598

119891dArr120598 lim

119891lim

119889119906119869

119891119869dArr120598119869=

dArr120588

119906

dArr120588Δ

119889

119906 lim

defines an absolute right lifting diagram Pasting the 2-cell on the right of (243) with the counit inthis way amounts to transposing the cone under 119906 lim across the adjunction 119891119869 ⊣ 119906119869

Wersquoll now observe that this transposed cone factors through the limit cone (lim 120588) in a canonicalway From the 2-functoriality of the simplicial cotensor in its exponent variable 119891119869Δ = Δ119891 and120598119869Δ = Δ120598 Hence the pasting diagram displayed above-left equals the one displayed above-centerand hence also by naturality of whiskering the diagram above-right⁸ This latter diagram is a pastedcomposite of two absolute right lifting diagrams and is hence an absolute right lifting diagram in itsown right this universal property says that any cone over 119889 whose summit factors through 119891 factorsuniquely through the limit cone (lim 120588) through a map that then transposes along the adjunction119891 ⊣ 119906 Hence all of the diagrams in the statement are absolute right lifting diagrams including inparticular the one on the right-hand side of (243)

By combining Theorem 242 with Proposition 2111 we have immediately that

244 Corollary Equivalences preserve limits and colimits

We can also prove a more refined result

245 Proposition If 119860 ≃ 119861 then any family of diagrams in 119860 admitting a limit or colimit in 119861 alsoadmits a limit or colimit in 119860 that is preserved by the equivalence

Proof By Proposition 2111 the equivalence 119861 ⥲ 119860 is both left and right adjoint to its equiva-lence inverse preserving both limits and colimits of the composite family of diagrams119863 rarr 119860119869 ⥲ 119861119869Via the invertible 2-cells of the equivalence 119860119869 ≃ 119861119869 constructed by applying (minus)119869 ∶ 120101119974 rarr 120101119974 to theequivalence 119860 ≃ 119861 the preserved diagram 119863 rarr 119860119869 ⥲ 119861119869 ⥲ 119860119869 is isomorphic to the original family

⁸By naturality of whiskering 120598119869119889 sdot 119891119869119906119869120588 = 120588 sdot 120598119869Δ lim and since 120598119869Δ = Δ120598 this composite equals 120588 sdot Δ120598 lim

of diagrams 119863 rarr 119860119869 Thus we conclude that a family of diagrams in 119860 has a limit or colimit if andonly if its image in an equivalentinfin-category 119861 does and such limits and colimits are preserved by theequivalence

The following definition makes sense between small quasi-categories or equally between arbitraryinfin-categories in a cartesian closedinfin-cosmos

246 Definition (initial and final functor) A functor 119896 ∶ 119868 rarr 119869 is final if 119869-indexed colimits existif and only if and in such cases coincide with the restricted 119868-indexed colimits That is 119896 ∶ 119868 rarr 119869 isfinal if and only if for anyinfin-category 119860 the square

119860 119860

119860119869 119860119868Δ Δ

119860119896

preserves and reflects all absolute left lifting diagramsDually a functor 119896 ∶ 119868 rarr 119869 is initial if this square preserves and reflects all absolute right lifting

diagrams or informally if a generalized element defines a limit of a 119869-indexed diagram if and only ifit defines a limit of the restricted 119868-indexed diagrams

Historically final functors were called ldquocofinalrdquo with no obvious name for the dual notion Ourpreferred terminology hinges on the following mnemonic the inclusion of an initial element definesan initial functor while the inclusion of a terminal (aka final) element defines a final functor Theseresults are special cases of a more general result we now establish using exactly the same tactics astaken to prove Theorem 242

247 Proposition Left adjoints define initial functors and right adjoints define final functors

Proof If 119896 ⊣ 119903 with unit 120578∶ id119868 rArr 119903119896 and counit 120598 ∶ 119896119903 rArr id119869 then cotensoring into 119860 yieldsan adjunction

119860119869 119860119868

119860119896perp119860119903

with unit 119860120578 ∶ id119860119868 rArr 119860119896119860119903 and counit 119860120598 ∶ 119860119903119860119896 rArr id119860119869 To prove that 119896 is initial we must show that for any (119889 lim 120588) as displayed below-left

119860 119860 119860

119863 119860119869 119863 119860119869 119860119868dArr120588

ΔdArr120588

Δ Δlim

119889

119889 119860119896

the left-hand diagram is an absolute right lifting diagram if and only if the right-hand diagram is anabsolute right lifting diagram

By Lemmas 236 and 241 the right-hand diagram is an absolute right lifting diagram if and onlyif the pasted composite displayed below-left

119860 119860 119860

119863 119860119869 119860119868 119863 119860119869

119860119869

dArr120588Δ Δ

dArr120588Δlim

119889119860119896

119860119903dArr119860120598

119889

is also an absolute right lifting diagram On noting that 119860119903Δ = Δ and 119860120598Δ = idΔ the left-hand sidereduces to the right-hand side which proves the claim

Exercises

24i Exercise Show that any left adjoint 119891∶ 119861 rarr 119860 between infin-categories admitting all 119869-shapedcolimits preserves them in the sense that the square of functors

119861119869 119860119869

119861 119860colim

119891119869

colimcong

119891

commutes up to isomorphism

24ii Exercise Prove Lemma 241

24iii Exercise Given a proof of Theorem 242 that does not appeal to Lemma 241 by directlyverifying that the diagram on the right of (243) is an absolute right lifting diagram

24iv Exercise Use Lemma 241 to give a new proof of Proposition 219

CHAPTER 3

Weak 2-limits in the homotopy 2-category

In Chapter 2 we introduced adjunctions between infin-categories and limits and colimits of dia-grams valued within aninfin-category through definitions that are particularly expedient for establishingthe expected interrelationships But neither 2-categorical definition clearly articulates the universalproperties of these notions Definition 237 does not obviously express the expected universal prop-erty of the limit cone namely that the limit cone over a diagram 119889 defines the terminal element oftheinfin-category of cones over 119889 yet-to-be-defined Nor have we understood how an adjunction 119891 ⊣ 119906induces an equivalence on as-yet-to-be-defined hom-spaces Hom119860(119891119887 119886) ≃ Hom119861(119887 119906119886) for a pair ofgeneralized elementssup1 In this section we make use of the completeness axiom in the definition of aninfin-cosmos to exhibit a general construction that will specialize to give a definition of thisinfin-categoryof cones and also specialize to define these hom-spaces This construction will also permit us to rep-resent a functor betweeninfin-categories as aninfin-category in dual ldquoleftrdquo or ldquorightrdquo fashions Using thiswe can redefine an adjunction to consist of a pair of functors 119891∶ 119861 rarr 119860 and 119906∶ 119860 rarr 119861 so that theleft representation of 119891 is equivalent to the right representation of 119906 over 119860 times 119861

Our vehicle for all of these new definitions is the commainfin-category associated to a cospan

119862 119860 119861119892 119891

Hom119860(119891 119892)

119862 times 119861

(11990111199010)

Our aim in this chapter is to develop the general theory of comma constructions from the point ofview of the homotopy 2-category of aninfin-cosmos Our first payoff for this work will occur in Chapter4 where we study the universal properties of adjunctions limits and colimits in the sense of the ideasjust outlined The comma construction will also provide the essential vehicle for establishing themodel-independence of the categorical notions we will introduce throughout this text

There is a standard definition of a ldquocomma objectrdquo that can be stated in any strict 2-categorydefined as a particular weighted limit (see Example 7117) Comma infin-categories do not satisfy thisuniversal property in the homotopy 2-category however Instead they satisfy a somewhat peculiarldquoweakrdquo variant of the usual 2-categorical universal property that to our knowledge has not been dis-covered elsewhere in the categorical or homotopical literature expressed in terms of something wecall a smothering functor To introduce these universal properties in a concrete rather than abstractframework we start in sect31 by considering smothering functors involving homotopy categories ofquasi-categories The intrepid and impatient reader may skip the entirety of sect31 if they wish to in-stead first encounter these notions in their full generality

sup1A 2-categorical version of this result mdash exhibiting a bijection between sets of 2-cells mdash appears as Lemma 236 butin aninfin-category wersquod hope for a similar equivalence of hom-spaces

31 Smothering functors

Let 119876 be a quasi-category Recall from Lemma 1112 that its homotopy category h119876 hasbull the elements 1 rarr 119876 of 119876 as its objectsbull the set of homotopy classes of 1-simplices of119876 as its arrows where parallel 1-simplices are homo-

topic just when they bound a 2-simplex with the remaining outer edge degenerate andbull a composition relation if and only if any chosen 1-simplices representing the three arrows bound

a 2-simplexFor a 1-category 119869 it is well-known in classical homotopy theory that the homotopy category of dia-grams h(119876119869) is not equivalent to the category (h119876)119869 of diagrams in the homotopy category mdash exceptin very special cases such as when 119869 is a set (see Warning 233) The objects of h(119876119869) are homotopycoherent diagrams of shape 119869 in 119876 while the objects of (h119876)119869 are mere homotopy commutative diagramsThere is however a canonical comparison functor

h(119876119869) rarr (h119876)119869

defined by applying h ∶ 119980119966119886119905 rarr 119966119886119905 to the evaluation functor 119876119869 times 119869 rarr 119876 and then transposing ahomotopy coherent diagram is in particular homotopy commutative

Our first aim in this section is to better understand the relationship between the arrows in thehomotopy category h119876 and what wersquoll refer to as the arrows of 119876 namely the 1-simplices in thequasi-category To study this wersquoll be interested in the quasi-category in which the arrows of119876 live aselements namely 119876120794 where 120794 = Δ[1] is the nerve of the ldquowalkingrdquo arrow Our notation deliberatelyimitates the notation commonly used for the category of arrows if 119862 is a 1-category then 119862120794 is thecategory whose objects are arrows in 119862 and whose morphisms are commutative squares regarded as amorphism from the arrow displayed vertically on the left-hand side to the arrow displayed verticallyon the right-hand side This notational conflation suggests our first motivating question how doesthe homotopy category of 119876120794 relate to the category of arrows in the homotopy category of 119876

311 Lemma The canonical functor h(119876120794) rarr (h119876)120794 isbull surjective on objectsbull full andbull conservative ie reflects invertibility of morphisms

but not injective on objects nor faithful

Proof Surjectivity on objects asserts that every arrow in the homotopy category h119876 is repre-sented by a 1-simplex in 119876 This is the conclusion of Exercise 11ii(iii) which outlines the proof ofLemma 1112

To prove fullness consider a commutative square in h119876 and choose arbitrary 1-simplices repre-senting each morphism and their common composite

bull bull

bull bull119891

ℓ 119892

119896

By Lemma 1112 every composition relation in h119876 is witnessed by a 2-simplex in 119876 choosing a pairof such 2-simplices defines a diagram 120794 rarr 119876120794 which represents a morphism from 119891 to 119892 in h(119876120794)proving fullness

Surjectivity on objects and fullness of the functor h(119876120794) rarr h(119876)120794 are special properties havingto do with the diagram shape 120794 Conservativity is much more general as a consequence of the secondstatement of Corollary 1121

The properties of the canonical functor h(119876120794) rarr h(119876)120794 will reappear frequently so are worthgiving a name

312 Definition A functor 119891∶ 119860 rarr 119861 between 1-categories is smothering if it is surjective onobjects full and conservative That is a functor is smothering if and only if it has the right liftingproperty with respect to the set of functors

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

empty 120793 + 120793 120794

120793 120794 120128

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

Some elementary properties of smothering functors are established in Exercise 31i The mostimportant of these is

313 Lemma Each fibre of a smothering functor is a non-empty connected groupoid

Proof Suppose 119891∶ 119860 rarr 119861 is smothering and consider the fiber

119860119887 119860

120793 119861

119891

119887

over an object 119887 of 119861 By surjectivity on objects the fiber is non-empty Its morphisms are definedto be arrows between objects in the fiber of 119887 that map to the identity on 119887 By fullness any twoobjects in the fiber are connected by a morphism indeed by morphisms pointing in both directionsBy conservativity all the morphisms in the fiber are necessarily invertible

The argument used to prove Lemma 311 generalizes to

314 Lemma If 119869 is a 1-category that is free on a reflexive directed graph and119876 is a quasi-category then thecanonical functor h(119876119869) rarr (h119876)119869 is smothering

Proof Exercise 31ii

Cotensors are one of the simplicial limit constructions enumerated in axiom 121(i) Other limitconstructions listed there also give rise to smothering functors

315 Lemma For any pullback diagram of quasi-categories in which 119901 is an isofibration

119864 times119861 119860 119864

119860 119861

119901

119891

the canonical functor h(119864 times119861 119860) rarr h119864 timesh119861 h119860 is smothering

Proof As h ∶ 119980119966119886119905 rarr 119966119886119905 does not preserve pullbacks the canonical comparison functor of thestatement is not an isomorphism It is however bijective on objects since the composite functor

119980119966119886119905 119966119886119905 119982119890119905h obj

is given by evaluation on the set of vertices of each quasi-category and this functor does preservepullbacks

For fullness note that a morphism in h119864timesh119861 h119860 is represented by a pair of 1-simplices 120572∶ 119886 rarr 119886primeand 120598 ∶ 119890 rarr 119890prime in 119860 and 119864 whose images in 119861 are homotopic a condition that implies in particularthat 119891(119886) = 119901(119890) and 119891(119886prime) = 119901(119890prime) By Lemma 119 we can arrange this homotopy however we likeand thus we choose a 2-simplex witness 120573 so as to define a lifting problem

Λ1[2] 119864 ni

Δ[2] 119861 ni

119901

120573

119890119890 119890prime

119901(119890)

119891(119886) 119891(119886prime) = 119901(119890prime)

120598

119901(120598)

119891(120572)

Since 119901 is an isofibration a solution exists defining an arrow ∶ 119890 rarr 119890prime in 119864 in the same homotopyclass as 120598 so that 119901() = 119891(120572) The pair (120572 ) now defines the lifted arrow in h(119864 times119861 119860)

Finally consider an arrow 120794 rarr 119864 times119861 119860 whose image in h119864 timesh119861 h119860 is an isomorphism whichis the case just when the projections to 119864 and 119860 define isomorphisms By Corollary 1116 we maychoose a homotopy coherent isomorphism 120128 rarr 119860 extending the given isomorphism 120794 rarr 119860 Thisdata presents us with a lifting problem

120794 119864 times119861 119860 119864

120128 119860 119861

119901

119891

which Exercise 11v tells us we can solve This proves that h(119864 times119861 119860) rarr h119864 timesh119861 h119860 is conservativeand hence also smothering

A similar argument proves

316 Lemma For any tower of isofibrations between quasi-categories

⋯ 119864119899 119864119899minus1 ⋯ 1198642 1198641 1198640the canonical functor h(lim119899 119864119899) rarr lim119899 h119864119899 is smothering

Proof Exercise 31iii

317 Lemma For any cospan between quasi-categories 119862 119860 119861119892 119891

consider the quasi-categorydefined by the pullback

Hom119860(119891 119892) 119860120794

119862 times 119861 119860 times 119860

(11990111199010)

(coddom)

119892times119891

The canonical functor hHom119860(119891 119892) rarr Homh119860(h119891 h119892) is smothering

Proof Here the codomain is the category defined by an analogous pullback

Homh119860(h119891 h119892) (h119860)120794

h119862 times h119861 h119860 times h119860

(coddom)

h119892timesh119891

in 119966119886119905 and the canonical functor factors as

hHom119860(119891 119892) rarr h(119860120794) timesh119860timesh119860 (h119862 times h119861) rarr (h119860)120794 timesh119860timesh119860 (h119862 times h119861)By Lemma 315 the first of these functors is smothering By Lemma 311 the second is a pullback of asmothering functor By Exercise 31i(i) it follows that the composite functor is smothering

In the sections that follow we will discover that the smothering functors just constructed expressparticular ldquoweakrdquo universal properties of arrow pullback and comma constructions in the homotopy2-category of anyinfin-cosmos It is to the first of these that we now turn

Exercises

31i Exercise Prove that(i) The class of smothering functors is closed under composition retract product and pullback(ii) The class of smothering functors contains all surjective equivalences of categories(iii) All smothering functors are isofibrations that is maps that have the right lifting property

with respect to 120793 120128(iv) Prove that if 119891 and 119892119891 are smothering functors then 119892 is a smothering functorsup2

31iii Exercise Prove Lemma 316

32 infin-categories of arrows

In this section we replicate the discussion from the start of the previous section using an arbitraryinfin-category 119860 in place of the quasi-category 119876 The analysis of the previous section could have beendeveloped natively in this general setting but at the cost of an extra layer of abstraction and moreconfusing notation mdash with a functor space Fun(119883119860) replacing the quasi-category 119876

Recall an element of aninfin-category is defined to be a functor 1 rarr 119860 Tautologically the elementsof 119860 are the vertices of the underlying quasi-category Fun(1 119860) of 119860 In this section we will define

sup2It suffices in fact to merely assume that 119891 is surjective on objects and arrows

and study an infin-category 119860120794 whose elements are the 1-simplices in the underlying quasi-category of119860 We refer to 119860120794 as theinfin-category of arrows in 119860 and call its elements simply arrows of 119860

In fact wersquove tacitly introduced this construction already Recall 120794 is our preferred notation forthe quasi-categoryΔ[1] as this coincides with the nerve of the 1-category 120794with a single non-identitymorphism 0 rarr 1321 Definition (arrow infin-category) Let 119860 be an infin-category The infin-category of arrows in 119860 isthe simplicial cotensor 119860120794 together with the canonical endpoint-evaluation isofibration

119860120794 ≔ 119860Δ[1] 119860120597Δ[1] cong 119860 times 119860(11990111199010)

induced by the inclusion 120597Δ[1] Δ[1] For conciseness we write 1199010 ∶ 119860120794 ↠ 119860 for the domain-evaluation induced by the inclusion 0∶ 120793 120794 and write 1199011 ∶ 119860120794 ↠ 119860 for the codomain-evaluationinduced by 1∶ 120793 120794

As an object of the homotopy 2-category 120101119974 the infin-category of arrows comes equipped with acanonical 2-cell that we now construct

322 Lemma For anyinfin-category 119860 theinfin-category of arrows comes equipped with a canonical 2-cell

119860120794 1198601199010

1199011

dArr120581 (323)

that we refer to as the generic arrow with codomain 119860

Proof The simplicial cotensor has a strict universal property described inDigression 124 namely119860120794 is characterized by the natural isomorphism

Fun(119883119860120794) cong Fun(119883119860)120794 (324)

By the Yoneda lemma the data of the natural isomorphism (324) is encoded by its ldquouniversal elementrdquowhich is defined to be the image of the identity at the representing object Here the identity functorid ∶ 119860120794 rarr 119860120794 is mapped to an element of Fun(119860120794 119860)120794 a 1-simplex in Fun(119860120794 119860) which representsa 2-cell in the homotopy 2-category defining (323)

To see that its source and target must be the domain-evaluation and codomain-evaluation mapsnote that the action of the simplicial cotensor 119860(minus) on morphisms of simplicial sets is defined so thatthe isomorphism (324) is natural in the cotensor variable as well Thus by restricting along theendpoint inclusion 120793+120793 120794 we may regard the isomorphism (324) as lying over Fun(119883119860times119860) congFun(119883119860) times Fun(119883119860)

There is a 2-categorical limit notion that is analogous to Definition 321 which constructs forany object 119860 the universal 2-cell with codomain 119860 namely the cotensor with the 1-category 120794 Itsuniversal property is analogous to (324) but with the hom-categories of the 2-category in place of thefunctor spaces In 119966119886119905 this constructs the arrow category associated to a 1-category

In the homotopy 2-category 120101119974 by the Yoneda lemma again the data (323) encodes a naturaltransformation

hFun(119883119860120794) rarr hFun(119883119860)120794

of categories but this is not a natural isomorphism nor even a natural equivalence of categories butdoes express the arrowinfin-category as a ldquoweakrdquo arrow object with a universal property of the followingform

325 Proposition (the weak universal property of the arrow infin-category) The generic arrow (323)with codomain 119860 has a weak universal property in the homotopy 2-category given by three operations

(i) 1-cell induction Given a 2-cell over 119860 as below-left

119883 119883

= 119860120794

119860 119860

119904119905120572lArr 119904119905

119886

11990101199011 120581lArr

there exists a 1-cell 119886 ∶ 119883 rarr 119860120794 so that 119904 = 1199010119886 119905 = 1199011119886 and 120572 = 120581119886(ii) 2-cell induction Given a pair of functors 119886 119886prime ∶ 119883 119860120794 and a pair of 2-cells 1205910 and 1205911 so that

119883 119883

119860120794 119860120794 = 119860120794 119860120794

119860 119860

119886119886prime1205911lArr

119886119886prime1205910lArr

1199011

1199010

120581lArr1199011

1199010120581lArr 1199010

there exists a 2-cell 120591∶ 119886 rArr 119886prime so that119883 119883 119883 119883

119860120794 119860120794 119860120794 119860120794 119860120794 119860120794

119860 119860 119860 119860

119886119886prime

1205911lArr

119886119886prime 120591lArr119886119886prime 120591lArr

119886119886prime

1205910lArr

1199011 11990111199011

= and1199010

1199010 1199010

(iii) 2-cell conservativity Any 2-cell119883

119860120794

119886119886prime 120591lArr

with the property that both 1199011120591 and 1199010120591 are isomorphisms is an isomorphism

Proof Let 119876 = Fun(119883119860) and apply Lemma 311 to observe that the natural map of hom-categories

hFun(119883119860120794) hFun(119883119860)120794

hFun(119883119860) times hFun(119883119860)((1199011)lowast(1199010)lowast) (ev1ev0)

over hFun(119883119860times119860) cong hFun(119883119860) times hFun(119883119860) is a smothering functor Surjectivity on objects isexpressed by 1-cell induction fullness by 2-cell induction and conservativity by 2-cell conservativity

Note that the functors 119883 rarr 119860120794 that represent a fixed 2-cell with domain 119883 and codomain119860 arenot unique However they are unique up to ldquofiberedrdquo isomorphisms that whisker with (1199011 1199010) ∶ 119860120794 ↠119860times119860 to an identity 2-cell

326 Proposition Whiskering with (323) induces a bijection between 2-cells with domain119883 and codomain119860 as displayed below-left

⎧⎪⎪⎨⎪⎪⎩119883 119860

119904

119905

dArr120572

⎫⎪⎪⎬⎪⎪⎭

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

119883

119860 119860

119860120794

119905 119904

119886

11990101199011

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭cong

and fibered isomorphism classes of functors119883 rarr 119860120794 as displayed above-right where the fibered isomorphismsare given by invertible 2-cells

119883

119860 119860

119860120794

119905 119904119886119886prime 120574

11990101199011

so that 1199010120574 = id119904 and 1199011120574 = id119905

Proof Lemma 313 proves that the fibers of the smothering functor of Proposition 325 are con-nected groupoids The objects of these fibers are functors 119883 rarr 119860120794 and the morphisms are invertible2-cells that whisker with (1199011 1199010) ∶ 119860120794 ↠ 119860 times 119860 to an identity 2-cell The action of the smotheringfunctor defines a bijection between the objects of its codomain and their corresponding fibers

Our final task is to observe that the universal property of Proposition 325 is also enjoyed by anyobject (1198901 1198900) ∶ 119864 ↠ 119860 times 119860 that is equivalent to the arrow infin-category (1199011 1199010) ∶ 119860120794 ↠ 119860 times 119860 inthe slice infin-cosmos 119974119860times119860 We have special terminology to allow us to concisely express the type ofequivalence we have in mind

327 Definition (fibered equivalence) A fibered equivalence over aninfin-category 119861 in aninfin-cosmos119974 is an equivalence

119864 119865

119861

sim(328)

in the slicedinfin-cosmos119974119861 We write 119864 ≃119861 119865 to indicate that that specified isofibrations with thesedomains are equivalent over 119861

By Proposition 1217(vii) a fibered equivalence is just a map between a pair of isofibrations over acommon base that defines an equivalence in the underlyinginfin-cosmos the forgetful functor119974119861 rarr119974preserves and reflects equivalences Note however that it does not create them it is possible for twoinfin-categories 119864 and 119865 to be equivalent without there existing any equivalence compatible with a pairof specified isofibration 119864 ↠ 119861 and 119865 ↠ 119861

329 Remark At this point there is some ambiguity about the 2-categorical data that presents afibered equivalence related to the question posed in Exercise 14iii But since Proposition 1217(vii)tells us that a mere equivalence in 120101119974 involving a functor of the form (328) is sufficient to guaranteethat this as-yet-unspecified 2-categorical data exists we defer a careful analysis of this issue to sect35

3210 Proposition (uniqueness of arrow infin-categories) For any isofibration (1198901 1198900) ∶ 119864 ↠ 119860 times 119860equipped with a fibered equivalence 119890 ∶ 119864 ⥲ 119860120794 the corresponding 2-cell

119864 1198601198900

1198901

dArr120598

satisfies the weak universal property of Proposition 325 Conversely if (1198891 1198890) ∶ 119863 ↠ 119860times119860 and (1198901 1198900) ∶ 119864 ↠119860 times 119860 are equipped with 2-cells

119863 119860 and 119864 1198601198890

1198891

dArr120575

1198900

1198901

dArr120598

satisfying the weak universal property of Proposition 325 then 119863 and 119864 are fibered equivalent over 119860 times 119860

Proof We prove the first statement By the definition equation of 1-cell induction 120598 = 120581119890 where120581 is the canonical 2-cell of (323) Hence pasting with 120598 induces a functor

hFun(119883 119864) hFun(119883119860120794) hFun(119883119860)120794

hFun(119883119860) times hFun(119883119860)

119890lowast

((1199011)lowast(1199010)lowast) (ev1ev0)

and our task is to prove that this composite functor is smothering We see that the first functordefined by post-composing with the equivalence 119890 ∶ 119864 rarr 119860120794 is an equivalence of categories and thesecond functor is smothering Thus the composite is clearly full and conservative To see that it isalso surjective on objects note first that by 1-cell induction any 2-cell

119883 119860119904

119905

dArr120572

is represented by a functor 119886 ∶ 119883 rarr 119860120794 over119860times119860 Composing with any fibered inverse equivalence119890prime to 119890 yields a functor

119883 119860120794 119864

119860 times 119860

119886

(119905119904)(11990111199010)

sim119890prime

(11989011198900)

whose image after post-composing with 119890 is isomorphic to 119886 over119860times119860 Because this isomorphism isfibered (see Proposition 326) the image of 119886119890prime under the functor hFun(119883 119864) rarr hFun(119883119860)120794 returnsthe 2-cell 120572 This proves that this mapping is surjective on objects and hence defines a smotheringfunctor as claimed

The converse is left to Exercise 32ii and proven in a more general context in Proposition 3311

3211 Convention On account of Proposition 3210 we extend the appellation ldquoinfin-category of ar-rowsrdquo from the strict model constructed in Definition 321 to anyinfin-category that is fibered equivalentto it

Via Lemma 314 the discussion of this section extends to establish corresponding weak universalproperties for the cotensors 119860119869 of an infin-category 119860 with a free category 119869 We leave the explorationof this to the reader

Exercises

32i Exercise(i) Prove that a parallel pair of 1-simplices in a quasi-category 119876 are homotopic if and only if

they are isomorphic as elements of 119876120794 via an isomorphism that projects to an identity along(1199011 1199010) ∶ 119876120794 ↠ 119876times119876

(ii) Conclude that a parallel pair of 1-arrows in the functor space Fun(119883119860) between twoinfin-categories119883 and119860 in anyinfin-cosmos represent the same natural transformation if and only if theyare isomorphic as elements of Fun(119883119860)120794 cong Fun(119883119860120794) via an isomorphism whose domainand codomain components are an identity

(iii) Conclude that a parallel pair of 1-arrows in the functor space Fun(119883119860) which may be en-coded as functors 119883 119860120794 represent the same natural transformation if and only if they areconnected by a fibered isomorphism

119883 119860120794

119860 times 119860

(11990111199010)

32ii Exercise Prove the second statement of Proposition 3210

33 The comma construction

The comma infin-category is defined by restricting the domain and codomain of the infin-category ofarrows 119860120794 along specified functors with codomain 119860

331 Definition (comma infin-category) Let 119862 119860 119861119892 119891

be a diagram of infin-categoriesThe commainfin-category is constructed as a pullback of the simplicial cotensor 119860120794 along 119892 times 119891

Hom119860(119891 119892) 119860120794

119862 times 119861 119860 times 119860

(11990111199010)

120601

(11990111199010)

119892times119891

This construction equips the commainfin-categorywith a specified isofibration (1199011 1199010) ∶ Hom119860(119891 119892) ↠119862 times 119861 and a canonical 2-cell

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010

120601lArr

119892 119891

in the homotopy 2-category called the comma cone

334 Example (arrowinfin-categories as commainfin-categories) The arrowinfin-category arises as a spe-cial case of the comma construction applied to the identity span This provides us with alternatenotation for the generic arrow of (323) which may be regarded as a particular instance of a commacone

Hom119860

119860 119860

119860

1199011 1199010

120601lArr

The following proposition encodes the homotopical properties of the comma construction Theproof is by a standard argument in abstract homotopy theory which can be found in Appendix C Ahint for this proof is given in Exercise 33i

335 Proposition (maps between commas) A commutative diagram

119862 119860 119861

119903

119892

119901 119902

119891

induces a map between the commainfin-categories

Hom119860(119891 119892) Hom( 119891 )

119862 times 119861 times

(11990111199010)

Hom119901(119902119903)

(11990111199010)119903times119902

Moreover if 119901 119902 and 119903 are all(i) equivalences(ii) isofibrations or(iii) trivial fibrations

then the induced map is again an equivalence isofibration or trivial fibration respectively

There is a 2-categorical limit notion that is analogous to Definition 331 which constructs theuniversal 2-cell inhabiting a square over a specified cospan In 119966119886119905 the category so-constructed isreferred to as a comma category from when we borrow the name As with the case ofinfin-categories ofarrow commainfin-categories do not satisfy this 2-universal property strictly Instead

336 Proposition (the weak universal property of the comma infin-category) The comma cone (333)has a weak universal property in the homotopy 2-category given by three operations

(i) 1-cell induction Given a 2-cell over 119862 119860 119861119892 119891

as below-left

119883

119862 119861

119860

119888 119887

120572lArr

119892 119891

119883

Hom119860(119891 119892)

119862 119861

119860

119888 119887119886

1199011 1199010

120601lArr

119892 119891

there exists a 1-cell 119886 ∶ 119883 rarr Hom119860(119891 119892) so that 119887 = 1199010119886 119888 = 1199011119886 and 120572 = 120601119886(ii) 2-cell induction Given a pair of functors 119886 119886prime ∶ 119883 Hom119860(119891 119892) and a pair of 2-cells 1205910 and 1205911 so

119883 119883

Hom119860(119891 119892) Hom119860(119891 119892) = Hom119860(119891 119892) Hom119860(119891 119892)

119862 119861 119862 119861

119860 119860

119886119886prime

1205911lArr

119886119886prime

1205910lArr

11990111199011 1199010

120601lArr

1199011 1199010

120601lArr

1199010

119892 119891 119892 119891

Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892)

119862 119862 119861 119861

119886119886prime

1205911lArr

119886119886prime

1205910lArr

1199011 11990111199011

and 1199010

1199010 1199010

Hom119860(119891 119892)

119886119886prime 120591lArr

Proof The cosmological functor Fun(119883 minus) ∶ 119974 rarr 119980119966119886119905 carries the pullback (332) to a pull-back

Fun(119883Hom119860(119891 119892)) cong HomFun(119883119860)(Fun(119883 119891) Fun(119883 119892)) Fun(119883119860)120794

Fun(119883 119862) times Fun(119883 119861) Fun(119883119860) times Fun(119883119860)

(11990111199010)

120601

(11990111199010)

Fun(119883119892)timesFun(119883119891)

of quasi-categories Now Lemma 317 demonstrates that the canonical 2-cell (333) induces a naturalmap of hom-categories

hFun(119883Hom119860(119891 119892)) HomhFun(119883119860)(hFun(119883 119891) hFun(119883 119892))

hFun(119883 119862) times hFun(119883 119861)((1199011)lowast(1199010)lowast) (ev1ev0)

over hFun(119883 119862times119861) cong hFun(119883 119862)timeshFun(119883 119861) that is a smothering functor The properties of 1-cellinduction 2-cell induction and 2-cell conservativity follow from surjectivity on objects fullness andconservativity of this smothering functor respectively

The 1-cells 119883 rarr Hom119860(119891 119892) that are induced by a fixed 2-cell 120572∶ 119891119887 rArr 119892119888 are unique up tofibered isomorphism over 119862 times 119861

337 Proposition Whiskering with the comma cone (333) induces a bijection between 2-cells as displayedbelow-left ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

119883

119862 119861

119860

119888 119887

120572lArr

119892 119891

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

119883

119862 119861

Hom119860(119891 119892)

119888 119887

119886

11990101199011

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭cong

and fibered isomorphism classes of maps of spans from 119862 to 119861 as displayed above-right where the fiberedisomorphisms are given by invertible 2-cells

119883

119862 119861

Hom119860(119891 119892)

119888 119887

119886119886prime 120574cong

11990101199011

so that 1199010120574 = id119887 and 1199011120574 = id119888

Proof Lemma 313 proves that the fibers of the smothering functor of Proposition 336 areconnected groupoids The objects of these fibers are functors 119883 rarr Hom119860(119891 119892) and the morphismsare invertible 2-cells that whisker with

(1199011 1199010) ∶ Hom119860(119891 119892) ↠ 119862 times 11986161

to an identity 2-cell The action of the smothering functor defines a bijection between the objects ofits codomain and their corresponding fibers

The construction of the comma infin-category is also pseudo-functorial in lax maps defined in thehomotopy 2-category

338 Observation By 1-cell induction a diagram

119862 119860 119861

119903

119892

lArr120574 119901 lArr120573 119902

119891

induces a map between commainfin-categories as displayed below-right

Hom119860(119891 119892) Hom119860(119891 119892)

119862 119861 Hom( 119891 )

119860

1199011 1199010

120601lArr

120573darr12057411990211990101199031199011

119892119903 119891 119902 =1199011 1199010

120601lArrlArr120574

119901

lArr120573

119891 119891

that is well-defined and functorial up to fibered isomorphism

One of many uses of comma infin-categories is to define the internal mapping spaces between twoelements of aninfin-category 119860 This is one motivation for our notation ldquoHom119860rdquo

339 Definition For any two elements 119909 119910 ∶ 1 119860 of an infin-category 119860 their mapping space isthe commainfin-category Hom119860(119909 119910) defined by the pullback diagram

Hom119860(119909 119910) 119860120794

1 119860 times 119860

(11990111199010)

120601

(11990111199010)

(119910119909)

The mapping spaces in anyinfin-category are discrete in the sense of Definition 1221

3310 Proposition (internal mapping spaces are discrete) For any pair of elements 119909 119910 ∶ 1 119860 of aninfin-category 119860 the mapping space Hom119860(119909 119910) is discrete

Proof Our task is to prove that for any infin-category 119883 the functor space Fun(119883Hom119860(119909 119910))is a Kan complex This is so just when hFun(119883Hom119860(119909 119910)) is a groupoid ie when any 2-cell withcodomain Hom119860(119909 119910) is invertible By 2-cell conservativity a 2-cell with codomain Hom119860(119909 119910) isinvertible just when its whiskered composite with the isofibration (1199011 1199010) ∶ Hom119860(119909 119910) ↠ 1 times 1 isan invertible 2-cell but in fact this whiskered composite is an identity since 1 is terminal

As in our convention forinfin-categories of arrows it will be convenient to weaken the meaning ofldquocommainfin-categoryrdquo to extend this appellation to any object of119974119862times119861 that is fibered equivalent (seeDefinition 327) to the strict model (1199011 1199010) ∶ Hom119860(119891 119892) ↠ 119862 times 119861 defined by 331 This is justifiedbecause such objects satisfy the weak universal property of Proposition 336 and conversely any twoobjects satisfying this weak universal property are equivalent over 119862 times 119861

3311 Proposition (uniqueness of comma infin-categories) For any isofibration (1198901 1198900) ∶ 119864 ↠ 119862 times 119861that is fibered equivalent to Hom119860(119891 119892) ↠ 119862 times 119861 the 2-cell

119864

119862 119861

119860

1198901 1198900

120598lArr

119892 119891

encoded by the equivalence 119864 ⥲ Hom119860(119891 119892) satisfies the weak universal property of Proposition 336 Con-versely if (1198891 1198890) ∶ 119863 ↠ 119862 times 119861 and (1198901 1198900) ∶ 119864 ↠ 119862 times 119861 are equipped with 2-cells

119863 119864

119862 119861 and 119862 119861

119860 119860

1198891 1198890

120575lArr

1198901 1198900

120598lArr

119892 119891 119892 119891

(3312)

Proof The proof of the first statement proceeds exactly as in the special case of Proposition3210 We prove the converse solving Exercise 32ii

Consider a pair of 2-cells (3312) satisfying the weak universal properties enumerated in Proposi-tion 336 1-cell induction supplies maps of spans

119863

119862 119861

119860

1198891 1198890120575lArr

119892 119891

119863

119864

119862 119861

119860

1198891 1198890119889

1198901 1198900120598lArr

119892 119891

119864

119862 119861

119860

1198901 1198900120598lArr

119892 119891

119864

119863

119862 119861

119860

1198901 1198900119890

1198891 1198890120575lArr

119892 119891

with the property that 120598119889119890 = 120598 and 120575119890119889 = 120575 By Proposition 337 it follows that 119889119890 cong id119864 over 119862 times 119861and 119890119889 cong id119863 over 119862 times 119861 This defines the data of a fibered equivalence 119863 ≃ 119864sup3

3313 Convention On account of Proposition 3311 we extend the appellation ldquocommainfin-categoryrdquofrom the strict model constructed in Definition 331 to any infin-category that is fibered equivalent toit and refer to its accompanying 2-cell as the ldquocomma conerdquo

sup3For the reader uncomfortable with Remark 329 Proposition 353 and Lemma 354 provides a small boost to finishthe proof

For example in sect43we define theinfin-category of cones over a fixed diagram as a commainfin-categoryProposition 3311 gives us the flexibility to use multiple models for thisinfin-category which will be use-ful in characterizing the universal properties of limits and colimits

Exercises

33i Exercise Prove Proposition 335 by observing that the map Hom119901(119902 119903) factors as a pullback ofthe Leibniz cotensor of 120597Δ[1] Δ[1] with 119901 followed by a pullback of 119903 times 119902

33ii Exercise Use Proposition 337 to justify the pseudofunctoriality of the comma constructionin lax morphisms described in Observation 338

34 Representable commainfin-categories

Definition 331 constructs a commainfin-category for any cospan Of particular importance are thespecial cases of this construction where one of the legs of the cospan is an identity

341 Definition (left and right representations) Any functor 119891∶ 119860 rarr 119861 admits a left representa-tion and a right representation as a commainfin-category displayed below-left and below-right respec-tively

Hom119861(119891 119861) Hom119861(119861 119891)

119861 119860 119860 119861

119861 119861

1199011 1199010

120601lArr

1199011 1199010

120601lArr

119891 119891

To save space we typically depict the left comma cone over 119901 displayed above-left and the right commacone over 119901 displayed above-right as inhabiting triangles rather than squares

By Proposition 3311 the weak universal property of the comma cone characterizes the commaspan up to fibered equivalence over the product of the codomain objects Thus

342 Definition A commainfin-category Hom119860(119891 119892) ↠ 119862 times 119861 isbull left representable if there exists a functor ℓ ∶ 119861 rarr 119862 so that Hom119860(119891 119892) ≃ Hom119862(ℓ 119862) over119862 times 119861 and

bull right representable if there exists a functor 119903 ∶ 119862 rarr 119861 so that Hom119860(119891 119892) ≃ Hom119861(119861 119903) over119862 times 119861

In this section we prove the first of many representability theorems demonstrating that a functor119892∶ 119862 rarr 119860 admits an absolute right lifting along 119891∶ 119861 rarr 119860 if and only if the comma infin-categoryHom119860(119891 119892) is right representable the representing functor then defining the postulated lifting Weprove this over the course of three theorems each strengthening the previous statement The firsttheorem characterizes 2-cells

119861

119862 119860dArr120588

119891

119892

119903

that define absolute right lifting diagrams via an induced equivalence Hom119861(119861 119903) ≃119862times119861 Hom119860(119891 119892)between commainfin-categories The second theorem proves that a functor 119903 defines an absolute rightlifting of 119892 through 119891 just when Hom119860(119891 119892) is right-represented by 119903 the difference is that no 2-cell120588∶ 119891119903 rArr 119892 need be postulated a priori to exist The final theorem gives a general right-representabilitycriterion that can be applied to construct a right representation to Hom119860(119891 119892) without a priori spec-ifying the representing functor 119903

343 Theorem The triangle below-left defines an absolute right lifting diagram if and only if the induced1-cell below-right

119861

119862 119860dArr120588

119891

119892

119903

Hom119861(119861 119903)

119862 119861

119860

1199011 1199010120601lArr

120588lArr

119903

119892 119891

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199010 1199011119910

1199011 1199010

120601lArr

119892 119891

defines a fibered equivalence Hom119861(119861 119903) ≃ Hom119860(119891 119892) over 119862 times 119861

Proof Suppose that (119903 120588) defines an absolute right lifting of 119892 through 119891 and consider the cor-responding unique factorization of the comma cone under Hom119860(119891 119892) through 120588 as displayed below-center

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010

120601lArr

119892 119891

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010120577lArr

120588lArr

119903

119892 119891

Hom119860(119891 119892)

Hom119861(119861 119903)

119862 119861

119860

1199011 1199010119911

1199011 1199010120601lArr

120588lArr

119903

119892 119891

(345)By 1-cell induction the 2-cell 120577 factors through the right comma cone over 119903 as displayed above-right Substituting the right-hand side of (344) into the bottom portion of the above-right diagramwe see that 119910119911 ∶ Hom119860(119891 119892) rarr Hom119860(119891 119892) is a 1-cell that factors the comma cone for Hom119860(119891 119892)through itself Applying the universal property of Proposition 337 it follows that there is a fiberedisomorphism 119910119911 cong idHom119860(119891119892) over 119862 times 119861

To prove that 119911119910 cong idHom119861(119861119903) it suffices to argue similarly that the right comma cone over 119903restricts along 119911119910 to itself Since 120588 is absolute right lifting it suffices to verify the equality 120601119911119910 = 120601

after pasting below with 120588 But now reversing the order of the equalities in (345) and (344) we have

Hom119861(119861 119903)

Hom119860(119891 119892)

Hom119861(119861 119903)

119862 119861

119860

1199011 1199010

119910

1199011 1199010

119911

1199011 1199010120601lArr

120588lArr

119903

119892 119891

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010119910

1199011 1199010120577lArr

120588lArr

119903

119892 119891

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010119910

1199011 1199010

120601lArr

119892 119891

Hom119861(119861 119903)

119862 119861

119860

1199011 1199010120601lArr

120588lArr

119903

119892 119891

which is exactly what we wanted to show Thus we see that if (119903 120588) is an absolute right lifting of 119892through 119891 then the induced map (344) defines a fibered equivalence Hom119861(119861 119903) ≃ Hom119860(119891 119892)

Now conversely suppose the 1-cell 119910 defined by (344) is a fibered equivalence and let us arguethat (119903 120588) is an absolute right lifting of 119892 through 119891 By Proposition 3311 via this fibered equivalencethe 2-cell displayed on the left-hand side of (344) inherits the weak universal property of a commacone from Hom119860(119891 119892) So Proposition 337 supplies a bijection displayed below-left-center⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

119883

119862 119861

119860

119888 119887

120572lArr

119892 119891

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

119883

119862 119861

Hom119861(119861 119903)

119888 119887

119886

11990101199011

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭cong

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

119883

119862 119861

119888 119887120585lArr

119903

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

between 2-cells over the cospan and fibered isomorphism classes of maps of spans that is implementedfrom center to left by whiskering with the 2-cell 1205881199011 sdot 119891120601∶ 1198911199010 rArr 1198921199011 in the center of (344)Proposition 337 also applies to the right comma cone 120601 over 119903 ∶ 119862 rarr 119861 giving us a second bijectiondisplayed above center-right between the same fibered isomorphism classes of maps of spans and2-cells over 119903 This second bijection is implemented from center to right by pasting with the rightcomma cone 120601∶ 1199010 rArr 1199031199011 Combining these yields a bijection between the 2-cells displayed onthe right and the 2-cells displayed on the left implemented by pasting with 120588 which is precisely theuniversal property that characterizes absolute right lifting diagrams

As a special case of this result we can now present several equivalent characterizations of fullyfaithful functors betweeninfin-categories

346 Corollary The following are equivalent and define what it means for a functor 119891∶ 119860 rarr 119861 betweeninfin-categories to be fully faithful

(i) The identity defines an absolute right lifting diagram

119860

119860 119861

119891

(ii) The identity defines an absolute left lifting diagram

119860

119860 119861

119891

(iii) For all 119883 isin 119974 the induced functor

119891lowast ∶ hFun(119883119860) rarr hFun(119883 119861)is a fully faithful functor of 1-categories

(iv) The functor induced by the identity 2-cell id119891 is an equivalence

119964120794

119860 119860

Hom119861(119891 119891)

1199011 1199010

sim id1198911199011 1199010

Proof The statement (iii) is an unpacking of the meaning of both (i) and (ii) Theorem 343specializes to prove (i)hArr(iv) or dually (ii)hArr(iv)

It is not surprising that post-composition with a fully faithful functor of infin-categories shouldinduce a fully-faithful functor of hom-categories in the homotopy 2-category What is surprising isthat this definition is strong enough This result together with the general case of Theorem 343should be provide some retroactive justification for our use of absolute lifting diagrams in Chapter 2

Having proven Theorem 343 our immediate aim is to strengthen it to show that a fibered equiv-alence Hom119861(119861 119903) ≃ Hom119860(119891 119892) over 119862 times 119861 implies that 119903 ∶ 119862 rarr 119861 defines an absolute right liftingof 119892 through 119891 without a previously specified 2-cell 120588∶ 119891119903 rArr 119892

347 Theorem Given a trio of functors 119903 ∶ 119862 rarr 119861 119891∶ 119861 rarr 119860 and 119892∶ 119862 rarr 119860 there is a bijection between2-cells as displayed below-left and fibered isomorphism classes of maps of spans as displayed below-right

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

119861

119862 119860dArr120588

119891

119892

119903

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

Hom119861(119861 119903)

119862 119861

Hom119860(119891 119892)

1199011 1199010

119910

11990101199011

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭cong

that is constructed by pasting the right comma cone over 119903 and then applying 1-cell induction to factor throughthe comma cone for Hom119860(119891 119892)

Hom119861(119861 119903)

119862 119861

119860

1199011 1199010120601lArr

120588lArr

119903

119892 119891

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010119910

1199011 1199010

120601lArr

119892 119891

Moreover a 2-cell 120588∶ 119891119903 rArr 119892 displays 119903 as an absolute right lifting of 119892 through 119891 if and only if the corre-sponding map of spans 119910∶ Hom119861(119861 119903) rarr Hom119860(119891 119892) is an equivalence

The second clause is the statement of Theorem 343 so it remains only to prove the first We showthe claimed construction is a bijection by exhibiting its inverse the construction of which involves arather mysterious lemma the significance of which will gradually reveal itself For instance Lemma348 figures prominently in the proof of the external Yoneda lemma in sect55 and is also the mainingredient in a ldquocheaprdquo version of the Yoneda lemma appearing as Corollary 3410

348 Lemma Let 119891∶ 119860 rarr 119861 be any functor and denote the right comma cone over 119891 by

Hom119861(119861 119891)

119860 119861

1199011 1199010120601lArr

119891

Then the codomain-projection functor 1199011 ∶ Hom119861(119861 119891) ↠ 119860 admits a right adjoint right inverse⁴ inducedfrom the identity 2-cell id119891 defining an adjunction

119860 perp Hom119861(119861 119891)

119860119894 1199011

1199011

over 119860 whose counit is an identity and whose unit 120578∶ idrArr 1198941199011 satisfies the conditions 120578119894 = id119894 1199011120578 = id1199011and 1199010120578 = 120601

⁴A functor admits a right adjoint right inverse just when it admits a right adjoint in an adjunction whose counit isthe identity When the original functor is an isofibration as is the case here it suffices to merely assume that the counit isinvertible see Lemma 359 and Appendix B

Proof This adjunctionwill be constructed using theweak universal properties of the right commacone over 119891 The identity 2-cell id119891 induces a 1-cell over the right comma cone over 119891

119860

Hom119861(119861 119891)

119860 119861

119891119894

1199011 1199010120601lArr

119891

=119860

119860 119861

119891=

119891

Note that 1199011119894 = id119860 so we may take the counit to be the identity 2-cell Since 120601119894 = id119891 we have apasting equality

Hom119861(119861 119891)

Hom119861(119861 119891) 119860

Hom119861(119861 119891) Hom119861(119861 119891) = Hom119861(119861 119891)

119860 119861 119860 119861

1199010

1199011

120601lArr

1198941199011

119894 119891

11990111199011 1199010120601

lArr1199011 1199010120601

119891 119891

while allows us to induce a 2-cell 120578∶ idrArr 1198941199011 with defining equations 1199011120578 = id1199011 and 1199010120578 = 120601 Thefirst of these conditions ensures one triangle identity for the other we must verify that 120578119894 = id119894 By2-cell conservativity 120578119894 is an isomorphism since 1199011120578119894 = id119860 and 1199010120578119894 = 120601119894 = id119891 are both invertibleBy naturality of whiskering we have

119894 119894

120578119894

120578119894 120578119894

1198941199011120578119894

and since 1199011120578 = id1199011 the bottom edge is an identity So 120578119894 sdot 120578119894 = 120578119894 and since 120578119894 is an isomorphismcancelation implies that 120578119894 = id119894 as required

One interpretation of Lemma 348 is best revealed though a special case

349 Corollary For any element 119887 ∶ 1 rarr 119861 the identity at 119887 defines a terminal element in Hom119861(119861 119887)

Proof By Lemma 348 the codomain-projection from the right representation of any functor ad-mits a right adjoint right inverse induced from its identity 2-cell In this case the codomain-projectionis the unique functor ∶ Hom119861(119861 119887) rarr 1 so by Definition 221 this right adjoint identifies a terminalelement of Hom119861(119861 119887) corresponding to the identity morphism id119887 in the homotopy category h119861

The general version of Lemma 348 has a similar interpretation in the slicedinfin-cosmos119974119860 theidentity functor at119860 defines the terminal object and Lemma 348 asserts that id119891 induces a terminalelement of Hom119861(119861 119891) ldquoover 119860rdquo

Proof of Theorem 347 The inverse to the function that takes a 2-cell 119891119903 rArr 119892 and producesan isomorphism class of maps Hom119861(119861 119903) rarr Hom119860(119891 119892) over 119862 times 119861 is constructed by applyingLemma 348 to the functor 119903 ∶ 119862 rarr 119861 given a map of spans restrict along the right adjoint 119894 ∶ 119862 rarrHom119861(119861 119903) and paste with the comma cone for Hom119860(119891 119892) to define a 2-cell 119891119903 rArr 119892

Starting from a 2-cell 120588∶ 119891119903 rArr 119892 the composite of these two functions constructs the 2-celldisplayed below-left

119862

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

119894

119903

1199011 1199010

119910

1199011 1199010

120601lArr

119892 119891

119862

Hom119861(119861 119903)

119862 119861

119860

119894119903

1199011 1199010120601lArr

120588lArr

119903

119892 119891

119862

119862 119861

119860

=119903

120588lArr

119903

119892 119891

which equals the above-center pasted composite by the definition of 119910 from 120588 and equals the above-right composite since 120601119894 = id119903 Thus when a 2-cell 120588∶ 119891119903 rArr 119892 is encoded as a map 119910∶ Hom119861(119861 119903) rarrHom119860(119891 119892) over 119862 times 119861 and then re-converted into a 2-cell the original 2-cell 120588 is recovered

For the converse starting with a map 119911 ∶ Hom119861(119861 119903) rarr Hom119860(119891 119892) over 119862 times 119861 the compositeof these two functions constructs an isomorphism class of maps of spans 119908 displayed below-left byapplying 1-cell induction for the comma cone Hom119860(119891 119892) to the composite 2-cell pasted below-center-left

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010

119908

1199011 1199010

120601lArr

119892 119891

Hom119861(119861 119903)

119862

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199011

1199010

120601lArr

119894

119903

1199011 1199010

119911

1199011 1199010

120601lArr

119892 119891

Hom119861(119861 119903)

119862

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199011

120578lArr119894

1199011 1199010

119911

1199011 1199010

120601lArr

119892 119891

Hom119861(119861 119903)

Hom119860(119891 119892)

119862 119861

119860

1199011 1199010

119911

1199011 1199010

120601lArr

119892 119891

Applying Lemma 348 there exists a 2-cell 120578∶ id rArr 1198941199011 so that 1199010120578 = 120601 mdash this gives the pastingequality above center mdash and 1199011120578 = id mdash which gives the pasting equality above right Proposition337 now implies that 119908 cong 119911 over 119862 times 119861

A dual version of Theorem 347 represents 2-cells 119892 rArr 119891ℓ as fibered isomorphism classes of mapsHom119861(ℓ 119861) rarr Hom119860(119892 119891) over 119861 times 119862 Specializing these results to the case where one of 119891 or 119892 isthe identity we immediately recover a ldquocheaprdquo form of the Yoneda lemma

3410 Corollary Given a parallel pair of functors 119891 119892 ∶ 119860 119861 there are bijections between 2-cells asdisplayed below-center and fibered isomorphism classes of maps between their left and right representations ascommainfin-categories as displayed below-left and below-right respectively⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

Hom119861(119892 119861)

119861 119860

Hom119861(119891 119861)

1199011 1199010

119886

11990101199011

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭cong

⎧⎪⎪⎪⎨⎪⎪⎪⎩119860 119861

119891

119892dArr120572

⎫⎪⎪⎪⎬⎪⎪⎪⎭

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

Hom119861(119861 119891)

119860 119861

Hom119861(119861 119892)

1199011 1199010

119886

11990101199011

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭cong

that are constructed by pasting with the left comma cone over 119892 and right comma cone over 119891 respectively

Hom119861(119892 119861)

119861 119860

1199011 1199010lArr120601

119892

119891uArr120572

Hom119861(119892 119861)

Hom119861(119891 119861)

119861 119860

1199011 1199010119886

1199011 1199010120601lArr

119891

Hom119861(119861 119891)

119860 119861

1199011 1199010

119891

119892dArr120572

lArr120601 =

Hom119861(119861 119891)

Hom119861(119861 119892)

119860 119861

1199011 1199010119886

1199011 1199010120601lArr

119892

and then applying 1-cell induction to factor through the left comma cone over 119891 in the former case or the rightcomma cone over 119892 in the latter

Combining the results of this section we prove one final representability theorem that allowsus to recognize when a comma infin-category is right representable in the absence of a predeterminedrepresenting functor This result specializes to give existence theorems for adjoint functors and limitsand colimits in the next chapter

3411 Theorem The commainfin-category Hom119860(119891 119892) associated to a cospan 119862 119860 119861119892 119891

isright representable if and only if its codomain-projection functor admits a right adjoint right inverse

Hom119860(119891 119892)

119862 119861

11990101199011perp119894

in which case the composite 1199010119894 ∶ 119862 rarr 119861 defines the representing functor and the 2-cell represented by thefunctor 119894 ∶ 119862 rarr Hom119860(119891 119892) defines an absolute right lifting of 119892 through 119891

Proof If Hom119860(119891 119892) is represented on the right by a functor 119903 ∶ 119862 rarr 119861 then Hom119860(119891 119892) ≃Hom119861(119861 119903) over 119862 times 119861 and the codomain-projection functor is equivalent to 1199011 ∶ Hom119861(119861 119903) ↠ 119862which admits a right adjoint right inverse 119894 by Lemma 348 The proof of Theorem 343 then showsthat 119894 represents an absolute right lifting diagram Thus it remains only to prove the converse

To that end suppose we are given a right adjoint right inverse adjunction 1199011 ⊣ 119894 Unpacking thedefinition this provides an adjunction

119862 perp Hom119860(119891 119892)

119862119894 1199011

1199011

over 119862 whose counit is an identity and whose unit 120578∶ id rArr 1198941199011 satisfies the conditions 120578119894 = id119894 and1199011120578 = id1199011 By Theorem 347 to construct the fibered equivalence Hom119861(119861 119903) ≃119862times119861 Hom119860(119891 119892)with 119903 ≔ 1199010119894 it suffices to demonstrate that the 2-cell defined by restricting the comma cone forHom119860(119891 119892) along 119894

119862

Hom119860(119891 119892)

119862 119861

119860

119903119894

1199011 1199010

120601lArr

119892 119891

defines an absolute right lifting diagramBy 1-cell induction any 2-cell as displayed below-left induces a 1-cell119898 as displayed below-center

119883 119861

119862 119860

119887

119888 dArr120594 119891

119892

=119883 Hom119860(119891 119892) 119861

119862 119860

119887

119898

1198881199011

1199010

dArr120601 119891

119892

=119883 Hom119860(119891 119892) Hom119860(119891 119892) 119861

119862 119862 119860

119887

119898

1198881199011

dArr120578 1199011

1199010

dArr120601 119891119894

119892

Inserting the triangle equality 1199011120578 = id1199011 as displayed above-right constructs the desired factorization1199010120578119898∶ 119887 rArr 119903119888 of 120594 through 120601119894

In fact by 2-cell induction for the comma cone 120601 any 2-cell 1205910 ∶ 119887 rArr 119903119888 defining a factorizationof 120594∶ 119891119887 rArr 119892119888 through 120601119894 must have the form 1205910 = 1199010120591 for some 2-cell 120591∶ 119898 rArr 119894119888 so that 1205871120591 =id119888 The pair (1205910 id119888) satisfies the compatibility condition of Proposition 336(ii) to induce a 2-cell120591∶ 119898 rArr 119894119888 Wersquoll argue that the 2-cell 120591 is unique proving that the factorization 1199010120591∶ 119887 rArr 119903119888 is alsounique

To see this note that the adjunction 1199011 ⊣ 119894 over119862 exhibits the right adjoint as a terminal elementof the object 1199011 ∶ Hom119860(119891 119892) ↠ 119862 in the slice 2-category (120101119974)119862 It follows as in Lemma 224 thatfor any object 119888 ∶ 119883 rarr 119862 and any morphism119898∶ 119883 rarr Hom119860(119891 119892) over119862 there exists a unique 2-cell

119898 rArr 119894119888 over 119862 Thus there is a unique 2-cell 120591∶ 119898 rArr 119894119888 with the property that 1199011120591 = id119888 and so thefactorization 1199010120591∶ 119887 rArr 119903119888 of 120594 through 120601119894 must also be unique

More concisely Theorem 3411 shows that a commainfin-category Hom119860(119891 119892) is right representablejust when its codomain-projection functor 1199011 ∶ Hom119860(119891 119892) ↠ 119862 admits a terminal element as anobject of the sliced infin-cosmos 119974119862 dually Hom119860(119891 119892) is left representable just when its domain-projection functor admits an initial element as an object of the sliced infin-cosmos 119974119861 see Corollary3510 There is a small gap between this statement and the version proven in Theorem 3411 havingto do with the discrepancy between the homotopy 2-category of 119974119862 and the slice of the homotopy2-category 120101119974 over 119862 This is the subject to which we now turn

Exercises

34i Exercise How might one encode the existence of an adjunction 119891 ⊣ 119906 between a given oppos-ing pair of functors using commainfin-categories

35 Sliced homotopy 2-categories and fibered equivalences

Theinfin-category 119860120794 of arrows in 119860 together with its domain- and codomain-evaluation functors(1199010 1199011) ∶ 119860120794 ↠ 119860 times 119860 satisfies a weak universal property in the homotopy 2-category that charac-terizes the infin-category up to equivalence over 119860 times 119860 see Proposition 3210 Similarly the commainfin-category is characterized up to fibered equivalence as defined in Definition 327

As commented upon in Remark 329 there is some ambiguity regarding the 2-categorical datarequired to specify a fibered equivalence that we shall now address head-on The issue is that foran infin-category 119861 in an infin-cosmos 119974 the homotopy 2-category 120101(119974119861) of the sliced infin-cosmos ofProposition 1217 is not isomorphic to the 2-category (120101119974)119861 of isofibrations functors and 2-cellsover 119861 in the homotopy 2-category 120101119974 of119974 see Exercise 14iii

However there is a canonical comparison functor relating this pair of 2-categories that satisfies aproperty we now introduce

351 Definition (smothering 2-functor) A 2-functor 119865∶ 119964 rarr ℬ is smothering if it isbull surjective on 0-cellsbull full on 1-cells for any pair of objects 119860119860prime in 119964 and 1-cell 119896 ∶ 119865119860 rarr 119865119860prime in ℬ there exists119891∶ 119860 rarr 119860prime in119964 with 119865119891 = 119896

bull full on 2-cells for any parallel pair 119891 119892 ∶ 119860 119860prime in 119964 and 2-cell 119865119860 119865119860prime119865119891

119865119892

dArr120573 in ℬ there

exists a 2-cell 120572∶ 119891 rArr 119892 in119964 with 119865120572 = 120573 andbull conservative on 2-cells for any 2-cell 120572 in119964 if 119865120572 is invertible in ℬ then 120572 is invertible in119964

352 Remark Note that smothering 2-functors are surjective on objects 2-functors that are ldquolocallysmotheringrdquo meaning that the action on hom-categories is by a smothering functor as defined in312

The prototypical example of a smothering 2-functor solves Exercise 14iii

353 Proposition Let 119861 be aninfin-category in aninfin-cosmos119974 There is a canonical 2-functor

120101(119974119861) rarr (120101119974)11986173

from the homotopy 2-category of the slicedinfin-cosmos119974119861 to the 2-category of isofibrations functors and 2-cellsover 119861 in 120101119974 and this 2-functor is smothering

This follows more-or-less immediately from Lemma 315 but we spell out the details nonetheless

Proof The 2-categories 120101(119974119861) and (120101119974)119861 have the same objects mdash isofibrations with codomain119861mdash and 1-cells mdash functors between the ldquototal spacesrdquo that commute with these isofibrations to 119861mdashso the canonical mapping may be defined to act as the identity on underlying 1-categories

By the definition of the sliced infin-cosmos given in Proposition 1217 a 2-cell between functors119891 119892 ∶ 119864 119865 from 119901∶ 119864 ↠ 119861 to 119902 ∶ 119865 ↠ 119861 is a homotopy class of 1-simplices in the quasi-categorydefined by the pullback of simplicial sets below-left

Fun119861(119864 119865) Fun(119864 119865) (hFun)119861(119864 119865) hFun(119864 119865)

120793 Fun(119864 119861) 120793 hFun(119864 119861)

119902lowast

119901 119901

Unpacking a 2-cell 120572∶ 119891 rArr 119892 is represented by a 1-simplex 120572∶ 119891 rarr 119892 in Fun(119864 119865) that whiskerswith 119902 to the degenerate 1-simplex on the vertex 119901 isin Fun(119864 119861) and two such 1-simplices representthe same 2-cell if and only if they bound a 2-simplex of the form displayed in (118) that also whiskerswith 119902 to the degenerate 2-simplex on 119901

By contrast a 2-cell in (120101119974)119861 is a morphism in the category defined by the pullback of categoriesabove-right Such 2-cells are represented by 1-simplices 120572∶ 119891 rarr 119892 in Fun(119864 119865) that whisker with 119902 to1-simplices in Fun(119864 119861) that are homotopic to the degenerate 1-simplex on 119901 and two such 1-simplicesrepresent the same 2-cell if and only if they are homotopic in Fun(119864 119865)

Applying the homotopy category functor h ∶ 119980119966119886119905 rarr 119966119886119905 to the above-left pullback produces acone over the above-right pullback inducing a canonical map

120101(Fun119861(119864 119865)) rarr (hFun)119861(119864 119865)which is the action on homs of the canonical 2-functor 120101(119974119861) rarr (120101119974)119861

The 2-functor just constructed is bijective on 0- and 1-cells To see that it is full on 2-cells we mustshow that any 1-simplex 120572∶ 119891 rarr 119892 in Fun(119864 119865) for which 119902120572∶ 119901 rarr 119901 is homotopic to 119901 sdot 1205900 ∶ 119901 rarr 119901in Fun(119864 119861) is homotopic in Fun(119864 119865) to a 1-simplex from 119891 to 119892 over 119901 sdot 1205900 By Lemma 119 anysuch 120572 defines a lifting problem

Λ1[2] Fun(119864 119865) ni

Δ[2] Fun(119864 119861) ni

119902lowast

119891

119891 119892

119901

119901 119901

120572

119902120572

A solution exists since 119902lowast ∶ Fun(119864 119865) ↠ Fun(119864 119861) is an isofibration proving that 120101(119974119861) rarr (120101119974)119861is full on 2-cells

Now suppose 120572∶ 119891 rarr 119892 represents a 2-cell in Fun119861(119864 119865) whose image in (hFun)119861(119864 119865) is anisomorphism A map in a 1-category defined by a pullback is invertible if and only if its projectionsalong the legs of the pullback cone are isomorphisms Thus the image of 120572 is invertible if and only if

120572∶ 119891 rarr 119892 defines an isomorphism in hFun(119864 119865) which by Definition 1113 is the case if and only if120572∶ 119891 rarr 119892 represents an isomorphism in Fun(119864 119865) Since 120572 is fibered over the degenerate 1-simplexat 119901 this presents us with a lifting problem

120794 Fun119861(119864 119865) Fun(119864 119865)

120128 120793 Fun(119864 119861)

120572

119902lowast

119901

which Exercise 11v tells us we can solve This proves that 120101(119974119861) rarr (120101119974)119861 reflects invertibility of2-cells and hence defines a smothering 2-functor

Smothering 2-functors are not strictly speaking invertible but nevertheless 2-categorical struc-tures from the codomain can be lifted to the domain

354 Lemma Smothering 2-functors reflect equivalences for any smothering 2-functor 119865∶ 119964 rarr ℬ and1-cell 119891∶ 119860 rarr 119861 in119964 if 119865119891∶ 119865119860 ⥲ 119865119861 is an equivalence in ℬ then 119891 is an equivalence in119964

Proof By fullness on 1-cells an equivalence inverse 119892prime ∶ 119865119861 ⥲ 119865119860 to 119865119891 lifts to a 1-cell 119892∶ 119861 rarr119860 in119964 By fullness on 2-cells the isomorphisms id119865119860 cong 119892prime ∘ 119865119891 and 119865119891 ∘ 119892prime cong id119865119861 also lift to119964 andby conservativity on 2-cells these lifted 2-cells are also invertible

Applying Lemma 354 to the smothering 2-functor

120101(119974119861) rarr (120101119974)119861we resolve the ambiguity about the 2-categorical data of a fibered equivalence

355 Proposition(i) Any equivalence in (120101119974)119861 lifts to an equivalence in 120101(119974119861) That is fibered equivalences over 119861 may

be specified by defining an opposing pair of 1-cells 119891∶ 119864 rarr 119865 and 119892∶ 119865 rarr 119864 over 119861 together withinvertible 2-cells id119864 cong 119892119891 and 119891119892 cong id119865 that lie over 119861 in 120101119974

(ii) Moreover if 119891∶ 119864 rarr 119865 is a map between isofibrations over 119861 that admits an not-necessarily fiberedequivalence inverse 119892∶ 119865 rarr 119864 with not-necessarily fibered 2-cells id119864 cong 119892119891 and 119891119892 cong id119865 then thisdata is isomorphic to a genuine fibered equivalence

Proof The first statement is proven by Lemma 354 and Proposition 353 The second statementasserts that the forgetful 2-functor (120101119974)119861 rarr 120101119974 reflects equivalences Exercise 35i shows that forany map between isofibrations over 119861 that admits an equivalence inverse in the underlying 2-categorythe inverse equivalence and invertible 2-cells can be lifted to also lie over 119861

This gives a 2-categorical proof of Proposition 1217(vii) that for anyinfin-category119861 in aninfin-cosmos119974 the forgetful functor119974119861 rarr119974 preserves and reflects equivalences

The smothering 2-functor 120101(119974119861) rarr (120101119974)119861 can also be used to lift adjunctions that are fibered2-categorically over 119861 to adjunctions in the slicedinfin-cosmos119974119861

356 Definition (fibered adjunction) A fibered adjunction over aninfin-category 119861 in aninfin-cosmos119974 is an adjunction

119864 perp 119865

119861

119891

119906

in the slicedinfin-cosmos119974119861 We write 119891 ⊣119861 119906 to indicate that specified maps over 119861 are adjoint over119861

357 Lemma (pullback and pushforward of fibered adjunctions)(i) A fibered adjunction over 119861 can be pulled back along any functor 119896 ∶ 119860 rarr 119861 to define a fibered

adjunction over 119860(ii) A fibered adjunction over 119860 can be pushed forward along any isofibration 119901∶ 119860 ↠ 119861 to define a

fibered adjunction over 119861

Proof By Proposition 133(v) pullback defines a cosmological functor 119896lowast ∶ 119974119861 rarr 119974119860 whichdescends to a 2-functor 119896lowast ∶ 120101(119974119861) rarr 120101(119974119860) that carries fibered adjunctions over 119861 to fibered ad-junctions over 119860 This proves (i)

Composition with an isofibration 119901∶ 119860 ↠ 119861 also defines a 2-functor 119901lowast ∶ 120101(119974119860) rarr 120101(119974119861) thereason we ask 119901 to be an isofibration is due to our convention that the objects in the slicedinfin-cosmoiare isofibrations over a fixed base Thus composition with an isofibration carries a fibered adjunctionover 119860 to a fibered adjunction over 119861 proving (ii)

In analogy with Lemma 354 we have

358 Lemma If 119865∶ 119964 rarr ℬ is a smothering 2-functor then any adjunction in ℬ may be lifted to anadjunction in119964

Proof Exercise 35ii

Many of the examples of fibered adjunctions we will encounter are right adjoint right inverses orleft adjoint right inverses to a given isofibration The next result shows that whenever an isofibration119901∶ 119864 ↠ 119861 admits a left adjoint with unit an isomorphism then this left adjoint may be modified soas to define a left adjoint right inverse making the adjunction fibered over 119861 The dual also holds

359 Lemma Let 119901∶ 119864 ↠ 119861 be any isofibration that admits a right adjoint 119903prime ∶ 119861 rarr 119864 with counit120598 ∶ 119901119903prime cong id119861 an isomorphism Then 119903prime is isomorphic to a functor 119903 that lies strictly over 119861 and defines a rightadjoint right inverse to 119901 Thus any such 119901 defines a fibered adjunction

119864 perp 119861

119861119901

119901

119903

in119974119861 whose right adjoint 119903 lies strictly over 119861 whose counit is the identity 2-cell and in which the unit 120578 liesover 119861 in the sense that 119901120578 = id119901

Since the identity on 119861 defines the terminal object of the slicedinfin-cosmos119974119861 Lemma 359 canbe summarized more compactly as follows

3510 Corollary An isofibration 119901∶ 119864 ↠ 119861 admits a right adjoint right inverse if and only if it admitsa terminal element as an object of119974119861 Dually 119901∶ 119864 ↠ 119861 admits a left adjoint right inverse if and only if itadmits an initial element as an object of119974119861

3511 Example Lemma 348 constructs an adjunction in the sliced 2-category 120101119974119860 Lemma 358now allows us to lift it to a genuine adjunction

119860 perp Hom119861(119861 119891)

119860119894 1199011

1199011

in the sliced infin-cosmos 119974119860 By Corollary 3510 this situation may be summarized by saying that1199011 ∶ Hom119861(119861 119891) ↠ 119860 admits a terminal element over 119860

By Lemma 357(i) we may pull back the fibered adjunction along any element 119886 ∶ 1 rarr 119860 to obtainan adjunction

1 perp Hom119861(119861 119891119886)

119894119886

that identifies a terminal element in the fiber Hom119861(119861 119891119886) of 1199011 ∶ Hom119861(119861 119891) ↠ 119860 over 119886 Thisgeneralizes the result of Corollary 349

3512 Example (the fibered adjoints to composition) For any infin-category 119860 the adjoints to theldquocompositionrdquo functor ∘ ∶ 119860120794 times

119860119860120794 rarr 119860120794 constructed in Lemma 2113 may be constructed by com-

posing a triple of adjoint functors that are fibered over the endpoint-evaluation functors

119860120795 119860120794

119860 times 119860

1198601198891

(ev2ev0)

1198601199040

1198601199041perp

(ev1ev0)

with an adjoint equivalence involving a functor119860120795 ⥲rarr 119860120794 times119860119860120794 which also lies over119860times119860 Lemma

359 and its dual implies that these adjoint equivalences can be lifted to fibered adjoint equivalencesover 119860 times 119860 and now both adjoint triples and hence also the composite adjunctions

119860120794 times119860119860120794 119860120794

119860 times 119860

∘perpperp

(ev2ev0)

(minusiddom(minus))

(idcod(minus)minus)(ev1ev0)

lie in119974119860times119860

This fibered adjunction figures in the proof of a result that will allow us to convert limit andcolimit diagrams into right and left Kan extension diagrams in the next chapter

3513 Proposition A cospan as displayed below-left admits an absolute right lifting if and only if the cospandisplayed below-right admits an absolute right lifting

119861 Hom119860(119891 119860)

119862 119860 119862 119860dArr120588

119891dArr120598

1199011

119892

119903

119892

119894

in which case the the 2-cell 120598 is necessarily an isomorphism and can be chosen to be an identity

Proof By Theorem 3411 a cospan admits an absolute right lifting if and only if the codomain-projection functor from the associated comma infin-category admits a right adjoint right inverse Ourtask is thus to show that this right adjoint right inverse exists for Hom119860(119891 119892) if and only if this rightadjoint right inverse exists for Hom119860(1199011 119892)

From the defining pullback (332) that constructs the comma infin-category Hom119860(1199011 119892) repro-duced below-left we have the below-right pullback square

Hom119860(1199011 119892) Hom119860(119860 119892) 119860120794

119862 times Hom119860(119891 119860) 119862 times 119860 119860 times 119860

Hom119860(119891 119860) 119860

(11990111199010)

119862times1199011120587

119892times119860120587

1199011

Hom119860(1199011 119892) Hom119860(119860 119892)

Hom119860(119891 119860) 119860

1199010

1199011

(3514)By Lemma 357 the composition-identity fibered adjunction of Example 3512 pulls back along 119892 times119891∶ 119862 times 119861 rarr 119860 times 119860 to define a fibered adjunction

Hom119860(1199011 119892) cong Hom119860(119860 119892) times119860 Hom119860(119891 119860) Hom119860(119891 119892)

119862 times 119861

(11990111199010)

which then pushes forward along the projection 120587∶ 119862 times 119861 ↠ 119862 to a fibered adjunction over 119862

Hom119860(1199011 119892) Hom119860(119891 119892)

119862

1199011

between the codomain-projection for Hom119860(1199011 119892) and the codomain projection for Hom119860(119891 119892)Now by Corollary 3510 1199011 ∶ Hom119860(119891 119892) ↠ 119862 admits a right adjoint right inverse just when theobject on the right admits a terminal element while 1199011 ∶ Hom119860(1199011 119892) ↠ 119862 similarly admits a rightadjoint right inverse just when the object on the left admits a terminal element By Theorem 242 aterminal element on either side is carried by the appropriate right adjoint to a terminal element onthe other side This proves the equivalence of these conditions

It remains only to prove that the 2-cell for the absolute right lifting of 119892 through 1199011 is invertible ByTheorem 3411 this 2-cell is constructed as by restricting the comma cone along the terminal elementso it is given by the composite

119862 Hom119860(1199011 119892) 119860120794 1198601199010

dArr1205811199011

where the left-hand map is the terminal element just constructed and the middle one comes comesfrom the defining pullback diagram displayed on the left of (3514) As just argued that terminalelement may be chosen to be in the image of the right adjoint Hom119860(119891 119892) rarr Hom119860(119860 119892) times119860Hom119860(119891 119860) cong Hom119860(1199011 119892) whose component on the left factor is the identity Simultaneouslythe pullback defining the map Hom119860(1199011 119892) rarr 119860120794 factors through the projection onto the left factorso we see that the 2-cell in the absolute right lifting diagram is represented by the composite

119862 Hom119860(119891 119892) Hom119860(119860 119892) times119860 Hom119860(119891 119860) Hom119860(119860 119892) 119860120794 119860119894 (idcod(minus)minus) 1205871199010

dArr1205811199011

and hence that this cell is invertible

Exercises

35i Exercise Let 119861 be an object in a 2-category 119966 and consider a map

119864 119865

119861

119891

between isofibrations over 119861 Prove that if 119891 is an equivalence in119966 then 119891 is also an equivalence in theslice 2-category119966119861 of isofibrations over 119861 1-cells that form commutative triangles over 119861 and 2-cellsthat lie over 119861 in the sense that they whisker with the codomain isofibration to the identity 2-cell onthe domain isofibration

35ii Exercise Let 119865∶ 119964 rarr ℬ be a smothering 2-functor Show that any adjunction in ℬ can belifted to an adjunction in 119964 Demonstrate furthermore that if we have previously specified a lift ofthe objects 1-cells and either the unit or counit of the adjunction in ℬ then there is a lift of theremaining 2-cell that combines with the previously specified data to define an adjunction in119964 Thisproves a more precise version of Lemma 358

CHAPTER 4

Adjunctions limits and colimits II

Comma infin-categories provide a vehicle for encoding the universal properties of categorical con-structions that restrict to define equivalences between the internal mapping spaces introduced in Def-inition 339 between suitable pairs of elements Using the theory developed in Chapter 3 we quicklyprove a variety of results of this type first for adjunctions in sect41 and then for limits and colimits insect43 In an interlude in sect42 we introduce the infin-categories of cones over or under a diagram as acomma infin-category and then give a second model for these infin-categories of cones in the case of dia-grams indexed by simplicial sets built from Joyalrsquos join construction Then we conclude in sect44 withan application constructing the loops ⊢ suspension adjunction for pointed infin-categories containingan element that is both initial and terminal

41 The universal property of adjunctions

Our first result shows that an adjunction between an opposing pair of functors can equally beencoded by a ldquotransposing equivalencerdquo between their left and right representations as commainfin-cat-egories

411 Proposition An opposing pair of functors 119906∶ 119860 rarr 119861 and 119891∶ 119861 rarr 119860 define an adjunction 119891 ⊣ 119906if and only if Hom119860(119891 119860) ≃ Hom119861(119861 119906) over 119860 times 119861

Proof This is a special case of Theorem 347 If 119891 ⊣ 119906 then Lemma 236 tells us that its counit120598 ∶ 119891119906 rArr id119860 defines an absolute right lifting diagram Theorem 347 then tells us that the 1-cellinduced by the left-hand pasted composite

Hom119861(119861 119906)

119860 119861

119860

1199011 1199010120601lArr

120598lArr

119906

119891

Hom119861(119861 119906)

Hom119860(119891 119860)

119860 119861

119860

1199011 1199010119910

1199011 1199010

120601lArr

119891

defines a fibered equivalence Hom119861(119861 119906) ⥲ Hom119860(119891 119860) over 119860 times 119861 We interpret this result assaying that in the presence of an adjunction 119891 ⊣ 119906 the right comma cone over 119906 transposes to definethe left comma cone over 119891sup1

sup1If desired an inverse equivalence can be constructed by applying the dual of Theorem 347 to the absolute left liftingdiagram presented by the unit

Conversely Theorem 347 tells us that from a fibered equivalence Hom119861(119861 119906) ⥲ Hom119860(119891 119860)over 119860 times 119861 one can extract a 2-cell that defines an absolute right lifting diagram

119861

119860 119860dArr120598

119891119906

Lemma 236 then tells us that this 2-cell defines the counit of an adjunction 119891 ⊣ 119906

412 Observation (the transposing equivalence) To justify referring to the induced functor

Hom119861(119861 119906) ⥲ Hom119860(119891 119860)as a transposing equivalence recall that the transpose of a 2-cell 120594∶ 119887 rArr 119906119886 across the adjunction119891 ⊣ 119906 is computed by the left-hand pasting diagram below

119883

119860 119861

119860

119887119886 120594lArr

120598lArr

119906

119891

119883

Hom119861(119861 119906)

119860 119861

119860

119909119887119886

1199011 1199010120601lArr

120598lArr

119906

119891

119883

Hom119861(119861 119906)

Hom119860(119891 119860)

119860 119861

119909

119887119886

1199011 1199010

1199011 1199010120601lArr

119891

By the weak universal property of the right comma cone over 119906 the 2-cell 120594 is represented by theinduced functor 119883 rarr Hom119861(119861 119906) which then composes with the transposing equivalence to definea functor119883 rarr Hom119860(119891 119860) that represents the transpose of120594 by the pasting diagram equalities fromright to left

413 Corollary An adjunction 119861 119860119891

perp119906

induces an equivalence Hom119860(119891119887 119886) ≃ Hom119861(119887 119906119886)

over 119883 times 119884 for any pair of generalized elements 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119884 rarr 119861

Proof By the pullback construction of comma infin-categories given in (332) the equivalenceHom119860(119891 119860) ≃ Hom119861(119861 119906) in119974119860times119861 pulls back along 119886times119887∶ 119883times119884 rarr 119860times119861 to define an equivalenceHom119860(119891119887 119886) ≃ Hom119861(119887 119906119886) in119974119883times119884

In particular the equivalence of Proposition 411 pulls back to define an equivalence of internalmapping spaces introduced in 339

414 Proposition (the universal property of units and counits) Consider an adjunction

119861 119860119891

perp119906

with unit 120578∶ id119861 rArr 119906119891 and counit 120598 ∶ 119891119906 rArr id119860

Then for each element 119886 ∶ 1 rarr 119860 the component 120598119886 defines a terminal element of Hom119860(119891 119886) and for eachelement 119887 ∶ 1 rarr 119861 the component 120578119887 defines an initial element of Hom119861(119887 119906)

Proof By Corollary 413 the fibered equivalence Hom119860(119891 119860) ≃119860times119861 Hom119861(119861 119906) of Proposition411 pulls back to define equivalences

Hom119860(119891 119886) ≃119861 Hom119861(119861 119906119886) and Hom119860(119891119887 119860) ≃119860 Hom119861(119887 119906)By Corollary 349 id119906119886 induces a terminal element of Hom119861(119861 119906119886) and by Observation 412 its imageacross the equivalence Hom119861(119861 119906119886) ⥲ Hom119860(119891 119886) is again a terminal element which represents thetransposed 2-cell the component of the counit 120598 at the element 119886 The proof that the unit componentdefines a terminal element of Hom119861(119887 119906) is dual

A more sophisticated formulation of the universal property of unit and counit components willappear in Proposition 832 where it will form a key step in the proof that any adjunction extends toa homotopy coherent adjunction

The universal property of unit and counit components captured in Proposition 414 gives the mainidea behind the adjoint functor theorems a functor 119891∶ 119861 rarr 119860 admits a right adjoint just when foreach element 119886 ∶ 1 rarr 119860 the infin-category Hom119860(119891 119886) admits a terminal element The image of thisterminal element under the domain-projection functor 1199010 ∶ Hom119860(119891 119886) ↠ 119861 then defines the element119906119886∶ 1 rarr 119861 and the comma cone defines the component of the counit at 119886 The universal property ofthese unit components is then used to extend the mapping on elements to a functor 119906∶ 119860 rarr 119861

The result just stated is true in the infin-cosmos of quasi-categories and in other infin-cosmoi whereuniversal properties are generated by the terminalinfin-category 1sup2 What is true in allinfin-cosmoi is theversion of the result just stated where the quantifier ldquofor each element 119886 ∶ 1 rarr 119860rdquo is replaced withldquofor each generalized element 119886 ∶ 119883 rarr 119860rdquo in which case the meaning of ldquoterminal elementrdquo should beenhanced to ldquoterminal element over 119883rdquo see the remark after Corollary 349 Since every generalizedelement factors through the universal generalized element namely the identity functor at119860 it sufficesto prove

415 Proposition A functor 119891∶ 119861 rarr 119860 admits a right adjoint if and only if Hom119860(119891 119860) admits aterminal element over 119860 Dually 119891∶ 119861 rarr 119860 admits a left adjoint if and only if Hom119860(119860 119891) admits aninitial element over 119860

Proof By Proposition 411 119891∶ 119861 rarr 119860 admits a right adjoint if and only if the commainfin-categoryHom119860(119891 119860) is right representable Theorem 3411 specializes to tell us that this is the case if and onlyif the codomain-projection functor 1199011 ∶ Hom119860(119891 119860) ↠ 119860 admits a right adjoint right inverse whichby Corollary 3510 is equivalent to postulating a terminal element over 119860

The same suite of results from sect34 specialize to theorems that encode the universal propertiesof limits and colimits Before proving these we first construct the infin-category of cones over a fixeddiagram and also construct alternate models for the infin-categories of cones over varying 119869-indexeddiagrams in the case where 119869 is a simplicial set

Exercises

41i Exercise Prove that the transposing equivalence of Proposition 411 as elaborated upon inObservation 412 is natural with respect to pre-composing with a 2-cell 120573∶ 119887prime rArr 119887 or post-composingwith a 2-cell 120572∶ 119886 rArr 119886prime

sup2We delay the discussion of ldquoanalytically-provenrdquo theorems about quasi-categories until we demonstrate in Part IVthat such results apply also in biequivalentinfin-cosmoi

42 infin-categories of cones

421 Definition (theinfin-category of cones) Let 119889∶ 1 rarr 119860119869 be a 119869-shaped diagram in aninfin-category119860 Theinfin-category of cones over 119889 is the commainfin-category Hom119860119869(Δ 119889)with comma cone displayedbelow-left while theinfin-category of cones under 119889 is the commainfin-category Hom119860119869(119889 Δ)with commacone displayed below-right

Hom119860119869(Δ 119889) Hom119860119869(119889 Δ)

1 119860 119860 1

119860119869 119860119869

1199011 1199010

120601lArr

1199011 1199010

120601lArr

119889 Δ Δ 119889

By replacing the ldquo119889rdquo leg of the cospans Definition 421 can be modified to allow 119889∶ 119863 rarr 119860119869 tobe a family of diagrams or to defineinfin-categories of cones over any diagram of shape 119869 an element ofHom119860119869(Δ119860119869) is a cone with any summit over any 119869-indexed diagram

In the case where the indexing shape 119869 is a simplicial set (and not an infin-category in a cartesianclosedinfin-cosmos) there is another model of theinfin-categories of cones over or under a diagram thatmay be constructed using Joyalrsquos join construction The reason for the equivalence is that joins ofsimplicial sets are known to be equivalent to so-called ldquofat joinsrdquo of simplicial sets and a particularinstance of the fat join construction gives the shape of the cones appearing in Definition 421 Wenow introduce these notions

422 Definition (fat join) The fat join of simplicial sets 119868 and 119869 is the simplicial set constructed bythe following pushout

(119868 times 119869) ⊔ (119868 times 119869) 119868 ⊔ 119869

119868 times 120794 times 119869 119868 ⋄ 119869

120587119868⊔120587119869

from which it follows that(119868 ⋄ 119869)119899 ≔ 119868119899 ⊔ ( 1114018

[119899]↠[1]119868119899 times 119869119899) ⊔ 119869119899

Note there is a natural map 119868 ⋄ 119869 ↠ 120794 induced by the projection 120587∶ 119868 times 120794 times 119869 ↠ 120794 so that 119868 is thefiber over 0 and 119869 is the fiber over 1

119868 ⊔ 119869 119868 ⋄ 119869

120793 + 120793 120794

423 Lemma For any simplicial set 119869 andinfin-category 119860 we have natural isomorphisms

Hom119860119869(Δ119860119869) cong 119860120793⋄119869 and Hom119860119869(119860119869 Δ) cong 119860119869⋄120793

Proof The simplicial cotensor119860(minus) ∶ 119982119982119890119905op rarr119974 carries the pushout of Definition 422 to thepullback squares that define the left and right representations ofΔ∶ 119860 rarr 119860119869 as a commainfin-category

119860120793⋄119869 (119860119869)120794 119860119869⋄120793 (119860119869)120794

119860119869 times 119860 119860119869 times 119860119869 119860 times 119860119869 119860119869 times 119860119869

(11990111199010)

idtimesΔ Δtimesid

424 Definition (join) The join of simplicial sets 119868 and 119869 is the simplicial set 119868 ⋆ 119869

119868 ⊔ 119869 119868 ⋆ 119869

120793 + 120793 120794

with(119868 ⋆ 119869)119899 ∶= 119868119899 ⊔ ( 1114018

0le119896lt119899119868119899minus119896minus1 times 119869119896) ⊔ 119869119899

and with the vertices of these 119899-simplices oriented so that there is a canonical map 119868 ⋆ 119869 rarr 120794 so that119868 is the fiber over 0 and 119869 is the fiber over 1 See [24 sect3] for more details

The join functor minus ⋆ 119869∶ 119982119982119890119905 rarr 119982119982119890119905 preserves connected colimits but not the initial object orother coproducts but this issue can be rectified by replacing the codomain by the slice category under119869 Contextualized in this way the join admits a right adjoint defined by Joyalrsquos slice construction

425 Proposition The join functors admit right adjoints

119982119982119890119905 119868119982119982119890119905 119982119982119890119905 119869119982119982119890119905

119868⋆minus

perpminusminus

minus⋆119869

perpminusminus

defined by the natural bijections⎧⎪⎪⎪⎨⎪⎪⎪⎩

119868

119868 ⋆ Δ[119899] 119883ℎ

⎫⎪⎪⎪⎬⎪⎪⎪⎭cong 1114107 Δ[119899] ℎ119883 1114110 and

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119869

Δ[119899] ⋆ 119869 119883119896

⎫⎪⎪⎪⎬⎪⎪⎪⎭cong 1114107 Δ[119899] 119883119896 1114110

Proof As in the statement the simplicial set 119883119896 is defined to have 119899-simplices correspondingto maps Δ[119899] ⋆ 119869 rarr 119883 under 119869 with the right action by the simplicial operators [119898] rarr [119899] givenby pre-composition with Δ[119898] rarr Δ[119899] Since the join functor minus ⋆ 119869∶ 119982119982119890119905 rarr 119869119982119982119890119905 preservescolimits this extends to a bijection between maps 119868 rarr 119883119896 and maps 119868 ⋆ 119869 rarr 119883 under 119869 that isnatural in 119868 and in 119896 ∶ 119869 rarr 119883

426 Notation For any simplicial set 119869 we write

119869◁ ≔ 120793 ⋆ 119869 and 119869▷ ≔ 119869 ⋆ 120793and write ⊤ for the cone vertex of 119869◁ and perp for the cone vertex of 119869▷ These simplicial sets areequipped with canonical inclusions

119869◁ 119869 119869▷

427 Proposition (an alternate model) For any simplicial sets 119868 and 119869 and anyinfin-category 119860 there is anatural equivalence

119860119868⋆119869 119860119868⋄119869

119860119868⊔119869

res res

In particular there are comma squares

119860119869◁ 119860119869▷

119860119869 119860 119860 119860119869

119860119869 119860119869

res ev⊤

120601lArr

evperp res

120601lArr

Proof There is a canonical map of simplicial sets

(119868 times 119869) ⊔ (119868 times 119869) 119868 ⊔ 119869

119868 times 120794 times 119869 119868 ⋄ 119869 119868 ⋆ 119869

120794

120587119868⊔120587119869

that commutes with the inclusions of the fibers 119868 ⊔ 119869 over the endpoints of 120794 This dashed mapdisplayed above is defined on those 119899-simplices over 119868 ⋄ 119869 that map surjectively onto 120794 to send a triple(120572 ∶ [119899] ↠ [1] 120590 isin 119868119899 120591 isin 119869119899) representing an 119899-simplex of 119868 ⋄ 119869 to the pair (120590|0hellip119896 isin 119868119896 120591|119896+1hellip119899 isin119869119899minus119896minus1) representing an 119899-simplex of 119868 ⋆ 119869 where 119896 isin [119899] is the maximal vertex in 120572minus1(0) Proposition of Appendix C proves that this map induces a natural equivalence119876119868⋆119869 ⥲ 119876119868⋄119869 of quasi-categoriesover119876119869times119876119868 Taking119876 to be the functor space Fun(119883119860) proves the claimed equivalence for generalinfin-categories

Now Proposition 3311 and Lemma 423 implies that this fibered equivalence equips119860119869◁ and119860119869▷

with comma cone squares The 2-cells in (428) are represented by the maps

119869 ⊔ 119869 120793 ⊔ 119869 119869 ⊔ 119869 119869 ⊔ 120793

119869 times 120794 120793 ⋄ 119869 119869◁ 119869 times 120794 119869 ⋄ 120793 119869▷

120794 120794

which yield 2-cells

119860119869◁ (119860119869)120794 119860119869 119860119869▷ (119860119869)120794 119860119869

Δ ev⊤

1199010

1199011

dArr120581

Δ evperp

1199010

1199011

dArr120581

upon cotensoring into 119860

Exercises

42i Exercise Compute Δ[119899] ⋆ Δ[119898] and Δ[119899] ⋄ Δ[119898] and define a section

Δ[119899] ⋆ Δ[119898] rarr Δ[119899] ⋄ Δ[119898]to the map constructed in the proof of Proposition 427

43 The universal property of limits and colimits

We now return to the general context of Definition 231 simultaneously considering diagramsvalued in aninfin-category119860 that are indexed either by a simplicial set or by anotherinfin-category in thecase where the ambientinfin-cosmos is cartesian closed As was the case for Proposition 411 Theorem347 specializes immediately to prove

431 Proposition (colimits represent cones) A family of diagrams 119889∶ 119863 rarr 119860119869 admits a limit if andonly if theinfin-category of cones Hom119860119869(Δ 119889) over 119889 is right representable

Hom119860119869(Δ 119889) ≃119863times119860 Hom119860(119860 ℓ)in which case the representing functor ℓ ∶ 119863 rarr 119860 defines the limit functor Dually 119889∶ 119863 rarr 119860119869 admits acolimit if and only if theinfin-category of cones Hom119860119869(119889 Δ) under 119889 is left representable

Hom119860119869(119889 Δ) ≃119860times119863 Hom119860(119888 119860)in which case the representing functor 119888 ∶ 119863 rarr 119860 defines the colimit functor

Theorem 3411 now specializes to tell us that such representations can be encoded by terminalor initial elements a result which is easiest to interpret in the case of a single diagram rather than afamily of diagrams

432 Proposition (limits as terminal elements) Consider a diagram 119889∶ 1 rarr 119860119869 of shape 119869 in aninfin-category 119860

(i) If 119889 admits a limit then the 1-cell 1 rarr Hom119860119869(Δ 119889) induced by the limit cone 120598 ∶ Δℓ rArr 119889 definesa terminal element of theinfin-category of cones

(ii) Conversely if theinfin-category of cones Hom119860119869(Δ 119889) admits a terminal element then the cone repre-sented by this element defines a limit cone

Dually 119889 admits a colimit if and only if the infin-category Hom119860119869(119889 Δ) of cones under 119889 admits an initialelement in which case the initial element defines the colimit cone

Proof By Definition 237 a limit cone defines an absolute right lifting diagram which by Theo-rem 343 induces an equivalence Hom119860(119860 ℓ) ⥲ Hom119860119869(Δ 119889) over119860 By Corollary 349 the identityat ℓ induces a terminal element of Hom119860(119860 ℓ) which the equivalence carries to a terminal element oftheinfin-category of cones this being the element that represents the limit cone 120598 ∶ Δℓ rArr 119889

Conversely if Hom119860119869(Δ 119889) admits a terminal element this defines a right adjoint right inverse tothe codomain-projection functor Hom119860119869(Δ 119889) Theorem 3411 then tells us that the cone representedby this element 1 rarr Hom119860119869(Δ 119889) defines an absolute right lifting of 119889 through Δ

433 Remark The proof of Proposition 432 extends without change to the case of a family of di-agrams 119889∶ 119863 rarr 119860119869 in place of a single diagram since Theorem 3411 applies at this level of gen-erality For a family of diagrams 119889 parametrized by 119863 the infin-category of cones defines an object1199011 ∶ Hom119860119869(Δ 119889) ↠ 119863 of the slicedinfin-cosmos119974119863 and the terminal elements referred to in both (i)and (ii) should be interpreted as terminal elements in119974119863

434 Proposition An infin-category 119860 admits a limit of a family of diagrams 119889∶ 119863 rarr 119860119869 indexed by asimplicial setsup3 119869 if and only if there exists an absolute right lifting of 119889 through the restriction functor

119860119869◁

119863 119860119869dArr120598

119889

When these equivalent conditions hold 120598 is necessarily an isomorphism and may be chosen to be the identity

Proof ByDefinition 237 the family of diagrams admits a limit if and only if 119889 admits an absoluteright lifting through Δ∶ 119860 rarr 119860119869 By Proposition 3513 such an absolute lifting diagram exists if andonly if 119889 admits an absolute right lifting through codomain-projection functor 1199011 ∶ Hom119860119869(Δ119860119869) ↠119860119869 in which case the 2-cell of this latter absolute right lifting diagram is invertible By Proposition427 the restriction functor res ∶ 119860119869◁ ↠ 119860119869 is equivalent to this codomain-projection functor soabsolute right liftings of 119889 through 1199011 are equivalent to absolute right liftings of 119889 through res If thisabsolute lifting diagram is inhabited by an invertible 2-cell the isomorphism lifting property of theisofibration proven in Proposition 1410 can be used to replace the functor ran ∶ 119863 rarr 119860119869◁ with anisomorphic functor making the triangle commute strictly

Recall from Lemma 236 that absolute lifting diagrams can be used to encode the existence ofadjoint functors Combining this with Definition 232 Proposition 434 specializes to prove

435 Corollary Aninfin-category119860 admits all limits indexed by a simplicial set 119869 if and only if the restrictionfunctor

119860119869◁ 119860119869res

perpran

sup3We have stated this result for diagrams indexed by simplicial sets because its means is easiest to interpret but weactually prove it with the codomain-projection functor 1199011 ∶ Hom119860119869(Δ119860119869) ↠ 119860119869 in place of the equivalent isofibration119860119869◁ ↠ 119860119869 and this proof applies equally in the case of diagrams indexed byinfin-categories 119869 in cartesian closedinfin-cosmoithat may or may not have a join operation available

admits a right adjoint Dually aninfin-category119860 admits all colimits indexed by a simplicial set 119869 if and only ifthe restriction functor

119860119869▷ 119860119869res

perplan

admits a left adjoint

436 Remark Since the restriction functor is an isofibration Lemma 359 applies and the adjunc-tions of Corollary 435 can be defined so as to be fibered over theinfin-category of diagrams 119860119869

The adjunctions of Corollary 435 are particular useful in the case of pullbacks and pushouts

437 Definition (pushouts and pullbacks) A pushout in an infin-category 119860 is a colimit indexed bythe simplicial set

⟔≔ Λ0[2]Dually a pullback in aninfin-category 119860 is a limit indexed by the simplicial set

⟓≔ Λ2[2]Cones over diagrams of shape ⟓ or cones under diagrams of shape ⟔ define commutative squaresdiagrams of shape

⊡ ≔ Δ[1] times Δ[1] cong⟔▷cong⟓◁ A pullback square in an infin-category 119860 is an element of 119860⊡ in the essential image of the functor

ran of Proposition 434 for some diagram of shape ⟓ When 119860 admits all pullbacks these are exactlythose elements of 119860⊡ at which the component of the unit of the adjunction res ⊣ ran of Corollary435 is an isomorphism Dually a pushout square in119860 is an element in the essential image of the dualfunctor lan for some diagram of shape ⟔ ie those elements for which the component of the counitof the adjunction lan ⊣ res is an isomorphism

An 119883-indexed commutative square in 119860 is a diagram 119883 rarr 119860⊡ or equivalently an element ofFun(119883119860)⊡ We label the 0- and 1-simplex components as follows

119889 119887

119888 119886

119906

119907 119908 119891

119892

isin Fun(119883119860)

The diagram also determines a pair of 2-simplices that witness commutativity 119891119906 = 119908 = 119892119907 inhFun(119883119860) but the names of these witnesses wonrsquot matter for this discussion

438 Lemma An119883-indexed commutative square valued in aninfin-category119860 in119974 as below-left is a pullbacksquare if and only if the induced 2-cell below-right is an absolute right lifting diagram in119974119883

119889 119887 Hom119860(119860 119887)

119888 119886 119883 Hom119860(119860 119886)

119883

119906

119907 119908 119891 dArr(id119886119907) 119891lowast

1199011119892 119892

119906

1199011

The statement requires some explanation The 1-simplex 119892∶ 119888 rarr 119886 represents a 2-cell 119883 119860119888

119886dArr119892

inducing the map 119892 ∶ 1 rarr Hom119860(119860 119886) The map 119906 ∶ 119883 rarr Hom119860(119860 119887) is defined similarly Themap 119891lowast ∶ Hom119860(119860 119887) rarr Hom119860(119860 119886) is characterized by the pasting diagram

Hom119860(119860 119887)

119883 119860

1199011 1199010120601lArr119887dArr119891119886

Hom119860(119860 119887)

Hom119860(119860 119886)

119883 119860

1199011 1199010119891lowast

1199011 1199010120601lArr

119886

By Proposition 337 the composite 119891lowast119906 is isomorphic to 119891119906 By 2-cell induction the 2-cell maybe constructed by specifying its domain and codomain components the former of which we take to

be 119883 119860119889

119888dArr119907 and the latter of which we take to be id119886 Note that the 2-cell just constructed lies

in 120101119974119883 and so can be lifted to 120101(119974119883) by Proposition 353

Proof We prove the result in the case 119883 = 1 and then deduce the result for families of pullbackdiagrams from this case By Proposition 427 the pullback

119860⊡119892or119891 119860⊡

1 119860⟓

119892or119891

is equivalent to the infin-category of cones over the cospan diagram 119892 or 119891 By Proposition 432 toshow that the commutative square defines a pullback diagram is to show that (119906 119907 119891 119892) ∶ 1 rarr 119860⊡119892or119891defines a terminal element in the pullback We will show that this pullback119860⊡119892or119891 is also equivalent tothe commainfin-category HomHom119860(119860119886)(119891lowast 119892) By Theorem 3411 the pair (119906 (id119886 119907)) defines anabsolute right lifting if and only if it represents a terminal element in this commainfin-category whichwill prove the claimed equivalence

To see this first consider the diagram which induces a map between the two pullbacks

119860119891 119860120795 119860120794

Hom119860(119860 119886) 119860120794

119883 119860120794 119860

119883 119860

119901119891 11990102

11990112

11990101

1199011

119891

1199011

1199010

119886

1199011

Since 119860120795 ≃ 119860Λ1[2] the right-hand back square is equivalent to a pullback Composing the pullbacksquares in the back face of the diagram we obtain an equivalence 119860119891 ⥲ Hom119860(119860 119887) and by in-spection see that the map 119901119891 ∶ 119860119891 ↠ Hom119860(119860 119886) is equivalent to the map 119891lowast ∶ Hom119860(119860 119887) rarrHom119860(119860 119886) over Hom119860(119860 119886)

By applying (minus)120794 to the pullback diagram that defines Hom119860(119860 119886) we obtain a pullback squarethat factors as

Hom119860(119860 119886)120794 119860120794⋄120793 119860120794times120794

1 119860 119860120794

1199011207941

119886 Δ

By the equivalence 119860120794⋄120793 ≃ 119860120794⋆120793 of Proposition 427 the left-hand pullback square shows thatHom119860(119860 119886)120794 is equivalent to the pullback of 1199012 ∶ 119860120795 ↠ 119860 along 119886 ∶ 1 rarr 119860 Modulo this equiv-alence the map 1199010 ∶ Hom119860(119860 119886)120794 ↠ Hom119860(119860 119886) is the pullback of the fibered map

119860120795 119860120794

119860

11990102

1199012 1199011

along 119886 ∶ 1 rarr 119860 and the codomain projection is similarly the pullback of the fibered map 11990112 ∶ 119860120795 ↠119860120794

Putting this together it follows that the pullback

HomHom119860(119860119886)(119891lowastHom119860(119860 119886)) Hom119860(119860 119886)120794

Hom119860(119860 119887) Hom119860(119860 119886)

1199010

119891lowast

is equivalently computed by forming the limit

bull 119860⊡ 119860120795 119860120794

119860120795 119860120794

1 119860120794

11990102

11990112

1199011211990102

119891

The codomain projection 1199011 ∶ HomHom119860(119860119886)(119891lowastHom119860(119860 119886)) ↠ Hom119860(119860 119886) is the pullback of thetop-horizontal composite in the above diagram along Hom119860(119860 119886) rarr 119860120794 So we see that the commainfin-category HomHom119860(119860119886)(119891lowast 119892) is equivalently computed by the limit below-left or equivalently

by the limit below right exactly as we claimed

bull bull 1

bull 119860⊡ 119860120795 119860120794

119860120795 119860120794

1 119860120794

119892

11990102

11990112

1199011211990102

119891

119860⊡119892or119891 119860⊡

1 119860⟓

119892or119891

The same computation proves the general case for 119883 ne 1 when the comma infin-category is con-structed in 119974119883 see Proposition 1217 for a description of the simplicial limits in sliced infin-cosmoiAlternatively a diagram 119904 ∶ 119883 rarr 119860⊡ in 119974 also defines a 119883-indexed diagram in the infin-cosmos 119974119883valued in the infin-category 120587∶ 119860 times 119883 rarr 119883 This takes the form of a functor (119889 id119883) ∶ 119883 rarr 119860⊡ times 119883over 119883 Itrsquos easy to verify that a diagram valued in 120587∶ 119860 times 119883 ↠ 119883 whose component at 119883 is theidentity has a limit in119974119883 if and only if the 119860-component of the diagram has a limit in119974 Since id119883is the terminal object of119974119883 this object is theinfin-category 1 isin 119974119883 so the proof just give applies toprove the general case of 119883-indexed families of commutative squares

There is an automorphism of the simplicial set 120794times 120794 that swaps the ldquointermediaterdquo vertices (0 1)and (1 0) which induces a ldquotranspositionrdquo automorphism of119860⊡ By symmetry a commutative squarein 119860 is a pullback if and only if its transposed square is a pullback This gives a dual form of Lemma438 with the roles of 119891 and 119892 and of 119906 and 119907 interchanged As a corollary we can easily prove thatpullback squares compose both ldquoverticallyrdquo and ldquohorizontallyrdquo and can be cancelled from the ldquorightrdquoand ldquobottomrdquo

439 Proposition (composition and cancelation of pullback squares) Given a composable pair of119883-indexed commutative squares in 119860 and their composite rectangle defined via the equivalence 119860120795times120794 ≃119860⊡ times

119860120794119860⊡

119901 119889 119887

119890 119888 119886

119909

119910 119911

119906

119907 119908 119891

ℎ 119892

if the right-hand square is a pullback then the left-hand square is a pullback if and only if the composite rectangleis a pullback

Proof By Lemma 438 we are given an absolute right lifting diagram in119974119883

Hom119860(119860 119888)

119883 Hom119860(119860 119886)dArr(id119886119906)

119892lowast

119891

119907

By Lemma 241 the composite diagram

Hom119860(119860 119890)

Hom119860(119860 119888)

119883 Hom119860(119860 119886)

dArr(id119888119909)ℎlowast

dArr(id119886119906)119892lowast

119891

119907

119910

is an absolute right lifting diagram in 119974119883 if and only if the top triangle is an absolute right liftingdiagram in119974119883 By Lemma 438 this is exactly what we wanted to show

Terminal elements are special cases of limits where the diagram shape is empty For anyinfin-category119860 theinfin-category of diagrams119860empty cong 1 which tells us that there is a uniqueempty-indexed diagram in119860In this context theinfin-categories of cones over or under the unique diagram constructed in Definition421 are isomorphic to119860 In the case of cones over an empty diagram the domain-evaluation functorcarrying a cone to its summit is the identity on119860 while in the case of cones under the empty diagramthe codomain-evaluation functor carrying a cone to its nadir is the identity on 119860 The followingcharacterization of terminal elements can be deduced as a special case of Proposition 431 though wefind it easier to argue from Proposition 411

4310 Proposition For an element 119905 ∶ 1 rarr 119860 of aninfin-category 119860(i) 119905 defines a terminal element of119860 if and only if the domain-projection functor 1199010 ∶ Hom119860(119860 119905) ↠ 119860

is a trivial fibration(ii) 119905 defines an initial element of119860 if and only if the codomain-projection functor 1199011 ∶ Hom119860(119905 119860) ↠ 119860

is a trivial fibration

Proof Recall from Definition 221 that an element is terminal if and only if it is right adjointto the unique functor

1 119860119905perp

By Proposition 411 ⊣ 119905 if and only if there is an equivalence Hom1( 1) ≃119860 Hom119860(119860 119905) By thedefining pullback (332) for the comma infin-category the left representation of ∶ 119860 rarr 1 is 119860 itselfwith domain-projection functor the identity So the component of the equivalence Hom119860(119860 119905) ⥲ 119860over 119860 must be the domain projection functor 1199010 ∶ Hom119860(119860 119905) ↠ 119860 and we conclude that 119905 is aterminal element if and only if this isofibration is a trivial fibration

4311 Digression (terminal elements of a quasi-category) In the infin-cosmos of quasi-categories theisofibration 1199010 ∶ Hom119860(119860 119905) ↠ 119860 is equivalent over119860 to the slice quasi-category119860119905 defined as a rightadjoint to the join construction of Definition 424 Proposition 4310 proves that 119905 is terminal if andonly if the projection 119860119905 ↠ 119860 is a trivial fibration in the sense of Definition 1123 which transposesto Joyalrsquos original definition of a terminal element of a quasi-category See Appendix F for a full proof

We conclude with two results that could have been proven in Chapter 2 were it not for one smallstep of the argument as we explain A functor 119891∶ 119860 rarr 119861 preserves limits if the image of a limit conein119860 also defines a limit cone in 119861 In the other direction a functor 119891∶ 119860 rarr 119861 reflects limits if a conein 119860 that defines a limit cone in 119861 is also a limit cone in 119860

4312 Proposition A fully faithful functor 119891∶ 119860 rarr 119861 reflects any limits or colimits that exist in 119860

Proof The statement for limits asserts that given any family of diagrams 119889∶ 119863 rarr 119860119869 of shape119869 in 119860 any functor ℓ ∶ 119863 rarr 119860 and cone 120588∶ Δℓ rArr 119889 as below-left so that the whiskered compositewith 119891119869 ∶ 119860119869 rarr 119861119869 displayed below is an absolute right lifting diagram

119860 119861

119863 119860119869 119861119869dArr120588

119891

119889 119891119869

then (ℓ 120588) defines an absolute right lifting of 119889∶ 119863 rarr 119860119869 through Δ∶ 119860 rarr 119860119869 Our proof strategymirrors the results of sect24 By Corollary 346(i) to say that 119891 is fully faithful is to say that id119860 ∶ 119860 rarr 119860defines an absolute right lifting of 119891 through itself So by Lemma 241 and the hypothesis just statedthe composite diagram below-left is an absolute right lifting diagram and by 2-functoriality of thesimplicial cotensor with 119869 the diagram below-left coincides with the diagram below-right

119860 119860

119860 119861 119860119869

119863 119860119869 119861119869 119863 119860119869 119861119869

119891 Δ

dArr120588Δ

119891Δ

=dArr120588

119891119869ℓ

119889 119891119869 119889

119891119869

By Corollary 346(iv) to say that 119891 is fully faithful is to say that id119891 ∶ 119860120794 ⥲ Hom119861(119891 119891) is a fiberedequivalence over119860times119860 Applying (minus)119869 ∶ 119974 rarr 119974 this maps to a fibered equivalence id119891119869 ∶ (119860119869)120794 ⥲Hom119861119869(119891119869 119891119869) over 119860119869 times 119860119869 proving that if 119891∶ 119860 rarr 119861 is fully faithful then 119891119869 ∶ 119860119869 rarr 119861119869 is also⁴Hence by Corollary 346(i) id119860119869 ∶ 119860119869 rarr 119860119869 defines an absolute right lifting of 119891119869 through itselfApplying Lemma 241 again we now conclude that (ℓ 120588) is an absolute right lifting of 119889 through Δas required

An alternate approach to proving this result is suggested as Exercise 43iiiOur final result proves that for 119868 and 119869 simplicial sets whenever we are given a 119869-indexed diagram

valued in theinfin-category 119860119868 of 119868-indexed diagrams in 119860 its limit may be computed pointwise in thevertices of 119868 as the limit of the corresponding 119869-indexed diagram in 119860 Our argument requires thefollowing representable characterization of absolute lifting diagrams whose proof again makes use ofthe fact that they are preserved by cosmological functors

4313 Proposition A natural transformation defined in aninfin-cosmos119974 as below-left is an absolute rightlifting diagram if and only if its ldquoexternalizationrdquo displayed below-right defines a right lifting diagram in119980119966119886119905

119861 Fun(119862 119861)

119862 119860 120793 Fun(119862119860)dArr120588

119891 dArr120588119891lowast

119903

119892

119903

119892

⁴This is the statement that we could not yet prove in Chapter 2

that is preserved by precomposition with any functor 119888 ∶ 119883 rarr 119862 in 119974 in the sense that the diagram below isalso right lifting

Fun(119862 119861) Fun(119883 119861) Fun(119883 119861)

120793 Fun(119862119860) Fun(119883119860) 120793 Fun(119883 119862)dArr120588

119891lowast

119888lowast

119891lowast = dArr120588119888119891lowast119903

119892 119888lowast

119903119888

119892119888

Moreover the externalized right lifting diagrams in 119980119966119886119905 are in fact absolute

Proof Since Fun(120793 minus) ∶ 119980119966119886119905 rarr 119980119966119886119905 is naturally isomorphic to the identity functor for anyinfin-categories 119883119860 isin 119974 we have Fun(119883119860) cong Fun(120793Fun(119883119860)) and hence

hFun(119883119860) cong hFun(120793Fun(119883119860)) (4314)

This justifies our use of the same name 120588 for the 2-cell in 120101119974 and the 2-cell in 120101119980119966119886119905 in the statementTo say that (119903 120588) defines a right lifting of 119892 through 119891 in 120101119974 asserts a bijection between 2-cells

in hFun(119862 119861) with codomain 119903 and 2-cells in hFun(119862119860) with codomain 119892 and domain factoringthrough 119891 implemented by pasting with 120588 Under the correspondence of (4314) this asserts equallythat 119903 defines a right lifting of 119892 through 119891lowast in 120101119980119966119886119905 To say that (119903 120588) defines an absolute rightlifting of 119892 through 119891 is to assert the analogous right lifting property for the pair (119903119888 120588119888) defined byrestricting along any 119888 ∶ 119883 rarr 119862 This is exactly the first claim of the statement

It remains only to argue that if (119903 120588) is an absolute right lifting diagram in119974 then its externalizedright lifting diagram of quasi-categories is also absolute To see this first note that the cosmologicalfunctor Fun(119862 minus) ∶ 119974 rarr 119980119966119886119905 preserves this absolute right lifting diagram yielding an absoluteright lifting diagram of quasi-categories as below left

Fun(119862 119861) Fun(119862 119861) Fun(119862 119861)

Fun(119862 119862) Fun(119862119860) 120793 Fun(119862119860) 120793 Fun(119862 119862) Fun(119862119860)dArrFun(119862120588)

119891lowast dArr120588119891lowast = dArrFun(119862120588)

119891lowast119903lowast

119892lowast

119903

119892id119862

119903lowast

119892lowast

We obtain the desired absolute right lifting diagram by evaluating at the identity

4315 Proposition Let 119868 and 119869 be simplicial sets and let119860 be aninfin-category Then a diagram as below-leftis an absolute right lifting diagram

119860119868 119860119868 119860

119863 119860119868times119869 119863 119860119868times119869 119860119869dArr120588 Δ119868 dArr120588 Δ119868

ev119894

119889

119889 ev119894

if for each vertex 119894 isin 119868 the diagram above right is an absolute right lifting diagram

Note this statement is not a biconditional Even in the case of strict 1-categories there may existcoincidental limits of diagram valued in functor categories that are not defined pointwise [9 21710]

Proof By Proposition 4313 we may externalize and instead show that the diagram of quasi-categories displayed below left

Fun(119863119860)119868 Fun(119863119860)119868 Fun(119863119860)

120793 Fun(119863119860)119868times119869 120793 Fun(119863119860)119868times119869 Fun(119863119860)119869dArr120588 Δ119868 dArr120588 Δ119868

ev119894

119889

119889 ev119894

is absolute right lifting lifting if the diagrams above-right are for each vertex 119894 isin 119868 By naturality ofour methods our proof will show that the absolute right lifting diagrams on the left are preserved bypostcomposition with the restriction functors induced by 119889∶ 119883 rarr 119863 if the same is true on the right

We simplify our notation and write119876 for the quasi-category Fun(119863119860) and assume that for each119894 isin 119868 the diagram

119876119868 119876

120793 119876119868times119869 119876119869dArr120588 Δ119868

ev119894

119889 ev119894

is absolute right lifting By 2-adjunction (minus times 119868) ⊣ (minus)119868 a diagram as below-left is absolute rightlifting if and only if the transposed diagram below-right is a right lifting diagram and this remains thecase upon restricting along functors of the form 120587∶ 119883 times 119868 rarr 119868⁵

119876119868 119876

120793 119876119868times119869 119868 119876119869dArr120588 Δ119868 dArr120588

119889 119889

lim (4316)

Wersquoll argue that this right-hand diagram is in fact absolute right lifting which implies that the left-hand diagram is absolute right lifting as well as desired

To see this we appeal to Corollary a consequence of a general theorem proven in Appendix Fthat universal properties in 119980119966119886119905 are detected pointwise Specifically this tells us that the triangleabove right is an absolute right lifting diagram if and only if the restricted diagram is absolute rightlifting for each vertex 119894 isin 119868

119876

120793 119868 119876119869dArr120588

119894119889

and this is exactly what we have assumed in (4316)

⁵If the reader is concerned that 119868 is not a quasi-category there are two ways to proceed One is to replace 119868 by aquasi-category 119868 prime by inductively attaching fillers for inner horns note that 119868 and 119868 prime will have the same sets of verticesBy Proposition 1127 the diagram quasi-categories 119876119868 prime and 119876119868 are equivalent The other option is to observe that itdoesnrsquot matter if 119868 is a quasi-category or not because we may define hFun(119868 119876) ≔ h(119876119868) and by Corollary 1121 119876119868 is aquasi-category regardless of whether 119868 is

Exercises

43i Exercise Prove that if 119860 has a terminal element 119905 then for any element 119886 the mapping spaceHom119860(119886 119905) is contractible ie is equivalent to the terminalinfin-category 1

43ii Exercise Prove that a square in 119860 is a pullback if and only if its ldquotransposedrdquo square definedby composing with the involution119860⊡ cong 119860⊡ induced from the automorphism of 120794times120794 that swaps theldquooff-diagonalrdquo elements is a pulllback square

43iii Exercise ([60 37]) Use Theorem 343 and Corollary 346(iv) to prove that a fully faithfulfunctor 119891∶ 119860 rarr 119861 reflects all limits or colimits that exist in 119860 Why does this argument not alsoshow that 119891∶ 119860 rarr 119861 preserves them

44 Loops and suspension in pointedinfin-categories

441 Definition (pointed infin-categories) An infin-category 119860 is pointed if it admits a zero elementan element lowast ∶ 1 rarr 119860 that is both initial and terminal

The counit of the adjunction lowast ⊣ that witnesses the initiality of the zero element defines a naturaltransformation 120588∶ lowast rArr id119860 that we refer to as the family of points of 119860 Dually the unit of theadjunction ⊣ lowast that witnesses the terminality of the zero element defines a natural transformation120585∶ id119860 rArr lowast that we refer to as the family of copoints

Cospans and spans in aninfin-category 119860 may be defined by gluing together a pair of arrows alongtheir codomains or domains respectively

119860⟓ 119860120794 119860⟔ 119860120794

119860120794 119860 119860120794 119860

1199011

1199010

1199011 1199010

For instance the family of points in a pointedinfin-category 119860 is represented by a functor 120588∶ 119860 rarr 119860120794whose domain-component is constant at lowast and whose codomain component is id119860 Gluing two copiesof this map along their codomain defines a diagram ∶ 119860 rarr 119860⟓ Dually there is a diagram ∶ 119860 rarr119860⟔ defined by gluing the functor 120585∶ 119860 rarr 119860120794 that represents the family of copoints to itself alongtheir domains

442 Definition (loops and suspension) A pointed infin-category 119860 admits loops if it admits a limitof the family of diagrams

119860

119860 119860⟓dArr

in which case the limit functorΩ∶ 119860 rarr 119860 is called the loops functor Dually a pointedinfin-category119860 admits suspensions if it admits a colimit of the family of diagrams

119860

119860 119860⟔uArr

in which case the colimit functor Σ∶ 119860 rarr 119860 is called the suspension functor

Importantly if119860 admits loops and suspensions then the loops and suspension functors are adjoint

443 Proposition (the loops-suspension adjunction) If 119860 is a pointed infin-category that admits loopsand suspensions then the loops functor is right adjoint to the suspension functor

119860 119860Ω

perpΣ

The main idea of the proof is easy to describe If 119860 admits all pullbacks and all pushouts thenCorollary 435 supplies adjunctions

119860⟓ 119860⊡ 119860⟔

that are fibered over 119860 times 119860 upon evaluating at the intermediate vertices of the commutative squareBy pulling back along (lowast lowast) ∶ 1 rarr 119860 times 119860 we can pin these vertices at the zero element Since thezero element is initial and terminal theinfin-categories of pullback and pushout diagrams of this formare both equivalent to 119860 and the pulled-back adjoints now coincide with the loops and suspensionfunctors

The only subtlety in the proof that follows is that we have assumed weaker hypotheses that 119860admits only loops and suspensions but perhaps not all pullbacks and pushouts

Proof The diagram lands in a subobject119860⟓lowast of119860⟓ defined below-left that is comprised of thosepullback diagrams whose source elements are pinned at the zero element lowast of 119860

119860 119860

119860⟓lowast 119860⟓ 119860⟓lowast Hom119860(lowast 119860)

1 119860 times 119860 Hom119860(lowast 119860) 119860

120588or120588

lowast

120588

lowast

120588sim

sim 1199011

(lowastlowast)sim1199011

From a second construction of 119860⟓lowast displayed above-right and the characterization of initiality givenin Proposition 4310 we may apply the 2-of-3 property of equivalences to conclude first that 120588∶ 119860 ⥲Hom119860(lowast 119860) and then that the induced diagram lowast ∶ 119860 ⥲ 119860⟓lowast are equivalences Dually the diagram ∶ 119860 rarr 119860⟔ defines an equivalence lowast ∶ 119860 ⥲ 119860⟔lowast when its codomain is restricted to the subobject ofpushout diagrams whose target elements are pinned at the zero element lowast

By Proposition 434 a pointed infin-category 119860 admits loops or admits suspensions if and only ifthere exist absolute lifting diagrams as below-left and below-right respectively

119860⊡ 119860⊡

119860 119860⟓ 119860 119860⟔congdArr

resconguArr

By Theorem 343 the absolute right lifting diagram defines a right representation Hom119860⊡(119860⊡ ran) ≃Hom119860⟓(res ) this being a fibered equivalence over119860times119860⊡ The represented commainfin-category maybe pulled back along the inclusion of the subobject 119860⊡lowast 119860⊡ of commutative squares in 119860 whoseintermediate vertices are pinned at the zero object

Hom119860⟓(res ) (119860⟓)120794

Hom119860⟓lowast (res lowast) (119860⟓lowast )120794

119860⊡ times 119860 119860⟓ times 119860⟓

119860⊡lowast times 119860 119860⟓lowast times 119860⟓lowast

restimes

restimeslowast

The right-hand face of this commutative cube is not strictly a pullback but the universal property ofthe zero element implies that the induced map from (119860⟓lowast )120794 to the pullback is an equivalence It followsthat Hom119860⟓lowast (res lowast) is equivalent over 119860⊡lowast times 119860 to the pullback of Hom119860⟓(res ) along this inclusionand so the fibered equivalence pulls back to define a right representation for Hom119860⟓lowast (res lowast) Duallythe left representation for Hom119860⟔( res) pulls back to a left representation for Hom119860⟔lowast (lowast res) ByTheorem 347 these unpack to define absolute lifting diagrams

119860⊡lowast 119860⊡lowast 119860⊡lowast 119860⊡lowast

119860 119860⟓lowast 119860⟓lowast 119860⟓lowast 119860 119860⟔lowast 119860⟔lowast 119860⟔lowastcongdArr

res congdArr

resconguArr

res conguArr

lowast

ran ran

lowast

lan lan

Restricting along the inverse equivalences 119860⟓lowast ⥲ 119860 and 119860⟔lowast ⥲ 119860 and pasting with the invertible2-cell we obtain absolute lifting diagrams whose bottom edge is the identity

By Lemma 236 these lifting diagrams define adjunctions

119860 ≃ 119860⟓lowast 119860⊡lowast 119860⟔lowast ≃ 119860

which compose to the desired adjunction Σ ⊣ Ω

444 Definition An arrow 119891∶ 1 rarr 119860120794 from 119909 to 119910 in a pointedinfin-category 119860 admits a fiber if 119860admits a pullback of the diagram defined by gluing 119891 to the component 120588119910 of the family of points ByProposition 434 such pullbacks give rise to a pullback square

119860⊡

1 119860⟓congdArr

resran

120588119910or119891

that is referred to as the fiber sequence for 119891 Dually 119891 admits a cofiber if 119860 admits a pushout of thediagram defined by gluing 119891 to the component 120585119909 of the family of copoints in which case the pushout

square119860⊡

1 119860⟔conguArr

reslan

120585119909or119891

defines the cofiber sequence for 119891

Fiber and cofiber sequences define commutative squares in 119860 whose lower-left vertex is the zeroelement lowast The data of such squares is given by a commutative triangle in 119860 mdash an element of 119860120795mdash together with a nullhomotopy of the diagonal edge a witness that this edge factors through thezero element in h119860 Borrowing a classical term from homological algebra a commutative square in119860whose lower-left vertex is the zero element is referred to as a triangle in 119860

445 Definition (stableinfin-category) A stableinfin-category is a pointedinfin-category 119860 in which(i) every morphism admits a fiber and a cofiber that is there exist absolute lifting diagrams

119860⊡ 119860⊡

119860120794 119860⟓ 119860120794 119860⟔congdArr

resconguArr

resran

120588orid

120585orid

(ii) and a triangle in 119860 defines a fiber sequence if and only if it also defines a cofiber sequenceSuch triangles are called exact

A stableinfin-category admits loops and admits suspensions formed by taking fibers of the arrowsin the family of points and cofibers of arrows in the family of copoints respectively

446 Proposition If 119860 is a stableinfin-category then Σ ⊣ Ω are inverse equivalences

Proof In the proof of Proposition 443 the adjunction Σ ⊣ Ω is constructed as a composite ofadjunctions

that construct fiber and cofiber sequences By Proposition 219 the unit and counit of this compositeadjunction are given by

119860 119860 119860

dArrcong ΣdArr120598 lan

res dArr120578

res res

dArrcong resran Ω

By Definition 437 the unit of res ⊣ ran restricts to an isomorphism on the subobject of pushoutsquares In a stable infin-category the cofiber sequences in the image of lan ∶ 119860 rarr 119860⊡lowast are pullbacksquares so this tells us that 120578 lan is an isomorphism Dually the fiber sequences in the image of

ran ∶ 119860 rarr 119860⊡lowast are pushout squares which tells us that 120598 ran is an isomorphism Hence the unit andcounit of Σ ⊣ Ω are invertible so these functors define an adjoint equivalence

447 Proposition (finite limits and colimits in stable infin-categories) A stable infin-category admits allpushouts and all pullbacks and moreover a square is pushout if and only if it is a pullback

Proof Given a cospan 119892 or 119891∶ 119883 rarr 119860⟓ in 119860 form the cofiber of 119891 followed by the fiber of thecomposite map 119888 rarr 119886 rarr coker 119891

ker(119902119892) 119887 lowast

119888 119886 coker(119891)

119906

119907

119891

119892 119902

By Definition 445(ii) the cofiber sequence 119887 rarr 119886 rarr ker 119891 is also a fiber sequence By the pullbackcancelation result of Proposition 439 we conclude that ker(119902119892) computes the pullback of the cospan119892 or 119891

To see that this pullback square is also a pushout form the fiber of the map 119907

ker(119907) ker(119902119892) 119887

lowast 119888 119886

119906

119907

119891

119892

By the pullback composition result of Proposition 439 ker(119907) is also the fiber of the map 119891 By Def-inition 445(ii) the fiber sequences ker(119907) rarr ker(119902119892) rarr 119888 and ker(119907) rarr 119887 rarr 119886 are also cofibersequences Now by the pushout cancelation result of Proposition 439 we see that the right-handpullback square is also a pushout square A dual argument proves that pushouts coincide with pull-backs

Exercises

44i Exercise Arguing in the homotopy category show that if aninfin-category 119860bull admits an initial element 119894bull admits a terminal element 119905 andbull there exists an arrow 119905 rarr 119894

then 119860 is a pointedinfin-category

CHAPTER 5

Fibrations and Yonedarsquos lemma

The fibers 119864119887 of an isofibration 119901∶ 119864 ↠ 119861 over an element 119887 ∶ 1 rarr 119861 are necessarilyinfin-categoriesso it is natural to ask where the isofibration also encodes the data of functors between the fibers in thisfamily of infin-categories Roughly speaking an isofibration defines a cartesian fibration just when thearrows of 119861 act contravariantly functorially on the fibers and a cocartesian fibration when the arrowsof 119861 act covariantly functorially on the fibers This action is not strict but rather pseudofunctorial orhomotopy coherent in a sense that will be made precise when we study the comprehension construc-tion

One of the properties that characterizes cocartesian fibrations is an axiom that says that for any2-cell with codomain 119861 and specified lift of its source 1-cell there is a lifted 2-cell with codomain 119864with that one cell as its source In particular this lifting property can be applied in the case where the2-cell in question is a whiskered composite of an arrow in the homotopy category of 119861 as below-leftand the lift of the source 1-cell is the canonical inclusion of its fiber

119864119886 119864 119864119886 119864

119864119887

1 119861 1 119861

ℓ119886

=119901120573lowast

120573lowast(ℓ119886)dArr120594120573

ℓ119886

119901ℓ119887119886

119887

dArr120573

119887

In this case the codomain 120573lowast(ℓ119886) of the lifted cell 120594120573 displayed above right lies strictly above thecodomain of the original 2-cell and thus factors through the pullback defining its fiber This de-fines a functor 120573lowast ∶ 119864119886 rarr 119864 the ldquoactionrdquo of the arrow 120573 on the fibers of 119901 The pseudofunctoriality ofthese action maps arises from a universal property required of the specified lifted 2-cells namely thatthey are cartesian in a sense we now define

51 The 2-category theory of cartesian fibrations

There is a standard notion of cartesian fibration in a 2-category developed by Street [55] thatrecovers the Grothendieck fibrations when specialized to the 2-category 119966119886119905 This is not the correctnotion of cartesian fibration betweeninfin-categories as the universal property the usual notion demandsof lifted 2-cells is too strict Instead of referring to the notions defined here as ldquoweakrdquo we would preferto refer to the classical notion of cartesian fibration in a 2-category as ldquostrictrdquo were we to refer to itagain which we largely will not

To remind the reader of the interpretation of the data in the homotopy 2-category of aninfin-cosmoswe refer to 1- and 2-cells as ldquofunctorsrdquo and ldquonatural transformationsrdquo Before defining the notion of

cartesian fibration we describe the weak universal property enjoyed by a certain class of ldquoupstairsrdquonatural transformations

511 Definition (119901-cartesian transformations) Let 119901∶ 119864 ↠ 119861 be an isofibration A natural trans-

formation 119883 119864119890prime

119890dArr120594 with codomain 119864 is 119901-cartesian if

(i) induction Given any natural transformations 119883 119864119890Prime

119890dArr120591 and 119883 119861

119901119890Prime

119901119890primedArr120574 so that 119901120591 =

119901120594 sdot 120574 there exists a lift 119883 119864119890Prime

119890primedArr of 120574 so that 120591 = 120594 sdot

119890Prime 119890

119890prime

119901119890Prime 119901119890

119901119890prime

120591

120594

119901120591

120574 119901120594

isin hFun(119883 119864)

isin hFun(119883 119861)

119901lowast

(ii) conservativity Any fibered endomorphism of 120594 is invertible if 119883 119864119890prime

119890primedArr120577 is any natural

transformation so that 120594 sdot 120577 = 120594 and 119901120577 = id119901119890prime then 120577 is invertible

512 Remark (why ldquocartesianrdquo) The induction property for a 119901-cartesian natural transformation120594∶ 119890prime rArr 119890 says that for any 119890Prime ∶ 119883 rarr 119864 there is a surjective function from the set hFun(119883 119864)(119890Prime 119890prime)of natural transformations from 119890Prime to 119890prime to the pullback induced in the commutative square

hFun(119883 119864)(119890Prime 119890prime) hFun(119883 119864)(119890Prime 119890)

120594∘minus

119901 119901

119901120594∘minus

Were the induction property strict and not weak this square would be a pullback ie a ldquocartesianrdquosquare

It follows that 119901-cartesian lifts of a given 2-cell with specified codomain are unique up to fiberedisomorphism

513 Lemma (uniqueness of cartesian lifts) If 119883 119864119890prime

119890dArr120594 and 119883 119864

119890Prime

119890dArr120594prime are 119901-cartesian lifts of

a common 2-cell 119901120594 then there exists an invertible 2-cell 119883 119864119890Prime

119890primecongdArr120577 so that 120594prime = 120594 sdot 120577 and 119901120577 = id

Proof By induction there exists 2-cells 120577∶ 119890Prime rArr 119890prime and 120577prime ∶ 119890prime rArr 119890Prime so that 120594prime = 120594 sdot 120577 and120594 = 120594prime sdot 120577prime with 119901120577 = 119901120577prime = id The composites 120577 sdot 120577prime and 120577prime sdot 120577 are then fibered automorphisms of 120594and 120594prime and thus invertible by conservativity Now the 2-of-6 property for isomorphisms implies that120577 and 120577prime are also isomorphisms (though perhaps not inverses)

We frequently make use of the isomorphism stability of the 119901-cartesian transformations given bythe following suite of observations

514 Lemma Let 119901∶ 119864 ↠ 119861 be an isofibration(i) Isomorphisms define 119901-cartesian transformations(ii) Any 119901-cartesian lift of an identity is a natural isomorphism(iii) The class of 119901-cartesian transformations is closed under pre- and post-composition with natural isomor-

phisms

Proof Exercise 51i

Furthermore

515 Lemma (more conservativity) If 119883 119864119890prime

119890dArr120594 119883 119864

119890Prime

119890dArr120594prime and 119883 119864

119890Prime

119890primedArr120577 are 2-cells so

that 120594 and 120594prime are 119901-cartesian 120594prime = 120594 sdot 120577 and 119901120577 is invertible then 120577 is invertible

Proof Given the data in the statement we can use the induction property for 120594prime applied to thepair (120594 119901120577minus1) to induce a candidate inverse for 120577 and then apply the conservativity property toconclude that 120577 sdot and sdot 120577 are both isomorphisms By the 2-of-6 property 120577 is an isomorphism asdesired

We now introduce the class of cartesian fibrations

516 Definition (cartesian fibration) An isofibration 119901∶ 119864 ↠ 119861 is a cartesian fibration if(i) Any natural transformation 120573∶ 119887 rArr 119901119890 as below-left admits a 119901-cartesian liftsup1120594120573 ∶ 120573lowast119890 rArr 119890 as

below-right

119883 119864 119883 119864

119861 119861

119890

119887

uArr120573 119901 =

119890

120573lowast119890

uArr120594120573

119901

(ii) The class of 119901-cartesian transformations is closed under restriction that is if 119883 119864119890prime

119890dArr120594 is

119901-cartesian and 119891∶ 119884 rarr 119883 is any functor then 119884 119864119890prime119891

119890119891

dArr120594119891 is 119901-cartesian

The lifting property (i) implies that a 119901-cartesian transformation 120594∶ 119890prime rArr 119890 is the ldquouniversalnatural transformation over 119901120594 with codomain 119890rdquo in the following weak sense any transformation

sup1To ask that 120594120573 ∶ 120573lowast119890 rArr 119890 is a lift of 120573∶ 119887 rArr 119901119890 asserts that 119901120594120573 = 120573 and hence 119901120573lowast119890 = 119887

120595∶ 119890prime rArr 119890 factors through a 119901-cartesian lift 120594120595 ∶ (119901120595)lowast119890 rArr 119890 of 119901120595 via a 2-cell 120574∶ 119890prime rArr (119901120595)lowast119890 overan identity and moreover 120595 is 119901-cartesian if and only if this factorization 120574 is invertible

The reason for condition (ii) will become clearer in sect53 For now note that since all 119901-cartesianlifts of a given 2-cell 120573 are isomorphic and the class of 119901-cartesian cells is stable under isomorphismto verify the condition (ii) it suffices to show that for any functor 119891∶ 119884 rarr 119883 there is some 119901-cartesianlift of 120573 that restricts along 119891 to another 119901-cartesian transformation

517 Lemma (composites of cartesian fibrations) If 119901∶ 119864 ↠ 119861 and 119902 ∶ 119861 ↠ 119860 are cartesian fibrations

then so is 119902119901 ∶ 119864 ↠ 119860 Moreover a natural transformation 119883 119864119890prime

119890dArr120594 is 119902119901-cartesian if and only if 120594 is

119901-cartesian and 119901120594 is 119902-cartesian

Proof The first claim follows immediately from the second for the lifts required by Definition516(i) can be constructed by first taking a 119902-cartesian lift 120594120573 and then taking a 119901-cartesian lift 120594120594120573 ofthis lifted cell

119883 119864 119883 119864 119883 119864

119861 = 119861 = 119861

119860 119860 119860

119890

119886

uArr120573 119901

119890

119887

uArr120594120573 119901

119890

120573lowast119890

uArr120594120594120573119901

119902 119902 119902

and the stability condition 516(ii) is then inherited from the stability of 119901- and 119902-cartesian transfor-mations

To prove the second claim first consider a natural transformation 119883 119864119890prime

119890dArr120594 that is 119901-cartesian

and so that 119901120594 is 119902-cartesian Given any natural transformations 119883 119864119890Prime

119890dArr120591 and 119883 119860

119902119901119890Prime

119902119901119890primedArr120574 so

that 119902119901120591 = 119902119901120594 sdot 120574 119902-cartesianness of 119901120594 induces a lift 119883 119861119890Prime

Now 119901-cartesianness of 120594 induces a further lift 119883 119864119890Prime

119890primedArr of so that 120591 = 120594 sdot Moreover if

119883 119864119890prime

119890primedArr120577 is any natural transformation so that 120594 sdot 120577 = 120594 and 119902119901120577 = id then 119901120594 sdot 119901120577 = 119901120594 and by

conservativity for 119901120594 119901120577 is invertible Applying Lemma 515 we conclude that 120577 is invertible Thisproves that 120594 is 119902119901-cartesian

Conversely if 120594 is 119902119901-cartesian then Lemma 513 implies it is isomorphic to all other 119902119901-cartesianlifts of 119902119901120594 The construction given above produces a 119902119901-cartesian lift of any 2-cell that is 119901-cartesian

and whose image under 119901 is 119902-cartesian By the isomomorphism stability of 119901- and 119902-cartesian trans-formations of Lemma 514 120594 must also have these properties

The following lemma proves that cartesian fibrations come equipped with a ldquogeneric 119901-cartesiantransformationrdquo

518 Lemma An isofibration 119901∶ 119864 ↠ 119861 is a cartesian fibration if and only if the right comma cone over 119901displayed below-left admits a lift 120594 as displayed below-right

Hom119861(119861 119901) 119864 Hom119861(119861 119901) 119864

119861 119861

1199011

1199010

uArr120601 119901 =

1199011

119903

uArr120594

119901 (519)

with the property that the restriction of 120594 along any 119883 rarr Hom119861(119861 119901) is a 119901-cartesian transformation

Proof If 119901∶ 119864 ↠ 119861 is cartesian then the right comma cone120601 admits a 119901-cartesian lift120594∶ 119903 rArr 1199011by 516(i) which by 516(ii) has the property that the restriction of this 119901-cartesian transformationalong any 119883 rarr Hom119861(119861 119901) is also 119901-cartesian

For the converse suppose we are given the generic 119901-cartesian transformation 120594∶ 119903 rArr 1199011 of thestatement and consider a 2-cell 120573∶ 119887 rArr 119901119890 as below-left

119883 119864 119883 Hom119861(119861 119901) 119864 119883 Hom119861(119861 119901) 119864

119861 119861 119861

119890

119887

dArr120573 119901 =

119890

119887

120573 1199011

1199010uArr120601 119901 =

119890

119887

1205731199011

119903uArr120594

119901

By 1-cell induction 120573 = 120601120573 for some functor 120573 ∶ 119883 rarr Hom119861(119861 119901) as above- center By substitut-ing the equation (519) as above-right we see that 120594120573 is a lift of 120573 whose codomain is 1199011120573 = 119890as required The hypothesis that restrictions of 120594 are 119901-cartesian implies that this lift is a 119901-cartesiantransformation

Now Lemma 513 implies that any 119901-cartesian natural transformation is isomorphic to a restric-tion of 120594 Thus restrictions of 119901-cartesian transformations are isomorphic to restrictions of 120594 and itfollows from Lemma 514 that the class of 119901-cartesian transformation is closed under restriction

The first major result of this section is an internal characterization of cartesian fibrations inspiredby a similar result of Street [55 58 59] see also [63] Before stating this result recall from Lemma 348that from a functor 119901∶ 119864 ↠ 119861 we can build a fibered adjunction 1199011 ⊣119864 119894 where the right adjoint isinduced from the identity 2-cell id119901

119864

Hom119861(119861 119901)

119864 119861

119901119894

1199011 1199010120601lArr

119901

=119864

119864 119861

119901=

119901

119864 perp Hom119861(119861 119901)

119864119894 1199011

1199011

Similarly 1-cell induction for the right comma cone over 119901 applied to the generic arrow for 119864 inducesa functor

119864120794

119864

119861

11990101199011 120581lArr

119901

119864120794

Hom119861(119861 119901)

119864 119861

1199011 1199011199010119896

1199011 1199010120601lArr

119901

(5110)

5111 Theorem (an internal characterization of cartesian fibrations) For an isofibration 119901∶ 119864 ↠ 119861the following are equivalent

(i) 119901∶ 119864 ↠ 119861 defines a cartesian fibration(ii) The functor 119894 ∶ 119864 rarr Hom119861(119861 119901) admits a right adjoint over 119861

119864 perp Hom119861(119861 119901)

119861

119894

119901 1199010119903

(iii) The functor 119896 ∶ 119864120794 rarr Hom119861(119861 119901) admits a right adjoint with invertible counit

119864120794 Hom119861(119861 119901)

119896

When these equivalent conditions hold then for a natural transformation 119883 119864119890prime

119890dArr120595 the following are

equivalent(iv) 120595 is 119901-cartesian(v) 120595 factors through a restriction of the 2-cell 1199011120598 where 120598 is the counit of the adjunction 119894 ⊣ 119903 via a

natural isomorphism so that 119901 = id

119883 119864 = 119883 Hom119861(119861 119901) Hom119861(119861 119901) 119864

119864

119890

119890primeuArr120595 119901120595

119890prime

119890

119903

uArr120598 1199011

conguArr 119894

(vi) The component

119883 119864120794 119864120794120595

119896of the unit for 119896 ⊣ is invertible that is 120595 is in the essential image of the right adjoint

The right adjoint of (ii) is the domain-component of the generic cartesian lift of (519) that carte-sian transformation is then recovered as 1199011120598 where 120598 is the counit of the fibered adjunction 119894 ⊣ 119903This explains the statement of (v) By 1-cell induction the generic cartesian lift 120594 can be representedby a functor ∶ Hom119861(119861 119901) rarr 119864120794 and this defines the right adjoint of (iii) and explains the statementof (vi)

Before proving Theorem 5111 we make two further remarks on these postulated adjunctions

5112 Remark By Lemma 348 the functor 119894 ∶ 119864 rarr Hom119861(119861 119901) is itself a right adjoint over 119861 to thecodomain-projection functor Since the counit of the adjunction 1199011 ⊣ 119894 is an isomorphism it followsformally that the unit of the adjunction 119894 ⊣ 119903must also be an isomorphism whenever the adjunctionpostulated in (ii) exists see Lemma B32

5113 Remark In the case where the infin-categories 119864120794 and Hom119861(119861 119901) are defined by the strictsimplicial limits of Definitions 321 and 331 the 1-cell 119896 induced in (5110) can be modeled by anisofibration

119864120794

Hom119861(119861 119901) 119861120794

119864 119861

119901120794

1199011

119896

1199011

119901

namely the Leibniz cotensor of the codomain inclusion 1∶ 120793 120794 and the isofibration 119901∶ 119864 ↠ 119861Now Lemma 359 can be used to rectify the adjunction 119896 ⊣ 119903 of (iii) to a right adjoint right inverseadjunction that is then fibered over Hom119861(119861 119901) So when Theorem 5111(iii) holds we may modelthe postulated adjunction by a right adjoint right inverse to the isofibration 119896

Proof Wersquoll prove (i)rArr(iii)rArr(ii)rArr(i) and demonstrate the equivalences (iv)hArr(vi) and (iv)hArr(v)in parallel

(i)rArr(iii) If 119901∶ 119864 ↠ 119861 is cartesian then the right comma cone over 119901 admits a cartesian lift along119901

Hom119861(119861 119901) 119864 Hom119861(119861 119901) 119864

119861 119861

1199011

1199010

uArr120601 119901 =

1199011

119903

uArr120594

119901

defining a functor 119903 ∶ Hom119861(119861 119901) rarr 119864 over 119861 together with a 119901-cartesian transformation 120594∶ 119903 rArr 1199011By 1-cell induction this generic cartesian transformation is represented by a functor

Hom119861(119861 119901) 119864 = Hom119861(119861 119901) 119864120794 119864119903

1199011

dArr120594 1199010

1199011

dArr120581

which we take as our definition of the putative right adjoint By the definition (5110) of 119896 120601119896 =119901120581 = 119901120594 = 120601 so Proposition 337 supplies a fibered isomorphism ∶ 119896 cong id with 1199010 = id1199010 and1199011 = id1199011

To prove that 119896 ⊣ it remains to define the unit 2-cell which we do by 2-cell induction from apair given by an identity 2-cell 1199011 = id1199011 and a 2-cell 1199010 that remains to be specified The requiredcompatibility condition of Proposition 336(ii) asserts that this 1199010must define a factorization of thegeneric arrow

1199010 1199011

1199010119896 = 119903119896 1199011119896

120581

1199010 1199011=id

120594119896

(5114)

through120581119896 = 120594119896 Note 119901120581 = 120601119896 has120594119896 as its 119901-cartesian lift so we define 1199010 by the induction prop-erty for the cartesian transformation 120594119896 applied to the generic arrow 120581∶ 1199010 rArr 1199011 By construction1199011199010 = id

By Lemma B41 once we verify that 119896 and are invertible then this data this together with ∶ 119896 cong id defines an adjunction with invertible counit We prove 119896 is invertible by 2-cell conserva-tivity 1199011119896 = 1199011 = id and 1199010119896 = 1199011199010 = id

Similarly by 2-cell conservativity to conclude that is invertible it suffices to prove that 1199010is an isomorphism Restricting (5114) along we see that 1199010 defines a fibered isomorphism of119901-cartesian transformations

119903 1199011

1199010119896 = 119903119896

120594

1199010 120594119896

so this follows from Lemma 515Finally note that a transformation 120595∶ 119890prime rArr 119890 is 119901-cartesian if and only if its factorization through

the generic 119901-cartesian lift of 119901120595 is invertible This factorization may be constructed by restrictingthe 2-cells of (5114) along 120595 ∶ 119883 rarr 119864120794 since 120581120595 = 120595 so we see that 120595 is 119901-cartesian if and onlyif 1199010120595 is invertible By 2-cell conservativity 120595 is invertible if and only if its domain componentis invertible This proves that (iv)hArr(vi)

(iii)rArr(ii) By Remark 5113 we can model the left adjoint of (iii) by an isofibration 119896 ∶ 119864120794 ↠Hom119861(119861 119901) and use Lemma 359 to rectify the adjunction 119896 ⊣ with invertible counit into a rightadjoint right inverse adjunction that is then fibered over Hom119861(119861 119901) Composing with the projection1199010 ∶ Hom119861(119861 119901) ↠ 119861 Lemma 357(ii) then gives us a fibered adjunction over 119861

119864120794 perp Hom119861(119861 119901)

119861

119896

1199011199010 1199010

By the dual of Lemma 348 the 1-cell 119895 ∶ 119864 rarr 119864120794 induced by the identity defines a left adjoint rightinverse to the domain projection

119864 119864120794119895

1199010perp

1199011perp

(5115)

supplying a fibered adjunction 119895 ⊣ 1199010 over 119864 that we push forward along 119901∶ 119864 ↠ 119861 to a fiberedadjunction over 119861

119864 perp 119864120794

119861119901

119895

1199010 1199011199010

This pair of fibered adjunctions composes to define a fibered adjunction over 119861 with left adjoint 119896119895and right adjoint 119903 ≔ 1199010 Proposition 337 supplies a fibered isomorphism 119894 cong 119896119895 since both 119894 and 119896119895define functors 119864 Hom119861(119861 119901) are induced by the identity id119901 over the right comma cone over 119901Composing with this fibered isomorphism we can replace the left adjoint of the composite adjunctionby 119894

119864 119864120794 Hom119861(119861 119901) = 119864 perp Hom119861(119861 119901)

119861 119861119901

119895

119894

perp1199010 1199011199010

119896

1199010

119894

119901 1199010119903

proving (ii)(ii)rArr(i) Now suppose given a fibered adjunction

119864 perp Hom119861(119861 119901)

119861

119894

119901 1199010119903

We will show that the codomain component of the counit

Hom119861(119861 119901) Hom119861(119861 119901) 119864

119864119903

uArr1205981199011

119894

satisfies the conditions of the generic 119901-cartesian transformation described in Lemma 518 This willthen also demonstrate the equivalence (iv)hArr(v)

The first thing to check is that 1199011120598 defines a lift of the right comma cone along 119901 To see thisconsider the horizontal composite

119864

Hom119861(119861 119901) Hom119861(119861 119901)

119864 119861

119901

119903

uArr120598

1199011

1199010

uArr120601

119894

Naturality of whiskering provides a commutative square

1199010119894119903 1199010

1199011199011119894119903 1199011199011

1199010120598

120601119894119903 120601

1199011199011120598

in which 1199010120598 = id as a fibered counit and 120601119894119903 = id since 120601119894 = id119901 Thus 1199011199011120598 = 120601 and we see that1199011120598 is a lift of 120601 along 119901

It remains only to verify that the restriction of 1199011120598 along any functor defines a 119901-cartesian trans-formation To that end consider 120573 ∶ 119883 rarr Hom119861(119861 119901) representing a 2-cell 120573∶ 119887 rArr 119901119890 our taskis to verify that 1199011120598120573 is 119901-cartesian Note that 1199011199011120598120573 = 120601120573 = 120573 so to prove the inductionproperty consider a 2-cell 120591∶ 119890Prime rArr 119890 and a factorization 119901120591 = 120573 sdot 120574 for some 120574∶ 119901119890Prime rArr 119887 Our taskis to define a 2-cell ∶ 119890Prime rArr 119903120573 so that the pasted composite

119883 Hom119861(119861 119901) Hom119861(119861 119901) 119864

119864120573

119890Prime

119890

119903

uArr120598 1199011

uArr 119894

is 120591 Transposing across the adjunction 119894 ⊣ 119903 it suffices instead to define a 2-cell ∶ 119894119890Prime rArr 120573 so that1199011 = 120591 We define by 2-cell induction from this condition and 1199010 = 120574 a pair which satisfies the2-cell induction compatibility condition

119901119890Prime 119901119890Prime

119887 119901119890120574=1199010 1199011199011=119901120591

120573

precisely on account of the postulated factorization 119901120591 = 120573 sdot 120574 This verifies the induction conditionof Definition 511(i)

Now consider an endomorphism 120577∶ 119903 rArr 119903 so that 1199011120598120573 sdot120577 = 1199011120598120573 and 119901120577 = id119887 Write ≔1199011120598 ∶ Hom119861(119861 119901) rarr 119864120794 for the functor induced by 1-cell induction from the generic 119901-cartesiantransformation Now the conditions defining 120577 allow us to induce a 2-cell ∶ 120573 rArr 120573 satisfying1199010 = 120577 and 1199011 = id To prove that 120577 is invertible will make use of the naturality of whiskeringsquare for the horizontal composite

Hom119861(119861 119901)

119883 119864120794 119864

Hom119861(119861 119901) Hom119861(119861 119901)

120573

120573dArr

1199010

119896dArr

119903

1199010119903120573 119903119896120573

1199010120573 119903119896120573

120573cong

120577=1199010 119903119896cong

120573cong

where is a special case of the 2-cell just given this name to be described momentarily for whichwe will demonstrate that is an isomorphism The composite 119896 is a 2-cell induced from 1199010119896 =1199011199010 = 119901120577 = id and 1199011119896 = 1199011 = id so by 2-cell conservativity this is an isomorphism Now 120577 is acomposite of three isomorphisms and hence is invertible

To complete the proof we must define and prove that is invertible Specializing the construc-tion just given we define to be the induced 2-cell satisfying 119901 = id and 120581 = 1199011120598119896 sdot Transposingacross the fibered adjunction 119894 ⊣ 119903 it suffices to define the transposed 2-cell ∶ 1198941199010 rArr 119896 so that1199011 = 120581 and 1199010 = id And we may transpose once more along an adjunction 119895 ⊣ 1199010 of (5115) con-structed by the dual of Lemma 348 The counit 120584∶ 1198951199010 rArr id of this adjunction satisfied the definingconditions that 1199010120584 = id and 1199011120584 = 120581 so to construct satisfying the conditions just described itsuffices to define instead a 2-cell 120585∶ 119894 rArr 119896119895 satisfying the conditions 1199011120585 = id and 1199010120585 = id Theseconditions are satisfied by the fibered isomorphism 120585∶ 119894 cong 119896119895 that arises by Proposition 337 sinceboth 1-cells 119864 Hom119861(119861 119901) are induced by the identity id119901 Unpacking these transpositions isdefined to be the composite

≔ 1199010 1199031198941199010 1199031198961198951199010 119903119896120578cong

120585cong

119903119896120584

To verify that is invertible it thus suffices to demonstrate that 119903119896120584 is invertible To see thiswe consider another pasting diagram

119864 119864

Hom119861(119861 119901) Hom119861(119861 119901) 119864120794 119864120794 Hom119861(119861 119901) 119864

119894dArr120598

119895dArr120584

119903

congdArr

1199031199010

119896 119903

By the definition (5110) of 119896 120601119896 = 119901120581 = 119901120594 = 120601 so Proposition 337 supplies a fibered isomor-phism ∶ 119896 cong id with 1199010 = id1199010 and 1199011 = id1199011

Now naturality of whiskering supplies a commutative diagram of 2-cells

1199031198961198951199010119894119903 119903119896119894119903 119903119894119903

1199031198961198951199010 119903119896 119903

119903119896119895119903120598 cong

119903119896120584119894119903

119903119896120598

119903119894119903cong

119903120598cong

119903119896120584cong119903

Since 120598 is the counit of an adjunction 119894 ⊣ 119903 with invertible unit 119903120598 is an isomorphism so we see that119903119896120584 is an isomorphism if and only if 119903119896120584119894119903 is And this is the case since 119896120584119894 is an isomorphism by2-cell conservativity 1199010119896120584119894 = id119901 while 1199011119896120584119894 = 1199011120598119894 which is an isomorphism again because 120598 isthe counit of an adjunction 119894 ⊣ 119903 with invertible unit

One of the myriad applications of Theorem 5111 is

5116 Corollary Cosmological functors preserve cartesian fibrations and cartesian natural transforma-tions

Proof By Theorem 5111(i)hArr(iii) an isofibration 119901∶ 119864 ↠ 119861 in an infin-cosmos 119974 is cartesian ifand only if the isofibration 119896 ∶ 119864120794 ↠ Hom119861(119861 119901) defined in Remark 5113 admits a right adjointright inverse A cosmological functor 119865∶ 119974 rarr ℒ preserves the class of isofibrations and the simpli-cial limits that define the domain and codomain of this 119896 Moreover cosmological functors preserveadjunctions and natural isomorphisms so if this adjoint exists in 119974 it also does in ℒ Similarly theinternal characterization of 119901-cartesian natural transformations given by Theorem 5111(iv)hArr(vi) isalso preserved by cosmological functors

Another application of Theorem 5111 is that it allows us to conclude that cartesianness is anequivalence-invariant property of isofibrations

5117 Corollary Consider a commutative square between isofibrations whose horizontals are equivalences

119865 119864

119860 119861

119902

119892

119901

sim119891

Then 119901 is a cartesian fibration if and only if 119902 is a cartesian fibration in which case 119892 preserves and reflectscartesian transformations 120594 is 119902-cartesian if and only if 119892120594 is 119901-cartesian

Proof The commutative square of the statement induces a commutative square up to isomor-phism whose horizontals are equivalences

119865120794 119864120794

Hom119860(119860 119902) Hom119861(119861 119901)

119892120794

119896 cong 119896

simHom119891(119891119892)

By the equivalence-invariance of adjunctions the left-hand vertical admits a right adjoint with invert-ible counit if and only if the right-hand vertical does these adjunctions being defined in such a waythat the mate of the given square is an isomorphism built by composing with the natural isomorphismsof the horizontal equivalences (see Proposition B31) By Theorem 5111(i)hArr(iii) it follows that 119901 iscartesian if and only if 119902 is cartesian

Supposing the postulated adjoints exist via their construction the whiskered composites 119892120794120578 and120578119892120794 of the units of the respective adjunctions are isomorphic Hence the component at an element120594 ∶ 119883 rarr 119865120794 of 119892120794120578 is invertible if and only if the component of 120578119892120794 is invertible since 119892120794 isan equivalence the former is the case if and only if the component of 120578 is invertible By Theorem5111(iv)hArr(vi) this proves that 120594 is 119901-cartesian if and only if 119892120594 is 119902-cartesian

In terminology we now introduce the square defined by the equivalences and also the squaredefined by their inversessup2 defines a cartesian functor from 119902 to 119901

5118 Definition (cartesian functor) Let 119901∶ 119864 ↠ 119861 and 119902 ∶ 119865 ↠ 119860 be cartesian fibrations Acommutative square

119865 119864

119860 119861

119902

119892

119901

119891

defines a cartesian functor if 119892 preserves cartesian transformations if 120594 is 119902-cartesian then 119892120594 is119901-cartesian

sup2For any inverse equivalences 119892prime and 119891prime to 119892 and 119891 there is a natural isomorphism 119902119892prime cong 119891prime119891119902119892prime = 119891prime119901119892119892prime cong 119891prime119901Using the isofibration property of 119902 of Proposition 1410 119892prime may be replaced by an isomorphic functor 119892Prime which alsodefines an inverse equivalence to 119892 and for which the square 119902119892Prime = 119891prime119901 commutes strictly

The internal characterization of cartesian functorsmakes use of amap between right representablecommainfin-categories induced by the commutative square 119891119902 = 119901119892 defined by Proposition 335

5119 Theorem (an internal characterization of cartesian functors) For a commutative square

119865 119864

119860 119861

119902

119892

119901

119891

between cartesian fibrations the following are equivalent(i) The square (119892 119891) defines a cartesian functor from 119902 to 119901(ii) The mate of the canonical isomorphism

119865 119864 119865 119864

Hom119860(119860 119902) Hom119861(119861 119901) Hom119860(119860 119902) Hom119861(119861 119901)

119892

119894 cong 119894

119892

Hom119891(119891119892) Hom119891(119891119892)

119903 119903

in the diagram of functors over 119891∶ 119860 rarr 119861 is an isomorphism(iii) The mate of the canonical isomorphism

119865120794 119864120794 119865120794 119864120794

Hom119860(119860 119902) Hom119861(119861 119901) Hom119860(119860 119902) Hom119861(119861 119901)

119892120794

119896 cong 119896

119892120794

Hom119891(119891119892) Hom119891(119891119892)

in the diagram of functors is an isomorphism

Proof We will prove (i)hArr(ii) and (i)hArr(iii) The idea in each case is similar Conditions (ii) and(iii) imply that 119892 preserves the explicitly chosen cartesian lifts up to isomorphism which by Lemma513 implies that 119892 preserves all cartesian lifts Conversely assuming (i) we need to show that awhiskered copy of the counit of 119894 ⊣ 119903 and of the unit of 119896 ⊣ are isomorphisms The counit of 119894 ⊣ 119903and unit of 119896 ⊣ each encode the data of the factorizations of a natural transformation through thecartesian lift of its projection It will follow from (i) that the cells in question are cartesian and thefactorizations live over identities so Lemma 515 will imply that these natural transformations areinvertible

(i)hArr(iii) For convenience we take the functors 119896 to be the isofibrations of Remark 5113 sothe square on the left hand side of (iii) commutes strictly and its mate is the 2-cell 119892120794 By Theorem5111(iv)hArr(vi) this component of is invertible if and only if 119892120594 is 119901-cartesian where 120594 is the generic119902-cartesian lift of Lemma 518 for the cartesian fibration 119902 ∶ 119865 ↠ 119860 recall = 120594 By that lemmaagain 119892120594 is 119901-cartesian if and only if 119892 preserves cartesian transformations since the other canonical119902-cartesian lifts are constructed as restrictions of 120594

(i)hArr(ii) Let us write for Hom119891(119891 119892) to save space Since the unit of 119894 ⊣ 119903 is an isomorphismby Remark 5112 the mate of the isomorphism on the left hand side of (ii) is isomorphic to 119903120598 soour task is to show that this natural transformation is invertible if and only if 119892 defines a cartesianfunctor Recall from the proof of Theorem 5111(ii)rArr(i) that 1199011120598 defines the generic 119902-cartesian lift of

Lemma 518 for the cartesian fibration 119902 ∶ 119865 ↠ 119860 whiskering with ≔ Hom119891(119891 119892) ∶ Hom119860(119860 119902) rarrHom119861(119861 119901) carries this to a 2-cell whose projection with 1199010 is an identity since 1199010120598 = id and whoseprojection along 1199011 is 1198921199011120598 by the commutativity of the left-hand portion of the diagram below

119865 119864 119864

Hom119860(119860 119902) Hom119860(119860 119902) Hom119861(119861 119901) Hom119861(119861 119901)

119865 119864 119864

119892

119894 119894 119894119903dArr120598

1199011

119903

1199011

dArr120598

1199011

119892

Now naturality of whiskering provides a commutative square of natural transformations

1199011119894119903119894119892119903 119903

1199011119894119903 1199011

119903120598

1199011120598119894119892119903 cong 1199011120598

1199011120598

where wersquove simplified some of the names since 1199011119894 = id Since 120598 is the counit of an adjunction 119894 ⊣ 119903with invertible unit 120598119894 is an isomorphism Note 119901119903120598 = 1199010120598 = id and the right-hand vertical 1199011120598 isa 119901-cartesian lift of the restriction of 120601 this 120601 being the right comma cone over 119901 which equals thewhiskered right comma cone 119891120601 this120601 being the right comma cone over 119902 by the definition of Thebottom horizontal 1199011120598 is similarly a lift of 119891120601 = 120601 So if 119892 is a cartesian functor the right-handvertical and bottom horizontal are both 119901-cartesian lifts of a common 2-cell and the conservativityproperty implies that 119903120598 is invertible Conversely if 119903120598 is invertible the 1199011120598 = 1198921199011120598 is isomorphicto a 119901-cartesian transformation and is consequently 119901-cartesian Since Lemma 518 constructs theother canonical 119902-cartesian lifts as restrictions of 1199011120598 this is the case if and only if 119892 is a cartesianfunctor

5120 Proposition (pullback stability) If

119865 119864

119860 119861

119902

119892

119901

119891

is a pullback square and 119901 is a cartesian fibration then 119902 is a cartesian fibration Moreover a natural transfor-mation 120594 with codomain 119865 is 119902-cartesian if and only if 119892120594 is 119901-cartesian and in particular the pullback squaredefines a cartesian functor

Proof We apply Theorem 5111(i)rArr(iii) in the form described in Remark 5113 to the cartesianfibration 119901 which yields a fibered adjunction

119864120794 perp Hom119861(119861 119901)

Hom119861(119861 119901)119896

119896

(5121)

We will now argue that this functor 119896 pulls back to the corresponding functor for 119902 To that endfirst note that the top face of the following cube is a pullback since the front back and bottom facesare

119865120794

119864120794

Hom119860(119860 119902) 119860120794

Hom119861(119861 119901) 119861120794

119865 119860

119864 119861

119892120794

119896

119902120794

119896Hom119891(119891119892)

1199011 119891120794 1199011

1199011

119901120794

119892119902

119891119901

1199011

The right adjoint (minus)120794 preserves pullbacks so 119865120794 is the pullback of 119901120794 along 119891120794 and since this pullbacksquare factors through the top face of the cube along the square inducing the maps 119896 we conclude thatthis last square is a pullback as claimed

Now pullback defines a cosmological functor Hom119891(119891 119892)lowast ∶ 119974Hom119861(119861119901) rarr 119974Hom119860(119860119902) that car-ries the fibered adjunction (5121) to a fibered adjunction

119865120794 perp Hom119860(119860 119902)

Hom119860(119860 119902)119896

119896

which by Theorem 5111(iii)rArr(i) proves that 119902 is a cartesian fibration Moreover by construction ofthe adjunction 119896 ⊣ as a pullback of the adjunction 119896 ⊣ both of the mates in Theorem 5119(iii)are identities proving that (119892 119891) defines a cartesian functor

To see that (119892 119891) creates cartesian natural transformations note that a natural transformation 120594with codomain119865 is represented by an element 120594 ∶ 119883 rarr 119865120794 and 119892120594 is represented by the image of thiselement under the functor 119892120794 ∶ 119865120794 rarr 119864120794 By Theorem 5111(iv)hArr(vi) 120594 is 119902-cartesian just when thecomponent 120594 of the unit of 119896 ⊣ This unit component 120578120594 is the pullback of the corresponding

unit component 120578119892120794120594 indexed by 119892120594 and by conservativity of the smothering functor

hFun(119883 119865120794) rarr hFun(119883Hom119860(119860 119902)) timeshFun(119883Hom119861(119861119901))

hFun(119883 119864120794)

if 120578119892120794120594 is invertible then so is 120578120594

Pullback squares provide a key instance of cartesian functors Another is given by the followinglemma which can be proven using Theorem 5119

5122 Lemma If

119865 119864

119861

119892

119902 119901

is a functor between cartesian fibrations that admits a left adjoint over 119861 then 119892 defines a cartesian functor

Proof If ℓ ⊣119861 119892 then the cosmological functor 119901lowast1 ∶ 119974119861 rarr119974119861120794 carries this fibered adjunctionto a fibered adjunction

Hom119861(119861 119902) Hom119861(119861 119901)

Homid119861(id119861119892)

Homid119861(id119861ℓ)

Now both horizontal functors in the commutative square

119865120794 119864120794

Hom119860(119860 119902) Hom119861(119861 119901)

119892120794

119896 cong 119896

Hom119891(119891119892)

admit left adjoints and a standard result from the calculus of mates tells us that the mate with respectto the vertical adjunctions 119896 ⊣ is an isomorphism if and only if the mate with respect to the horizon-tal adjunctions is an isomorphism the latter natural transformation between left adjoints being thetranspose of the former natural transformation between their right adjoints This is the case becausethe mate with respect to the left adjoints lies is the fiber of the smothering functor of Proposition 325for 119865120794

Examples of cartesian fibrations are overdue

5123 Proposition (domain projection) For anyinfin-category119860 the domain-projection functor 1199010 ∶ 119860120794 ↠119860 defines a cartesian fibration Moreover a natural transformation 120594 with codomain119860120794 is 1199010-cartesian if andonly if 1199011120594 is invertible

Before giving the proof we explain the idea A 2-cell

119883 119860120794

119860

120573

119886

1199010uArr120572

defines a composable pair of 2-cells 120572∶ 119886 rArr 119909 and 120573∶ 119909 rArr 119910 in hFun(119883119860) Composing these we

induce a 2-cell 119883 119860120794120573∘120572

120573

dArr120594 representing the commutative square

119886 119909

119910 119910

120572

120573∘120572 120573

so that 1199010120594 = 120572 as required and 1199011120594 = idsup3

Proof We use Theorem 5111(i)hArr(iii) and prove that 1199010 is cartesian by constructing an appro-priate adjoint to the functor

(119860120794)120794 120794 times 120794

Hom119860(119860 1199010) 119860120794 120795 120794

119860120794 119860 120794 120793

1199011207940

1199011

119896

1199011

0times120794

1199010

120794times1

defined by cotensoring with the 1-categories displayed above right⁴To construct a right adjoint with invertible counit to the map 119896 it suffices to construct a left

adjoint left inverse to the inclusion of 1-categories 120795 120794 times 120794 with image (0 0) rarr (0 1) rarr (1 1)The left adjoint ℓ ∶ 120794 times 120794 rarr 120795 is a left inverse on the image of 120795 and sends (1 0) to the terminalelement of 120795

120794 times 120794 ni

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(0 0) (0 1)

(1 0) (1 1)

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

isin 120795

(119860120794)120794 119860120795 ≃ Hom119860(119860 1199010)

119896

119860ℓ

defines the desired right adjoint with invertible counit

sup3While the weak universal property of the arrowinfin-category can be used to induce 2-cells with codomain119860120794 it cannotbe used to prove equations between the induced 2-cells as required to demonstrate that induction condition of Definition511 Thus some sort ofinfin-cosmos-level argument is necessary to establish this result

⁴The cotensor 119860(minus) carries pushouts of simplicial sets to pullbacks of infin-categories and the pushout of 120794 cup120793 120794 ofsimplicial sets is Λ1[2] not 120795 = Δ[2] However on account of the equivalence of infin-categories 119860120795 ≃ 119860Λ1[2] no harmcomes from making the indicated substitution

The characterization of 1199010-cartesian transformations now follows from Theorem 5111(iv)hArr(vi)A cell 120594 with codomain 119860120794 is 1199010-cartesian if and only if it its representing element of (119860120794)120794 is in theessential image of the right adjoint 119860ℓ ∶ 119860120795 rarr 119860120794times120794 Clearly if 120594 is in the essential image then itscodomain component must be invertible

Conversely suppose the codomain component

119883 119860120794times120794 119860120794120594 ev(1minus)

represents a natural isomorphism Applying Lemma 1112 to Fun(119883119860) one can build a diagram inFun(119883119860)120794times120794 cong Fun(119883119860120794times120794) whose top edge is the given isomorphism whose vertical edges areisomorphisms and whose bottom edge is an identity This glues onto 120594 to define a diagram 119883 rarr119860Λ1[2]times120794 and now the composite

119883 119860Λ1[2]times120794 119860120795times120794 119860120794times120794times120794sim 119860119888

where 119888 ∶ 120794times120794 rarr 120795 is the surjective functor that sends (0 0) and (0 1) to 0 witnesses an isomorphismbetween 120594 and a diagram in the image of 119860ℓ whose codomain component is the identity

The same argument proves that for any 119892∶ 119862 rarr 119860 the domain-projection 1199010 ∶ Hom119860(119860 119892) ↠119860 is a cartesian fibration For 119891∶ 119861 rarr 119860 the domain-projections 1199010 ∶ Hom119860(119891 119860) ↠ 119861 and1199010 ∶ Hom119860(119891 119892) ↠ 119861 are obtained by pullback via Proposition 5120 These cartesian fibrationsfigures in a final important lemma about the class of 119901-cartesian transformations

5124 Lemma Let 119901∶ 119864 ↠ 119861 be a cartesian fibration and consider a composable pair of natural transfor-

mations 119883 119864119890prime

119890dArr120595 and 119883 119864

119890Prime

119890primedArr120595prime with codomain 119864

(i) If 120595 and 120595prime are 119901-cartesian then so is 120595 sdot 120595prime(ii) If 120595 and 120595 sdot 120595prime are 119901-cartesian then so is 120595prime

Proof For (i) recall from the proof of Lemma 518 that a 119901-cartesian lift of 119901120595 is given by thecomposite

119883 Hom119861(119861 119901) 119864119901120595

119903

1199011

dArr120594

with the natural transformation 120594∶ 119903 rArr 1199011 of (519) whose domain 119903 is the right adjoint of Theorem5111(ii) Since we are assuming that120595 is 119901-cartesian 120595 is isomorphic to this whiskered natural trans-formation so we may redefine 120595 to equal it and redefine 120595prime to absorb the isomorphism By Lemma514 this new 120595prime is still 119901-cartesian since wersquove assumed 120595prime is This modification does not change thecomposition transformation 120595 sdot 120595prime that we desire to show is 119901-cartesian

Now by 2-cell induction the diagram as below-left defines a 2-cell as below-right

119901119890Prime 119901119890prime

119901119890 119901119890

119901120595prime

119901(120595sdot120595prime) 119901120595 119883 Hom119861(119861 119901)

119901(120595sdot120595prime)

119901120595

dArr120574

By the generalization of Proposition 5123 120574 is a 1199010-cartesian cell By Lemma 5122 the fibered rightadjoint 119903

Hom119861(119861 119901) perp 119864

1198611199010 119903

119894

119901

carries 120574 to a 119901-cartesian transformation 119903120574Now the horizontal composite

119883 Hom119861(119861 119901) 119864

119901(120595sdot120595prime)

119901120595

dArr120574

119903

1199011

dArr120594

provides a commutative diagram of natural transformations

119890prime

119890 119890

120594119901(120595sdot120595prime)

119903120574

120595

In particular the composite 120595 sdot 119903120574 is a 119901-cartesian natural transformation Since 119901119903120574 = 1199010120574 = 119901120595prime119903120574 is a 119901-cartesian lift of the cartesian transformation 120595prime so 120595prime and 119903120574 are isomorphic But now120595 sdot 120595prime is isomorphic to the 119901-cartesian transformation 120594119901(120595sdot120595prime) and Lemma 514 proves that 120595 sdot 120595prime is119901-cartesian

Now for (ii) suppose 120595 and 120595 sdot 120595prime are 119901-cartesian and let 120594prime ∶ rArr 119890prime denote a 119901-cartesian lift of119901120595prime Consider the factorization 120595prime = 120594prime sdot 120579 of 120595prime through its 119901-cartesian lift with 119901120579 = id Nowpart (i) implies that 120595 sdot 120594prime is 119901-cartesian and 120595 sdot 120595prime = 120595 sdot 120594prime sdot 120579 so Lemma 515 implies that 120579 isan isomorphism Now 120595prime is isomorphic to a 119901-cartesian cell so Lemma 514 implies that 120595prime must be119901-cartesian

For infin-categories admitting pullbacks the codomain-projection functor also defines a cartesianfibration

5125 Proposition (codomain projection) Let 119860 be an infin-category that admits pullbacks in the senseof Definition 437 Then the codomain-projection functor 1199011 ∶ 119860120794 ↠ 119860 is a cartesian fibration and the1199011-cartesian arrows are just those 2-cells that represent pullback squares

Proof Via Theorem 5111(i)hArr(iii) we desire a right adjoint right inverse to the functor 119896 definedbelow-left applying 119860minus to the diagram of simplicial sets appearing below-right

(119860120794)120794 120794 times 120794

Hom119860(119860 1199011) 119860120794 ⟓ 120794

119860120794 119860 120794 120793

1199011207941

1199011

119896

1199011

1times120794

1199010

120794times1

This is done in Corollary 435

119860⊡ 119860⟓(119860120794)120794 congres

perpran

cong Hom119860(119860 1199011)

Exercises

51ii Exercise Attempt to prove directly from Definition 516 that cosmological functors preservecartesian fibrations and explain what goes wrong

51iii Exercise Categorify the intuition that cartesian fibrations 119901∶ 119864 ↠ 119861 and 119902 ∶ 119865 ↠ 119861 defineldquocontravariant 119861-indexed functors valued ininfin-categoriesrdquo by proving that a cartesian functor

119864 119865

119861

119892

119901 119901

defines a ldquonatural transformationrdquo show that there exists a natural isomorphism in the square of fibers

119864119887 119865119887

119864119886 119865119886

120573lowast

119892

existcong 120573lowast

119892

where the action of an arrow 120573 in the homotopy category of 119861 on the fibers is defined by factoring thedomain of a 119901- or 119902-cartesian lift of 120573

119864119887 119864 119864119887 119864

119864119886

1 119861 1 119861

ℓ119887

=119901 120573lowast 120573lowast(ℓ119887)

uArr120594120573

ℓ119887

119901ℓ119886119887

119886

uArr120573

119886

51iv Exercise Use Proposition 5120 to prove that cartesian functors pull back

52 Cocartesian fibrations and bifibrations

By the dual of Proposition 5123 the codomain-projection functor is also a cocartesian fibrationa notion we now introduce

521 Definition (119901-cocartesian transformations) Let 119901∶ 119864 ↠ 119861 be an isofibration A natural trans-

formation 119883 119864119890

119890primedArr120594 with codomain 119864 is 119901-cocartesian if

(i) induction Given any natural transformations 119883 119864119890

119890PrimedArr120591 and 119883 119861

119901119890prime

119901119890PrimedArr120574 so that 119901120591 =

120574 sdot 119901120594 there exists a lift 119883 119864119890prime

119890PrimedArr of 120574 so that 120591 = sdot 120594

119890 119890Prime

119890prime

119901119890 119901119890Prime

119901119890prime

120591

120594

119901120591

119901120594 120574

isin hFun(119883 119864)

isin hFun(119883 119861)

119901lowast

522 Definition (cocartesian fibration) An isofibration 119901∶ 119864 ↠ 119861 is a cocartesian fibration if(i) Any natural transformation 120573∶ 119901119890 rArr 119887 as below-left admits a lift 120594120573 ∶ 119890 rArr 120573lowast119890 as below-right

119883 119864 119883 119864

119861 119861

119890

119887

dArr120573 119901 =

119890

120573lowast119890

dArr120594120573

119901

that is a 119901-cartesian transformation so that 119901120594 = 120573(ii) The class of 119901-cocartesian transformations is closed under restriction along any functor that

is if 119883 119864119890

119890primedArr120594 is 119901-cocartesian and 119891∶ 119884 rarr 119883 is any functor then 119884 119864

119890119891

119890prime119891

dArr120594119891 is 119901-co-

cartesian

The dual to Theorem 5111 asks for left adjoints to 119894 ∶ 119864 rarr Hom119861(119901 119861) and 119896 ∶ 119864120794 rarr Hom119861(119901 119861)in place of right adjoints See Exercise 52i This result can be deduced immediately by consideringthe isofibration 119901∶ 119864 ↠ 119861 as a map in the dualinfin-cosmos

523 Observation (dual infin-cosmoi) There is an identity-on-objects functor (minus)∘ ∶ 120491 rarr 120491 thatreverses the ordering of the elements in each ordinal [119899] isin 120491 The functor (minus)∘ sends a face map120575119894 ∶ [119899minus1] ↣ [119899] to the face map 120575119899minus119894 ∶ [119899minus1] ↣ [119899] and sends the degeneracy map 120590119894 ∶ [119899+1] ↠ [119899]to the degeneracy map 120590119899minus119894 ∶ [119899 + 1] ↠ [119899] Precomposition with this involutive automorphisminduces an involution (minus)op ∶ 119982119982119890119905 rarr 119982119982119890119905 that sends a simplicial set 119883 to its opposite simplicialset 119883op with the orientation of the vertices in each simplex reversed This construction preserves allconical limits and colimits and induces an isomorphism (119884119883)op cong (119884op)119883op

on exponentialsFor anyinfin-cosmos119974 there is a dualinfin-cosmos119974co with the same objects but with functor spaces

defined byFun119974co(119860 119861) ≔ Fun119974(119860 119861)op

The isofibrations equivalences and trivial fibrations in119974co coincide with those of119974Conical limits in 119974co coincide with those in 119974 while the cotensor of 119860 isin 119974 with 119880 isin 119982119982119890119905 is

defined to be 119860119880op In particular the cotensor of aninfin-category with 120794 is defined to be 119860120794op

whichexchanges the domain and codomain projections from arrow and commainfin-categories

524 Definition An isofibration 119901∶ 119864 ↠ 119861 defines a bifibration if 119901 is both a cartesian fibrationand a cocartesian fibration

Projections give trivial examples of bifibrations

525 Example For anyinfin-categories119860 and 119861 the projection functor 120587∶ 119860times119861 ↠ 119861 is a bifibrationin which a 2-cell with codomain119860times119861 is120587-cocartesian or120587-cartesian if and only if its composite withthe projection 120587∶ 119860 times 119861 ↠ 119860 is an isomorphism

526 Proposition Let 119901∶ 119864 ↠ 119861 be a bifibration Then any arrow 119883 119861119886

119887

dArr120573 induces a fibered

adjunction

119864119886 perp 119864119887 119864119886 119864 119864119887

119883 119883 119861 119883

120573lowast

119901119886 120573lowast 119901119887

ℓ119886

119901119886

119901

ℓ119887

119901119887

119886 119887between the fibers of 119901 over 119886 and 119887

As will be remarked upon following the proof of this result the left adjoint 120573lowast is the covariantlypseudofunctorial action of the arrow 120573 on the fibers of 119901 defined in (501) while the right adjoint 120573lowastis the dual contravariantly pseudofunctorial action

Proof Write 120573 ∶ 119883 rarr 119861120794 for the functor induced by 120573 Note that the pullbacks defining thefibers over its domain edge 119886 and codomain edge 119887 factor as

119864119886 Hom119861(119901 119861) 119864 Hom119861(119861 119901) 119864119887

119883 119861120794 119861 119861120794 119883

ℓ119886

119901119886

119902

1199010119901 119902

1199011

ℓ119887

119901119887

119886

120573 1199010 1199011 120573

119887

Now via Remark 5113 Theorem 5111(i)rArr(iii) and its dual provide a right adjoint right inverseto 119896 ∶ 119864120794 rarr Hom119861(119861 119901) and a left adjoint right inverse to 119896 ∶ 119864120794 rarr Hom119861(119901 119861) Composingthe former fibered adjunction with 119902 ∶ Hom119861(119861 119901) ↠ 119861120794 and the latter fibered adjunction with119902 ∶ Hom119861(119901 119861) ↠ 119861120794 we obtain a composable pair of adjunctions

Hom119861(119901 119861) 119864120794 Hom119861(119861 119901)

119861120794

119902119896 119901120794

119896

119902

fibered over 119861120794 note in both cases that 119902119896 = 119901120794 Pulling back the composite adjunction along120573 ∶ 119883 rarr 119861120794 yields the fibered adjunction of the statement

527 Remark (action of arrows on the fibers of a cocartesian fibration) If 119901∶ 119864 ↠ 119861 is a cocartesianfibration but not a cartesian fibration the construction in the proof of Proposition 526 still definesthe functor 120573lowast ∶ 119864119886 rarr 119864119887 Examining the details of this construction we see that it produces thefunctor given this name in (501) The functor just constructed as the pullback over 120573 of 119896ℓ isinduced by the composite commutative square

119864119886 Hom119861(119901 119861) 119864

119883 119861120794 119861

119901119886

119902

1199011ℓ

119901

120573 1199011

which defines a cone over the pullback defining 119864119887 The top-horizontal functor takes the codomain-component of the 119901-cocartesian lift of 120573 with domain ℓ119886 This recovers the description given at thestart of this chapter

Exercises

52i Exercise Formulate the dual to Theorem 5111 providing an internal characterization of co-cartesian fibrations

52ii Exercise Prove that for any infin-category 119860 the codomain-projection functor 1199011 ∶ 119860120794 ↠ 119860defines a cocartesian fibration

53 The quasi-category theory of cartesian fibrations

In this section we reinterpret the notion of cartesian fibration and cartesian natural transforma-tion from the point of view of the infin-cosmos 119974 rather than its quotient homotopy 2-category 120101119974In doing so we recall that the functors 119890 ∶ 119883 rarr 119864 between infin-categories are precisely the verticesor 0-arrows in the quasi-categorical functor space Fun(119883 119864) The 1-simplices or 1-arrows 120594∶ 119890prime rarr 119890

of Fun(119883 119864) represent natural transformations 119883 119864119890prime

119890dArr120594 Every natural transformation from 119890prime

to 119890 is represented by a 1-arrow 119890prime rarr 119890 and a parallel pair of 1-arrows represent the same naturaltransformation if and only if they are homotopic bounding a 2-arrow in Fun(119883 119864) whose 0th or 2ndedge is degenerate

Before analyzing the 0- and 1-arrows in the functor spaces in particular we consider the collectionof functor spaces of an infin-cosmos globally and prove another important corollary of the internalcharacterization of cartesian fibrations The notion of cartesian fibrations are representably definedin the following sense

531 Proposition Let 119901∶ 119864 ↠ 119861 be an isofibration betweeninfin-categories in aninfin-cosmos119974 Then 119901 isa cartesian fibration if and only if

(i) For all 119883 isin 119974 the isofibration 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861) is a cartesian fibration between quasi-categories

(ii) For all 119891∶ 119884 rarr 119883 isin 119974 the square

Fun(119883 119864) Fun(119884 119864)

Fun(119883 119861) Fun(119884 119861)

119901lowast

119891lowast

119901lowast

119891lowast

is a cartesian functor

Proof If 119901∶ 119864 ↠ 119861 is a cartesian fibration then Theorem 5111(i)rArr(iii) constructs a right ad-joint right inverse to 119896 ∶ 119864120794 ↠ Hom119861(119861 119901) The simplicial bifunctor Fun(minus minus) ∶ 119974op times 119974 rarr 119980119966119886119905defines a 2-functor Fun(minus minus) ∶ 120101119974op times 120101119974 rarr 120101119980119966119886119905 which transposes to a Yoneda-type embeddingFun(minus minus) ∶ 120101119974 rarr 120101119980119966119886119905120101119974

opfrom the homotopy 2-category of 119974 to the 2-category of 2-functors

2-natural transformations and modifications This 2-functor carries the adjunction 119896 ⊣ to an ad-junction in the 2-category 120101119980119966119886119905120101119974

op This latter adjunction defines for each 119883 isin 119974 a right adjoint

right inverse adjunction

Fun(119883 119864)120794 Fun(119883 119864120794) perp Fun(119883Hom119861(119861 119901)) HomFun(119883119861)(Fun(119883 119861) 119901lowast)cong119896lowast

conglowast

and for each 119891∶ 119884 rarr 119883 in119974 a strict adjunction morphism⁵ commuting strictly with the left adjointsand with the right adjoints

Fun(119883 119864)120794 cong Fun(119883 119864120794) Fun(119884 119864120794) cong Fun(119884 119864)120794

Hom(Fun(119883 119861) 119901lowast) cong Fun(119883Hom119861(119861 119901)) Fun(119884Hom119861(119861 119901)) cong Hom(Fun(119884 119861) 119901lowast)

119891lowast

119896lowast⊢ 119896lowast ⊣

119891lowast

lowast lowast

(532)By Theorems 5111(iii)rArr(i) and 5119(iii)rArr(i) this demonstrates the two conditions of the statement

Conversely supposing 119901∶ 119864 ↠ 119861 satisfies conditions (i) and (ii) by Theorems 5111(i)rArr(iii) and5119(i)rArr(iii) there is a commutative square 119896lowast119891lowast = 119891lowast119896lowast where both verticals 119896lowast admit right adjointright inverses 119896lowast ⊣ and the mate of the identity 119896lowast119891lowast = 119891lowast119896lowast defines an isomorphism 119891lowast cong 119891lowastBy Proposition in Appendix B this data suffices to internalize the right adjoints to the repre-sentable functors 119896lowast ∶ Fun(119883 119864120794) rarr Fun(119883Hom119861(119861 119901)) to a functor lowast ∶ Fun(119883Hom119861(119861 119901)) rarr

⁵A strict adjunctionmorphism is given by a pair of functors the horizontals of (532) that define strictly commutativesquares with both the left and with the right adjoints and so that the units of each adjunction whisker along these functorsto each other and the counits of each adjunction whisker along these functors to each other See Appendix B for more

Fun(119883 119864120794) arising from post-composition with some ∶ Hom119861(119861 119901) rarr 119864120794 This right adjoint isextracted as the image of the identity element

Fun(Hom119861(119861 119901)Hom119861(119861 119901)) Fun(Hom119861(119861 119901) 119864120794)

and the unit and counit are internalized similarly the condition on mates is used to verify the triangleequalities that demonstrate that 119896 ⊣ Now Theorem 5111(iii)rArr(i) proves that 119901∶ 119864 ↠ 119861 is acartesian fibration

An easier argument along the same lines demonstrates

533 Corollary A commutative square between cartesian fibrations as displayed below-left

119865 119864 Fun(119883 119865) Fun(119883 119864)

119860 119861 Fun(119883119860) Fun(119883 119861)

119902

119892

119901 119902lowast

119892lowast

119901lowast

119891 119891lowast

defines a cartesian functor in aninfin-cosmos 119974 if and only if for all 119883 isin 119974 the square displayed above rightdefines a cartesian functor between cartesian fibrations of quasi-categories

Proof Exercise 53i

Our aim is now to characterize those 1-arrows in Fun(119883 119864) that represent 119901-cartesian naturaltransformation for some cartesian fibration 119901∶ 119864 ↠ 119861 Recall that the 1-arrows in Fun(119883 119864) are inbijection with the 0-arrows of Fun(119883 119864)120794 cong Fun(119883 119864120794)

534 Definition (119901-cartesian 1-arrow) Let 119901∶ 119864 ↠ 119861 be a cartesian fibration and consider a 1-arrow120594 in Fun(119883 119864) defining an element 120594 isin Fun(119883 119864)120794 cong Fun(119883 119864120794) Say 120594 is a 119901-cartesian 1-arrow ifit is isomorphic to some object in the image of the right adjoint right inverse functor

Fun(119883Hom119861(119861 119901)) Fun(119883 119864120794)lowast

of Theorem 5111(iii)

The new notion of 119901-cartesian 1-arrow coincides exactly with the previous notion of 119901-cartesiannatural transformation

535 Lemma Consider a cartesian fibration 119901∶ 119864 ↠ 119861 betweeninfin-categories For a 1-arrow 120594∶ 119890prime rarr 119890 inFun(119883 119864) the following are equivalent

(i) 120594 is a 119901-cartesian 1-arrow

(ii) 120594 represents a 119901-cartesian natural transformation 119883 119864119890prime

119890dArr120594

(iii) 120594 represents a natural transformation

120793 Fun(119883 119864)119890prime

119890

dArr120594

that is cartesian for 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861)

Conversely a natural transformation 119883 119864119890prime

119890dArr120594 is 119901-cartesian if and only if any representing 1-arrow

120594 ∶ 119883 rarr 119864120794 satisfies all of these equivalent conditions

Proof We start with the final clause Note from Exercise 32i that homotopic 1-arrows are iso-morphic as objects of Fun(119883 119864120794) so if some 1-arrow representing a 119901-cartesian transformation is a119901-cartesian 1-arrow then all representatives of that natural transformation are

Now the equivalence (i)hArr(ii) follows fromTheorem 5111(iv)hArr(vi) oncewe establish that a 1-arrowof Fun(119883 119864) when encoded as a functor 120594∶ 119883 rarr 119864120794 is in the essential image of lowast if and only if thecomponent 120594 of the unit of 119896 ⊣ is invertible

If 120594∶ 119890prime rarr 119890 is a 119901-cartesian 1-arrow then by definition there exists some 120573∶ 119883 rarr Hom119861(119861 119901)and an invertible 2-cell

119883 119864120794

Hom119861(119861 119901)

120594

120573 dArrcong

The unit of the adjunction 119896 ⊣ of Theorem 5111(iii) has the property that is invertible so thecomponent 120573 is invertible and so 120594 is also invertible By Theorem 5111(vi)rArr(iv) this implies that120594 represents a 119901-cartesian natural transformation

Conversely if 120594 defines a 119901-cartesian natural transformation then Theorem 5111(iv)rArr(vi) tellsus that for any representing 1-arrow 120594 ∶ 119883 rarr 119864120794 the component 120594 is an isomorphism Inparticular 120594 is isomorphic to 119896120594 which proves that 120594 is in the essential image of lowast and thusdefines a 119901-cartesian 1-arrow

Finally via the characterization of cartesian transformations given in Theorem (vi) and the factthat the adjunction 119896 ⊣ in 120101119974 induces an adjunction 119896lowast ⊣ lowast in 120101119980119966119886119905 the conditions (ii) and(iii) are tautologically equivalent The former refers to the adjunction between categories hFun(119883 119864)and hFun(119883 119861) while the latter refers to the adjunction between categories hFun(120793Fun(119883 119864)) andhFun(120793Fun(119883 119861)) and these are isomorphic

Combining Lemma 535with Proposition 531 we arrive at a new equivalent definition of cartesianfibrations

536 Corollary An isofibration 119901∶ 119864 ↠ 119861 is a cartesian fibration if and only if(i) Any 1-arrow with codomain 119861 admits a 119901-cartesian lift with specified codomain 0-arrow(ii) 119901-cartesian 1-arrows are stable under precomposition with 0-arrows

537 Lemma Let 119901∶ 119864 ↠ 119861 be a cartesian fibration and consider a 2-arrow

119890prime

119890Prime 119890

120595120595prime

120595Prime

in Fun(119883 119864)(i) If 120595 and 120595prime are 119901-cartesian 1-arrows so is 120595Prime(ii) If 120595 and 120595Prime are 119901-cartesian 1-arrows so is 120595prime

(iii) If 120595 and 120595Prime are 119901-cartesian 1-arrows and 119901120595prime is invertible then 120595prime is invertible

Proof Via Lemma 535 (i) and (ii) are Lemmas 5124(i) and (ii) while (iii) is Lemma 515

538 Lemma A 2-cell as below left

119876 Fun(119883 119864) 120793 119876 Fun(119883 119864) 119883 119864119890prime

119890

dArr120594119902

119890prime

119890

dArr120594

119890prime119902

119890119902dArr120594119902

is 119901lowast-cartesian if and only if each of its components 120594119902 is 119901-cartesian

Proof If 120594 is 119901lowast-cartesian then so is the restriction 120594119902 along any element 119902 ∶ 120793 rarr 119876 By Lemma535 this tells us that 120594119902 defines a 119901-cartesian transformation

Conversely if120594119902 is a 119901-cartesian transformation then Lemma 535 tells us that120594119902 is a 119901lowast-cartesiantransformation Now consider the factorization120594 = 120594120594sdot120579 through 119901lowast-cartesian lift120594120594 of 119901lowast120594 Becausethe components 120594119902 of 120594 are 119901lowast-cartesian the components 120579119902 of 120579 are isomorphisms By Lemma an arrow in an exponetial Fun(119883 119864)119876 is an isomorphism if and only if it is a pointwise isomorphismso this implies that 120594 By isomorphism stability of cartesian transformations we thus conclude that120594 is 119901lowast-cartesian

Exercises

53i Exercise Prove Corollary 533

54 Discrete cartesian fibrations

Recall from Definition 1221 that an object 119864 in an infin-cosmos 119974 is discrete if for all 119883 isin 119974the functor-space Fun(119883 119864) is a Kan complex Since a quasi-category is a Kan complex just when itshomotopy category is a groupoid (see Corollary 1115) equivalently 119864 is discrete if and only if everynatural transformation with codomain 119864 is invertible

From this definition it follows that an isofibration 119901∶ 119864 ↠ 119861 considered as an object of 119974119861 isdiscrete if and only if any 2-cell with codomain 119864 that whiskers with 119901 to an identity is invertible Infact the discrete objects are exactly those isofibrations that define conservative functors in 120101119974

541 Lemma An isofibration 119901∶ 119864 ↠ 119861 is a discrete object of119974119861 if and only if 119901∶ 119864 ↠ 119861 is a conservative

functor meaning any 119883 119864119886

119887

dArr120574 for which 119901120574 is an isomorphism is invertible

Proof Exercise 54i

Our aim in this section is to study a special class of cartesian fibrations and cocartesian fibrations

542 Definition An isofibration 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration if it is a cartesian fibra-tion and if it is discrete as an object of119974119861 Dually an isofibration 119901∶ 119864 ↠ 119861 is a discrete cocartesianfibration if it is a cocartesian fibration and if it is discrete as an object of119974119861

The fibers of a discrete object 119901∶ 119864 ↠ 119861 in 119974119861 are discrete infin-categories in infin-cosmoi whoseobjects model (infin 1)-categories the discrete cartesian fibrations and discrete cocartesian fibrationsare ldquoinfin-groupoid-valued pseudofunctorsrdquo

There is also a direct 2-categorical characterization of the discrete cartesian fibrations whichreveals that unlike the case for cartesian and cocartesian fibrations for their discrete analogues thereare no special classes of 119901-cartesian or 119901-cocartesian cells

543 Proposition(i) If 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration every natural transformation with codomain 119864 is119901-cartesian

(ii) An isofibration 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration if and only if every 2-cell 120573∶ 119887 rArr 119901119890 hasan essentially unique lift given 120594∶ 119890prime rArr 119890 and 120595∶ 119890Prime rArr 119890 so that 119901120594 = 119901120595 = 120573 then there existsan isomorphism 120574∶ 119890Prime rArr 119890prime with 120594 sdot 120574 = 120595 and 119901120574 = id

Note that (i) implies immediate that any commutative square

119865 119864

119860 119861

119902

119892

119901

119891

from a cartesian fibration 119902 to a discrete cartesian fibration 119901 defines a cartesian functor

Proof By the definition of cartesian fibration any 2-cell 120595 with codomain 119864 factors through a119901-cartesian lift of 119901120595 along a 2-cell 120574 so that 119901120574 = id The discrete objects of 119974119861 are exactly thoseisofibrations with the property that any 2-cell with codomain 119864 that whiskers with 119901 to an identity isinvertible In particular120574 is an isomorphism and now120595 is isomorphic to a 119901-cartesian transformationand hence by Lemma 514 itself 119901-cartesian

By (i) and Lemma 513 itrsquos now clear that if 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration thenany 2-cell 120573∶ 119887 rArr 119901119890 has an essentially unique lift For the converse note first that any 119901∶ 119864 ↠ 119861satisfying this hypothesis is a discrete object if 120595∶ 119890prime rArr 119890 is so that 119901120595 = id then id ∶ 119890 rArr 119890 isanother lift of 119901120595 and essential uniqueness provides an inverse isomorphism 120595minus1 ∶ 119890 rArr 119890prime

To complete the proof we now show that any 2-cell 120594∶ 119890prime rArr 119890 is cartesian for 119901 and to that endconsider a pair 120591∶ 119890Prime rArr 119890 and 120574∶ 119901119890Prime rArr 119901119890prime so that 119901120591 = 119901120594 sdot 120574 By the hypothesis that every 2-celladmits an essentially unique lift we can construct a lift 120583∶ rArr 119890prime so that 119901120583 = 120574 Now 120591 and 120594 sdot 120583are two lifts of 119901120591 with the same codomain so there exists an isomorphism 120579∶ 119890Prime rArr with 119901120579 = idThe composite 120583 sdot 120579 then defines the desired lift of 120574 to a cell so that 120591 = 120594 sdot 120583 sdot 120579

544 Example (domain projection from an element) For an element 119887 ∶ 1 rarr 119861 the domain-projectionfunctor 1199010 ∶ Hom119861(119861 119887) ↠ 119861 is a discrete cartesian fibration Cartesianness was established in Propo-

sition 5123 and discreteness follows immediately from 2-cell conservativity If 119883 Hom119861(119861 119887)119886

119887

dArr120574

is a natural transformation for which 1199010120574 is an identity then since 1199011120574 is also an identity 120574 must beinvertible

Dually the codomain-projection functor 1199011 ∶ Hom119861(119887 119861) ↠ 119861 is a discrete cocartesian fibration

545 Lemma (pullback stability) If

119865 119864

119860 119861

119902

119892

119901

119891

is a pullback square and 119901 is a discrete cartesian fibration then 119902 is a discrete cartesian fibration

Proof In light of Proposition 5120 it remains only to verify that 119902 is discrete Consider a 2-cell

119883 119865119886

119887

dArr120574 so that 119902120574 is invertible Then 119891119902120574 = 119901119892120574 is invertible and conservativity of 119901 implies

that 119892120574 is invertibleBy Lemma 315 the pullback square of functor spaces

Fun(119883 119865) Fun(119883 119864)

Fun(119883119860) Fun(119883 119861)

119902lowast

119892lowast

119901lowast

119891lowast

induces a smothering functor

hFun(119883 119865) rarr hFun(119883 119864) timeshFun(119883119861)

hFun(119883119860)

Wersquove just verified that the image of 120574 is an isomorphism so conservativity implies that 120574 is alsoinvertible

In analogy with Theorem 5111 there is an internal characterization of discrete cartesian fibra-tions which in the discrete case takes a much simpler form Recall any functor 119901∶ 119864 rarr 119861 inducesfunctors 119896 ∶ 119864120794 rarr Hom119861(119861 119901) and 119896 ∶ 119864120794 rarr Hom119861(119901 119861) as in (5110) by applying 119901 to the genericarrow for 119864

546 Proposition (internal characterization of discrete fibrations) An isofibration 119901∶ 119864 ↠ 119861 is adiscrete cartesian fibration if and only if the functor 119896 ∶ 119864120794 rarr Hom119861(119861 119901) is an equivalence and a discretecocartesian fibration if and only if the functor 119896 ∶ 119864120794 rarr Hom119861(119901 119861) is an equivalence

Recall from Theorem 5111(iii) that 119901∶ 119864 ↠ 119861 defines a cartesian fibration if and only if 119896 ∶ 119864120794 rarrHom119861(119901 119861) admits a right adjoint with invertible counit Proposition 546 asserts that 119901 defines a dis-crete cartesian fibration if and only if the unit of that adjunction a natural transformation that definesthe factorization of any natural transformation with codomain119864 through the canonical 119901-cartesian liftof its image under 119901 is an isomorphism in which case that adjunction defines an adjoint equivalenceand all natural transformations with codomain 119864 are 119901-cartesian

Proof Assume first that 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration By Theorem 5111(i)rArr(iii)119896 ∶ 119864120794 rarr Hom119861(119861 119901) then admits a right adjoint with invertible counit ∶ 119896 cong id We will showthat in this case the unit ∶ id rArr 119896 is also invertible proving that 119896 ⊣ defines an adjoint equiva-lence

Since the counit of 119896 ⊣ is invertible 119896 is an isomorphism Thus 1199011119896 = 1199011 and 1199010119896 = 1199011199010are both isomorphisms By conservativity of the discrete fibration 119901∶ 119864 ↠ 119861 proven in Lemma 541this implies that 1199010 is invertible and now 2-cell conservativity for119864120794 reveals that is an isomorphism

Conversely if 119896 ∶ 119864120794 ⥲ Hom119861(119861 119901) is an equivalence by Proposition 2111 we may choose a rightadjoint equivalence inverse The counit of this adjoint equivalence is necessarily an isomorphismso by Theorem 5111(iii)rArr(i) we know that 119901∶ 119864 ↠ 119861 is a cartesian fibration Since the unit of119896 ⊣ is also an isomorphism Theorem 5119(vi)rArr(iv) tells us that every natural transformationwith codomain 119864 is 119901-cartesian and now the conservativity property for cartesian transformations ofLemma 515 tells us that 119901∶ 119864 ↠ 119861 defines a conservative fun ctor and in particular is discrete

Since equivalences and simplicial limits in aninfin-cosmos are representably-defined notions it fol-lows immediately from Proposition 546 that

547 Proposition An isofibration 119901∶ 119864 ↠ 119861 in aninfin-cosmos 119974 defines a discrete cartesian fibration ifand only if for all 119883 isin 119974 the functor 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861) defines a discrete cartesian fibration ofquasi-categories

Using the internal characterization it is straightforward to verify that discrete cartesian fibrationscompose and cancel on the left

548 Lemma(i) If 119901∶ 119864 ↠ 119861 and 119902 ∶ 119861 ↠ 119860 are discrete cartesian fibrations so is 119902119901 ∶ 119864 ↠ 119860(ii) If 119901∶ 119864 ↠ 119861 is an isofibration and 119902 ∶ 119861 ↠ 119860 and 119902119901 ∶ 119864 ↠ 119860 are discrete cartesian fibrations

then so is 119901∶ 119864 ↠ 119861

Proof By considering the defining pullback diagrams the map 119864120794 ↠ Hom119860(119860 119902119901) that testswhether 119902119901 ∶ 119864 ↠ 119860 is a discrete cartesian fibration factors as the map 119864120794 ↠ Hom119861(119861 119901) thattests whether 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration followed by a pullback of the map 119861120794 ↠Hom119860(119860 119902) that tests whether 119902 ∶ 119861 ↠ 119860 is a discrete cartesian fibration

119864120794

Hom119861(119861 119901) 119861120794 119860120794

Hom119860(119860 119902119901) Hom119860(119860 119902)

119864 119861 119860

11990111199011

119901 119902

Both parts now follow from the 2-of-3 property

The internal characterization of discrete cartesian fibrations is useful for establishing further ex-amples

549 Lemma A trivial fibration 119901∶ 119864 ⥲rarr 119861 is a discrete bifibration

Proof Recall fromRemark 5113 that the canonical functors 119896 ∶ 119864120794 ↠ Hom119861(119861 119901) and 119896 ∶ 119864120794 ↠Hom119861(119901 119861) can be constructed as the Leibniz cotensor of the monomorphism 1∶ 120793 120794 in the firstcase and 0∶ 120793 120794 in the second with the trivial fibration 119901∶ 119864 ⥲rarr 119861 By Lemma 1210 both mapsare trivial fibrations and in particular equivalences Now Proposition 546 proves that 119901 is a discretecartesian fibration and also a discrete cocartesian fibration

A final important family of examples of discrete cartesian fibrations areworth establishing Propo-sition 5123 proves that for any infin-category 119860 the domain-projection functor 1199010 ∶ 119860120794 ↠ 119860 definesa cartesian fibration Thus functor does not define a discrete cartesian fibration in theinfin-cosmos119974but recall that 1199010-cartesian lifts can be constructed to project to identity arrows along 1199011 ∶ 119860120794 ↠ 119860This suggests that we might productively consider the domain-projection functor as a map over119860 inwhich case we have the following result

5410 Proposition The functor

119860120794 119860 times 119860

119860

(11990111199010)

1199011 120587(5411)

defines a discrete cartesian fibration in the sliceinfin-cosmos119974119860

Proof Note that 2-cell conservativity implies that (5411) is a discrete object in (119974119860)120587 ∶ 119860times119860↠119860 cong119974119860times119860 so it remains only to prove that this functor defines a cartesian fibration We prove this usingTheorem 5111(i)hArr(ii) The first step is to compute the right representable comma object for thefunctor (5411) by interpreting the formula (332) in the sliceinfin-cosmos119974119860 using Proposition 1217The 120794-cotensor of the object 120587∶ 119860 times 119860 ↠ 119860 is 120587∶ 119860 times 119860120794 ↠ 119860 so this right representable commais computed by the left-hand pullback in119974119860 below

Hom119860(119860 1199010) 119860 times 119860120794 119860120794

119860120794 119860 times 119860 119860

119860

1199012 id119860 times1199011

120587

1199011

(11990111199010)

120587

Pasting with the right-hand pullback in119974 we recognize that theinfin-category so-constructed coincideswith the right representable comma object for the functor 1199010 ∶ 119860120794 ↠ 119860 considered as a map in119974 Under the equivalence Hom119860(119860 1199010) ≃ 119860120795 established in the proof of Proposition 5123 theisofibration 1199012 ∶ Hom119860(119860 1199010) ↠ 119860 is evaluation at the final element 2 isin 120795 in the composable pair ofarrows Similarly the canonical functor 119894 ∶ 119860120794 rarr Hom119860(119860 1199010) induced by id1199010 in119974 coincides withthe canonical functor 119894 ∶ 119860120794 rarr Hom119860(119860 1199010) over 119860 induced by id(11990111199010) in119974119860

Now applying Proposition 5123 and Theorem 5111(i)rArr(ii) in 119974 this functor 119894 admits a rightadjoint 119903 over the domain-projection functor

119860120794 perp Hom119860(119860 1199010)

119860

119894

1199010 1199010119903

By the proofs of Theorem 5111(iii)rArr(ii) and Proposition 5123 this adjunction can be constructed bycotensoring 119860(minus) the composite adjunction of categories

120794 120794 times 120794 120795 119860120794 perp 119860120795

120793 + 120793 119860 times 119860

120794times0⊤

1205751

120794times⊤ℓ

119896

1205900119894

(11990111199010) (11990121199010)119903

(20)(10)

where ℓ ⊣ 119896 is described in the proof of Proposition 5123 The composite right adjoint is the functor1205900 ∶ 120795 ↠ 120794 that sends 0 and 1 to 0 and 2 to 1 while the composite left adjoint is the functor 1205751 ∶ 120794 ↣120795 that sends 0 to 0 and 1 to 2 In particular this adjunction lies in the strict slice 2-category underthe inclusion of the ldquoendpointsrdquo of 120794 and 120795

It follows that upon cotensoring into 119860 we obtain a fibered adjunction over 119860 times 119860 which byTheorem 5111(ii)rArr(i) implies that (5411) is a cartesian fibration in119974119860 completing the proof

Combining Propositions 5123 and Proposition 5410 we can now generalize both results to ar-bitrary commainfin-categories

5412 Corollary For any functors 119862 119860 119861119892 119891

betweeninfin-categories in aninfin-cosmos119974(i) The domain-projection functor 1199010 ∶ Hom119860(119891 119892) ↠ 119861 is a cartesian fibration Moreover a natural

transformation 120594 with codomain Hom119860(119891 119892) is 1199010-cartesian if and only if 1199011120594 is invertible(ii) The codomain-projection functor 1199011 ∶ Hom119860(119891 119892) ↠ 119862 is a cocartesian fibration Moreover a natu-

ral transformation 120594 with codomain Hom119860(119891 119892) is 1199011-cartesian if and only if 1199010120594 is invertible(iii) The functor

Hom119860(119891 119892) 119862 times 119861

119862

(11990111199010)

1199011 120587

defines a discrete cartesian fibration in119974119862(iv) The functor

Hom119860(119891 119892) 119862 times 119861

119861

(11990111199010)

1199010 120587

defines a discrete cocartesian fibration in119974119861

Proof We prove (i) and (iii) and leave the dualizations to the reader For (iii) we first use thecosmological functor 119892lowast ∶ 119974119860 rarr119974119862 which preserves discrete cartesian fibrations to establish that

Hom119860(119860 119892) 119862 times 119860

119862

(11990111199010)

1199011 120587

defines a discrete cartesian fibration in 119974119862 this argument works because 1199011 ∶ Hom119860(119860 119892) ↠ 119862 isthe pullback of 1199011 ∶ 119860120794 ↠ 119860 along 119892 Now

Hom119860(119891 119892) Hom119860(119860 119892)

119862 times 119861 119862 times 119860

119862

(11990111199010)1199011 1199011 (11990111199010)

120587

119862times119891

120587

is a pullback square in119974119862 so Lemma 545 now implies that the pullback is also a discrete cartesianfibration

Using (iii) we can now prove (i) This follows directly from a general claim that if

119864 119862 times 119861

119862119902

(119902119901)

120587

defines a cartesian fibration in 119974119862 then 119901∶ 119864 ↠ 119861 defines a cartesian fibration in 119974 By Theorem5111(i)hArr(ii) this functor defines a cartesian fibration in119974119862 if and only if the functor 119894

119864 perp Hom119861(119861 119901)

119862 times 119861

119894

(119902119901) (11990211990111199010)119903

admits a right adjoint 119903 over 119862 times 119861 Composing with 120587∶ 119862 times 119861 ↠ 119861 this fibered adjunction definesan adjunction over 119861 and Theorem 5111(i)hArr(ii) applied this time in 119974 allows us to conclude that119901∶ 119864 ↠ 119861 is a cartesian fibration

Note that the domain projection 1199010 ∶ Hom119860(119891 119860) ↠ 119861 is the pullback of 1199010 ∶ 119860120794 ↠ 119860 along119891∶ 119861 rarr 119860 so Proposition 5120 proves directly from Proposition 5123 that this functor is a cartesianfibration but 1199010 ∶ Hom119860(119860 119892) ↠ 119860 is not similarly a pullback of 1199010 ∶ 119860120794 ↠ 119860 This is why a morecircuitous argument to the general result is needed

Exercises

55 The external Yoneda lemma

Let 119887 ∶ 1 rarr 119861 be an element of aninfin-category 119861 and consider its right representation Hom119861(119861 119887)as a comma infin-category In this case there is no additional data given by the codomain-projectionfunctor but Example 544 observes that the domain-projection functor 1199010 ∶ Hom119861(119861 119887) ↠ 119861 hasa special property it defines a discrete cartesian fibration The fibers of this map over an element119886 ∶ 1 rarr 119861 are the internal mapping spaces Hom119861(119886 119887) of Definition 339 In this way the rightrepresentation of the element 119887 encodes the contravariant functor represented by 119887 which is why allalong wersquove been referring to the commainfin-categories Hom119861(119861 119887) as ldquorepresentablerdquo

Our aim in this section is to state and prove the Yoneda lemma in this setting where contravari-ant representable functors are encoded as discrete cartesian fibrations A dual statement applies tocovariant representable functors encoded as discrete cocartesian fibration 1199011 ∶ Hom119861(119887 119861) ↠ 119861 butfor ease of exposition we leave the dualization to the reader Informally the Yoneda lemma asserts thatldquoevaluation at the identity defines an equivalencerdquo so the first step towards the statement of the Yonedalemma is to introduce this identity element which in fact is something wersquove already encountered

The identity arrow id119887 induces an element id119887 ∶ 1 rarr Hom119861(119861 119887) which Corollary 349 provesis terminal in the infin-category Hom119861(119861 119887) of arrows in 119861 with codomain 119887 The identity elementinclusion defines a functor over 119861

1 Hom119861(119861 119887)

119861119887

id119887

1199010(551)

Technically this functor does not live in the slicedinfin-cosmos119974119861 because the domain object 119887 ∶ 1 rarr 119861is not an isofibration but nevertheless for any isofibration 119901∶ 119864 ↠ 119861 restriction along id119887 inducesa functor between sliced quasi-categorical functor spaces

Fun119861(Hom119861(119861 119887)1199010minusminusrarrrarr 119861119864

119901minusrarrrarr 119861)

evid119887minusminusminusminusminusrarr Fun119861(1119887minusrarr 119861 119864

Here the codomain is the quasi-category defined by the pullback

Fun119861(119887 119901) Fun(1 119864)

120793 Fun(1 119861)

119901lowast

119887

which is isomorphic to Fun(1 119864119887) the underlying quasi-category of the fiber 119864119887 of 119901∶ 119864 ↠ 119861 over 119887If a discrete cartesian fibration over 119861 is thought of as a 119861-indexed discrete infin-category valued

contravariant functor then maps of discrete cartesian fibrations over 119861 are ldquonatural transformationsrdquothe ldquonaturality in 119861rdquo arises because we only allow functors over 119861 This leads to our first statement ofthe fibrational Yoneda lemma

552 Theorem (external Yoneda lemma discrete case) If 119901∶ 119864 ↠ 119861 is a discrete cartesian fibrationthen

119901minusrarrrarr 119861) cong Fun(1 119864119887)

is an equivalence of Kan complexes

Theorem 552 is subsumed by a generalization that allows 119901∶ 119864 ↠ 119861 to be any cartesian fibra-tion not necessarily discrete In this case 119901 encodes an ldquoinfin-category-valued contravariant 119861-indexedfunctorrdquo as does 1199010 ∶ Hom119861(119861 119887) ↠ 119861 The correct notion of ldquonatural transformationrdquo between twosuch functors is now given by a cartesian functor over 119861 see Exercise 51iii To that end for a pair ofcartesian fibration 119902 ∶ 119865 ↠ 119861 and 119901∶ 119864 ↠ 119861 we write

Funcart119861 (119865119902minusrarrrarr 119861119864

119901minusrarrrarr 119861) sub Fun119861(119865

119902minusrarrrarr 119861119864

for the sub quasi-category containing all those simplices whose vertices define cartesian functors from119902 to 119901⁶

553 Theorem (external Yoneda lemma) If 119901∶ 119864 ↠ 119861 is a cartesian fibration then

Funcart119861 (Hom119861(119861 119887)1199010minusminusrarrrarr 119861119864

is an equivalence of quasi-categories

The proofs of these theorems overlap significantly and we develop them in parallel The basicidea is to use the universal property of id119887 as a terminal element of Hom119861(119861 119887) to define a rightadjoint to evid119887 and prove that when 119901∶ 119864 ↠ 119861 is discrete or when the domain is restricted to thesub-quasi-category of cartesian functors this adjunction defines an adjoint equivalence Note that thefunctor evid119887 is the image of the functor id119887 under the 2-functor Fun119861(minus 119901) ∶ 120101(119974119861)op rarr 119980119966119886119905 Ifthe adjunction ⊣ id119887 lived in the sliceinfin-cosmos119974119861 this would directly construct a right adjointto evid119887 The main technical difficulty in following the outline just given is that the adjunction thatwitnesses the terminality of id119887 does not live in the slice of the homotopy 2-category 120101119974119861 but ratherin a lax slice of the homotopy 2-category that we now introduce

554 Definition Consider a 2-category 120101119974 and an object 119861 isin 120101119974 The lax slice 2-category 120101119974⫽119861is the strict 2-category whosebull objects are maps 119891∶ 119883 rarr 119861 in 120101119974 with codomain 119861bull 1-cells are diagrams

119883 119884

119861119891

119896

120572rArr 119892

in 120101119974 andbull 2-cells from the 1-cell displayed above to the 1-cell below-right are 2-cells 120579∶ 119896 rArr 119896prime so that

119883 119884 119883 119884

119861 119861119891

119896uArr120579119896prime

120572rArr 119892 = 119891

119896prime

120572primerArr 119892

556 Lemma The identity functor (551) is right adjoint to the right comma cone

Hom119861(119861 119887) 1

1198611199010

120601rArr 119887

in 120101119974⫽119861

⁶For any quasi-category 119876 and any subset 119878 of its vertices there is a ldquofullrdquo sub-quasi-category 119876119878 sub 119876 containingexactly those vertices and all the simplices of 119876 that they span

Proof Since 1 is the terminalinfin-category we take the counit of the postulated adjunction to bethe identity By Definition 554 to define the unit we must provide a 2-cell

Hom119861(119861 119887) Hom119861(119861 119887) Hom119861(119861 119887) 1 Hom119861(119861 119887)

119861 1198611199010

id119887uArr120578

1199010 = 1199010

id119887

119887120601rArr

1199010

so that 1199010120578 = 120601 This is the defining property of the unit in Lemma 348 The forgetful 2-functor120101119974⫽119861 rarr 120101119974 is faithful on 1- and 2-cells so the verification of the triangle equalities in Lemma 348proves that they also hold in 120101119974⫽119861

Using somewhat non-standard 2-categorical techniques we will transfer the adjunction of Lemma556 to an adjunction between the quasi-categories Fun119861(119887 119901) and Fun119861(1199010 119901) see Proposition 5513Because our initial adjunction lives in the lax rather than the strict slice the construction will besomewhat delicate passing through a pair of auxiliary 2-categories that we now introduce

557 Definition Let 120101119974 be the homotopy 2-category of aninfin-cosmos and write 120101119974⟓ for the strict2-category whosebull objects are cospans

119860 119861 119864119891 119901

in which 119901 is a cartesian fibrationbull 1-cells are diagrams of the form

119860prime 119861prime 119864prime

119860 119861 119864

119886

119891prime

uArr120601 119887

119901prime

119890

119891 119901

bull and whose 2-cells consist of triples 120572∶ 119886 rArr 120573∶ 119887 rArr and 120598 ∶ 119890 rArr between the verticals ofparallel 1-cell diagrams so that 119901120598 = 120573119901prime and sdot 119891120572 = 120573119891prime sdot 120601

559 Definition Let 120101119974 be the homotopy 2-category of aninfin-cosmos and write 120101119974⊡ for the strict2-category whosebull objects are pullback squares

119865 119864

119860 119861

119902

119892

119901

119891

whose verticals are cartesian fibrations

bull 1-cells are cubes

119865prime 119864prime

119865 119864

119860 119861

119892prime

119902primeℓ

uArr120594 119890

119901prime

119892

119901

119886

119891prime

uArr120601 119887

119891

119902(5510)

whose vertical faces commute and in which 120594∶ 119892ℓ rArr 119890119892prime is a 119901-cartesian lift of 120601119902prime andbull whose 2-cells are given by quadruples 120572∶ 119886 rArr 120573∶ 119887 rArr 120598 ∶ 119890 rArr and 120582∶ ℓ rArr ℓ in which 120598

and 120582 are respectively lifts of 120573119901prime and 120572119902prime and so that 120601 sdot 119891120572 = 120573119891prime sdot 120601 and sdot 119892120582 = 120598119892prime sdot 120594

These definitions are arranged so that there is an evident forgetful 2-functor 120101119974⊡ rarr 120101119974⟓

5511 Lemma The forgetful 2-functor 120101119974⊡ rarr 120101119974⟓ is a smothering 2-functor

Proof Proposition 5120 tells us that 120101119974⊡ rarr 120101119974⟓ is surjective on objects To see that it is fullon 1-cells first form the pullbacks of the cospans in (558) then define 120594 to be any 119901-cartesian lift of120601119902prime with codomain 119890119892prime By construction the domain of 120594 lies strictly over 119891119886119902prime and so this functorfactors uniquely through the pullback leg 119892 defining the map ℓ of (5510)

To prove that 120101119974⊡ rarr 120101119974⟓ is full on 2-cells consider a parallel pair of 1-cells in 120101119974⊡ For oneof these we use the notation of (5510) and for the other we denote the diagonal functors by and ℓ and denote the 2-cells by and the requirement that these 1-cells be parallel implies that thepullback faces are necessarily the same Now consider a triple 120572∶ 119886 rArr 120573∶ 119887 rArr and 120598 ∶ 119890 rArr satisfying the conditions of Definition 557 Our task is to define a fourth 2-cell 120582∶ ℓ rArr ℓ so that119902120582 = 120572119902prime and sdot 119892120582 = 120598119892prime sdot 120594

To achieve this we first define a 2-cell 120574∶ 119892ℓ rArr 119892ℓ using the induction property of the 119901-cartesiancell ∶ 119892ℓ rArr 119892prime applied to the composite 2-cell 120598119892prime sdot 120594 ∶ 119892ℓ rArr 119892prime and the factorization 119901120598119892prime sdot 119901120594 =119902prime sdot 119891120572119902prime By construction 119901120574 = 119891120572119902prime so the pair 120572119902prime and 120574 induces a 2-cell 120582∶ ℓ rArr ℓ so that119902120582 = 120572119902prime and 119892120582 = 120574 The quadruple (120572 120573 120598 120582) now defines the required 2-cell in 120101119974⊡

Finally for 2-cell conservativity suppose 120572 120573 and 120598 as above are isomorphisms By the conser-vativity property for pullbacks to show that 120582 is an isomorphism it suffices to prove that 119902120582 = 120572119902primeis which we know already and that 119892120582 = 120574 is invertible But 120574 was constructed as a factorization120598119892prime sdot 120594 = sdot 120574 with 119901120574 = 119891120572119902prime Since 120598 is an isomorphism 120598119892prime sdot 120594 is 119901-cartesian so Lemma 515proves that 120574 is an isomorphism

5512 Remark While we cannot directly define a pullback 2-functor 120101119974⟓ rarr 120101119974 in the homotopy2-category because the 2-categorical universal property of pullbacks in 120101119974 is weak and not strict thezig zag of 2-functors 120101119974⟓ larr 120101119974⊡ rarr 120101119974 in which the backwards map is a smothering 2-functor andthe forwards map evaluates at the pullback vertex defines a reasonable replacement

Using Lemma 5511 we can now construct the desired adjunction

5513 Proposition For any element 119887 ∶ 1 rarr 119861 and any cartesian fibration 119901∶ 119864 ↠ 119861 the evaluation atthe identity functor admits a right adjoint

Fun119861(1199010 119901) Fun119861(119887 119901)

evid119887

119877

defined by the domain-component of the 119901lowast-cartesian lift of the right comma cone over 119887

Fun119861(119887 119901) Fun(1 119864)

Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864)

120793 Fun(1 119861)

120793 Fun(Hom119861(119861 119887) 119861)

uArr120594119877

119901lowast

lowast

119901lowastuArr120601

119887lowast

1199010

(5514)

The idea will be to transfer the adjunction of Lemma 556 through a sequence of 2-functors

120101119980119966119886119905⊡ 120101119980119966119886119905

120101119974op⫽119861 120101119980119966119886119905⟓

using Lemma 358 to lift along the middle smothering 2-functor

Proof Fixing a cartesian fibration 119901∶ 119864 ↠ 119861 in119974 we define a 2-functor⁷119974op⫽119861 rarr 120101119980119966119886119905⟓ that

carries a 1-cell (555) to120793 Fun(119884 119861) Fun(119884 119864)

120793 Fun(119883 119861) Fun(119883 119864)

119892

uArr120572 119896lowast

119901lowast

119890

119891 119901lowast

and a 2-cell 120579∶ 119896 rArr 119896prime to the 2-cell that acts via pre-whiskering with 120579 in its two non-identitycomponents By Corollary 5116 the functors 119901lowast are cartesian fibration of quasi-categories

We now apply the 2-functor 119974op⫽119861 rarr 120101119980119966119886119905⟓ to the adjunction of Lemma 556 to obtain an

adjunction in 120101119980119966119886119905⟓ and then use the smothering 2-functor of Lemma 5511 and Lemma 358 to liftthis to an adjunction in 120101119980119966119886119905⊡ As elaborated on in Exercise 35ii the lifted adjunction in 120101119980119966119886119905⊡can be constructed using any lifts of the objects 1-cells and either the unit or counit of the adjunctionin 120101119980119966119886119905⟓

⁷To explain the variance recall that the 2-functor Fun(minus 119861) ∶ 120101119974op rarr 120101119980119966119886119905 is contravariant on 1-cells but covarianton 2-cells Such 2-functors like all 2-functors preserve adjunctions though in this case the left and right adjoints areinterchanged while the units and counits retain the same roles

In particular we may take the left adjoint of the lifted adjunction in 120101119980119966119886119905⊡ to be any lift of theimage in 120101119980119966119886119905⟓ of the right adjoint of the adjunction ⊣ id119887 in 120101119974⫽119861 and so our left adjoint is

Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864)

Fun119861(119887 119901) Fun(1 119864)

120793 Fun(Hom119861(119861 119887) 119861)

120793 Fun(1 119861)

=evid119887

119901lowast

id119887lowast

119901lowast=

1199010id119887

lowast

119887

We may also take the right adjoint to be any lift of the image of the right adjoint of the adjunctionThis proves that the right adjoint is defined by (5514) Since the counit of ⊣ id119887 is an identitythe counit of the lifted adjunction may also be taken to be an identity

Finally we compose with the forgetful 2-functor 120101119980119966119886119905⊡ rarr 120101119980119966119886119905 that evaluates at the pullbackvertex to project our adjunction in 120101119980119966119886119905⊡ to the desired adjunction in 120101119980119966119886119905

The proof of the discrete case of the Yoneda lemma is now one line

Proof of Theorem 552 If 119901∶ 119864 ↠ 119861 is discrete then Fun119861(1199010 119901) and Fun119861(119887 119901) are Kan com-plexes so the adjunction defined in Proposition 5513 is an adjoint equivalence

Specializing to the case of two right representable discrete cartesian fibrations we conclude thatthe Kan complex of natural transformations is equivalent to the underlying quasi-category of theirinternal mapping space

5515 Corollary (external Yoneda embedding) For any elements 119909 119910 ∶ 1 119860 in aninfin-category 119860evaluation at the identity of 119909 induces an equivalence of Kan complexes

Fun119860(Hom119860(119860 119909)Hom119860(119860 119910)) Fun(1Hom119860(119909 119910))sim

evid119909

It remains to prove the general case of Theorem 553 The next step is to observe that the rightadjoint in the adjunction of Proposition 5513 lands in the sub quasi-category of cartesian functorsfrom 1199010 to 119901 Lemma 5516 proves this after which it is short work to complete the proof of Theorem553 by arguing that this restricted adjunction defines an adjoint equivalence

5516 Lemma For each vertex in Fun119861(119887 119901) below-left

Fun119861(119887 119901) Fun119861(1199010 119901)119877

1 119864 Hom119861(119861 119887) 119864

119861 119861119887

119890

119901 ↦ 1199010

119877119890

119901

the map 119877119890 in Fun119861(1199010 119901) above-right defines a cartesian functor in119974119861

Proof From the definition of the right adjoint in (5514) and Lemma 535 we see that 119877119890 is thedomain component of a 119901-cartesian lift 120594 of the composite natural transformation below-left

Hom119861(119861 119887) 1 119864 Hom119861(119861 119887) 119864

119861 119861

uArr1206011199010

119887

119890

119901 =

119890

119877119890

uArr120594

1199010 119901

Since 1199010 ∶ Hom119861(119861 119887) ↠ 119861 is discrete every natural transformation 120595 with codomain Hom119861(119861 119887) is1199010-cartesian so to prove that 119877119890 defines a cartesian functor we must show that 119877119890120595 is 119901-cartesianTo that end consider the horizontal composite

119883 Hom119861(119861 119887) 119864

119910

119909

uArr120595

119890

119877119890

uArr120594

By naturality of whiskering we have 120594119910 sdot 119877119890120595 = 119890120595 sdot 120594119909 = 120594119909 since 1 is the terminal infin-categoryand hence 119890120595 is an identity Now Lemma 5124(ii) implies that 119877119890120595 is 119901-cartesian

Proof of Theorem 553 By Lemma 5516 the adjunction of Proposition 5513 restricts to de-fine an adjunction

Funcart119861 (1199010 119901) Fun119861(119887 119901)

evid119887

119877Since the counit of the original adjunction ⊣ id119887 is an isomorphism and smothering 2-functorsare conservative on 2-cells the counit of the adjunction of Proposition 5513 and hence also of therestricted adjunction is an isomorphism As in the proof of Theorem 552 we will prove that evid119887is an equivalence by demonstrating that the unit of the restricted adjunction is also invertible ByLemma it suffices to verify this elementwise proving that the component of the unit indexed by acartesian functor

Hom119861(119861 119887) 119864

1198611199010

119891

119901

is an isomorphismUnpacking the proof of Proposition 5513 the unit of evid119887 ⊣ 119877 is defined to be a factorization

Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864) Fun(1 119864)

Fun119861(1199010 119901) Fun119861(119887 119901) Fun(1 119864) Fun(Hom119861(119861 119887) 119864)

Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864)

id119887lowast

lowastevid119887

119877 uArr120594

lowast

uArrFun(120578119864)

of the pre-whiskering 2-cell Fun(120578 119864) through the 119901lowast-cartesian lift 120594 The component of the pre-whiskering 2-cell Fun(120578 119864) at the cartesian functor 119891 is 119891120578 Since 1199010 ∶ Hom119861(119861 119887) ↠ 119861 is a discretecartesian fibration any 2-cell such as 120578 which has codomain Hom119861(119861 119887) is 1199010-cartesian and since 119891is a cartesian functor we then see that 119891120578 is 119901-cartesian

By Lemma 538 the components of the 119901lowast-cartesian cell 120594 define 119901-cartesian natural transforma-tions in119974 As is a natural transformation with codomain Fun119861(1199010 119901) its components project along119901 to the identity In this way we see that 119891 is a factorization of the 119901-cartesian transformation 119891120578through a 119901-cartesian lift of 120601 over an identity and Lemma 515 proves that 119891 is an isomorphism asdesired

Exercises

55i Exercise Given an element 119891∶ 1 rarr Hom119860(119909 119910) in the internal mapping space between a pairof elements in aninfin-category 119860 use the explicit description of the inverse equivalence to the map ofCorollary 5515 to construct a map

Hom119860(119860 119909) Hom119860(119860 119910)

1198601199010

119891lowast

1199010

which represents the ldquonatural transformationrdquo defined by post-composing with 119891⁸

⁸Hint this construction is a special case of the construction given in the first half of the proof of Lemma 5516

Part II

Homotopy coherent category theory

CHAPTER 6

Simplicial computads and homotopy coherence

Consider a diagram 119889∶ 1 rarr 119860119869 in aninfin-category119860 indexed by a 1-category 119869 Via the isomorphismFun(1 119860119869) cong Fun(1 119860)119869 an element in the infin-category of diagrams 119860119869 equally defines a functor119889∶ 119869 rarr Fun(1 119860) valued in the underlying quasi-category of 119860 Applying h ∶ 119980119966119886119905 rarr 119966119886119905 thisdescends to a diagram h119889∶ 119869 rarr h119860 of shape 119869 in the homotopy category of 119860 such diagrams arecalled homotopy commutative But the original diagram 119889 has a much richer property defining what iscalled a homotopy coherent diagram of shape 119869 in the quasi-category Fun(1 119860)

To make the data involved in defining a homotopy coherent diagram most explicit we first in-troduce a general notion of homotopy coherent diagram as a simplicial functor valued in a simplicialcategory whose hom-spaces are Kan complexes What makes such diagrams ldquohomotopy coherentrdquo andnot just ldquosimplicially enrichedrdquo is that their domains are required to be ldquofreerdquo simplicial categories ofa particular form that we refer to by the name simplicial computads Because every quasi-category canbe presented up to equivalence by a Kan-complex enriched category it will follow that ldquoall diagramsvalued in quasi-categories are homotopy coherentrdquo

To build intuition for the general notion of a homotopy coherent diagram it is helpful to considera special case of diagrams indexed by the category

120654 ≔ 0 1 2 3 ⋯

whose objects are finite ordinals and with a morphism 119895 rarr 119896 if and only if 119895 le 119896 and valued in theKan-complex enriched category of spaces 119982119901119886119888119890 A 120654-shaped graph in 119982119901119886119888119890 is comprised of spaces119883119896 for each 119896 isin 120654 together with continuous maps 119891119895119896 ∶ 119883119895 rarr 119883119896 whenever 119895 lt 119896sup1 This data definesa homotopy commutative diagram just when 119891119894119896 ≃ 119891119895119896 ∘ 119891119894119895 whenever 119894 lt 119895 lt 119896sup2

To extend this data to a homotopy coherent diagram120654 rarr 119982119901119886119888119890 requiresbull Chosen homotopies ℎ119894119895119896 ∶ 119891119894119896 ≃ 119891119895119896 ∘ 119891119894119895 whenever 119894 lt 119895 lt 119896 This amounts to specifying a path

in Map(119883119894 119883119896) from the vertex 119891119894119896 to the vertex 119891119895119896 ∘ 119891119894119895 which is obtained as the composite ofthe two vertices 119891119894119895 isin Map(119883119894 119883119895) and 119891119895119896 isin Map(119883119895 119883119896)

sup1To simplify somewhat we adopt the convention that 119891119895119895 is the identity making this data into a reflexive directedgraph with implicitly designated identities

sup2This data defines a strictly commutative diagram (aka a functor 120654 rarr 119982119901119886119888119890) just when 119891119894119896 = 119891119895119896 ∘ 119891119894119895 whenever119894 lt 119895 lt 119896 Strictly commutative diagrams are certainly homotopy commutative Homotopy coherent category theoryarose from the search for conditions under which something like the converse implication held a homotopy commutativediagram is realized by (ie naturally isomorphic to up to homotopy) a strictly commutative diagram if and only if itextends to a homotopy coherent diagram [18 25]

bull For 119894 lt 119895 lt 119896 lt ℓ the chosen homotopies provide four paths in Map(119883119894 119883ℓ)

119891119894ℓ 119891119896ℓ ∘ 119891119894119896

119891119895ℓ ∘ 119891119894119895 119891119896ℓ ∘ 119891119895119896 ∘ 119891119894119895

ℎ119894119896ℓ

ℎ119894119895ℓ 119891119896ℓ∘ℎ119894119895119896

ℎ119895119896ℓ∘119891119894119895

We then specify a higher homotopy mdash a 2-homotopy mdash filling in this squarebull For 119894 lt 119895 lt 119896 lt ℓ lt 119898 the previous choices provide 12 paths and six 2-homotopies in

Map(119883119894 119883119898) that assemble into the boundary of a cube We then specify a 3-homotopy a ho-motopy between homotopies between homotopies filling in this cube

bull EtcEven in this simple case of the category 120654 this data is a bit unwieldy Our task is to define a

category to index this homotopy coherent data arising from120654 the objects119883119894 the functions119883119894 rarr 119883119895the 1-homotopies ℎ119894119895119896 the 2-homotopies and so on This data will assemble into a simplicial categorywhose objects are the same as the objects of 120654 but which will have 119899-morphisms in each dimension119899 ge 0 to index the 119899-homotopies Importantly this simplicial category will be ldquofreely generatedrdquo froma much smaller collection of data We begin by studying such ldquofreely generatedrdquo simplicial categoriesunder the name simplicial computads

61 Simplicial computads

611 Definition (free categories and atomic arrows) An arrow 119891 in a 1-category is atomic if it is notan identity and if it admits no non-trivial factorizations ie if whenever 119891 = 119892 ∘ ℎ either 119892 or ℎ is anidentity

A 1-category is free if every arrow may be expressed uniquely as a composite of atomic arrowswith the convention that empty composites correspond to identity arrowssup3

612 Digression (on free categories and reflexive directed graphs) The category of presheaves onthe truncation

120491le1 ≔ bull bull119904

119905119894 119894119904 = 119894119905 = id

defines the category 119970119901ℎ of (reflexive directed) graphs for us a graph consists of a set of vertices aset of edges each with a specified source and target vertex and a distinguished ldquoidentityrdquo endo-edgefor each vertex Any category has an underlying reflexive directed graph and this forgetful functoradmits a left adjoint defining the free category whose atomic and identity arrows are the arrows inthe given graph

119966119886119905 119970119901ℎ119880

perp119865

Recall fromDigression 122 that a simplicial category119964may be presented by a family of 1-categories119964119899 of119899-arrows for119899 ge 0 eachwith a common set of objects that assemble into a diagram119964bull ∶ 120491op rarr

sup3Alternatively a 1-category is free if every non-identity arrow may be expressed uniquely as a non-empty compositeof atomic arrows and if identity arrows admit no non-trivial factorizations

119966119886119905 comprised of identity-on-objects functors The notion of ldquofreerdquo simplicial category was first in-troduced by Dwyer and Kan [19 45]

613 Definition (simplicial computad) A simplicial category119964 is a simplicial computad if and onlyifbull Each category119964119899 of 119899-arrows is freely generated by the graph of atomic 119899-arrowsbull If 119891 is an atomic 119899-arrow in119964119899 and 120590∶ [119898] ↠ [119899] is an epimorphism in 120491 then the degenerated119898-arrow 119891 sdot 120590 is atomic in119964119898

By the Eilenberg-Zilber lemma⁴ a simplicial category119964 is a simplicial computad if and only if allof its non-identity arrows can be expressed uniquely as a composite

119891 = (1198911 sdot 1205721) ∘ (1198912 sdot 1205722) ∘ ⋯ ∘ (119891ℓ sdot 120572ℓ)in which each 119891119894 is non-degenerate and atomic and each 120572119894 is a degeneracy operator in 120491

614 Example A 1-category 119860 may be regarded as a simplicial category 119860bull in which 119860119899 ≔ 119860 forall 119899 In the construction the hom-spaces of 119860 coincide with the hom-sets of 119860 Such ldquoconstantrdquosimplicial categories define simplicial computads if and only if the 1-category 119860 is free

615 Example For any simplicial set 119880 let 120794[119880] denote the simplicial category with two objectsldquominusrdquo and ldquo+rdquo and with hom-spaces defined by

120794[119880](+ minus) ≔ empty 120794[119880](+ +) ≔ 120793 ≕ 120794[119880](minus minus) 120794[119880](minus +) ≔ 119880This simplicial category is a simplicial computad because there are no composable sequences of arrowsin 120794[119880] containing more than one non-identity arrow Every arrow from minus to + is atomic

616 Definition A simplicial functor 119866∶ 119964 rarr ℬ between simplicial computads defines a sim-plicial computad morphism if it maps every atomic arrow 119891 in 119964 to an arrow 119866119891 which is eitheratomic or an identity in ℬ Write 119982119982119890119905-119966119901119905119889 sub 119982119982119890119905-119966119886119905 for the non-full subcategory of simplicialcomputads and their morphisms

The axioms of Definition 613 assert that the atomic and identity 119899-arrows of a simplicial com-putad assemble into a diagram in 119970119901ℎ120491epi

opand a simplicial computad morphism restricts to define

a natural transformation between the underlying 120491epiop-indexed graphs of atomic and identity ar-

rows in this way restricting to the atomic or identity arrows defines a functor atom ∶ 119982119982119890119905-119966119901119905119889 rarr119970119901ℎ120491epi

op The next lemma tells us that the category 119982119982119890119905-119966119901119905119889 is canonically isomorphic to the

intersection of 119970119901ℎ120491epiop

and 119982119982119890119905-119966119886119905 in 119966119886119905120491epiop

617 Lemma The functor that carries a simplicial computad to its underlying diagram of atomic and identityarrows and the inclusion of simplicial computads into simplicial categories define the legs of a pullback cone

119982119982119890119905-119966119901119905119889 119982119982119890119905-119966119886119905

119970119901ℎ120491epiop

119966119886119905120491epiop

119865120491epiop

⁴The Eilbenberg-Zilber lemma asserts that any degenerate simplex in a simplicial set may be uniquely expressed as adegenerated image of a non-degenerate simplex see [21 II31 pp 26-27]

Moreover since 119982119982119890119905-119966119886119905 and 119970119901ℎ120491epiop

have colimits the functors to 119966119886119905120491epiop

preserve them and 119865120491epiop

is an isofibration it follows that 119982119982119890119905-119966119901119905119889 has colimits created by either of the functors to 119982119982119890119905-119966119886119905 or to119970119901ℎ120491epi

Proof If119964 is a simplicial category presented as a simplicial object119964bull ∶ 120491op rarr 119966119886119905 then119964 isa simplicial computad if and only if there exists a dotted lift as below-left

120491epiop 119970119901ℎ 120491epi

op 119970119901ℎ

120491op 119966119886119905 120491op 119966119886119905

119865

dArrexist

119865119964bull

119964bull

ℬbull

dArr119866

in which case this lift is necessarily unique Correspondingly as simplicial functor 119866∶ 119964 rarr ℬ de-fines a computad morphism if and only if the restricted natural transformation above-right also liftsthrough the free category functor again necessarily uniquely These facts verify that the category119982119982119890119905-119966119901119905119889 is captured as the stated pullback

Now consider a diagram 119863∶ 119869 rarr 119982119982119890119905-119966119901119905119889 and form its colimit cone in 119982119982119890119905-119966119886119905 and in119970119901ℎ120491epi

op The functors to 119966119886119905120491epi

opcarry these to a pair of isomorphic colimit cones under the same

diagram and since 119865120491epiop

is an isofibration (any category isomorphic to a free category is itself a freecategory and this isomorphism necessarily restricts to underlying graphs of atomic arrows) there existsa colimit cone under 119863 in 119970119901ℎ120491epi

opwhose image under 119865120491epi

opis equal to the image of the colimit

cone under 119863 in 119982119982119890119905-119966119886119905 Now the universal property of the pullback allows us to lift this cone to119982119982119890119905-119966119901119905119889 and a similar argument using the 2-categorical universal property of the pullback diagramof categories demonstrates that the lifted cone is a colimit cone

618 Definition (simplicial subcomputads) A simplicial computad morphism 119964 ℬ that is in-jective on objects and faithful mdash ie carries every atomic arrow in 119964 to an atomic arrow in ℬ mdashdisplays119964 as a simplicial subcomputad of ℬ

The simplicial subcomputad 119878 generated by a set of arrows 119878 in a simplicial computad 119964 is thesmallest simplicial subcomputad of 119964 containing those arrows The objects of 119878 are those objectsthat appear as domains or codomains of arrows in 119878 and its set of morphisms is the smallest subset ofmorphisms containing 119878 that have the following closure properties

bull if 119891 isin 119878 and 120572∶ [119899] rarr [119898] isin 120491 then 119891 sdot 120572 isin 119878bull If 119891 119892 isin 119878 are composable then 119891 ∘ 119892 isin 119878

Lemma 617 proves that simplicial computads and computad morphisms are closed under colimitsformed in the category of simplicial categories For certain special colimit shapes the property of beinga simplicial subcomputad is also preserved

619 Lemma A simplicial subcategory 119964 ℬ of a simplicial computad ℬ displays 119964 as a simplicialsubcomputad of ℬ just when 119964 is closed under factorizations if 119891 and 119892 are composable arrows of ℬ and119891 ∘ 119892 isin 119964 then 119891 and 119892 are in119964

Proof Exercise 61i

6110 Lemma Simplicial subcomputads are closed under coproduct pushout and colimit of countable se-quences

Proof Simplicial subcomputad inclusions are precisely those morphisms in 119982119982119890119905-119966119901119905119889 whoseimages in 119970119901ℎ120491epi

opare pointwise monomorphisms Lemma 617 proves that colimits in 119982119982119890119905-119966119901119905119889

are crated in 119970119901ℎ120491epiop

As colimits in this presheaf category are formed pointwise and as monomor-phisms are stable under coproduct pushout and colimit of countable sequences the result follows

6111 Definition (relative simplicial computad) The class of all relative simplicial computads is theclass of all simplicial functors that can be expressed as a countable composite of pushouts of coproductsofbull the unique simplicial functor empty 120793 andbull the simplicial subcomputad inclusion 120794[120597Δ[119899]] 120794[Δ[119899]] for 119899 ge 0

The next lemma reveals that relative simplicial computads differ from simplicial subcomputadinclusions only in the fact that the domains of relative simplicial computads need not be simplicialcomputadsmdashbutwhen they are the codomain is also a simplicial computad and themap is a simplicialsubcomputad inclusion

6112 Lemma If119964 is a simplicial computad then an inclusion of simplicial categories119964 ℬ is a simplicialcomputad inclusion if and only if it is a relative simplicial computad In particular a simplicial category ℬ isa simplicial computad if and only if empty ℬ is a relative simplicial computad

Proof First note that empty 120793 and 120794[120597Δ[119899]] 120794[Δ[119899]] are simplicial subcomputad inclusionsthe first case is trivial and for the latter all the non-identity arrows in 120794[119880] atomic By Lemma 6110to prove that any relative simplicial computad 119964 ℬ whose domain is a simplicial computad is asubcomputad inclusion it suffices to prove that for any simplicial computad 119964 and any simplicialfunctor 119891∶ 120794[120597Δ[119899]] rarr 119964 not necessarily a computad moprhism the pushout of 120794[120597Δ[119899]] 120794[Δ[119899]] along 119891 is a simplicial computad containing119964 as a simplicial subcomputad

The subcomputad inclusion 120794[120597Δ[119899]] 120794[Δ[119899]] is full on 119903-arrows for 119903 lt 119899 for 119903 gt 119899 thecodomain is constructed by adjoining one atomic arrow from minus to+ for each epimorphism [119903] ↠ [119899]It follows that the pushout119964119964prime is similarly full on 119903-arrows for 119903 lt 119899 and for 119903 ge 119899 the categoryof 119903-arrows of 119964prime is obtained from that of 119964 by adjoining one atomic 119903-arrow for each degeneracyoperator [119903] ↠ [119899] with boundary specified by the attaching map 119891 If we adjoin an arrow to a freecategory along its boundary we get a free category with that arrow as an extra generator so each ofthe categories of 119903-arrows of 119964prime are freely generated and it is clear from this description that ondegenerating one of these extra adjoined arrows we just map it to one of the generating arrows wersquoveadjoined at a higher dimension This proves that 119964 119964prime is a simplicial subcomputad inclusionas claimed It follows inductively that the codomain of a relative simplicial computad is a simplicialcomputad whenever its domain is in which case the inclusion is a subcomputad inclusion

For the converse we inductively present any simplicial subcomputad inclusion 119964 ℬ as asequential composite of pushouts of coproducts of the mapsempty 120793 and 120794[120597Δ[119899]] 120794[Δ[119899]] Notethat 120794[Δ[119899]] contains a single non-degenerate atomic arrow not present in 120794[120597Δ[119899]] namely theunique non-degenerate 119899-arrow representing the 119899-simplex

At stage ldquominus1rdquo use the inclusion empty 120793 to attach any object in ℬ119964 to 119964 At stage ldquo0rdquo usethe inclusion 120794[empty] 120794[Δ[0]] to attach each atomic 0-arrows of ℬ that is not in 119964 Iteratively at

stage ldquo119903rdquo use the inclusion 120794[120597Δ[119903]] 120794[Δ[119903]] to attach each atomic 119903-arrows of ℬ that is not in119964 There is a canonical map from the codomain of the relative simplicial computad defined in thisway to ℬ which by construction is bijective on objects and sends the unique atomic arrow attachedby each cell to an atomic arrow in ℬ in particular this comparison functor which by constructionlies in 119982119982119890119905-119966119901119905119889 is in fact a simplicial subcomputad inclusion The comparison is surjective onatomic 119899-arrows by construction and hence surjective on all arrows because the unique factorizationany arrow into non-degenerate atomics is present in the colimit at the stage corresponding to thedimension of that arrow Thus we have presented119964 ℬ as a relative simplicial computad inclusion

The morphisms listed in Definition 6111 are the generating cofibrations in the Bergner modelstructure on simplicial categories [4] Hence the relative simplicial computads are precisely the cellu-lar cofibrations those that are built as sequential composites of pushouts of coproducts of generatingcofibrations (without closing under retracts) For non-cofibrant domains the notion of Bergner cofi-bration is more general than the notion of relative simplicial computad However

6113 Lemma Every retract of a simplicial computad is a simplicial computad

Proof A simplicial category 119964 is a retract of a simplicial computad ℬ if there exist simplicialfunctors

119964 ℬ 119964119878 119877

so that 119877119878 = id The category 119964 is then recovered as the coequalizer (or the equalizer) of 119878119877 andthe identity so if we knew that this idempotent defined a computad morphism we could appeal toLemma 617 and be done but we do not know this so we must argue further

First we demonstrate that every retract of a free category is free To see this wersquoll first show thatthe inclusion 119964119899 ℬ119899 satisfies the ldquo2-of-3rdquo property of 119891 and 119892 are composable morphisms of ℬ119899so that two of three of 119891 119892 and 119891 ∘ 119892 lie in119964119899 then so does the third This is clear in the case where119891 119892 isin 119964119899 so suppose that 119891 119891 ∘119892 isin 119964119899 Then 119891 and 119891 ∘119892 are fixed points for the idempotent 119878119877 andso we have 119891 ∘ 119892 = 119878119877(119891 ∘ 119892) = 119878119877(119891) ∘ 119878119877(119892) = 119891 ∘ 119878119877(119892) In the free category ℬ119899 all morphismsare both monic and epic so 119892 = 119878119877(119892) lies in 119964119899 as well Now by induction it is easy to verify thatevery arrow in 119964119899 factors uniquely as a product of composites of atomic arrows in ℬ119899 with each ofthese composites defining an atomic arrow in119964119899

This verifies the first condition of Definition 613 It remains only to verify that degenerate imagesof atomic arrows in 119964119899 are atomic To that end consider an epimorphism 120572∶ [119898] ↠ [119899] and anatomic 119899-arrow 119891 in 119964119899 If 119891 sdot 120572 = 119892 ∘ ℎ is a non-trivial factorization in 119964119898 ℬ119898 then since ℬis a simplicial computad we must have 119878(119891) = 119892prime ∘ ℎprime with 119892prime sdot 120572 = 119878(119892) and ℎprime sdot 120572 = 119878(ℎ) Since119891 = 119877119878(119891) = 119877119892prime ∘119877ℎprime is atomic we must have 119877119892prime or 119877ℎprime an identity but then 119877119892prime sdot 120572 = 119877119878(119892) = 119892or 119877ℎprime sdot 120572 = 119877119878(ℎ) = ℎ is an identity and so the factorization 119891 sdot 120572 = 119892 ∘ ℎ is trivial after all

Consequently

6114 Corollary The simplicial computads are precisely the cofibrant simplicial categories in the Bergnermodel structure

Exercises

61ii Exercise Prove that the 119903-skeleton of a simplicial computad defined by discarding all arrows ofdimension⁵ greater than 119903 is the simplicial subcomputad generated by the atomic arrows of dimension119903

62 Free resolutions and homotopy coherent simplices

The original example of a simplicial computad also due to Dwyer and Kan [19] is given by thefree resolution of a 1-category 119862

621 Definition (free resolutions) Write 119865 ⊣ 119880 for the free category and underlying graph functorsof Digression 612 Note that the components of the counit and comultiplication of the comonad

(119865119880∶ 119966119886119905 rarr 119966119886119905 120598 ∶ 119865119880 rArr id 119865120578119880∶ 119865119880 rArr 119865119880119865119880)define identity-on-objects functors

For any 1-category 119862 we will define a simplicial category 119865119880bull119862 with the same objects and withthe category of 119899-arrows defined to be (119865119880)119899+1119862 A 0-arrow is a sequence of composable arrows in119862 A non-identity 119899-arrow is a sequence of composable arrows in 119862 with each arrow in the sequenceenclosed in exactly 119899 pairs of well-formed parentheses

The simplicial object 120491op rarr 119966119886119905 is formed by evaluating the comonad resolution at 119862 isin 119966119886119905

119865119880119862 (119865119880)2119862 (119865119880)3119862 (119865119880)4119862 ⋯ (622)

For 119895 ge 1 the face maps(119865119880)119896120598(119865119880)119895 ∶ (119865119880)119896+119895+1119862 rarr (119865119880)119896+119895119862

remove the parentheses that are contained in exactly 119896 others while 119865119880⋯119865119880120598 composes the mor-phisms inside the innermost parentheses For 119895 ge 1 the degeneracy maps

119865(119880119865)119896120578(119880119865)119895119880∶ (119865119880)119896+119895minus1119862 rarr (119865119880)119896+119895119862double up the parentheses that are contained in exactly 119896 others while 119865⋯119880119865120578119880 inserts parenthesesaround each individual morphism

623 Example (free resolution of a group) A classically important special case is given by the freeresolution of a 1-object groupoid whose automorphisms are the elements of a discrete group 119866 Inthis case each category of 119899-arrows is again a 1-object groupoid The category of 0-arrows is the groupof words in the non-identity elements of 119866 The category of 1-arrows is the group of words of wordsand so on

We now explain the sense in which free resolutions are ldquoresolutionsrdquo of the original 1-category Asdiscussed in Example 614 a 1-category119862 can be regarded as a constant simplicial category119862bull whosehom-spaces coincide with the hom-sets of119862 There is a canonical ldquoaugmentationrdquo map 120598 ∶ 119865119880bull119862 rarr 119862in 119982119982119890119905-119966119886119905 that is determined by its degree zero component 120598 ∶ 119865119880119862 rarr 119862 which is just given bycomposition in 119862

⁵An 119899-arrow 119891 in a simplicial category119964 has dimension 119903 if there exists a non-degenerate 119903-arrow 119892 and an epimor-phism 120572∶ [119899] ↠ [119903] so that 119891 = 119892 sdot 120572

624 Proposition The functor 120598 ∶ 119865119880bull119862 rarr 119862 is a local homotopy equivalence of simplicial sets That isfor any pair of objects 119909 119910 isin 119862 the map 120598 ∶ 119865119880bull119862(119909 119910) rarr 119862(119909 119910) is a simplicial homotopy equivalence119865119880bull119862(119909 119910) is homotopy equivalent to the discrete set 119862(119909 119910) of arrows in 119862 from 119909 to 119910

Proof The augmented simplicial object

119862 (119865119880)119862 (119865119880)2119862 (119865119880)3119862 (119865119880)4119862 ⋯

is split at the level of reflexive directed graphs (ie after applying119880) These splittings are not functorsbut that wonrsquot matter These directed graph morphisms displayed here are all identity on objectswhich means that for any 119909 119910 isin 119862 there is a split augmented simplicial set

119862(119909 119910) (119865119880)119862(119909 119910) (119865119880)2119862(119909 119910) (119865119880)3119862(119909 119910) (119865119880)4119862(119909 119910)⋯

and now some classical simplicial homotopy theory of Meyer [42] reviewed in Appendix C proves that120598 ∶ 119865119880bull119862(119909 119910) rarr 119862(119909 119910) is a simplicial homotopy equivalence

As the name suggests free resolutions are simplicial computads

625 Proposition (free resolutions are simplicial computads) The free resolution of any 1-categorydefines a simplicial computad in which the atomic 119899-arrows are those enclosed in precisely one pair of outermostparentheses

Proof In the free resolution of a 1-category 119862 the category of 119899-arrows is (119865119880)119899+1119862 The cate-gory 119865119880119862 is the free category on the underlying graph of 119862 Its arrows are sequences of composablenon-identity arrows of 119862 the atomic 0-arrows are the non-identity arrows of 119862 An 119899-arrow is asequence of composable arrows in 119862 with each arrow in the sequence enclosed in exactly 119899 pairs ofparentheses The atomic 119899-arrows are those enclosed in precisely one pair of parentheses on the out-side Since composition in a free category is by concatenation the unique factorization property isclear Since degeneracy arrows ldquodouble uprdquo on parentheses these preserve atomics as required

The atomic 119899-arrows in the free resolution of a 1-category index the generating 119899-homotopies ina homotopy coherent diagram such as enumerated for the homotopy coherent 120596-simplex at the startof this chapter

626 Definition (the homotopy coherent 120596-simplex) The homotopy coherent 120596-simplex ℭΔ[120596]is a simplicial category defined to be the free resolution of the category

120654 ≔ 0 1 2 3 ⋯

The objects of ℭΔ[120596] are natural numbers 119896 ge 0 Unpacking Definition 621 we can completelydescribe its arrows

bull A non-identity 0-arrow from 119895 to 119896 is a sequence of non-identity composable morphisms from 119895to 119896 the data of which is uniquely determined by the objects being passed through In particularthere are no 0-arrows from 119895 to 119896 if 119895 gt 119896 and the only 0-arrow from 119896 to 119896 is the identity If 119895 lt 119896the non-identity 0-arrows from 119895 to 119896 correspond to subsets

119895 119896 sub 1198790 sub [119895 119896]of the closed internal [119895 119896] = 119905 isin 120654 ∣ 119895 le 119905 le 119896 containing both endpoints

bull A 1-arrow from 119895 to 119896 is a once bracketed sequence of non-identity composable morphisms from119895 to 119896 This data is specified by two nested subsets

119895 119896 sub 1198790 sub 1198791 sub [119895 119896]the larger one 1198791 specifying the underlying unbracketed sequence and the smaller one 1198790 speci-fying the placement of the brackets

bull A 119903-arrow from 119895 to 119896 is an 119903 times bracketed sequence of non-identity composable morphismsfrom 119895 to 119896 the data of which is specified by nested subsets

119895 119896 sub 1198790 sub ⋯ sub 119879 119903 sub [119895 119896] (627)

indicating the locations of all of the parentheses⁶The face and degeneracy maps of (622) are the obvious ones either duplicating or omitting one ofthe sets 119879 119894 In particular the 119903-arrows just enumerated are non-degenerate if and only if each of theinclusions 1198790 sub ⋯ sub 119879 119903 is proper

We now describe the geometry of the mapping spaces ℭΔ[120596](119895 119896)

628 Lemma (homs in the homotopy coherent 120596-simplex are cubes) The mapping spaces of the homo-topy coherent 120596-simplex are defined for 119895 119896 isin 120654 by

ℭΔ[120596](119895 119896) cong

⎧⎪⎪⎪⎨⎪⎪⎪⎩

empty 119895 gt 119896Δ[0] 119896 = 119895 or 119896 = 119895 + 1Δ[1]119896minus119895minus1 119895 lt 119896

Proof Because120654 has no arrows from 119895 to 119896 when 119895 gt 119896 these hom-spaces of ℭΔ[120596] are similarlyempty When 119896 = 119895 or 119896 = 119895 + 1 we have 119895 119896 = [119895 119896] using the notation of Definition 626 soℭΔ[120596](119895 119896) cong Δ[0] is comprised of a single point

For 119896 gt 119895 there are 119896minus119895minus1 elements of [119895 119896] excluding the endpoints and so we see thatℭΔ[120596](119895 119896)has 2119896minus119895minus1 vertices The 119903-simplices ofℭΔ[120596](119895 119896) are given by specifying 119903+1 vertices mdash each a subset119895 119896 sub 119879 119894 sub [119895 119896]mdash that respect the ordering of subsets relation From this we see that

ℭΔ[120596](119895 119896) cong Δ[1]119896minus119895minus1

⁶The nesting is because parenthezations should be ldquowell formedrdquo with open brackets closed in the reverse order to thatin which they were opened

is the nerve of the poset of subsets 119895 119896 sub 119879 sub [119895 119896] ordered by inclusion as displayed for instancein the case 119895 = 0 and 119896 = 4

ℭΔ[120596](0 4) ∶=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 4 0 1 4

0 3 4 0 1 3 4

0 2 4 0 1 2 4

0 2 3 4 0 1 2 3 4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Proposition 625 proves that the homotopy coherent 120596-simplex is a simplicial computad and itsproof identifies its atomic arrows

629 Lemma The simplicial category ℭΔ[120596] is a simplicial computad whose atomic 119903-arrows are those witha single outermost parenthesis ie those sequences of subsets

119895 119896 = 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]for which 1198790 = 119895 119896 Geometrically the atomic arrows from 119895 to 119896 are precisely the simplices in the hom-cubeℭΔ[120596](119895 119896) cong Δ[1]119896minus119895minus1 that contain the initial vertex 119895 119896

The finite ordinals define full subcategories of120654 In this way the homotopy coherent 120596-simplexrestricts to define homotopy coherent simplices in each finite dimension

6210 Definition (the homotopy coherent 119899-simplex) The homotopy coherent 119899-simplex ℭΔ[119899]is the full subcategory of the homotopy coherent 120596-simplex ℭΔ[120596] spanned by the objects 0hellip 119899Equivalently it is the free resolution of the ordinal category with 119899 + 1 objects

Explicitly the homotopy coherent 119899-simplex has mapping spaces given for 119895 119896 isin [119899] by cubes

ℭΔ[119899](119895 119896) cong

⎧⎪⎪⎪⎨⎪⎪⎪⎩

each of which may be understood as the nerve of the poset of subsets 119895 119896 sub 119879 sub [119895 119896] ordered byinclusion An 119903-arrow may be represented as a nested sequence of subsets

119895 119896 sub 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]The homotopy coherent 119899-simplex is a simplicial computad whose atomic 119903-arrows are those se-

quences for which 1198790 = 119895 119896 or those simplices that contain the initial vertex in the hom-cube

Exercises

62i Exercise Compute the free resolution of the commutative square category 120794times120794 and compareit with the product ℭΔ[1] times ℭΔ[1] of two copies of the free resolution of 120794 This computationimplies that the functorℭ∶ 119982119982119890119905 rarr 119982119982119890119905-119966119901119905119889 to be introduced inDefinition 632 does not preserveproducts

63 Homotopy coherent realization and the homotopy coherent nerve

Our aim now is to introduce the homotopy coherent realization of any simplicial set 119883 which willdefine a simplicial computad ℭ119883 whose objects are the vertices of 119883 The homotopy coherent real-ization of Δ[119899] will be the homotopy coherent 119899-simplex ℭΔ[119899] of Definition 6210 The homotopycoherent realization of 119883 will be defined by ldquogluing togetherrdquo homotopy coherent 119899-simplices in acanonical way succinctly ℭ119883 is defined as a colimit in 119982119982119890119905-119966119901119905119889 of a diagram of homotopy coher-ent simplices indexed by the category of simplices of 119883 We will then leverage Lemma 629 into anexplicit presentation of ℭ119883 as a simplicial computad stated as Theorem 6310 recovering a result ofDugger and Spivak

The homotopy coherent realization and homotopy coherent nerve functors are determined by thecosimplicial object

120491 119982119982119890119905-119966119886119905

[119899] ℭΔ[119899]

ℭΔ[bull]

where a simplicial operator 120572∶ [119899] rarr [119898] acts on an 119903-arrow from 119895 to 119896 as described in Definition6210 by taking the direct image of the sequence of subsets (627)

We introduce these functors in turn

631 Definition (homotopy coherent nerve) The homotopy coherent nerve of a simplicial category119964 is the simplicial set 120081119964 whose 119899-simplices

120081119964119899 ≔ 119982119982119890119905-119966119886119905(ℭΔ[119899]119964)are defined to be diagrams ℭΔ[119899] rarr 119964 the simplicial operators act contravariantly on 120081119964 by pre-composition

Explicitly a homotopy coherent 119899-simplex in119964 is given bybull a sequence of objects 1198860 hellip 119886119899 isin 119964 andbull a sequence of simplicial maps

119886119894119895 ∶ Δ[1]119896minus119895minus1 rarr119964(119886119895 119886119896)for each 0 le 119895 lt 119896 le 119899

bull satisfying the simplicial functoriality condition

Δ[1]119896minus119895minus1 times Δ[1]119895minus119894minus1 Δ[1]119896minus119894minus1

119964(119886119895 119886119896) times 119964(119886119894 119886119895) 119964(119886119894 119886119896)

119886119895119896times119886119894119895

or119895

119886119894119896∘

Δ[1]119896minus119895minus1 times Δ[1]119895minus119894minus1 cong Δ119896minus119894minus2 Δ[1]119896minus119894minus1

ℭΔ[119899](119895 119896) times ℭΔ[119899](119894 119895) ℭΔ[119899](119894 119896)

or119895

is the map that sends a pair of 119903-simplices

119894 119895 sub 1198780 sub ⋯ sub 119878119903 sub [119894 119895] and 119895 119896 sub 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]

to their union119894 119896 sub 1198780 cup 1198790 sub ⋯ sub 119878119903 cup 119879 119903 sub [119894 119896]

If the 0 1-valued coordinates of the cubeΔ[1]119896minus119894minus1 are indexed by integers 119894 lt 119905 lt 119896 then the imageof or119895 is the 119895 = 1 face of the cube

632 Definition (homotopy coherent realization) The homotopy coherent realization functor ℭ isthe pointwise left Kan extension of the cosimplicial object ℭΔ[bull] along the Yoneda embedding

119982119982119890119905

120491 119982119982119890119905-119966119886119905

ℭよ

ℭΔ[bull]

uArrcong

The value of a pointwise left Kan extension at an object 119883 isin 119982119982119890119905 can be computed as a colimitindexed by the comma category Hom119982119982119890119905(よ 119883) [48 621] This comma category is better know asthe category of simplices of 119883 whose objects are simplices of 119883 and in which a morphism from an119899-simplex 119909 to an 119898-simplex 119910 is a simplicial operator 120572∶ [119899] rarr [119898] so that 119910 sdot 120572 = 119909 In this casethe colimit formula gives

ℭ119883 ≔ colim[119899]isin120491119909isin119883119899

ℭΔ[119899]

By general abstract nonsense

633 Proposition The homotopy coherent realization functor is left adjoint to the homotopy coherent nerve

119982119982119890119905 119982119982119890119905-119966119886119905

perp120081

Proof The homotopy coherent nerve was defined so that this adjoint correspondencewould holdfor the standard simplices and the general result follows since every simplicial set is a colimit indexedby its category of simplices of standard simplices See [48 659] for more details

634 Lemma The homotopy coherent realization functor takes its values in the subcategory of simplicialcomputads and computad morphisms

119982119982119890119905 119982119982119890119905-119966119886119905

119982119982119890119905-119966119901119905119889

Proof By Lemma 617 which proves that the category of simplicial computads is closed undercolimits in the category of simplicial categories it suffices to demonstrate that the cosimplicial objectℭΔ[bull] is valued in the subcategory of simplicial computads and simplicial computad morphisms Weknow already that the homotopy coherent simplices are simplicial computads so we need only verifythat the simplicial operators act by computad morphisms

A simplicial operator 120572∶ [119899] rarr [119898] acts on the 119903-arrow from 119895 to 119896 described in Definition 6210by taking the direct image of the sequence of subsets (627) The condition that characterizes theatomic arrows 119895 119896 = 1198790 is preserved by direct images so we see that 120572 defines a computad mor-phism ℭ120572∶ ℭΔ[119899] rarr ℭΔ[119898] As the subcategory 119982119982119890119905-119966119901119905119889 119982119982119890119905-119966119886119905 is closed under colimits

it follows that every homotopy coherent realization is a simplicial computad and any simplicial map119883 rarr 119884 induces a morphism of simplicial computads ℭ119883 rarr ℭ119884

635 Lemma For any inclusion119883 119884 of simplicial sets the morphism ℭ119883 ℭ119884 is a simplicial subcom-putad inclusion Moreover in the case of 119883 Δ[119899] an atomic 119903-arrow

119895 119896 = 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]from 119895 to 119896 of ℭΔ[119899] lies in the subcomputad ℭ119883 if and only if the simplex spanned by the vertices of 119879 119903 liesin 119883

Proof Recall that everymonomorphism of simplicial sets119883 119884 admits a canonical decomposi-tion as a sequential composite of pushouts of coproducts of the simplex boundary inclusions 120597Δ[119899] Δ[119899] Since all colimits are preserved by the left adjoint ℭ and Lemma 6110 proves that simplicialsubcomputads are closed under colimits of this form it suffices to prove that ℭ120597Δ[119899] ℭΔ[119899] isa simplicial subcomputad inclusion Wersquoll argue more generally that for 119883 sub Δ[119899] ℭ119883 ℭΔ[119899]is a simplicial subcomputad inclusion where the atomic arrows of ℭ119883 are as described in the secondclause of the statement

We argue using ideas from Reedy category theory reviewed in Appendix C Our task is to showthat the image of 119883 Δ[119899] under the functor

119982119982119890119905 119982119982119890119905-119966119901119905119889 119970119901ℎ120491epiopℭ atom

that takes a simplicial set to its 120491epiop-indexed graph of atomic and identity arrows is a pointwise

monomorphismThe simplicial subset 119883 sub Δ[119899] can be described as a colimit of certain faces of Δ[119899] glued along

their common faces the functor just described preserves these colimits Our claim asserts that thecomposite cosimplicial object ℭΔ[bull] ∶ 120491 rarr 119982119982119890119905 rarr 119982119982119890119905-119966119901119905119889 rarr 119970119901ℎ120491epi

opis Reedy monomorphic

meaning that every atomic 119903-arrow 119879bull in the homotopy coherent 119899-simplex is uniquely expressible inthe form 120572 sdot 119878bull where 120572∶ [119898] ↣ [119899] is a monomorphism in 120491 and 119878bull is ldquonon-co-degeneraterdquo ie notin the image of any monomorphism This is clear take 119898 = |119879 119903| minus 1 and define 120572∶ [119898] ↣ [119899] tobe the inclusion with image 119879 119903 sub [119899] Then take 119878bull to be the atomic 119903-arrow from 0 to 119898 in ℭΔ[119898]whose direct image under 120572 is 119879bull It is clear that 119878bull is not in the image of any smaller face map Thisargument also reveals that the atomic 119903-arrows 119879bull of ℭΔ[119899] that are present in the subcomputad ℭ119883are exactly those for which the vertices of 119879bull are contained in one of its faces

Lemma 635 allows us to compute the following subcomputads of the homotopy coherent sim-plex Before stating the results of these computations we introduce notation that suggests the correctgeometric intuition

636 Notation (cubes boundaries and cubical horns) We introduce special notation for the fol-lowing simplicial setsbull Write 119896 for the simplicial cube Δ[1]119896bull Write 120597119896 for the boundary of the 119896-cube Formally 120597119896 is the domain of the iterated Leibniz

product (120597Δ[1] Δ[1])times119896 If an 119903-simplex in 119896 is represented by a 119896-tuple of maps 120588119894 ∶ [119903] rarr[1] then that 119903-simplex lies in 120597119896 if and only if there is some 119894 for which 120588119894 is constant at eithervertex of [1]

bull Write ⊓119896119895119890 sub 120597119896 for the cubical horn containing only the face 119890 isin [1] in direction 1 le 119895 le 119896Formally ⊓119896119895119890 is the domain of the iterated Leibniz product

(120597Δ[1] Δ[1])times119895minus1times(Δ[0] 119890minusrarr Δ[1])times(120597Δ[1] Δ[1])times119896minus119895

An 119903-simplex in 119896 represented by a 119896-tuple of map 120588119894 ∶ [119903] rarr [1] lies in ⊓119896119895119890 if and only if forsome 119894 ne 119895 the map 119901119894 is constant or 119901119895 is the constant operator at 119890 isin [1]

637 Lemma (coherent subsimplices) The homotopy coherent realizations of the simplicial sphere and innerand outer horns define subcomputads of the homotopy coherent 119899-simplex containing all of the objects anddefined on homs by

(i) spheres ℭ120597Δ[119899] ℭΔ[119899] is full on all arrows except those from 0 to 119899 and has

ℭ120597Δ[119899](0 119899) ℭΔ[119899](0 119899)

120597119899minus1 119899minus1cong cong

(ii) inner horns For 0 lt 119896 lt 119899 ℭΛ119896[119899] ℭΔ[119899] is full on all arrows except those from 0 to 119899 andhas

ℭΛ119896[119899](0 119899) ℭΔ[119899](0 119899)

⊓119899minus11198961 119899minus1

cong cong(iii) outer horns ℭΛ119899[119899] ℭΔ[119899] is full on all arrows except those from 0 to 119899 minus 1 or 119899 and has

ℭΛ119899[119899](0 119899 minus 1) ℭΔ[119899](0 119899 minus 1) ℭΛ119899[119899](0 119899) ℭΔ[119899](0 119899)

120597119899minus2 119899minus2 ⊓119899minus1119899minus10 119899minus1

cong cong cong cong

Similarly ℭΛ0[119899] ℭΔ[119899] is full on all arrows except those from 0 or 1 to 119899 and has

ℭΛ0[119899](1 119899) ℭΔ[119899](1 119899) ℭΛ0[119899](0 119899) ℭΔ[119899](0 119899)

120597119899minus2 119899minus2 ⊓119899minus110 119899minus1

cong cong cong cong

Proof For (i) the only non-degenerate simplex of Δ[119899] that is not present in 120597Δ[119899] is the topdimensional 119899-simplex Consequently the only atomic 119903-arrows119879bull that are not present inℭ120597Δ[119899] arethose with 119879 119903 = [0 119899] The atomic 119903-arrows must also have 1198790 = 0 119899 and consequently correspondprecisely to those 119903-simplices of the cube 119899minus1 that contain both the first and last vertex Thus wesee that ℭ120597Δ[119899](0 119899) is isomorphic to the cubical boundary 120597119899minus1

For (ii) the only non-degenerate simplex of 120597Δ[119899] that is not present in an inner horn Λ119896[119899] isthe 119896-th face of the top dimensional simplex Consequently the only atomic 119903-arrows 119879bull that are notpresent in ℭΛ119896[119899] but are present in ℭ120597Δ[119899] are those with 119879 119903 = [0 119899]119896 The atomic 119903-arrowsmust also have 1198790 = 0 119899 and consequently correspond precisely to those 119903-simplices of the cube119899minus1 that contain both the first vertex and last vertex of the 119896 = 0 face of the cube 119899minus1 Thus wesee that ℭΛ119896[119899](0 119899) is isomorphic to the cubical horn ⊓119899minus11198961

For (iii) the only non-degenerate simplex of 120597Δ[119899] that is not present in the outer horn Λ119899[119899] isthe 119899-th face of the top dimensional simplex Consequently the only atomic 119903-arrows 119879bull that are not

present in ℭΛ119899[119899] but are present in ℭ120597Δ[119899] are those with 119879 119903 = [0 119899 minus 1] note that the source ofsuch 119903-arrows is 0 and the target is 119899minus1 The atomic 119903-arrows must also have1198790 = 0 119899minus1 and so asin the proof of (i) this identifies the inclusionℭΛ119899[119899](0 119899minus1) ℭΔ[119899](0 119899minus1) as 120597119899minus2 119899minus2The subcomputad ℭΛ119899[119899] is also missing non-atomic 119903-arrows that are present in ℭ120597Δ[119899] namelythose composites of the unique arrow from 119899 minus 1 to 119899 with the atomic 119903-arrows from 0 to 119899 minus 1 justdescribed Such non-atomic 119903-arrows are represented by sets of inclusions with 1198790 = 0 119899minus1 119899 and119879 119903 = [0 119899] and thus are found in the 119899minus1 = 1 face of the cube 119899minus1 Thus we see that ℭΛ119899[119899](0 119899)is isomorphic to the cubical horn ⊓119899minus1119899minus10

638 Definition (bead shapes) We call the atomic 119903-arrows of ℭΔ[119899](0 119899) that are not present inℭ120597Δ[119899](0 119899) bead shapes By Lemma 637 an 119903-dimensional bead shape corresponds to a sequenceof subsets

0 119899 = 1198790 sub 1198791 sub ⋯ sub 119879 119903minus1 sub 119879 119903 = [0 119899] (639)and is non-degenerate if and only if each of the inclusions is proper

Putting this together we can now explicitly present the simplicial computad structure on the ho-motopy coherent realization of any simplicial set

6310 Theorem (homotopy coherent realizations explicitly) For any simplicial set 119883 the homotopycoherent realization ℭ119883 is a simplicial computad in whichbull The objects of ℭ119883 are the vertices of the simplicial set 119883bull The atomic 0-arrows are non-degenerate 1-simplices of 119883 with the initial vertex of the simplex defining

the source of the 0-arrow and the final vertex of the simplex defining the target of the 0-arrowbull The atomic 1-arrows are non-degenerate 119899-simplices of 119883 with the initial vertex of the simplex defining

the source of the 0-arrow and the final vertex of the simplex defining the target of the 0-arrowbull The atomic 119896-arrows are pairs comprised of a non-degenerate 119899-simplex in119883 together with a 119896-dimensional

bead shape in ℭΔ[119899] This source of this 119896-arrow is the initial vertex of the 119899-simplex while the targetis the final vertex of the 119899-simplex and the arrow is non-degenerate if and only if the bead shape is non-degenerate

Note that the description of atomic 119896-arrows subsumes those of the atomic 0-arrows and atomic1-arrows as there is a unique 1-dimensional bead shape inℭΔ[119899] and a 0-dimensional bead shape existsonly in the case 119899 = 1 The data of a non-degenerate atomic 119896-arrow from 119909 to 119910 in ℭ119883 is given bya ldquobeadrdquo that is a non-degenerate 119899-simplex in 119883 from 119909 to 119910 together with the additional data of aldquobead shaperdquo sequence of proper subset inclusions (639) which Dugger and Spivak refer to as a ldquoflagof vertex datardquo [16] Non-atomic 119896-arrows are then ldquonecklacesrdquo that is strings of beads in 119883 joinedhead to tail together with accompanying ldquovertex datardquo for each simplex

Proof As observed in the proof of Lemma 635 using the canonical skeletal decomposition ofthe simplicial set 119883

∐119883119899119871119899119883

120597Δ[119899] ∐119883119899119871119899119883

Δ[119899]

empty sk0119883 ⋯ sk119899minus1119883 sk119899119883 ⋯ colim = 119883

the homotopy coherent realization ℭ119883 is constructed iteratively by a process that adjoints one copyof ℭΔ[119899] along a map of its boundary ℭ120597Δ[119899] for each non-degenerate 119899-simplex of 119883 Thus eachatomic 119896-arrow arises from a unique pushout of this form as the image of an atomic 119896-arrow in ℭΔ[119899]that is not present in the subcomputad ℭ120597Δ[119899]

6311 Remark From the description of Theorem 6310 the subcomputad inclusions ℭ119883 ℭ119884induced by monomorphisms of simplicial sets 119883 119884 are easily understood an 119903-arrow in ℭ119884 lies inℭ119883 if and only if each bead in its representing necklace in119884 lies in119883 This generalizes the descriptiongiven to subcomputads of homotopy coherent simplices in Lemma 635

Our convention is to identify 1-categories with their nerves The homotopy coherent realizationof these simplicial sets then produces a simplicial computad But we have already encountered a wayto produce a simplicial computad from a 1-category namely via the free resolution of Definition 621This would lead to a potential source of ambiguity were it not for the happy coincidence that thesetwo constructions are isomorphic⁷

6312 Proposition (free resolutions are homotopy coherent realizations) For any 1-category the freeresolution is naturally isomorphic to the homotopy coherent realization of its nerve

Proof Proposition 625 and Theorem 6310 present both simplicial categories as simplicial com-putads We will argue that they have the same objects and non-degenerate atomic 119896-arrows

Both have the same set of objects the objects of the 1-category coinciding with the vertices in itsnerve Atomic 0-arrows of the free resolution are morphisms in the category while atomic 0-arrowsin the coherent realization are non-degenerate 1-simplices of the nerve mdash these are the same thingAtomic non-degenerate 1-arrows of the free resolution are sequences of at least two morphisms (en-closed in a single set of outer parentheses) while atomic 1-arrows of the coherent realization are non-degenerate simplices of dimension at least two mdash again these are the same Finally a non-degenerateatomic 119896-arrow is a sequence of 119899 composable morphisms with (119896minus1) non-repeating bracketings thisnon-degeneracy necessitates 119899 gt 119896 This data defines a 119899-simplex in the nerve together with a non-degenerate atomic 119896-arrow in ℭΔ[119899](0 119899) ie an atomic 119896-arrow in the coherent realization

We now exploit the description of the subcomputads of the homotopy coherent 119899-simplex inLemma 637 to prove a key source of examples of quasi-categories a result first stated in this form in[13]

6313 Theorem The homotopy coherent nerve 120081119982 of a Kan-complex enriched category 119982 is a quasi-category

Proof By adjunction to extend along an inner horn inclusion Λ119896[119899] Δ[119899] mapping into thehomotopy coherent nerve120081119982 is to extend along simplicial subcomputad inclusionsℭΛ119896[119899] ℭΔ[119899]mapping into the Kan complex enriched category 119982 By Lemma 637(ii) the only missing 119903-arrows

⁷Note the isomorphism between the homotopy coherent realization of the 119899-simplex and the free resolution of theordinal category [119899] is tautologous The left Kan extension along the Yoneda embedding is defined so as to agree withℭΔ[bull] ∶ 120491 rarr 119982119982119890119905-119966119901119905119889 on the subcategory of representables Many arguments involving simplicial sets can be reducedto a check on representables with the extension to the general case following formally by ldquotaking colimitsrdquo This resulthowever is not one of them since we are trying to prove something for all categories and the embedding 119966119886119905 119982119982119890119905 doesnot preserve colimits

are in the mapping space from 0 to 119899 so we are asked to solve a single lifting problem

ℭΛ119896[119899](0 119899) cong ⊓119899minus11198961 Map(1198830 119883119899)

ℭΔ[119899](0 119899) cong 119899minus1

Cubical horn inclusions can be filled in the Kan complex Map(1198830 119883119899) completing the proof

We conclude by giving a precise meaning to the notion that motivated this chapter

6314 Definition Let 119883 be a simplicial set A homotopy coherent diagram of shape 119883 in a Kancomplex enriched category119982 is a simplicial functorℭ119883 rarr 119982 from the homotopy coherent realizationof 119883 to 119982

Exercise 62i reveals that there are two possible notions of natural transformation between homo-topy coherent diagram We opt for the more structured of the two

6315 Definition Let 119865119866∶ ℭ119883 119982 be homotopy coherent diagrams of shape 119883 in a Kan com-plex enriched category 119982 A homotopy coherent natural transformation 120572∶ 119865 rArr 119866 is a homotopycoherent diagram of shape 119883 times 120794

ℭ119883

ℭ[119883 times 120794] 119982

ℭ119883

1198650120572

1198661

that restricts to 119865 and to 119866 along the edges of the cylinder

These notions are originally due to Boardman and Vogt in [8] who observed that homotopy coher-ent natural transformations do not define a the morphisms of a 1-category Instead as they observedthey define the 1-arrows of a quasi-category

6316 Corollary Homotopy coherent diagrams of shape 119883 valued in a Kan complex enriched category119982 and homotopy coherent natural transformations between them define the objects and 1-arrows of a quasi-category Coh(119883 119982) whose 119899-simplices are homotopy coherent diagram ℭ[119883 times Δ[119899]] rarr 119982

Proof By adjunctionℭ ⊣ 120081 the simplicial set Coh(119883 119982) is isomorphic to (120081119982)119883 As the quasi-categories form an exponential ideal in the category of simplicial sets this follows immediately fromTheorem 6313

6317 Remark (all diagrams in homotopy coherent nerves are homotopy coherent) Corollary 6316explains that any homotopy coherent diagram ℭ119883 rarr 119982 of shape 119883 in a Kan complex enriched cat-egory 119982 transposes to define a map of simplicial sets 119883 rarr 120081119982 valued in the quasi-category definedas the homotopy coherent nerve of 119982 While not every quasi-category is isomorphic to a homotopycoherent nerve of a Kan complex enriched category once consequence of the internal Yoneda lemmato be proven much later is that every quasi-category is equivalent to a homotopy coherent nerve Thisexplains the slogan introduced at the beginning of this chapter and the title for this part of the bookall diagrams in quasi-categories are homotopy coherent thus quasi-category theory can be understoodas ldquohomotopy coherent category theoryrdquo

Exercises

63i Exercise Compare the simplicial computad structure of the homotopy coherent 120596-simplex asgiven by Theorem 6310 with the simplicial computad structure of Lemma 629

CHAPTER 7

Weighted limits ininfin-cosmoi

71 Weighted limits and colimits

Let (119985times 120793) denote a complete and cocomplete cartesian closed monoidal category The ex-amples we have in mind are (119982119982119890119905 times 120793) its cartesian closed subcategory (119966119886119905 times 120793) or its furthercartesian closed subcategory (119982119890119905 times 120793)

Ordinary limits and colimits are objects representing the functor of cones with a given apex over orunder a fixed diagram Weighted limits and colimits are defined analogously except that the cones overor under a diagram might have exotic ldquoshapes rdquo which are allowed to vary with the objects indexingthe diagram More formally in the119985-enriched context the weight defining the ldquoshaperdquo of a cone overa diagram indexed by119964 or under a diagram indexed by119964op takes the form of a functor in119985119964

Before introducing the general notion of weighted limit and colimit we first reacquaint ourselvesan example that we have seen already in Digression 124

711 Example (tensors and cotensors) A diagram indexed by the category 120793 valued in a119985-enrichedcategoryℳ is just an object119860 inℳ In this case the weight is just an object119880 of119985 The119880-weightedlimit of the diagram119860 is an object ofℳ denoted119860119880 mdash or denoted 119880119860 whenever superscripts areinconvenient mdash called the the cotensor of 119860 isin ℳ with 119880 isin 119985 defined by the universal property

ℳ(119883119860119880) cong 119985(119880ℳ(119883119860))this isomorphism between mapping spaces in 119985 Dually the 119880-weighted colimit of 119860 is an object119880 otimes 119860 isin ℳ called the tensor of 119860 isin ℳ with 119880 isin 119985 defined by the universal property

ℳ(119880 otimes 119860119883) cong 119985(119880ℳ(119860119883))this isomorphism again between mapping spaces in119985 Assuming such objects exist the cotensor andtensor define119985-enriched bifunctors

119985op timesℳ ℳ 119985timesℳ ℳminusminus minusotimesminus

in a unique way making the defining isomorphisms natural in 119880 and 119860 as wellSince 119985 is cartesian closed it is tensored and cotensored over itself with 119880 otimes 119881 ≔ 119880 times 119881

and 119881119880 ≔ 119985(119880119881) In particular the defining natural isomorphisms characterizing tensors andcotensors inℳ can be rewritten as

ℳ(119883119860119880) cong ℳ(119883119860)119880 and ℳ(119880 otimes 119860119883) cong ℳ(119860119883)119880

The fact that the natural isomorphisms defining tensors and cotensors are required to exist in119985(and not merely in 119982119890119905) has the following consequence

712 Lemma (associativity of tensors and cotensors) If ℳ is a 119985-category with tensors and cotensorsthen for any 119880119881 isin 119985 and 119860 isin ℳ there exist natural isomorphisms

119880 otimes (119881 otimes 119860) cong (119880 times 119881) otimes 119860 and (119860119881)119880 cong 119860119880times119881165

Proof By the defining universal property

ℳ(119883 (119860119881)119880) cong 119985(119880ℳ(119883119860119881)) cong 119985(119880119985(119881ℳ(119883119860)))cong 119985(119880 times 119881ℳ(119883119860)) cong ℳ(119883119860119880times119881)

for all 119883 isin ℳ By the Yoneda lemma (119860119881)119880 cong 119860119880times119881 The case for tensors is similar

We now introduce the general notions of weighted limit and weighted colimit from three differentviewpoints We introduce these perspectives in the reverse of the logical order because we find thisroute to be the most intuitive We first describe the axioms that characterize the weighted limit andcolimit bifunctors whenever they exist We then explain how weighted limits and colimits can beconstructed again assuming these exist We then finally introduce the general universal property thatdefines a particular weighted limit or colimit which tells us when the notions just introduced do infact exist

713 Definition (weighted limits and colimits axiomatically) For a small 119985-enriched category 119964and a large119985-enriched categoryℳ the weighted limit and weighted colimit bifunctors

lim119964minus minus∶ (119985119964)op timesℳ119964 rarrℳ and colim119964

minus minus∶ 119985119964 timesℳ119964op rarrℳare characterized by the following pair of axioms whenever they exist

(i) Weighted (co)limits with representable weights evaluate at the representing object

lim119964119964(119886minus) 119865 cong 119865(119886) and colim119964

119964(minus119886)119866 cong 119866(119886)

(ii) The weighted (co)limit bifunctors are cocontinuous in the weight for any diagrams 119865 isin ℳ119964

and 119866 isin ℳ119964op the functor colim119964

minus 119866 preserves colimits while the functor lim119964minus 119865 carries

colimits to limitssup1We interpret axiom (ii) to mean that weights can be ldquomade-to-orderrdquo a weight constructed as a colimitof representables mdash as all119985-valued functors are mdash will stipulate the expected universal property

714 Definition (weighted limits and colimits constructively) The limit of 119865 isin ℳ119964 weighted by119882 isin 119985119964 is computed by the functor cotensor product

lim119964119882 119865 ≔ 1114009

119886isin119964119865(119886)119882(119886) ≔ eq

⎛⎜⎜⎜⎜⎝ prod119886isin119964

119865(119886)119882(119886) prod119886119887isin119964

119865(119887)119964(119886119887)times119882(119886)⎞⎟⎟⎟⎟⎠ (715)

where the product and equalizer should be interpreted as conical limits see Digression 124 or Defini-tion 7114 below The maps in the equalizer diagram are induced by the actions 119964(119886 119887) times 119882(119886) rarr119882(119887) and 119865(119886) rarr 119865(119887)119964(119886119887) of the hom-object 119964(119886 119887) on the 119985-functors 119882 and 119865 the latter case

makes use of the natural isomorphism 119865(119887)119964(119886119887)times119882(119886) cong (119865(119887)119964(119886119887))119882(119886)

of Lemma 712

sup1More precisely as will be proven in Proposition 7112 the weighted colimit functor colim119964minus 119866 preserves weighted

colimits while the weighted limit functor lim119964minus 119865 carries weighted colimits to weighted limits

Dually the colimit of 119866 isin ℳ119964opweighted by 119882 isin 119985119964 is computed by the functor tensor

product

colim119964119882119866 ≔ 1114009

119886isin119964119882(119886) otimes 119866(119886) ≔ coeq

⎛⎜⎜⎜⎜⎝ ∐119886119887isin119964

(119882(119886) times 119964(119886 119887)) otimes 119866(119887) ∐119886isin119964

119882(119886) otimes 119866(119886)⎞⎟⎟⎟⎟⎠

(716)where the coproduct and coequalizer should be interpreted as conical colimits One of the maps in thecoequalizer diagram is induced by the action119964(119886 119887) otimes 119866(119887) rarr 119866(119886) of119964(119886 119887) on the contravariant119985-functor 119866 and the natural isomorphism (119882(119886) times 119964(119886 119887)) otimes 119866(119887) cong 119882(119886) otimes (119964(119886 119887) otimes 119866(119887)) ofLemma 712 the other uses the covariant action of119964(119886 119887) on119882 as before

717 Definition (weighed limits and colimits the universal property) The limit lim119964119882 119865 of the dia-

gram 119865 isin ℳ119964 weighted by119882 isin 119985119964 and the colimit colim119964119882119866 of119866 isin ℳ119964op

weighted by119882 isin 119985119964

are characterized by the universal properties

ℳ(119883 lim119964119882 119865) cong 119985119964(119882ℳ(119883 119865)) and ℳ(colim119964

119882119866119883) cong 119985119964(119882ℳ(119866119883)) (718)

each of these defining an isomorphism between objects of119985

When the indexing category 119964 is clear from context as is typically the case we frequently dropit from the notation for the weighted limit and weighted colimit We now argue that these threedefinitions characterize the same objects Along the way we obtain results of interest in their ownright that we record separately

719 Lemma The category119985 admits all weighted limits as defined by the formula of (715) satisfying thenatural isomorphism of (718) Explicitly for a weight 119882∶ 119964 rarr 119985 and a diagram 119865∶ 119964 rarr 119985 theweighted limit

lim119964119882 119865 ≔ 119985119964(119882 119865)

is the119985-object of119985-natural transformations from119882 to 119881

Proof The119985-functor119985(120793 minus) ∶ 119985 rarr 119985 represented by the monoidal unit is naturally isomor-phic to the identity functor So taking119883 = 120793 in the universal property of (718) in the case where thediagram 119865 isin 119985119964 is valued in the119985-category119985 we have

lim119964119882 119865 cong 119985119964(119882 119865)

Simultaneously the formula (715) computes the119985-object119985119964(119882 119865) of119985-natural transformationsfrom119882 to 119865 defined in Definition A38

The119985-object of119985-natural transformations satisfies the natural isomorphism119985(119881119985119964(119882 119865)) cong119985119964(119882119985(119881 119865)) for any 119881 isin 119985 Applying the observation that 119882-weighted limits of 119985-valuedfunctors are 119985-objects of natural transformations to the functor ℳ(119883 119865minus) and ℳ(119866minus119883) in thecase of 119865 isin ℳ119964 and 119866 isin ℳ119964op

we may re-express the natural isomorphism (718) as

7110 Corollary The weighted limits and weighted colimits of (718) are representably defined as weightedlimits in119985 for119882 isin 119985119964 and 119865 isin ℳ119964 and 119866 isin ℳ119964op

the weighted limit and colimit are characterizedby isomorphisms

ℳ(119883 lim119882 119865) cong lim119882ℳ(119883 119865) and ℳ(colim119882119866119883) cong lim119882ℳ(119866119883) (7111)

natural in 119883 in119985

We now unify the Definitions 713 714 and 717

7112 Proposition When the limits and colimits of (715) and (716) exist they define objects satisfyingthe universal properties (718) or equivalently (7111) and bifunctors satisfying the axioms of Definition 713

Proof The proofs are dual so we confine our attention to the limit case The general case of theimplication Definition 714 rArr 717 mdash for either weighted limits or weighted colimits mdash is a directconsequence of the special case of this implication for weighted limits valued in ℳ = 119985 proven asLemma 719 and Corollary 7110 The limits of (715) inℳ are also defined representably in terms ofthe analogous limits in119985 So the objected defined by (715) represents the119985-functor lim119882ℳ(minus 119865)that defines the weighted limit lim119882 119865

It remains to prove that the weighted limits of Definitions 714 and 717 satisfy the axioms ofDefinition 713 In the case of a119985-valued diagram 119865 isin 119985119964 axiom (i) is the119985-Yoneda lemma

119985119964(119964(119886 minus) 119865) cong 119865(119886)proven in Theorem A311 Once again the general case for 119865 isin ℳ119964 follows from the special casefor 119985-valued diagrams for to demonstrate an isomorphism lim119964(119886minus) 119865 cong 119865(119886) in ℳ it suffices todemonstrate an isomorphismℳ(119883 lim119964(119886minus) 119865) cong ℳ(119883 119865(119886)) in119985 for all 119883 isin ℳ and have such anatural isomorphism by applying (7111) and the observation just made to the functor ℳ(119883 119865minus) isin119985119964

For the axiom (ii) consider a diagram 119870∶ 119973op rarr 119985119964 of weights and a weight 119881 isin 119985119973 so thatcolim119973

119881 119870 cong 119882 For any 119865 isin ℳ119964 we will show that the 119985-functor lim119964minus 119865∶ (119985119964)op rarr ℳ carries

the 119881-weighted colimit of 119870 to the 119881-weighted limit of the composite diagram lim119964119870 119865∶ 119973 rarrℳ

The universal property (718) applied first to the colim119973119881 119870-weighted limit of the diagram 119865 and

the object 119883 and then to the 119881-weighted colimit of the diagram 119870 and the objectℳ(119883 119865) suppliesisomorphisms

ℳ(119883 lim119964colim119973119881 119870

119865) cong 119985119964(colim119973119881 119870ℳ(119883 119865)) cong 119985119973(119881119985119964(119870ℳ(119883 119865)))

Applying (718) twice more first for the weights 119870119895 for each 119895 isin 119973 and then for the weight119881 and thediagram lim119964

119870 119865∶ 119973 rarrℳ we have

cong 119985119973(119881ℳ(119883 lim119964119870 119865)) cong ℳ(119883 lim119973

119881 lim119964119870 119865)

By the Yoneda lemma this proves that

lim119964colim119973119881 119870

119865 cong lim119973119881 lim119964

119867 119865

ie that the weighted limit functor lim119964minus 119865 is carries a weighted colimit of weights to the analogous

weighted limit of weights

7113 Remark (for unenriched indexing categories) When the indexing category is unenriched thelimit and colimit formulas from Definition 714 simplify

lim119964119882 119865 cong eq

⎛⎜⎜⎜⎜⎝ prod119886isin119964

119865(119886)119882(119886) prod119964(119886119887)

119865(119887)119882(119886)⎞⎟⎟⎟⎟⎠

colim119964119882119866 cong coeq

⎛⎜⎜⎜⎜⎝ ∐119964(119886119887)

(119882(119886) otimes 119866(119887) ∐119886isin119964

119882(119886) otimes 119866(119886)⎞⎟⎟⎟⎟⎠

and in fact it suffices to consider only non-identity arrows or even just atomic arrows

7114 Definition (conical limits and colimits) The unit for the cartesian product defines a terminalobject 120793 isin 119985 The constant diagram at the terminal object then defines a terminal object 120793 isin 119985119964 Alimit weighted by the terminal weight is called a conical limit and a colimit weighted by the terminalweight is called a conical colimit It is common to use the simplified notation lim 119865 ≔ lim119964

120793 119865 andcolim119866 ≔ colim119964

120793 119866Conical limits and colimits satisfy the defining universal properties

ℳ(119883 lim 119865) cong 119985119964(120793ℳ(119883 119865)) and ℳ(colim119866119883) cong 119985119964(120793ℳ(119866119883))which say that lim 119865 and colim119866 represent the functors of119985-enriched conical cones over 119865 or under119866 respectively

We can now properly understand the formulae for weighted limits and colimits given in Definition714 In particular these formulae give criteria under which weighted limits or colimits are guaranteedto exist

7115 Corollary Ifℳ is a119985-enriched category that admits cotensors and conical limits of all unenricheddiagram shapes then ℳ admits all weighted limits Dually if ℳ admits tensors and conical colimits of allunenriched diagram shapes thenℳ admits all weighted colimits

7116 Remark Ifℳ is a119985-category whose underlying unenriched category admits all small limitsthen if ℳ admits cotensors and tensors over 119985 then ℳ admits all weighted limits Via the Yonedalemma the presence of tensors suffices to internalize the isomorphism of sets expressing the unen-riched universal property of limits to an isomorphism in 119985 that expresses the universal property ofconical limits See Exercise 71i

7117 Example (commas) The comma infin-category is the limit in the infin-cosmos 119974 of the diagram⟓rarr 119974 given by the cospan

119862 119860 119861119892 119891

weighted by the diagram ⟓rarr 119982119982119890119905 given by the cospan

120793 120794 1207931 0

Under the simplification of Remark 7113 the formula for the weighted limit reduces to the equalizer

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119860120794

119862 times 119860120794 times 119861 119860 times 119860119862 times 119861

(11990111199010)120587

120587 119892times119891

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

which computes the pullback of (332) The universal property (718) says that functors119883 rarr Hom119860(119891 119892)in119974 correspond to simplicial natural transformations the data of which is given by the three dashedvertical maps that fit into two commutative squares

120793 120794 120793

Fun(119883 119862) Fun(119883119860) Fun(119883 119861)

119888 120572

119887

119892lowast 119891lowast

7118 Example (Bousfield-Kan homotopy limits) In their classic book on homotopy limits and col-imits [11] Bousfield and Kan define the homotopy limit of a diagram indexed by a 1-category 119860 andvalued in a Kan-complex enriched categoryℳ to be the limit weighted by the functor

119860 119982119982119890119905

119886 119860119886which carries each object 119886 isin 119860 the the nerve of the slice category over 119886

7119 Example (Kan extensions as weighted colimits) The usual colimit or limit formula that com-putes the value of a pointwise left or right Kan extension of an unenriched functor 119865∶ 119862 rarr 119864 along119870∶ 119862 rarr 119863 at an object 119889 isin 119863 can be succinctly expressed by the weighted colimit or weighted limit

lan119870 119865(119889) ∶= colim119863(119870minus119889) 119865 and ran119870 119865(119889) ∶= lim119863(119889119870minus) 119865

We conclude with a few results from the general theory of weighted limits and colimits Immedi-ately from their defining universal properties it can be verified that

7120 Lemma (weighted limits of restricted diagrams) Suppose given a119985-functor119870∶ 119964 rarr ℬ a weight119882∶ 119964 rarr 119985 and diagrams 119865∶ ℬ rarr ℳ and 119866∶ ℬop rarrℳ Then the119882-weighted limit or colimit of therestricted diagram is isomorphic to the lan119870119882-weighted limit or colimit of the original diagram

lim119964119882 (119865 ∘ 119870) cong limℬ

lan119870119882 119865 and colim119964119882 (119866 ∘ 119870) cong colimℬ

lan119870119882119866

An enriched adjunction is comprised of a pair of 119985-functors 119865∶ ℬ rarr 119964 and 119880∶ 119964 rarr ℬtogether with a family of isomoprphisms119964(119865119887 119886) cong ℬ(119887119880119886) that are119985-natural in both variablessee Definition A315 The usual Yoneda-style argument enriches to show

7121 Proposition (weighted RAPLLAPC) A119985-enriched right adjoint functor119880∶ 119964 rarr ℬ preservesall weighted limits that exist in119964 while itrsquos119985-enriched left adjoint 119865∶ ℬ rarr 119964 preserves all weighted colimitsthat exist in ℬ

Proof Exercise 71iv

By the axioms ofDefinition 121infin-cosmoiwill admit a large class of simplicially-enrichedweight-ed limits built from the simplicial cotensors and conical simplicial limits enumerated in 121(i) Inpractice infin-cosmoi often arise as subcategories (of ldquofibrant objectsrdquo) in a larger category that is alsoadmits simplicially-enriched weighted colimits which can then be reflected back into theinfin-cosmosto defined weighted bicolimits This is a story for much later so we will confine our attention to thecase of weighted limits for the rest of this chapter

Exercises

71i Exercise Suppose ℳ is a tensored and 119985-enriched category whose underlying unenrichedcategory admits limits of all unenriched diagram shapes Show that ℳ admits conical limits of allunenriched diagram shapes proving the extension of Corollary 7115 described in Remark 7116

71ii Exercise Taking the base for enrichment119985 to be 119982119890119905 compute the following weighted limitsof a simplicial set 119883 regarded as a diagram in 119982119890119905120491

op weighted by

(i) the standard 119899-simplex Δ[119899] isin 119982119890119905120491op

(ii) the spine of the 119899-simplex the simplicial subset Γ[119899] Δ[119899] obtained by gluing together the

119899 edges from 119894 to 119894 + 1 into a composable path(iii) the 119899-simplex boundary 120597Δ[119899] isin 119982119890119905120491

opsup2

71iv Exercise Prove Proposition 7121

72 Flexible weighted limits and the collage construction

Our aim in this section is to introduce a special class of 119982119982119890119905-valued weights whose associatedweighted limit notions are homotopically well-behaved Borrowing a term from 2-category theorywe refer to these weights as flexible All of the limits enumerated in 121(i) are flexible limits In factwe will prove that infin-cosmoi admit all flexible weighted limits because these can be built out of thelimits enumerated in 121(i) In sect74 we will use this observation to help us verify the limit axiom fornewly constructedinfin-cosmoi in a more systematic way

In this section we will characterize the class of flexible weights as precisely those whose associatedcollages define relative simplicial computads which will allow us to readily produce examples Insect73 we will establish the homotopical properties of flexible weighted limits and make precise therelationships between this class of the limits and limits assumed present in any infin-cosmos by axiom121(i)

721 Definition (flexible weights and projective cell complexes) Let119964 be a simplicial categorybull A simplicial natural transformation of the form

120597Δ[119899] times 119964(119886 minus) Δ[119899] times 119964(119886 minus)is called a projective 119899-cell at 119886 isin 119964

bull A natural transformation 120572∶ 119881 119882 in 119982119982119890119905119964 that can be expressed as a countable compositeof pushouts of coproducts of projective cells is called a projective cell complex

bull A weight119882 isin 119982119982119890119905119964 is flexible just when empty 119882 is a projective cell complex

722 Remark (on generalized projective cells) Since any monomorphism of simplicial sets 119880 119881 can be decomposed as a sequential composite of pushouts of coproducts of boundary inclusions120597Δ[119899] Δ[119899] the class of projective cell complexes may be also be described as the class of maps in119982119982119890119905119964 that can be expressed as a countable composite of pushouts of coproducts of monomorphisms119880 times119964(119886 minus) 119881 times119964(119886 minus) for some 119886 isin 119964

723 Example Since any simplicial set can be decomposed as a sequential composite of pushouts ofcoproducts of boundary inclusions 120597Δ[119899] Δ[119899] simplicial cotensors are flexible weights

724 Example Conical products also define flexible weighted limits built by attaching one projec-tive 0-cell for each object in the indexing set

725 Non-Example Conical limits indexed by any 1-category that contains non-identity arrows arenot flexible because the legs of a conical cone over the domain and codomain of each arrow in thediagram are required to define a strictly commutative triangle of 0-arrows The specifications for a

sup2The limit of a simplicial object weighted by 120597Δ[119899] is called the 119899th-matching object see Appendix C

flexible weight allow us to free attach 119899-arrows of any dimension but do not provide a mechanism fordemanding strict commutativity of any diagram of 119899-arrows mdash only commutativity up to the presenceof a higher cell

726 Digression (on flexible limits in 2-category theory) Simplicial limits weighted by flexibleweights should be thought of as analogous to flexible 2-limits ie 2-limits built out of products insert-ers equifiers and retracts (splittings of idempotents) [6] More precisely simplicial limits weightedby flexible weights are analogous to the PIE limits those built just from products inserters and equi-fiers but we choose to adopt the moniker from the slight larger class of weights because we find itto be more evocative The PIE limits also include iso-inserters descent objects comma objects andEilenberg-Moore objects as well as all pseudo lax and oplax limits Many important 2-categoriessuch as the 2-category of accessible categories and accessible functors fail to admit all 2-categoricallimits but do admit all PIE-limits [40]

The weights for flexible limits are the cofibrant objects in a model structure on the diagram2-category 119966119886119905119964 that is enriched over the folk model structure on 119966119886119905 the PIE weights are exactlythe cellular cofibrant objects Correspondingly the projective cell complexes of Definition 721 areexactly the cellular cofibrations in the projective model structure on 119982119982119890119905119964

Recall that a weight is intended to describe the ldquoshaperdquo of cones over diagrams indexed by a par-ticular category In the case a weight 119882∶ 119964 rarr 119982119982119890119905 valued in simplicial sets we can describe theshape of119882-cones directly as a simplicial category called the collage of the weight

727 Definition (collage construction) The collage of a weight119882∶ 119964 rarr 119982119982119890119905 is a simplicial cat-egory coll119882 which contains 119964 as a full subcategory together with one additional object ⊤ whoseendomorphism space is the point The mapping spaces from any object 119886 isin 119964 to ⊤ are empty whilethe mapping spaces from ⊤ to the image of119964 coll119882 are defined by

coll119882(⊤ 119886) ≔ 119882(119886)with the action maps119964(119886 119887) times119882(119886) rarr 119882(119887) from the simplicial functor119882 used to define compo-sition This defines a simplicial category together with a canonical inclusion 120793 +119964 coll119882 that isbijective on objects and fully faithful on 120793 and119964 separately

728 Proposition (collage adjunction)(i) The collage construction defines a fully faithful functor

119982119982119890119905119964 120793+119964119982119982119890119905-119966119886119905coll

from the category of 119964-indexed weights to the category of simplicial categories under 120793 + 119964 whoseessential image is comprised of those ⟨119890 119865⟩ ∶ 120793 + 119964 rarr ℰ that are bijective on objects fully faithfulwhen restricted to 120793 and119964 and have the property that there are no arrows in ℰ from the image of 119865to 119890

(ii) The collage functor admits a right adjoint

119982119982119890119905119964 120793+119964119982119982119890119905-119966119886119905

perpwgt

which carries a pair ⟨119890 119865⟩ ∶ 120793 +119964 rarr ℰ to the weight ℰ(119890 119865minus) ∶ 119964 rarr 119982119982119890119905172

Proof The construction of the collage functor is straightforward and left to the reader Thecharacterization of its essential image follows from the observation that to define a simplicial functor119882∶ 119964 rarr 119982119982119890119905 requires no more and no less thanbull the specification of simplicial sets119882(119886) for each 119886 isin 119964 andbull and the specification of simplicial maps119964(119886 119887) times 119882(119886) rarr 119882(119887) for each 119886 119887 isin 119964

so that this action is associative in sense that the diagram

119964(119887 119888) times 119964(119886 119887) times 119882(119886) 119964(119886 119888) times 119882(119886)

119964(119887 119888) times 119882(119887) 119882(119888)

commutes This is the same as what is required to extend 120793 +119964 to a simplicial category in which allof the additional maps start at ⊤ and end at an object in119964

The adjunction asserts that simplicial functors

120793 +119964

coll119882 ℰ⟨119890119865⟩

119866

from coll119882 to ℰ under 120793 + 119964 stand in natural bijective correspondence with simplicial naturaltransformations 120574∶ 119882 rArr ℰ(119890 119865minus) Since the inclusion 1 + 119964 coll119882 is bijective on objectsand full on most homs the data of the simplicial functor requires only the specification of the mapscoll119882(⊤ 119886) = 119882(119886) rarr ℰ(119890 119865119886) These define the components of the simplicial natural transforma-tion 120574 and functoriality of 119866 corresponds to naturality of 120574

The collage adjunction has a useful and important interpretation

729 Corollary The collage of a weight119882∶ 119964 rarr 119982119982119890119905 realizes the shape of119882-weighted cones in thesense that simplicial functors 119866∶ coll119882 rarr ℰ with domain coll119882 stand in bijection to 119882-cones over thediagram 119866|119964 with summit 119866(⊤)

On account of Lemma 7120 wersquoll be interested in computing left Kan extensions of weightsencoded as collages

7210 Lemma For anyweight119882∶ 119964 rarr 119982119982119890119905 and simplicial functor119870∶ 119964 rarr ℬ the pushout of simplicialcategories

120793 +119964 120793 + ℬ

coll119882 coll(lan119870119882)

120793+119870

computes the collage of the weight lan119870119882∶ ℬ rarr 119982119982119890119905

Proof By the defining universal property a simplicial functor out of the pushout is given by apair of functors ⟨119890 119865⟩ ∶ 120793 + ℬ rarr ℰ and 119866∶ coll119882 rarr ℰ so that

120793 +119964

coll119882 ℰ⟨119890119865119870⟩

119866

By Corollary 729 this data defines a 119882-cone with summit 119890 over the diagram 119865119870∶ 119964 rarr ℰ ByLemma 7120 such data equivalently describes a lan119870119882-cone with summit 119890 over the diagram 119865∶ ℬ rarrℰ Applying Corollary 729 again we conclude that the pushout is given by the simplicial categorycoll(lan119870119882) as claimed

In analogy with Corollary 435 we can encode simplicial 119882-weighted limits as a right Kan ex-tension from the indexing simplicial category to the simplicial category that describes the shape of119882-cones

7211 Lemma For any simplicial functor 119865∶ 119964 rarr ℰ and any weight119882∶ 119964 rarr 119982119982119890119905 the weighted limitlim119882 119865 exists if and only if the pointwise right Kan extension of 119865 along 119964 coll119882 exists in which caselan 119865(⊤) cong lim119882 119865

Proof Since119964 coll119882 is fully faithful pointwise right Kan extensionsmay be chosen to definegenuine extensions

119964

coll119882 ℰ119865

ran 119865ByCorollary 729 this data defines a119882-cone over119865with summit ran 119865(⊤) thatwe denote by120582∶ 119882 rArrℰ(ran 119865(⊤) 119865(minus))

By Corollary 729 again the data of a cone over the right Kan extension diagram displayed below-left

119964 119964

coll119882 ℰ coll119882 ℰ119865 = 119865

119866

uArr120572

119866existuArr120578

ran 119865

defines a119882-cone119882 ℰ(119866(⊤) 119866|119964(minus)) ℰ(119866(⊤) 119865(minus))

120574 120572lowast

over 119865 The universal property of the right Kan extension depicted above-right says this cone factorsuniquely through the 119882-cone 120582 along a map 120578⊤ ∶ 119866(⊤) rarr ran 119865(⊤) Thus the right Kan extensionof 119865 along 119964 coll119882 equips the resulting 119882-cone with the universal property of the 119882-weightedlimit

A particularly convenient aspect of the collage construction is that it allows us to detect the classof flexible weights

7212 Theorem (flexible weights and collages) A natural transformation 120572∶ 119881 119882 between weightsin119982119982119890119905119964 is a projective cell complex if and only if coll(120572) ∶ coll119881 coll119882 is a relative simplicial computadIn particular119882 is a flexible weight if and only if 120793 +119964 coll119882 is a relative simplicial computad

Proof If 120572∶ 119881 119882 is a projective cell complex then it can be presented as a countable com-posite of pushouts of coproducts of projective cells of varying dimensions indexed by the objects119886 isin 119964 Since the collage construction is a left adjoint it preserves these colimits and hence themap coll(120572) ∶ coll119881 coll119882 as a transfinite composite of pushouts of coproducts of simplicial func-tors coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) in 120793+119964119982119982119890119905-119966119886119905 This composite colimit dia-gram is connected mdash note collempty = 120793 + 119964 so this cell complex presentation is also preserved by the

forgetful functor 120793+119964119982119982119890119905-119966119886119905 rarr 119982119982119890119905-119966119886119905 and the simplicial functor coll(120572) ∶ coll119881 coll119882can be understood as a transfinite composite of pushouts of coproducts of coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) in 119982119982119890119905-119966119886119905

This is advantageous because there is a pushout square in 119982119982119890119905-119966119886119905

120794[120597Δ[119899]] coll(120597Δ[119899] times 119964(119886 minus)) 120597Δ[119899] 120597Δ[119899] times 119964(119886 119886)

120794[Δ[119899]] coll(Δ[119899] times 119964(119886 minus)) Δ[119899]] Δ[119899] times 119964(119886 119886)

id119886

(7213)

whose horizontals sends the two objects minus and+ of the simplicial computads defined in Example 615to ⊤ and 119886 and act on the non-trivial hom-spaces via the inclusions whose component in 119964(119886 119886) isconstant at the identity element at 119886 The fact that coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) isa pushout of 120794[120597Δ[119899]] 120794[Δ[119899]] can be verified by transposing across the adjunction of Proposition728 and applying the Yoneda lemma From this we see that coll(120572) ∶ coll119881 coll119882 is a transfinitecomposite of pushouts of coproducts of simplicial functors 120794[120597Δ[119899]] 120794[Δ[119899]] which proves thatthis map is a relative simplicial computad

Conversely if coll(120572) ∶ coll119881 coll119882 is a relative simplicial computad then it can be presentedas a countable composite of pushouts of coproducts of simplicial functors 120794[120597Δ[119899]] 120794[Δ[119899]]since this inclusion is bijective on objects the inclusion empty 120793 is not needed Since the only arrowsof coll119882 that are not present in coll119881 have domain ⊤ and codomain 119886 isin 119964 the characterization ofthe essential image of the collage functor of Proposition 728(i) allows us to identify each stage of thecountable composite

coll119881 coll(1198821) ⋯ coll(119882 119894) coll(119882 119894+1) ⋯ coll119882

as the collage of some weight 119882 119894 ∶ 119964 rarr 119982119982119890119905 Each attaching map 120794[120597Δ[119899]] rarr coll119882 119894 in the cellcomplex presentation acts on objects by mapping minus and + to ⊤ and 119886 for some 119886 isin 119964 and hencefactors through the top horizontal of the pushout square (7213) Hence the inclusion coll(119882 119894) coll(119882 119894+1) is a pushout of a coproduct of the maps coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus))one for each cell 120794[120597Δ[119899]] 120794[Δ[119899]] whose attaching map sends + to 119886 isin coll(119882 119894) As the collagefunctor is fully faithful we have now expressed coll(120572) ∶ coll119881 coll119882 as a countable compositeof pushouts of coproducts of simplicial functors coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) Afully faithful functor that preserves colimits also reflects them so in this way we see that 120572∶ 119881 119882is a countable composite of pushouts of coproducts of projective cells proving that it is a projectivecell complex as claimed

As the first of many applications we introduce the weights for pseudo limits by constructing theircollages and observe immediately that this class of weights is flexible

7214 Definition (weights for pseudo limits) For any simplicial set 119883 the coherent realization ofthe canonical inclusion 120793 + 119883 119883◁ defines a collage 120793 + ℭ119883 ℭ(119883◁) By Lemma 635 120793 +ℭ119883 ℭ(119883◁) is a simplicial subcomputad inclusion and hence by Lemma 6112 a relative simplicialcomputad Thus Theorem 7212 tells us that the collage 120793+ℭ119883 ℭ(119883◁) encodes a flexible weight119882119883 ∶ ℭ119883 rarr 119982119982119890119905 which we call the weight for the pseudo limit of a homotopy coherent diagramof shape 119883 The 119882119883-weighted limit of a homotopy coherent diagram of shape 119883 is then referred toas the pseudo weighted limit of that diagram

Since the left adjoint of the collage adjunction is fully faithful its unit is an isomorphism and thispermits us to define the weight119882119883 explicitly for a vertex 119909 isin 119883

119882119883(119909) ≔ ℭ(119883◁)(⊤ 119909)

We call119882119883 the weight for a ldquopseudordquo limit because we anticipate considering homtopy cohererentdiagrams valued in Kan-complex enriched categories in which all arrows in positive dimension areautomatically invertible By Corollary 733 119882119883-weighted limits also exist for diagrams valued ininfin-cosmoi which are only quasi-categorically enriched In such contexts it would bemore appropriateto refer to119882119883 as the weight for lax limits since in that context the 1-arrows of ℭ(119883◁) will likely mapto non-invertible morphisms

Exercises

72i Exercise Compute the collage of the weight 119880∶ 120793 rarr 119982119982119890119905 and use Theorem 7212 to give asecond proof that simplicial cotensors are flexible weighted limits Compare this argument with thatgiven in Example 723

72ii Exercise The inclusion into the join 120793+119883 119883◁ cong 120793⋆119883 is bijective on vertices The com-plement of the image contains a non-degenerate (119899 + 1)-simplex for each non-degenerate 119899-simplexof 119883 whose initial vertex is ⊤ and whose 0th face is isomorphic to the image of that simplex UseTheorem 6310 to describe the atomic arrows of the simplicial computad ℭ(119883◁) that are not in theimage of 120793 + ℭ119883

73 Homotopical properties of flexible weighed limits

In a119985-model categoryℳ the fibrant objects are closed under weighted limits whose weights areprojective cofibrant For instance the fibrant objects in a 119966119886119905-enriched model structure are closedunder flexible weighted limits [33 54] in the sense of [6] Specializing this argument to the case ofinfin-cosmoi we obtain the following result

731 Proposition (flexible weights are homotopical) Let119882∶ 119964 rarr 119982119982119890119905 be a flexible weight and let119974 be aninfin-cosmos

(i) The weighted limit lim119964119882 119865 of any diagram 119865∶ 119964 rarr 119974 may be expressed as a countable inverse limit

of pullbacks of products of isofibrations

119865119886Δ[119899] 119865119886120597Δ[119899] (732)

one for each projective 119899-cell at 119886 in the given projective cell complex presentation of119882(ii) If 119881 119882 isin 119982119982119890119905119964 is a projective cell complex between flexible weights then for any diagram

119865∶ 119964 rarr 119974 the induced map between weighted limits

lim119882 119865 ↠ lim119881 119865is an isofibration

(iii) If 120572∶ 119865 rArr 119866 is a simplicial natural transformation between a pair of diagrams 119865119866∶ 119964 119974 whosecomponents 120572119886 ∶ 119865119886 ⥲ 119866119886 are equivalences then the induced map

lim119882 119865 lim119882119866sim120572

is an equivalence

Proof To begin observe that the axioms of Definition 713 imply that the limit of 119865 weightedby the weight 119880 times 119964(119886 minus) for 119880 isin 119982119982119890119905 and 119886 isin 119964 is the cotensor 119865119886119880 Consequently the mapof weighted limits induced by the projective 119899-cell at 119886 is the isofibration (732) By definition anyflexible weight is built as a countable composite of pushouts of coproducts of these projective cellsand the weighted limit functor lim119964

minus 119865 carries each of these conical colimits to the correspondinglimit notion So it follows that lim119964

119882 119865 may be expressed as a countable inverse limit of pullbacks ofproducts of the maps (732) This proves 92

The same argument proves (ii) By definition a relative cell complex119881 119882 is built as a countablecomposite of pushouts of coproducts of these projective cells and the weighted limit functor lim119964

minus 119865carries each of these conical colimits to the corresponding limit notion So it follows that lim119964

119882 119865 isthe limit of a countable tower of isofibrations whose base in lim119964

119881 119865 where each of these isofibrationsis the pullback of products of the maps (732) appearing in the projective cell complex decompositionof119881 119882 As products pullbacks and limits of towers of isofibrations are isofibrations (ii) follows

In 92 we have decomposed each weighted limit lim119964119882 119865 as the limit of a tower of isofibrations

in which each of these isofibrations is the pullback of a product of the isofibrations (732) We argueinductively that if 120572∶ 119865 rArr 119866 is a componentwise equivalence then the map induced between thetowers of isofibrations for 119865 and for 119866 by the projective cell complex presentation of 119882 is a level-wise equivalence It follows from the standard argument in abstract homotopy theory reviewed inAppendix C that the inverse limit is then an equivalence proving (iii)

The bottom of the tower of isofibrations is limempty 119865 cong 1 cong limempty119866 which is certainly an equiva-lence For the inductive step observe that upon taking the map of weighted limits induced by eachprojective 119899-cell at 119886 in119882 we obtain a commutative square

119865119886Δ[119899] 119866119886Δ[119899]

119865119886120597Δ[119899] 119866119886120597Δ[119899]

sim120572Δ[119899]119886

sim120572120597Δ[119899]119886

defining a pointwise equivalence between the isofibrations the simplicial cotensor as cosmologi-cal functor preserves equivalences Now the product of squares of this form gives a commutativesquare whose horizontals are isofibrations and whose verticals are equivalences A pullback of thissquare forms the next layer in the tower of isofibrations by the inductive hypothesis the map be-tween the codomains of the pulled back isofibrations is already known to be an equivalence Nowthe equivalence-invariance of pullbacks of isofibrations established in Appendix C completes theproof

Immediately from the construction of Proposition 73192

733 Corollary (infin-cosmoi admit all flexible weighted limits) infin-cosmoi admit all flexible weightedlimits and cosmological functors preserve them

Our aim is now to describe a converse of sorts to Proposition 73192 which proves that the flexibleweighted limit of any diagram in an infin-cosmos can be constructed out of the limits of diagrams ofisofibrations axiomatized in 121(i) Over a series of lemmas we will construct each of the limitslisted there as instances of flexible weighted limits It will follow that any quasi-categorically enrichedcategory equipped with a class of representably-defined isofibrations that possesses flexible weighted

limits will admit all of the simplicial limits of 121(i) This will help us identify new examples ofinfin-cosmoi

To start simplicial cotensors are flexible weighted limits For any simplicial set 119880 the collage of119880∶ 120793 rarr 119982119982119890119905 is the simplicial computad 120794[119880] As 120793 + 120793 120794[119880] is a simplicial subcomputadinclusion Theorem 7212 tells us that 119880∶ 120793 rarr 119982119982119890119905 is a flexible weight this solves Exercise 72i

This leaves only the conical limits The weights for products are easily seen to be flexible directlyfrom Definition 721 However the weights for conical pullbacks or limits of towers of isofibrationsare not flexible because the definition of a cone over either diagram shape imposes composition rela-tions on 0-arrows

734 Example (the collage of the conical pullback) Let ⟓ denote the 1-category 119888 rarr 119886 larr 119887 Itscollage is the 1-category with four objects and five non-identity 0-arrows as displayed

⊤ 119887

119888 119886regarded as a constant simplicial category as in Example 614 Because the square commutes thiscategory is not free and hence does not define a simplicial computad though the subcategory 120793+ ⟓is free and hence is a simplicial computad Lemma 6111 tells us that the inclusion is not a relativesimplicial computad and so by Theorem 7212 the weight for the conical pullback is not flexible

Our strategy is to modify the weights for pullbacks and for limits of countable towers so that eachcomposition equation involved in defining cones over such diagrams is replaced by the insertion ofan ldquoinvertiblerdquo arrow of one dimension up where we must also take care to define this ldquoinvertibilityrdquowithout specifying any equations between arrows in the next dimension We have a device for speci-fying just this sort of isomorphism recall from Exercise 11iv(i) a diagram 120128 rarr Fun(119860 119861) specifies aldquohomotopy coherent isomorphismrdquo between a pair of 0-arrows 119891 and 119892 from 119860 to 119861 given bybull a pair of 1-arrows 120572∶ 119891 rarr 119892 and 120573∶ 119892 rarr 119891bull a pair of 2-arrows

119892 119891

119891 119891 119892 119892

120573Φ

120572Ψ

120572 120573

bull a pair of 3-arrows whose outer faces are Φ andΨ and whose inner faces are degeneratebull etc

We now introduce the weight for pullback diagrams whose cone shapes are given by squares in-habited by a homotopy coherent isomorphism

735 Definition (iso-commas) The iso-comma object 119862 ⨰119860119861 of a cospan

119862 119860 119861119892 119891

in a simplicially-enriched and cotensored category ℳ is the limit weighted by a weight 119882⨰ ∶ ⟓rarr119982119982119890119905 defined by the cospan

120793 120128 1207931 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119860120128

(11990211199020)120587

120587 119892times119891

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

where the maps (1199021 1199020) ∶ 119860120128 rarr 119860times119860 are defined by restricting along the endpoint inclusion 120793+120793 =120597120128 120128 In aninfin-cosmos this map is an isofibration and the equalizer defining the iso-comma objectis computed by the pullback

119862 ⨰119860119861 119860120128

119862 times 119861 119860 times 119860

(11990211199020)

119892times119891

737 Lemma Iso-comma objects are flexible weighted limits and in particular exist in anyinfin-cosmos

Proof Reprising the notation for the category ⟓ used in Example 734 the weight 119882⨰ is con-structed by the pushout

120597120128 times ⟓(119886 minus) ⟓ (119887 minus)⊔ ⟓(119888 minus)

120128 times ⟓(119886 minus) 119882⨰

where the attaching map picks out the two arrows in the cospan ⟓ As a projective cell complex119882⨰ is built from a project 0-cell at 119887 a projective 0-cell at 119888 and two projective 119896-cells at 119886 for each119896 gt 0 correspnoding to the non-degenerate simplices of 120128 As described by Remark 722 these maybe attached all at once In this way we see that119882⨰ is a flexible weight so Corollary 733 tells us thatiso-comma objects exist in anyinfin-cosmos a fact that is also evident from the pullback (736)

738 Remark In the homotopy 2-category of an infin-cosmos there is a canonical invertible 2-celldefining the iso-comma cone

119862 ⨰119860119861

119862 119861

119860

1199021 1199020120601cong

119892 119891

that has a weak universal property analogous to that of the comma cone presented in Proposition 336The proof like the proof of that result makes use of the fact that 119860120128 is the weak 120128-cotensor in thehomotopy 2-category The proof of this fact is somewhat delicate making use of marked simplicialsets as appeared already in the proof of Corollary 1116 which gives 1-cell induction

Our notation for iso-commas is deliberately similar to the usual notation for pullbacks In aninfin-cosmos iso-commas can be used to compute ldquohomotopy pullbacksrdquo of diagrams in which neithermap is an isofibration When at least one map of the cospan is an isofibration these constructions areequivalent

739 Lemma (iso-commas and pullbacks) In an infin-cosmos 119974 pullbacks and iso-commas of cospans inwhich at least one map is an isofibration are equivalent More precisely given a pullback square as below-leftand an iso-comma square as below-right

119875 119861 119862 ⨰119860119861 119861

119862 119860 119862 119860

119888

119887

119891 1199021 120601cong

1199020

119891

119892 119892

119875 ≃ 119862 ⨰119860119861 over 119862 and up to isomorphism over 119861

Proof Applying Lemma 1212 to the functor 119887 ∶ 119875 rarr 119861 we can replace the span (119888 119887) ∶ 119875 rarr119862times119861 by a span (119888119902 119901) ∶ 119875119887 ↠ 119862times119861whose legs are both isofibrations that is related via an equivalence119904 ∶ 119875 ⥲ 119875119887 that lies over 119862 on the nose and over 119861 up to isomorphism We will show that under thehypothesis that 119891 is an isofibration this new span is equivalent to the iso-comma span

To see this note that the factorization constructed in (1213) is in fact defined using an iso-commaconstructed via the pullback in the top square of the diagram below-left Since the map 119887 is itselfdefined by a pullback the bottom square of the diagram below-left is also a pullback defining theleft-hand pullback rectangle

119875119887 119861120128

119875 times 119861 119861 times 119861

119862 times 119861 119860 times 119861

(119902119901) (11990211199020)

119887times119861

119888times119861 119891times119861

119892times119861

119862 ⨰119860119861 119860 ⨰

119860119861 119860120128

119862 times 119861 119860 times 119861 119860 times 119860

(11990211199020)

119892times119861 119860times119891

Now the iso-comma is constructed by a similar pullback rectangle displayed above-right And because119891 is an isofibration Lemma 1210 tells us that the Leibniz tensor 1198940 1114023⋔ 119891∶ 119861120128 ⥲rarr 119860⨰

119860119861 of 1198940 ∶ 120793 120128

with 119891∶ 119861 ↠ 119860 is a trivial fibration This equivalence commutes with the projections to 119860 times 119861 andhence the maps (119888119902 119901) ∶ 119875119887 ↠ 119862 times 119861 and (1199021 1199020) ∶ 119862 ⨰119860 119861 ↠ 119862 times 119861 defined as pullbacks of anequivalence pair of isofibrations along 119892 times 119861 are equivalent as claimed

We now introduce a flexible weight diagrams given by a countable tower of 0-arrows whose coneshapes will have a homotopy coherent isomorphism in the triangle over each generating arrow in thediagram

7310 Definition (iso-towers) Recall the category120654 whose objects are natural numbers and whosemorphisms are freely generated by maps 120580119899119899+1 ∶ 119899 rarr 119899 + 1 for each 119899

The iso-tower of a diagram 119865∶ 120654op rarrℳ in a simplicially enriched and cotensored categoryℳis the limit weighted by the diagram119882larr ∶ 120654op rarr 119982119982119890119905 defined by the pushout

∐119899isin120654

120597120128 times 120654(minus 119899) ∐119899isin120654

120654(minus 119899)

∐119899isin120654

120128 times 120654(minus 119899) 119882larr

(id119899120580119899119899+1)

(7311)

in 119982119982119890119905120654op

By Definition 713(ii) in aninfin-cosmos the iso-tower of a diagram

119865 ≔ ⋯ 119865119899+1 119865119899 ⋯ 1198651 1198650119891119899+2119899+1 119891119899+1119899 119891119899119899minus1 11989121 11989110

is constructed by the pullback

lim119882larr 119865 prod119899isin120654

119865120128119899

prod119899isin120654

119865119899 prod119899isin120654

119865119899 times 119865119899

120588

120601

prod(11990211199020)

(119891119899+1119899id119865119899)

(7312)

The limit cone is generated by a 0-arrow120588119899 ∶ lim119882larr 119865 rarr 119865119899 for each119899 isin 120654 together with a homotopycoherent isomorphism 120601119899 in each triangle over a generating arrow 119865119899+1 rarr 119865119899 in the 120654op-indexeddiagram

7313 Lemma Iso-towers are flexible weighted limits and in particular exist in anyinfin-cosmos

Proof The weight 119882larr is a projective cell complex built by attaching one projective 0-cell ateach 119899 isin 120654 mdash forming the coproduct appearing in the upper right-hand corner of (7311) mdash andthen by attaching a projective 119896-cell at each 119899 isin 120654 for each non-degenerate 119896-simplex of 120128 Ratherthan attach each projective 119896-cell for fixed 119899 isin 120654 in sequence by Remark 722 these can all beattached at once by taking a single pushout of the ldquogeneralized projective cell at 119899rdquo defined by themap 120597120128 times 120654(minus 119899) 120128 times 120654(minus 119899) These are the maps appearing as the left-hand vertical of (7311)Now Corollary 733 or the formula (7312) make it clear that such objects exist in anyinfin-cosmos

7314 Lemma (iso-towers and inverse limits) In aninfin-cosmos 119974 the inverse limit of a countable towerof isofibrations is equivalent to the iso-pullback of that tower

Proof We will rearrange the formula (7312) to construct the iso-tower lim119882larr 119865 as an inverselimit of a countable tower of isofibrations 119875∶ 120654op rarr 119974 that is pointwise equivalent to the diagram119865∶ 120654op rarr119974 In the case where the diagram 119865 is also given by a tower of isofibrations

lim119875 cong ⋯ 119875119899+1 119875119899 ⋯ 1198751 1198750

lim 119865 cong ⋯ 119865119899+1 119865119899 ⋯ 1198651 1198650

119901119899+2119899+1 119901119899+1119899

sim 119890119899

119901119899119899minus1

sim 119890119899

11990121 11990110

sim 1198901 sim 1198900

119891119899+2119899+1 119891119899+1119899 119891119899119899minus1 11989121 11989110

(7315)

the equivalence invariance of the inverse limit of a diagram of isofibrations will imply that the limitslim119882larr 119865 cong lim119875 and lim 119865 are equivalent as claimed

Theinfin-categories 119875119899 will be defined as conical limits of truncated versions of the diagram (7312)To start define 1198750 ≔ 1198650 and 1198900 to be the identity then define 1198751 11990110 and 1198901 via the pullback

1198751 1198651201280 1198650

1198651 1198650

11990110

sim1198901

sim 1199021

sim1199020

11989110

Note that 1198751 cong 1198651 ⨰11986501198650 computes the iso-comma objects of the cospan given by id1198650 and 11989110

Now define 1198752 11990121 and 1198902 using the composite pullback

1198752 bull 1198751 1198651201280 1198650

bull 1198651201281 1198651 1198650

1198652 1198651

11990121

sim1198902

sim1198901

sim 1199021

sim1199020

sim 1199021

sim1199020 11989110

11989121

Continuing inductively 119875119899 119901119899119899minus1 and 119890119899 are defined by appending the diagram

119865120128119899minus1 119865119899minus1

119865119899 119865119899minus1

sim1199020

sim 1199021

119891119899119899minus1

to the limit cone defining 119875119899minus1 and taking the limit of this composite diagramThere is one small problem with the construction just given it defines a diagram (7315) in which

each square commutes up to isomorphism mdash the isomorphism encoded by the map 119875119899 rarr 119865120128119899minus1 mdashnot on the nose But because the maps 119891119899+1119899 are isofibrations this is no problem The isomorphisminhabiting the square 119890011990110 cong 119891101198901 can be lifted along 11989110 to define a new map 119890prime1 ∶ 1198751 ⥲ 1198651isomorphic to 1198901 as observed in the proof of Theorem 147 this 119890prime1 is then also an equivalence sowe replace 1198901 with 119890prime1 and then continue inductively to lift away the isomorphisms in the square119890prime111990121 cong 119891211198902

Since inverse limits of towers of isofibrations are equivalence-invariant it follows that lim119875 ≃lim 119865 By construction lim119875 cong lim119882larr 119865 so it follows that lim119882larr 119865 ≃ lim 119865 which is what wewanted to show

Exercises

73i Exercise Verify mdash either directly from Definition 721 or by applying Theorem 7212 mdash thatconical products are flexible weighted limits

74 Moreinfin-cosmoi

Our aim in this section is to introduce further examples ofinfin-cosmoiOur first example is a special case of a more general result that will appear in Appendix E that we

nonetheless spell out in detail to illustrate the ideas involved in this sort of argument The walkingarrow category 120794 is an inverse Reedy category where the domain of the non-identity arrow is assignedldquodegree 1rdquo and the codomain is assigned ldquodegree zerordquo This Reedy structure motivates the definitionsin theinfin-cosmos of isofibrations that we now introduce

741 Proposition (infin-cosmoi of isofibrations) For anyinfin-cosmos119974 there is aninfin-cosmos119974120794 whose(i) objects are isofibrations 119901∶ 119864 ↠ 119861 in119974(ii) functor-spaces say from 119902 ∶ 119865 ↠ 119860 to 119901∶ 119864 ↠ 119861 are defined by pullback

Fun(119865119902minusrarrrarr 119860119864

119901minusrarrrarr 119861) Fun(119865 119864)

Fun(119860 119861) Fun(119865 119861)

119901lowast

119902lowast

(iii) isofibrations from 119902 to 119901 are commutative squares

119865 119864bull

119860 119861

119902

119892

119901

119891

in which the horizontals and the induced map from the initial vertex to the pullback of the cospan areisofibrations in119974

(iv) limits are defined pointwise in119974(v) and in which a map

119865 119864

119860 119861

119902

119892

119901

sim119891

is an equivalence in theinfin-cosmos119974120794 if and only if 119892 and 119891 are equivalences in119974Relative to these definitions the domain codomain and identity functors

119974120794 119974dom

are all cosmological

Proof The diagram category119974120794 inherits its simplicially enriched limits defined pointwise from119974 The functor-spaces described in (ii) are the usual ones for an enriched category of diagrams Thisverifies 121(i)

For axiom 121(ii) note that the product and simplicial cotensor functors carry pointwise isofibra-tions to isofibrations The pullback of an isofibration as in (iii) along a commutatative square from an

isofibration 119903 to 119901 may be formed in119974 Our task is to show that the induced map 119905 is an isofibrationand also that the square from 119905 to 119903 is an isofibration in the sense of (iii)

119866 times119864119865 119865

119866 119864bull bull

119862 times119861119860 119860

119862 119861

119905 119902

119911

119901 119903

The map 119905 factors as a pullback of 119911 followed by a pullback of 119903 as displayed above and is thus an isofi-bration as claimed This observation also verifies that the square from 119905 to 119903 defines an isofibrationA similar argument verifies the Leibniz stability of the isofibrations and that the limit of a tower ofisofibration is an isofibartion This proves that 119974120794 defines an infin-cosmos in such a way so that thedomain codomain and identity functors are cosmological

Finally since pullbacks of isofibrations in 119980119966119886119905 are invariant under equivalences a pair of equiv-alences (119892 119891) induces an equivalence between the functor-spaces defined in (ii) The converse that anequivalence in119974120794 defines a pair of equivalences in119974 follows from the fact that domain and codomain-projection functors are cosmological and Lemma 132

In close analogy with Proposition 353 we have a smothering 2-functor that relates the homo-topy 2-category of 119974120794 to the 2-category of isofibrations commutative squares and parallel naturaltransformations in the homotopy 2-category of119974

743 Lemma There is an identity on objects and 1-cells smothering 2-functor 120101(119974120794) rarr (120101119974)120794 whosecodomain is the 2-category whosebull objects are isofibrations in119974bull 1-cells are commutative squares between suchbull 2-cells are pairs of 2-cells in 120101119974

119865 119864

119860 119861

119892

119892prime119902

dArr120572

119901119891

119891prime

dArr120573

Proof Exercise 74i

For anyinfin-cosmos119974 and any subcategory of its underlying 1-category mdash that is for any subset ofits objects and subcategory of its 0-arrows mdash one can form a quasi-categorically enriched subcategoryℒ sub 119974 that contains exactly those objects and 0-arrows and all higher dimensional arrows that theyspan We call such subcategoriesℒ full on positive-dimensional arrows note the functor spaces ofℒare quasi-categories because all inner horn inclusions are bijective on vertices We will take particularinterest in subcategories that satisfy a further ldquorepletenessrdquo condition

744 Definition Let 119974 be an infin-cosmos A subcategory ℒ sub 119974 is replete in 119974 if it is full onpositive-dimensional arrows and moreover

(i) Everyinfin-category in119974 that is equivalent to an object inℒ lies inℒ(ii) Any equivalence in119974 between objects inℒ lies inℒ(iii) Any arrow in119974 that is isomorphic in119974 to an arrow inℒ lies inℒ

745 Lemma Suppose ℒ sub 119974 is a replete subcategory of an infin-cosmos Then any map 119901∶ 119864 rarr 119861 inℒ that defines an isofibration in 119974 is a representably-defined isofibration in ℒ that is for all 119883 isin ℒ119901lowast ∶ Funℒ(119883 119864) ↠ Funℒ(119883 119861) is an isofibration of quasi-categories

Proof Since 119974 is an infin-cosmos axiom 121(ii) requires that 119901lowast ∶ Fun119974(119883 119864) ↠ Fun119974(119883 119861)is an isofibration of quasi-categories Because the inner horn inclusions are bijective on vertices andFunℒ(119883 119864) Fun119974(119883 119864) is full on positive-dimensional arrows it follows immediately that therestricted map 119901lowast ∶ Funℒ(119883 119864) ↠ Funℒ(119883 119861) lifts against the inner horn inclusions Thus it remainsonly to solve lifting problems of the form displayed below-left

120793 Funℒ(119883 119864) Fun119974(119883 119864)

120128 Funℒ(119883 119861) Fun119974(119883 119861)

119890

120573

The lifting problem defines a 0-arrow 119890 ∶ 119883 rarr 119864 in ℒ and an isomorphism 120573∶ 119887 cong 119901119890 in ℒ Itssolution in119974 defines a 0-arrow 119890prime ∶ 119883 rarr 119864 in119974 so that 119901119890prime = 119887 together with an isomorphism 119890 cong 119890primein119974 By fullness on positive-dimensional arrows to show that this lift factors through the inclusionFunℒ(119883 119864) Fun119974(119883 119864) we need only argue that the map 119890prime lies in ℒ but this is the case bycondition (iii) of Definition 744

The following result describes a condition under which a replete subcategory ℒ sub 119974 inherits aninfin-cosmos structure created from119974

746 Proposition Suppose ℒ sub 119974 is a replete subcategory of aninfin-cosmos If ℒ is closed under flexibleweighted limits in 119974 then ℒ defines an infin-cosmos with isofibrations equivalences trivial fibrations andsimplicial limits created by the inclusionℒ 119974 which then defines a cosmological functor

When these conditions hold we refer to ℒ as a replete sub infin-cosmos of 119974 and ℒ 119974 as acosmological embedding

Proof To say that a replete subcategory ℒ 119974 is closed under flexible weighted limits meansthat for any diagram in ℒ and any limit cone in119974 that limit cone lies in ℒ and satisfies appropriatesimplicially-enriched universal property of Definition 717 in there We must verify that each of thelimits of axiom 121(i) exist in ℒ Immediately ℒ has a terminal object products and simplicialcotensors since all of these are flexible weighted limits By Lemmas 737 and 7313ℒ also admits theconstruction of iso-comma objects and of iso-towers

Define the class of isofibrations in ℒ to be those maps in ℒ that define isofibrations in 119974 ByLemmas 739 and 7314 pullbacks and limits of towers of isofibrations are equivalent in 119974 to theiso-commas and iso-towers formed over the same diagrams Since these latter limit cones lie in ℒ byhypothesis so do the equivalence former cones by repleteness ofℒ in119974

There is a little more still to verify namely that pullbacks and limits of towers of isofibrationssatisfy the simplicially-enriched universal property as conical limits in ℒ In the case of a pullbackdiagram

119875 119861

119862 119860

119888

119887

119891

119892

inℒ we must show that for each 119883 isin ℒ the functor-space Funℒ(119883 119875) is isomorphic to the pullbackFunℒ(119883 119862)timesFunℒ(119883119860)Funℒ(119883 119861) of functor spaces We have such an isomorphism for functor spacesin119974 and on account of the commutative diagram

Funℒ(119883 119875) Funℒ(119883 119862) timesFunℒ(119883119860)

Funℒ(119883 119861)

Fun119974(119883 119875) Fun119974(119883 119862) timesFun119974(119883119860)

Fun119974(119883 119861)cong

and fullness on positive-dimensional arrows we need only verify surjectivity of the dotted map on0-arrows So consider a cone (ℎ ∶ 119883 rarr 119861 119896 ∶ 119883 rarr 119862) over the pullback diagram inℒ By the universalproperty of the isocomma119860⨰

119861119864 there exists a factorization 119910∶ 119883 rarr 119862⨰

119860119861 inℒ Composing with the

equivalence119862⨰119860119861 ≃ 119875 this map is equivalent to the factorization 119911 ∶ 119883 rarr 119875 of the cone (ℎ 119896) through

the limit cone (119887 119888) in119974 that exists on account of the strict universal property of the pullback in thereBy repleteness the isomorphism between 119911 and the composite of 119910 with the equivalence suffices toshow that 119911 lies in ℒ Hence the functor spaces in ℒ are isomorphic A similar argument invokingLemma 7314 proves that inverse limits of towers of isofibrations define conical limits in ℒ Thiscompletes the proof of the limit axiom 121(i)

Since the isofibrations in ℒ are a subset of the isofibrations in 119974 and the limit constructionsin both contexts coincide most of the closure properties of 121(ii) are inherited from the closureproperties in119974 The one exception is the requirement that the isofibrations inℒ define isofibrationsof quasi-categories representably which was proven for any replete subcategory in Lemma 745 Thisproves thatℒ defines aninfin-cosmos

Finally we argue that the equivalences in ℒ coincide with those of 119974 which will imply that thetrivial fibrations inℒ coincide with those of119974 as well Condition (ii) of Definition 744 implies thatfor any arrow inℒ that defines an equivalence in119974 its equivalence inverse and witnessing homotopiesof Lemma 1215 lie inℒ Because we have already shown thatℒ admits cotensors with 120128 preserved bythe inclusion ℒ 119974 Lemma 1215 implies that this data defines an equivalence in ℒ Converselyany equivalence inℒ extends to the data of (1216) and sinceℒ 119974 preserves 120128-cotensors this datadefines an equivalence in 119974 Thus by construction the infin-cosmos structure of ℒ is preserved andreflected by the inclusionℒ 119974 as claimed

In practice the repleteness condition of Definition 744 is satisfied by any subcategory of objectsand 0-arrows that is determined by someinfin-categorical property so the main task in verifying that asubcategory defines aninfin-cosmos is verifying the closure under flexible weighted limits

747 Proposition For anyinfin-cosmos119974 let119974⊤ denote the quasi-categorically enriched category whose(i) objects areinfin-categories in119974 that possess a terminal object

(ii) functor spaces Fun⊤(119860 119861) sub Fun(119860 119861) are the sub-quasi-categories whose 0-arrows preserve termi-nal objects and containing all 119899-arrows they span

Then the inclusion119974⊤ 119974 creates aninfin-cosmos structure on119974⊤ from119974 and moreover for each object of119974⊤ defined as a flexible weighted limit of some diagram in119974⊤ its terminal element is created by the 0-arrowlegs of the limit cone

Proof We apply Proposition 746 Lemma 226 and Proposition 2110 verify the repletenesscondition so it remains only to prove closure under flexible weighted limits which we do by inductionover the tower of isofibrations constructed in Proposition 73192 which expresses a flexible weightedlimit lim119882 119865 as the inverse limit of a tower of isofibrations

lim119882 119865 ⋯ lim119882119896+1 119865 lim119882119896 119865 ⋯ lim1198820 119865 1

each of which is a pullback of products of maps of the form (732) indexed by the projective cells ofthe flexible weight119882 Wersquoll argue inductively that eachinfin-category in this tower possesses a terminalelement thatrsquos created by the legs of the tower of isofibrations

For the base case note that if (119860119894)119894isin119868 is a family of infin-categories possessing terminal elements119905119894 ∶ 1 rarr 119860119894 then the product of the adjunctions ⊣ 119905119894 defines an adjunction

1 cong prod119894isin119868 1 prod119894isin119868119860119894perp

(119905119894)119894isin119868congprod119894 119905119894

exhibiting (119905119894)119894isin119868 as a terminal element of prod119894isin119868119860119894 By construction this terminal element is jointlycreated by the legs of the limit cone Note that by construction the product-projection functors pre-serve this terminal element and the map into the productinfin-categoryprod119894isin119868119860119894 induced by any familyof terminal element preserving functors (119891 ∶ 119883 rarr 119860119894)119894isin119868 will preserve terminal elements This verifiesthat the subcategory119974⊤ is closed under products

For the inductive step consider a pullback diagram

lim119882119896+1 119865 119860Δ[119899]

lim119882119896 119865 119860120597Δ[119899]

that arises from the attaching map for a projective 119899-cell The inductive hypothesis tells us thatlim119882119896 119865 admits a terminal element 119905119896 and for each vertex of 119894 isin 120597Δ[119899] the corresponding com-ponent ℓ119894 ∶ lim119882119896 119865 rarr 119860 of the limit cone preserves it Since 119865 is a diagram valued in119974⊤ and119860 is aninfin-category in its image we know that 119860 must possess a terminal element 119905 ∶ 1 rarr 119860 By Proposition217(iii) the constant diagram at 119905 then defines a terminal element in 119860120597Δ[119899] and 119860Δ[119899] which wealso denote by 119905 By terminality there is a 1-arrow 120572∶ ℓ(119905119896) rarr 119905 isin 119860120597Δ[119899] whose components at each119894 isin 120597Δ[119899] are isomorphisms in 119860 By Lemma it follows that 120572 is also an isomorphism which tellsus that ℓ(119905119896) is also a terminal element of 119860120597Δ[119899] The same argument demonstrates that terminalelements in simplicial cotensors in this case by 120597Δ[119899] are jointly created by the 0-arrow componentsof the limit cone namely by evaluation on each of the vertices of the cotensoring simplicial set Theproof is completed now by the following lemma

748 Lemma Consider a pullback diagram

119865 119864

119860 119861

119892

119902

119901

119891

in which the infin-categories 119860 119861 and 119864 possess a terminal element and the functors 119891 and 119901 preserve themThen 119865 possesses a terminal element that is created by the legs of the pullback cone 119902 and 119892

Proof If 119890 ∶ 1 rarr 119864 and 119886 ∶ 1 rarr 119860 are terminal then this implies that 119891(119886) cong 119901(119890) isin 119861 Using thefact that 119901 is an isofibration there is a lift 119890prime cong 119890 of this isomorphism along 119901 that then defines anotherterminal element of 119864 The pair (119886 119890prime) now induces an element 119905 of 119865 that we claim is terminal

To see this wersquoll apply Proposition 4310 which proves that 119905 is a terminal element of 119865 if andonly if the domain-projection functor 1199010 ∶ Hom119865(119865 119905) ↠ 119865 is a trivial fibration By construction of119905 we know that the domain-projection functors for the elements 119892119905 119902119905 and 119901119892119905 = 119891119902119905 are all trivialfibrations and moreover the top and bottom faces of the cube

Hom119865(119865 119905) Hom119864(119864 119892119905)

Hom119860(119860 119902119905) Hom119861(119861 119891119902119905)

119865 119864

119860 119861

1199010

sim 1199010

sim1199010119902

119892

119901

119891

1199010≀

are pullbacks Since homotopy pullbacks are homotopical the fact that the three maps between thecospans are equivalences implies that the map between their pullbacks is also an equivalence as re-quired

Applying the result of Proposition 747 to 119974co constructs an infin-cosmos 119974perp whose objects areinfin-categories in119974 that possess an initial object and 0-arrows are initial-element-preserving functorsCombining these we get an infin-cosmos for the pointed infin-categories of Definition 441 those thatpossess a zero element

749 Proposition For anyinfin-cosmos119974 let119974lowast denote the quasi-categorically enriched category of pointedinfin-categories ieinfin-categories that possess a zero element and functors that preserve them Then the inclusion119974lowast 119974 creates aninfin-cosmos structure on119974⊤ from119974

Proof The natural inclusions define a pullback diagram of quasi-categorically enriched cate-gories

119974perp⊤ 119974⊤

119974perp 119974

where the objects of 119974perp⊤ are infin-categories that possess both an initial and terminal element andfunctors that preserve them separately

Now repleteness of 119974perp⊤ follows from repleteness of 119974⊤ and 119974perp as does closure under flexibleweighted limits Given a diagram in 119974perp⊤ admitting a flexible weighted limit in 119974 that limit conelies in both 119974⊤ and 119974perp and hence in their intersection Since the functor spaces of 119974perp⊤ are theintersections of the functor spaces of 119974⊤ and 119974perp in 119974 the simplicial universal property in thoseinfin-cosmoi restricts to the required simplicially-enriched universal property in119974perp⊤

Now theinfin-cosmos119974lowast of interest is the full subcategory of119974perp⊤ spanned by thoseinfin-categoriesin which the initial and terminal element coincide To prove that 119974lowast is an infin-cosmos created bycosmological embedding 119974lowast 119974perp⊤ it remains only to show that this subcategory is closed un-der flexible weighted limits To that end consider a diagram 119865∶ 119964 rarr 119974perp⊤ and a flexible weight119882∶ 119964 rarr 119982119982119890119905 We have shown that the limit lim119882 119865 admits an initial element 119894 ∶ 1 rarr lim119882 119865 andalso a terminal element 119905 ∶ 1 rarr lim119882 119865 each of which is preserved the the 0-arrow legs of the limitcone The elements 119894 and 119905 are vertices in the underlying quasi-category Fun(1 lim119882 119865) and remainrespectively initial and terminal in there Appealing to the universal property of either there is a1-simplex 120572∶ 119894 rarr 119905 isin Fun(1 lim119882 119879) representing a natural transformation 120572∶ 119894 rArr 119905 in 120101119974 andlim119882 119865 is a pointedinfin-category if and only if 120572 is invertible

Invertibility of 1-simplices in quasi-categories can be detected by applying the cosmological func-tor (minus) ∶ 119980119966119886119905 rarr 1-119966119900119898119901 of Example 137 a 1-simplex in Fun(1 lim119882 119865) is an isomorphism if andonly if its image in Fun(1 lim119882 119865) is marked Corollary 733 proves that the cosmological functorFun(1 minus) ∶ 119974 rarr 1-119966119900119898119901 presrves flexible weighted limits so

Fun(1 lim119882 119865) cong lim119882 Fun(1 119865minus)and thus it remains only to show that a 1-simplex in a flexibleweighted limit of a diagramof 1-complicialsets whose image under each of the legs of the limit cone is marked is marked in the weighted limitIndeed the markings for all simplicially enriched limits in 1-119966119900119898119901 are defined in this manner a1-simplex is marked if and only if each of its components is marked So this proves that119974lowast 119974perp⊤is closed under flexible weighted limits completing the proof

Applying the result of Proposition 747 or its dual to the infin-cosmos 119974119861 of isofibrations over119861 isin 119974 we obtain two newinfin-cosmoi of interest

7410 Corollary For anyinfin-category119861 in aninfin-cosmos119974 the slicedinfin-cosmos119974119861 admits subinfin-cosmoi

ℛ119886119903119894(119974)119861 119974119861 ℒ119886119903119894(119974)119861whosebull objects are isofibrations over 119861 admitting a right adjoint right inverse or left adjoint right inverse respec-

tively andbull 0-arrows are functors over 119861 that commute with the respective right or left adjoints up to fibered isomor-

phismwith theinfin-cosmos structures created by the inclusions

Proof Theseinfin-cosmoi are defined byℛ119886119903119894(119974)119861 ≔ (119974119861)⊤ andℒ119886119903119894(119974)119861 ≔ (119974119861)perp

Leveraging Corollary 7410 we can establish similar cosmological embeddings

ℛ119886119903119894(119974) 119974120794 ℒ119886119903119894(119974)

The quasi-categorically enriched subcategoriesℛ119886119903119894(119974) andℒ119886119903119894(119974) are replete in119974120794 so by Propo-sition 746 we need only check closure under flexible weighted limits We argue separately for coten-sors which are easy and for the conical limits which are harder For this we make use of a general1-categorical result making use of the fact that the codomain-projection functor cod ∶ ℛ 119886119903119894(119974) rarr 119974is a Grothendieck fibration of underlying 1-categories defined by restricting the Grothendieck fibra-tion cod ∶ 119974120794 rarr119974

7411 Lemma Let 119875∶ ℰ rarr ℬ be a Grothendieck fibration between 1-categories Suppose that 119973 is a smallcategory that 119863∶ 119973 rarr ℰ is a diagram and that

(i) the diagram 119875119863∶ 119973 rarr ℬ has a limit 119871 in ℬ with limiting cone 120582∶ Δ119871 rArr 119863(ii) the diagram 120582lowast119863∶ 119973 rarr ℰ119871

119973 ℰ 119973 ℰ

ℬ ℬ

119863

Δ119871uArr120582 119875 =

119863

120582lowast119863

uArr120594

119875

constructed by lifting the cone 120582 to a cartesian natural transformation 120594∶ 120582lowast119863 rArr 119863 has a limit119872in the fibre ℰ119871 with limiting cone 120583∶ Δ119872 rArr 120582lowast119863 and

(iii) the limit 120583∶ Δ119872 rArr 120582lowast119863 is preserved by the re-indexing functor 119906lowast ∶ ℰ119871 rarr ℰ119861 associated with anyarrow 119906∶ 119861 rarr 119871 in ℬ

Then the composite cone

Δ119872 120582lowast119863 119863120583 120594

displays119872 as a limit of the diagram 119863 in ℰ

Proof Any arrow 119891∶ 119864 rarr 119864prime in the domain of a Grothendieck fibration 119875∶ ℰ rarr ℬ factorsuniquely up to isomorphism through a ldquoverticalrdquo arrow in the fiber ℰ119875119864 followed by a ldquohorizontalrdquocartesian lift of 119875119891 with codomain 119864prime

Given a cone 120572∶ Δ119864 rArr 119863 with summit 119864 isin ℰ over 119863 by (i) its image 119875120572∶ Δ119875119864 rArr 119875119863 factorsuniquely through the limit cone 120582∶ Δ119871 rArr 119863 via a map 119887 ∶ 119875119864 rarr 119871 isin ℬ By the universal property ofthe cartesian lift 120594 of 120582 constructed in (ii) it follows that 120572 factors uniquely through 120594 via a naturaltransformation 120573∶ Δ119864 rArr 120582lowast119863 so that 119875120573 = Δ119887 This arrow factors uniquely up to isomorphism vialdquoverticalrdquo natural transformation 120574∶ Δ119864 rarr 120572lowast119863 cong 119887lowast120574lowast119863 followed by a ldquohorizontalrdquo cartesian lift of119887 By (iii) the limit cone 120583∶ Δ119872 rArr 120582lowast119863 in ℰ119871 pulls back along 119887 to a limit cone in ℰ119875119864 throughwhich the pullback of 120573 factors via a map 119896 ∶ 119864 rarr 119887lowast119872 Thus 120573 itself factors uniquely through 120583 viathe composite of this map 119896 ∶ 119864 rarr 119887lowast119872 with the cartesian arrow 119887lowast119872rarr119872 lifting 119887 ∶ 119875119864 rarr 119871

7412 Proposition For anyinfin-cosmos119974 theinfin-cosmos of isofibrations admits subinfin-cosmoi

ℛ119886119903119894(119974) 119974120794 ℒ119886119903119894(119974)

whosebull objects are isofibrations admitting a right adjoint right inverse or left adjoint right inverse respectively

andbull 0-arrows are commutative squares between the right or left adjoints respectively whose mates are isomor-

phisms

with theinfin-cosmos structures created by the inclusions

We refer to a commutative square between right adjoints whose mate is an isomorphism as anexact square

Proof The quasi-categorically enriched subcategoriesℛ119886119903119894(119974) andℒ119886119903119894(119974) are replete in119974120794so by Proposition 746 we need only check that ℛ119886119903119894(119974) 119974120794 is closed under flexible weightedlimits the argument forℒ119886119903119894(119974) 119974120794 is dual We argue separately for cotensors and for the conicallimits

If 119901∶ 119864 ↠ 119861 is an isofibration admitting a right adjoint right inverse in 119974 and 119880 is a simplicialset then the cosmological functor (minus)119880 ∶ 119974 rarr 119974 carries this data to a right adjoint right inverseto 119901119880 ∶ 119864119880 ↠ 119861119880 which proves that the simplicial cotensor in 119974120794 of an object in ℛ119886119903119894(119974) liesin ℛ119886119903119894(119974) The limit cone for the cotensor is given by the canonical map of simplicial sets 119880 rarr

Fun(119864119880119901119880minusminusrarrrarr 119861119880 119864

119901minusrarrrarr 119861) defined on each vertex 119906∶ 120793 rarr 119880 by the commutative square

119864119880 119864

119861119880 119861

119901119880

119906lowast

119901

119906lowast

(7413)

The maps 119906lowast define the components of a simplicial natural transformation from (minus)119880 to the iden-tity functor and thus the mate of this commutative square is an identity so the limit cone for the119880-cotensor lies inℛ119886119903119894(119974) Finally to verify the universal property of the cotensor inℛ119886119903119894(119974) wemust show that for any commutative square whose domain is an isofibration admitting a right adjointright inverse

119865 119864119880

119860 119861119880119902 119901119880

that composes with each of the squares (7413) to an exact square is itself exact To see this takethe mate to define a 1-arrow in Fun(119860 119864119880) cong Fun(119860 119864)119880 and note that the hypothesis says thatthe components of this 1-arrow are invertible for each vertex of 119880 Lemma then tells us that this1-arrow is invertible as required

Taking 119880 to be a set the argument just given proves also that ℛ119886119903119894(119974) is closed in 119974120794 underproducts It remains only to show that it is closed under the remaining conical limits By pullbackstability of fibered adjunctions the Grothendieck fibration of 1-categories cod ∶ 119974120794 rarr 119974 restrictsto cod ∶ ℛ 119886119903119894(119974) rarr 119974 so we may appeal to Lemma 7411 to calculate 1-categorical limit cones inℛ119886119903119894(119974) sub 119974120794 as composites of cartesian cells with limit cones of fiberwise diagrams By Corol-lary 7410 these fiberwise limits in 119974119861 of diagrams in ℛ119886119903119894(119974)119861 lie in ℛ119886119903119894(119974)119861 ℛ119886119903119894(119974)Moreover these 1-categorical limits are preserved by the simplicial cotensor which by PropositionA56 implies that their universal property enriches to define conical limits In this way we see thatℛ119886119903119894(119974) 119974120794 is closed under flexible weighted limits and thus defines a cosmological embeddingas claimed

Proposition 7412 allows us to construct furtherinfin-cosmoi of interest

7414 Proposition For anyinfin-cosmos119974 and simplicial set 119869 there exist subinfin-cosmoi

119974⊤119869 119974 119974perp119869

whosebull objects areinfin-categories in119974 that admit all limits of shape 119869 or all colimits of shape 119869 respectivelybull 0-arrows are the functors that preserve them

with theinfin-cosmos structures created by the inclusions Moreover for each object of 119974⊤119869 or 119974perp119869 defined asa flexible weighted limit of some diagram in that infin-cosmos its 119869-shaped limits or colimits arecreated by the0-arrow legs of the limit or colimits cones respectively

Proof First note that the quasi-categorically enriched subcategories119974⊤119869 and119974perp119869 are replete in119974 so by Proposition 746 we need only confirm that the inclusions are closed under flexible weightedlimits We prove this in the case of colimits the other case being dual

For any fixed simplicial set 119869 there is a cosmological functor 119865119869 ∶ 119974 rarr 119974120794 defined on objectsby mapping an infin-category 119860 to the isofibration 119860119869▷ ↠ 119860119869 in the notation of 426 and a functor119891∶ 119860 rarr 119861 to the commutative square

119860119869▷ 119861119869▷

119860119869 119861119869

119891119869▷

119891119869

By Corollary 435 119860 admits colimits of shape 119869 if and only if this isofibration admits a left adjointright inverse and now it is clear that 119891∶ 119860 rarr 119861 preserves these colimits if and only if the squaredisplayed above is exact In summary the quasi-categorically enriched subcategory119974perp119869 is defined bythe pullback

119974perp119869 ℒ119886119903119894(119974)

119974 119974120794

119865119869

Proposition 7412 proves that ℒ119886119903119894(119974) 119974120794 is closed under flexible weighted limits and 119865119869 ∶ 119974 rarr119974120794 preserves them so it follows as in the proof of Lemma 617 that119974perp119869 is closed in119974 under flexibleweighted limits Now Proposition 746 proves that the inclusion 119974perp119869 119974 creates an infin-cosmosstructure

7415 Proposition Theinfin-cosmos of isofibrations admits subinfin-cosmoi

119966119886119903119905(119974) 119974120794 119888119900119966119886119903119905(119974)

whose objects are cartesian or cocartesian fibrations respectively and whose 0-arrows are cartesian functorswith theinfin-cosmos structures created by the inclusions Similarly for anyinfin-category 119861 in aninfin-cosmos119974the slicedinfin-cosmos119974119861 admits subinfin-cosmoi

119966119886119903119905(119974)119861 119974119861 119888119900119966119886119903119905(119974)119861

whose objects are cartesian or cocartesian fibrations over 119861 respectively and whose 0-arrows are cartesianfunctors with theinfin-cosmos structures created by the inclusions

Proof The quasi-categorically enriched subcategories119966119886119903119905(119974) and 119888119900119966119886119903119905(119974) are replete in119974120794

by Corollary 5117 By Theorems 5111 and 5119 the quasi-categorically enriched category 119966119886119903119905(119974)is defined by the pullback

119966119886119903119905(119974) ℛ 119886119903119894(119974)

119974120794 119974120794

119870

along the simplicial functor that sends an isofibration 119901∶ 119864 ↠ 119861 to the isofibration 119896 ∶ 119864120794 ↠Hom119861(119861 119901) defined by Remark 5113 The simplicial functor 119870 is constructed out of weighted limitsand thus preserves all weighted limits and the replete subcategory inclusion ℛ119886119903119894(119974) 119974120794 cre-ates flexible weighted limits by Proposition 7412 Hence as in the proof of Lemma 617 119966119886119903119905(119974)is closed in 119974120794 under flexible weighted limits and now Proposition 746 proves that the inclusion119966119886119903119905(119974) 119974120794 creates aninfin-cosmos structure

The result for cartesian fibrations with a fixed base can be proven directly by a similar argumentor deduced by considering 119974119861 sub 119974120794 as the (non-replete) subcategory whose 119899-arrows have id119861 astheir codomain components

Exercises

74ii Exercise Use an argument similar to that given in the proof of Proposition 749 to prove thatanyinfin-cosmos119974 admits a subinfin-cosmos119982119905119886119887(119974) 119974 whose objects are the stableinfin-categories ofDefinition 445 and whose morphisms are the exact functors which preserve the zero elements andthe exact triangles

74iii Exercise Consider a functor between isofibrations

119864 119865

119861

119901

119892

119902

119903⊢

119904⊣

in which 119901 admits a right adjoint right inverse 119903 and 119902 admits a right adjoint right inverse 119904 Prove thatif 119892119903 cong 119904 over119861 then the mate of the identity 119902119892 = 119901 is an isomorphism This proves that the 0-arrowsin theinfin-cosmosℛ119886119903119894(119974)119861 are exact transformations between right adjoint right inverse adjunctions

74iv Exercise Prove thatℒ119886119903119894(119974) cong ℛ 119886119903119894(119974co)co

74v Exercise Use Exercise 12iv to show that ifℒ 119974 is a replete subinfin-cosmos then an object119860 isin ℒ is discrete if and only if 119860 is discrete as an object of119974

75 Weak 2-limits revisited

To wrap up this chapter on weighted limits we briefly switch the base of enrichment from sim-plicial sets to categories and reconsider the weak 2-limits introduced in Chapter 3 We give a general

definition that unifies the weak 2-limits introduced in special cases there and prove their essentialuniqueness in a uniform manner

Before turning our attention to weak 2-limits we describe the explicit construction of the weightedlimit of any 119966119886119905-valued diagram Let 119964 be a small 2-category and consider any pair of 2-functors119865119882∶ 119964 119966119886119905 the first regarded as the diagram and the second as the weight

751 Lemma (on the construction of weighted limits in119966119886119905) For any diagram 119865∶ 119964 rarr 119966119886119905 and weight119882∶ 119964 rarr 119966119886119905 the weighted limit lim119882 119865 isin 119966119886119905 exists defined to be the category whosebull objects are 2-natural transformations 120572∶ 119882 rArr 119865 andbull morphisms are modifications

Proof By Definition 717 the 119882-weighted limit of 119865 is a category lim119882 119865 isin 119966119886119905 characterizedby a natural isomorphism of categories

119966119886119905(119883 lim119882 119865) cong 119966119886119905119964(119882119966119886119905(119883 119865minus))

for any category119883 The 2-functor 119966119886119905(120793 minus) ∶ 119966119886119905 rarr 119966119886119905 is the identity so taking119883 = 120793 this tells usthat

lim119882 119865 cong 119966119886119905119964(119882 119865)

the category of 2-natural transformations and modifications from119882 to 119865

752 Definition (weak 2-limits in a 2-category) Consider a 2-functor 119865∶ 119964 rarr 119966 indexed by a small2-category119964 and a weight119882∶ 119964 rarr 119966119886119905 A119882-cone with summit 119875 isin 119966

119882 119966(119875 119865minus)120582

displays 119875 as a weak 2-limit of 119865 if and only if for all 119883 isin 119966 the functor induced by composition with120582

119966(119883 119875) lim119882 119966(119883 119865minus)120582lowast

to the119882-weighted limit of the diagram119966(119883 119865minus) ∶ 119964 rarr 119966119886119905 is smothering in the sense of Definition312

The weak universal property encoded by a smothering functor is sufficiently strong to characterizethe limit objects up to equivalence in the ambient 2-category

753 Proposition (uniqueness of weak 2-limits) For any fixed diagram and fixed weight any pair ofweak 2-limits are equivalent via an equivalence that commutes with the legs of the limit cones

Proof If119882 119966(119875 119865minus) and 119882 119966(119875prime 119865minus)120582 120582prime

both define weak 2-limit 119882-weighted cones over a 2-functor 119865∶ 119964 rarr 119966 then for any 119883 isin 119966 wehave a pair of smothering functors

119966(119883 119875) lim119882 119966(119883 119865minus) 119966(119883 119875prime)120582lowast 120582primelowast

Taking 119883 = 119875 the identity id119875 isin 119966(119875 119875) maps to the cone 120582 isin lim119882 119966(119875 119865minus) which then liftsalong the right-hand smothering functor to define a 1-cell 119906∶ 119875 rarr 119875prime a 1-cell 119907∶ 119875prime rarr 119875 is definedsimilarly as the lift of 120582prime isin lim119882 119966(119875prime 119865minus) along 120582lowast By construction both 119906 and 119907 commute withthe legs of the limit cones 120582 and 120582prime

Now 120582lowast carries the composite 119907119906 isin 119966(119875 119875) to the cone 119882 rArr 119966(119875 119865minus) whose component at119886 isin 119964 is the composite

119875 119875prime 119875 119865119886119906

120582119886

119907

120582prime119886

120582119886

which equals simply the cone leg 120582119886 Thus id119875 and 119907119906 lie in the same fiber of the smothering functor120582lowast and so by Lemma 313 must be isomorphic via an isomorphism that whiskers to identities alongthe legs of the limit cone Similarly 119906119907 cong id119875prime proving that 119875 ≃ 119875prime as claimed

Exercises

75i Exercise An inserter is a limit of a diagram indexed by the parallel pair category bull bullweighted by the weight

120793 120794 isin 1199661198861199050

1Prove that the homotopy 2-category of aninfin-cosmos has weak inserters of any parallel pair of functors119891 119892 ∶ 119860 119861 constructed by the pullback

Ins(119891 119892) 119861120794

119860 119861 times 119861

(11990111199010)

(119892119891)

CHAPTER 8

Homotopy coherent adjunctions and monads

Bar and cobar resolutions are ubiquitous in modern homotopy theory defining for instances var-ious completions of spaces and spectra [11] and free resolutions such as given in Definition 621 For-mally these bar or cobar constructions are associated to the monad or comonad of an adjunction

119860 119861119906perp119891

120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860

between infin-categories The monad and comonad resolutions associated to an adjunction are dualBy the triangle equalities the unit and counit maps give rise to a coaugmented cosimplicial object inhFun(119861 119861) the ldquomonad resolutionrdquo

id119861 119906119891 119906119891119906119891 119906119891119906119891119906119891 ⋯120578120578119906119891

119906119891120578

119906120598119891 (801)

an augmented simplicial object in hFun(119860119860) the ldquocomonad resolutionrdquo

id119860 119891119906 119891119906119891119906 119891119906119891119906119891119906 ⋯120598 119891120578119906

119891119906120598

120598119891119906

and augmented simplicial objects in hFun(119860 119861) and hFun(119861119860) admitting forwards and backwardscontracting homotopies

119906 119906119891119906 119906119891119906119891119906 119906119891119906119891119906119891119906 ⋯

119891 119891119906119891 119891119906119891119906119891 119891119906119891119906119891119906119891 ⋯

120578119906

119906120598 119906119891120578119906

120578119906119891119906

119906120598119891119906

119906119891119906120598

120598119891

119891120578 119891119906120598119891

120598119891119906119891

119891120578119906119891

119891119906119891120578

A classical categorical observation tells a richer story relating the four resolutions displayed aboveThere is a strict 2-category 119964119889119895 containing two objects and an adjunction between them mdash the free2-category containing an adjunction mdash and collectively these four diagrams display the image of a2-functor whose domain is119964119889119895 [53] More precisely each diagram is the image of one of the four hom-categories of this two object 2-category a 2-functor 119964119889119895 rarr 120101119974 extending the adjunction 119891 ⊣ 119906 isdefined by a pair of objects119860119861 isin 120101119974 the monad and comonad resolutions in the functor categorieshFun(119861 119861) and hFun(119860119860) and the dual pair of split augmented simplicial objects in hFun(119860 119861) andhFun(119861119860) The fact that these resolutions assemble into a 2-functor says that eg that the image of

the comonad resolution under 119906 is an augmented simplicial object in hFun(119860 119861) that admits ldquoextradegeneraciesrdquo

In this chapter we will prove that any adjunction in the homotopy 2-category of an infin-cosmosmdash that is any adjunction between infin-categories mdash can be lifted to a homotopy coherent adjunction intheinfin-cosmos The data of a homotopy coherent adjunction is indexed by a simplicial computad thatis uncannily closely related to the free adjunction 119964119889119895 In fact we define the free homotopy coherentadjunction to be the 2-category119964119889119895 regarded as a simplicial category by identifying its hom-categorieswith their nerves Section 81 is spent justifying this definition by introducing a graphical calculusthat allows us to precisely understand homotopy coherent adjunction data and prove that 119964119889119895 is asimplicial computad

In a homotopy coherent adjunction the resolutions (801) (802) and (803) lift to homotopycoherent diagrams

120491+ rarr Fun(119861 119861) 120491op+ rarr Fun(119860119860) 120491⊤ rarr Fun(119860 119861) 120491perp rarr Fun(119861119860)

indexed by the 1-categories introduced in Definition 239 and valued in the functor quasi-categories oftheinfin-cosmos In the case of the split augmented simplicial objects the contracting homotopies alsocalled ldquosplittingsrdquo or ldquoextra degeneraciesrdquo are given by the bottom and top 120578rsquos respectively ApplyingProposition 2311 it follows that the geometric realization or homotopy invariant realization of thesimplicial objects spanned by maps in the image of 119891119906119891 and 119906119891119906 are simplicial homotopy equivalentto 119891 and 119906 Dual results apply to the (homotopy invariant) totalization of the cosimplicial objectspanned by these same objects in this case the ldquoextra codegeneraciesrdquo are given by the top and bottom120598rsquos

The main theorem of this chapter proves that homotopy coherent adjunctions are abundant in-deed any adjunction ofinfin-categories extends to a homotopy coherent adjunction Homotopy coherentadjunctions extending a subcomputad of generating adjunction data are not unique on the nose How-ever whenever the subcomputad of generating adjunction is ldquoparentalrdquo mdash loosely generating from theuniversal property of either the right adjoint or the left adjoint exclusively mdash then extensions to a fullhomotopy coherent adjunction define the vertices of a contractible Kan complex proving appropri-ately generated extensions are ldquohomotopically uniquerdquo

All of the results in this chapter apply to any adjunction defined in (the homotopy 2-category of) aquasi-categorically enriched category119974 Yet since wersquoll typically apply these results toinfin-cosmoi weretain the usual notation Fun(119860 119861) for the hom quasi-categories of119974 to trigger the correct intuitionin these contexts

81 The free homotopy coherent adjunction

In this section we present a strict 2-category 119964119889119895 introduced by Schanuel and Street under thename ldquothe free adjunctionrdquo [53] which has the universal property that it is the free 2-category contain-ing an adjunction Immediately after introducing this classical object we take the unorthodox stepof reconsidering it as a simplicial category via a mechanism that we shall describe We develop a newpresentation of 119964119889119895 by introducing a graphical calculus that allows us to prove the surprising factthat this simplicial category is a simplicial computad This justifies referring to it as the free homotopycoherent adjunction The remainder of this chapter will explore the consequences of this definition

811 Definition (the free adjunction) Let119964119889119895 denote the 2-category with two objects + and minus andthe four hom-categories

119964119889119895(+ +) ≔ 120491+ 119964119889119895(minus minus) ≔ 120491op+ 119964119889119895(minus +) ≔ 120491⊤ 119964119889119895(+ minus) ≔ 120491perp

displayed in the following cartoon

+ minus120491perpcong120491⊤op

120491+ ⟂ 120491op+

120491⊤cong120491perpop

Here 120491⊤ 120491perp sub 120491 sub 120491+ are the subcategories of order-preserving maps that preserve the top orbottom elements respectively in each ordinal as described in Definition 239 Their intersection

120491perp⊤ ≔ 120491perp cap 120491⊤ cong 120491op+

is the subcategory of order-preserving maps that preserve both the top and bottom elements in eachordinal This identifies 120491op

+ with the subcategory 120491perp⊤ sub 120491+ of ldquointervalsrdquo as is elaborated upon inDigression 818

The horizontal composition maps in 119964119889119895 are defined in 119964119889119895(minus minus)op cong 119964119889119895(+ +) cong 120491+ by theordinal sum operation

120491+ times 120491+ 120491+

[119899] [119898] [119899 + 119898 + 1]

[119899prime] [119898prime] [119899prime + 119898prime + 1]

120572120573 ↦ 120572oplus120573 120572 oplus 120573(119894) ≔ 1114108120572(119894) 119894 le 119899120573(119894 minus 119899 minus 1) + 119899prime + 1 119894 gt 119899

The object [minus1] isin 120491+ serves as the identity for ordinal sum and thus represents the identity 1-cellson minus and + in119964119889119895 Ordinal sum restricts to the subcategories 120491perp 120491⊤ sub 120491+ to give bifunctors

120491+ times 120491⊤ 120491⊤ 120491perp times 120491+ 120491perp

119964119889119895(+ +) times 119964119889119895(minus +) 119964119889119895(minus +) 119964119889119895(+ minus) times 119964119889119895(+ +) 119964119889119895(+ minus)

cong cong

cong cong∘ ∘

defining these horizontal composition operations that wersquoll later come to think of as ldquoactionsrdquo of119964119889119895(+ +) on the left and right of119964119889119895(minus +) and119964119889119895(+ minus) The opposites of these functors definesthe action of119964119889119895(minus minus) on the right and left of119964119889119895(minus +) and119964119889119895(+ minus)

812 Lemma The 2-category119964119889119895 contains a distinguished adjunction

+ minus[0]

⟂[0]

with unit ∶ [minus1] rarr [0] isin 120491+ cong 119964119889119895(+ +) and counit given by the same map in 120491op+ cong 119964119889119895(minus minus)

Proof We must verify the triangle equalities in 119964119889119895(minus +) and 119964119889119895(+ minus) these categories areopposites and the calculation in each case is dual so we focus on the case of119964119889119895(minus +) The whiskeredcomposite of the unit ∶ [minus1] rarr [0] isin 119964119889119895(+ +) with the right adjoint [0] isin 119964119889119895(minus +) is the map1205750 ∶ [0] rarr [1] isin 119964119889119895(minus +) which is indeed top-element preserving The whiskered composite of thecounit in 119964119889119895(minus minus) with the right adjoint in 119964119889119895(minus +) is defined by whiskering the opposite map

∶ [minus1] rarr [0] isin 119964119889119895(+ +) cong 119964119889119895(minus minus)op with [0] isin 119964119889119895(+ minus) cong 119964119889119895(minus +)op This composite isthe map 1205751 ∶ [0] rarr [1] isin 119964119889119895(+ minus) cong 119964119889119895(minus +)op which is indeed top-element preserving Underthe isomorphism119964119889119895(minus +) cong 119964119889119895(+ minus)op this corresponds to the map 1205900 ∶ [1] rarr [0] isin 119964119889119895(minus +)Now the composite 1205900 sdot 1205750 ∶ [0] rarr [0] isin 119964119889119895(minus +) is the identity as required

The 2-categorical universal property of 119964119889119895 mdash that 2-functors 119964119889119895 rarr 119966 correspond to adjunc-tions in the 2-category119966mdashis statedwithout proof in [53] Wewill take a roundabout route to verifyingit in Proposition 8113 that uses the simplicial computads of Chapter 6

Throughout this text we have found it convenient to identify 1-categories with their nerves whichdefine simplicial sets The 1-categories can be characterized as those simplicial sets that admit uniqueextensions along any inner horn inclusion or spine inclusion or as those 2-coskeletal simplicial setsthat admit unique extensions along inner horn or spine inclusions in dimensions 2 and 3 see Remark115 Similarly in developing homotopy coherent category theory it will be convenient to identity2-categories with the simplicial categories obtained by identifying each of the hom-categories withits nerve mdash a categorification of the previous construction As a corollary of the characterization ofnerves of 1-categories we obtain a characterization of the simplicially enriched categories that arisein this way

813 Lemma (2-categories as simplicial categories) A 2-category119964may be regarded as a quasi-categoricallyenriched category whosebull objects are the objects of119964bull 0-arrows in119964(119909 119910) are the 1-cells of119964 from 119909 to 119910

bull 1-arrows in119964(119909 119910) from 119891 to 119892 are the 2-cells of119964 of the form 119909 119910119891

119892dArr

bull and in which there exists a 2-arrow 120590 in119964(119909 119910) whose faces 120590119894 ≔ 120590 sdot 120575119894 are 1-arrows in119964(119909 119910)

119892

119891 ℎ

12059001205902

1205901

120590 119909 119910 = 119909 119910119892

119891

dArr1205902

ℎdArr1205900

119891

ℎdArr1205901

if and only if 1205901 is the vertical composite 1205900 sdot 1205902

with the higher-dimensional arrows determined by the property that each of the hom-spaces is 2-coskeletalConversely a simplicially-enriched category 119964 is isomorphic to a 2-category if and only if each of its hom-spaces are 2-coskeletal simplicial sets that admit unique extensions along the spine inclusions in dimensions 2and 3

We now give the first of two presentations of the free homotopy coherent adjunction Since we usethe same notation for 1-categories and their nerves we also adopt the same notation for a 2-categoryand its corresponding simplicial category under the embedding of Lemma 813

814 Definition (the free homotopy coherent adjunction as a 2-category) The free homotopy co-herent adjunction to be the free adjunction119964119889119895 regarded as a simplicial category Explicitly119964119889119895 hastwo objects + and minus and the four hom quasi-categories defined by

with the composition maps defined in 811200

This presentation of the free homotopy coherent adjunction is not particularly enlightening ViaLemma 813 and Definition 811 we could in principle describe the 119899-arrows in 119964119889119895 but itrsquos trickyto get a real feel for them We will now reintroduce this simplicial category in a different guise thatachieves just this Before doing so note

815 Observation To specify a simplicial category thought of as an identity-on-objects simplicialobject in 119966119886119905 it suffices to specifybull a set of objectsbull for each 119899 ge 0 a set of 119899-arrows whose domains and codomains are among the specified object

setbull a right action of the morphisms in 120491 on this graded set of arrows that preserves domains and

codomainsbull a ldquohorizontalrdquo composition operation for the 119899-arrows with compatible (co)domains that pre-

serves the simplicial action

We will now reintroduce119964119889119895 following this outline by exhibiting its graded set of 119899-arrows be-tween the objects + and minus

816 Definition (strictly undulating squiggles) Define a graded set of arrows between objects minus and+ whose 119899-arrows are strictly undulating squiggles on 119899 + 1 lines such as displayed below in the case119899 = 5

012345

+ 2 3 1 4 2 + minus 4 minus 3 2 + minus

The lines are labeled 0 1 hellip 119899 and the gaps between them are labeled minus 1 hellip 119899 + A squiggle muststart on the right-hand side and end on the left-hand side in either the gap minus or + The right-handstarting gap becomes the domain of the squiggle and the left-hand ending gap becomes its codomainthese conventions chosen to follow the usual composition order Each turning point of the squigglemust lie entirely within a single gap The qualifier ldquostrict undulationrdquo refers to the requirement thatadjacent turning points should be distinct and that they should oscillate up and down as we proceedfrom right to left

Formally the data of a strictly undulating squiggle on 119899 + 1 lines can be encoded by a string119886 = (1198860 1198861 hellip 119886119903minus1 119886119903) of letters in the set minus 1 2 hellip 119899 + corresponding to the gaps in which eachsuccessive turning point occurs whose width is the integer 119908(119886) ≔ 119903 subject to the following condi-tions

(i) The domain 119886119908(119886) and codomain 1198860 of 119886 are both in minus +(ii) If 1198860 = minus then for all 0 le 119894 lt 119908(119886) we have 119886119894 lt 119886119894+1 whenever 119894 is even and 119886119894 gt 119886119894+1

whenever 119894 is odd and if 1198860 = + then for all 0 le 119894 lt 119908(119886) we have 119886119894 gt 119886119894+1 whenever 119894 iseven and 119886119894 lt 119886119894+1 whenever 119894 is odd

817 Lemma The graded set of strictly undulating squiggles admits a right action of the morphisms in 120491 thatpreserves domains and codomains and a ldquohorizontalrdquo composition operation for arrows in the same degree with

compatible (co)domains that preserves the simplicial action Relative to the horizontal composition an 119899-arrowis atomic if and only if there are no instances of + or minus occurring in its interior While the faces of an atomicarrow need not be atomic the degeneracies of an atomic arrow always are

Proof The horizontal composition of 119899-arrows is given by horizontal juxtaposition of strictlyundulating squiggles on 119899 + 1 lines which produces a well-formed squiggle just when the codomainof the right-hand squiggle matches the domain of the left-hand squiggle

+ 3 4 1 3 2 +

+ 2 3 1 3 minus

+ 3 4 1 3 2 + 2 3 1 3 minus

This operation is clearly associative Moreover any strictly undulating squiggle admits a unique de-composition into squiggles that do not contain+ orminus in their interior sequences of gaps which provesthat any squiggle 119886 = (1198860 1198861 hellip 119886119903minus1 119886119903) with the property that 1198861 hellip 119886119903minus1 isin 1 hellip 119899 is atomic

The right action of the simplicial operators on strictly undulating squiggles is best described intwo cases If 120572∶ [119898] ↠ [119899] is an epimorphism then a strictly undulating squiggle on lines 0hellip 119899becomes a strictly undulating squiggle on lines 0hellip 119898 by replacing each line labeled 119894 isin [119899]with lineslabeled by each element of the fiber 120572minus1(119894) and then ldquopulling these lines apartrdquo to create new gaps

[5] [2]

2 3 4 1

120572

119886 ≔12

+ 1 + 2 + minus 1 minus

119886 sdot 120572 ≔

012345

Note that ldquopulling apart linesrdquo does not create instances of + or minus in the interior sequence of gaps sothe action by degeneracy operators preserves atomic arrows

If 120572∶ [119898] ↣ [119899] is a monomorphism then a strictly undulating squiggle on lines 0hellip 119899 becomesa strictly undulating squiggle on lines 0hellip 119898 by removing the lines labelled by elements 119894 isin [119899] notin the image of 120572 and renumbering the lines that are in the image in sequence The original squigglewill still undulate between the new lines but may not do so ldquostrictlyrdquo mdash it is possible for the squiggleto turn around mutliple times in the same gap mdash but this is easily corrected by ldquopulling the string

tautrdquo

[2] [5]

0 11 42 5

120572

119886 ≔

012345

+ 2 3 1 4 2 + minus 5 2 4 3 + minus

119886 sdot 120572 ≔12

+ 1 1 minus 1 1 + minus 2 1 1 1 + minus

+ minus + minus 2 1 + minus

Note that in both of these cases the actions by epimorphisms and monomorphisms preserve domainsand codomains of squiggles and respect horizontal concatenations

Now in general a simplicial operator 120572∶ [119898] rarr [119899] can be factored uniquely in 120491 as an epimor-phism followed by a monomorphism so it acts on a strictly undulating squiggle on 119899+1 lines by firstldquoremoving lines and pulling tautrdquo and then by ldquoduplicating lines and pulling apartrdquo

We leave the formalization of these geometric descriptions of the actions of the simplicial opera-tors on strictly undulating squiggles presented as sequences satisfying the axioms of Definition 816(i)and (ii) to Exercise 81iii with the hint that a combinatorial description of this action can easily bedefined using the ldquointerval representationrdquo of 120491op

818 Digression (the interval representation) There is a faithful interval representation of the cat-egory 120491op

+ as a subcategory of 120491 in the form of a functor

120491op+ 120491

[119899 minus 1] [119899]

[119899] [119899 + 1]

[119899 + 1] [119899 + 2]

120575119894 ↦

120590119894

120575119894+1120590119894

whose image is the subcategory 120491perp⊤ ≔ 120491perp cap 120491⊤ sub 120491 of simplicial operators that preserve both thetop and bottom elements in each ordinal

If we think of the elements of [119899] as labelling the lines in the graphical representation of an119899-arrow then the elements of ir[119899] ≔ [119899 + 1] label the gaps with the bottom element 0 relabeled asminus and the top element 119899+1 relabeled as + If the elementary face operators 120575119894 and elementary degen-eracy operators 120590119894 act by removing and duplicating the lines in a squiggle diagram then the functorsir 120575119894 and ir 120590119894 describe the corresponding actions on the gaps

The graphical description of its 119899-arrows of119964119889119895 clearly exhibits a simplicial computad structureon a simplicial category we now more formally introduce following the outline of Observation 815

819 Definition (the free homotopy coherent adjunction as a simplicial computad) The free ho-motopy coherent adjunction119964119889119895 is the simplicial computad with

bull two objects + and minusbull whose 119899-arrows are strictly undulating squiggles on 119899 + 1 lines of Definition 816 oriented from

right to left with the right-hand starting position defining the domain object and the left-handending position defining the codomain object

bull with the simplicial operators acting as described in Lemma 817 andbull with horizontal composition defined by horizontal juxtaposition of squiggle diagrams

which means that the atomic 119899-arrows are those squiggles each of whose interior undulations occursbetween the lines 0 and 119899

We now reconcile our two descriptions of the free homotopy coherent adjunction comparingDefinition 819 with Definition 814 Zaganidis discovered a similar proof in his PhD thesis [65sect233] that improves upon some aspects of the authorsrsquo original argument

8110 Proposition The simplicial computad119964119889119895 is isomorphic to the 2-category119964119889119895

Proof We explain how the 119899-arrows in each of the four homs of the simplicial category119964119889119895 cor-respond to sequences of 119899 composable arrows in the corresponding hom-categories 120491+ 120491perp 120491⊤ and120491perp⊤ of the 2-category119964119889119895 Here it is most convenient to think of120491perp120491⊤ and120491perp⊤ as subcategoriesof 120491 the latter via the interval representation of Digression 818

To easily visualize the isomorphism shade the region under a strictly undulating squiggle on 119899+1lines to define a topological space 119878 embedded in the plane In the case of a squiggle from minus to minusthis shaded region includes both the left and right boundary regions of the squiggle diagram Foreach 119894 isin [119899] the connected components of the intersection 119878119894 of the shaded region with the regionabove the labelled 119894 define a finite linearly ordered set ordered from left to right by the order in whichthe intervals defined as the intersection of each component with the line 119894 appear on that line Theseordinals define the objects in the composable sequence of arrows in120491+ or one of its subcategory120491perp⊤120491perp or 120491⊤ corresponding to the squiggle diagramsup1 Formally the ordinal representing the 119894th objectin the sequence of 119899 arrows counts the number of ldquomaximal convex subsequencesrdquo of the sequence119886 = (1198860 hellip 119886119908(119886)) that encodes the squiggle a maximal convex subsequence is a maximal sequence ofconsecutive entries 119886119895 satisfying the condition 119886119895 le 119894

The intersections of the shaded region 119878 with the regions above the lines 119894 = 0hellip 119899 define anested sequence of subspaces

1198780 1198781 ⋯ 119878119899minus1 119878119899Taking path components yields a composable sequence

12058701198780 rarr 12058701198781 rarr⋯rarr 1205870119878119899minus1 rarr 1205870119878119899As explained above each of the sets 1205870119878119894 is linearly ordered from left to right by the positions inwhich each component of 119878119894 intersects the line labeled 119894 and the functions induced by the inclusionsare order-preserving This defines the composable sequence of arrows in 120491+ 120491perp 120491⊤ or 120491perp⊤ corre-sponding to a strictly undulating squiggle on 119899 + 1-lines

For instance suppose the shaded region under the squiggle diagram intersects the line labelled119894 in 119898 + 1 components corresponding to the ordinal [119898] and similarly suppose the shaded region

sup1In the case of a squiggle from + to + it is possible that the squiggle diagram does not intersect the line labeled 0 inwhich case the subspace 1198780 is empty if this is the case the squiggle may also fail to cross the next several lines In each ofthe other three hom-categories the ordinals defined by taking the connected components of the intersection of each linewith the shaded region are non-empty because either the left or right boundary of the squiggle diagram is also shaded

under the squiggle diagram intersects the line labelled 119894+1 in 119896+1 components corresponding to theordinal [119896] We identify the corresponding simplicial operator 120572∶ [119898] rarr [119896] by taking the fiber over119905 isin [119896] to be the subset of shaded regions 119904 isin [119898] above the 119894th line that belong to the same connectedcomponent as 119905 in the shaded region above the line 119894 + 1 Formally the element 119905 isin [119896] representsa maximal convex subsequence of 119886 = (1198860 hellip 119886119908(119886)) comprised of those 119886119895 le 119894 + 1 This convexsubsequence is partitioned into possibly smaller maximally convex subsequences satisfying the morerestrictive condition 119886119895 le 119894 and the elements 119904 isin [119898] indexing these subsequences form the fiber of 120572over 119905 Note that if the domain of the squiggle is minus then the top element of each ordinal is necessarilypreserved because the right-hand boundary of the squiggle diagram defines a connected shaded regionsimilarly if the codomain of the squiggle is minus then the bottom element of each ordinal is preservedThe constructs a map from the simplicial computad119964119889119895 of Definition 819 to the 2-category119964119889119895 ofDefinition 814

The converse map can be constructed by iterating a splicing operation that we now introduceThis splicing operation proves that each of the hom-spaces of the simplicial computad 119964119889119895 satisfiesa ldquostrict Segal conditionrdquo that says that the set of 119899 + 119898-arrows is isomorphic to the set of pairs of119899-arrows and 119898-arrows whose last and first vertices coincide In more detail if 119886 and 119887 are strictlyundulating squiggles on 119899 + 1 and 119898 + 1 lines that lie in the same hom-space with the property thatthe 119899th vertex of 119886 coincides with the 0th vertex of 119887 then these squiggle diagrams can be uniquelyspliced to form a strictly undulating squiggle 119888 on 119899 +119898+ 1 lines whose face spanned by the vertices0hellip 119899 is 119886 and whose face spanned by the vertices 119899hellip 119898 + 119899 is 119887

119886 ≔123

minus + 1 3 minus 2 1 + 2 +

119887 ≔

minus 2 1 3 minus 4 1 + 2 3 minus +

119888 ≔

1234567

01234567

minus 5 4 6 1 3 minus 2 1 7 4 + 5 6 2 +

This splicing operation is defined graphically by separating the squiggle diagrams for 119886 and for 119887 intoan ordered sequence of components by cutting the squiggle 119886 each time it enters or leaves the gapmarked ldquo+rdquo removing the dotted arcs at the bottom of the squiggle diagram and cutting the squiggle119887 each time it enters or leaves the gap marked ldquominusrdquo removing the dotted arcs at the top of the depictedsquiggle diagram This requires two cuts for each occurrence of ldquo+rdquo or ldquominusrdquo in the interior of 119886 or 119887respectively and one cut for each occurrence of ldquo+rdquo or ldquominusrdquo as the source or target of 119886 or 119887 respectivelyThe condition that the 119899th vertex of 119886 coincides with the 0th vertex of 119887 ensures that the numbers ofcuts made to 119886 (the number of times 119886 intersects the 119899th line) and 119887 (the number of times 119887 intersectsthe 0th line) are the same Now form 119888 by sewing together these squiggle components in order to formthe crossings of the line labeled ldquo119899rdquo

By iterating the splicing operation we can construct a strictly undulating squiggle on 119899 + 1 linesfrom a sequence of 119899 composable arrows once we know how to encode a single 1-arrow of 119964119889119895 as astrictly undulating squiggle on 2 lines We give full details in the case of a 1-arrow from + to + A

simplicial operator 120572∶ [119898] rarr [119896] defines a strictly undulating squiggle on 2 lines from + to + by asequence 119886 with ldquominusrdquo occurring 119898 + 1 times (each occurrence of which corresponds to an intersectionof the shaded region and the 0th line) and ldquo+rdquo occurring 119896 + 2 times (each consecutive pair boundingan intersection of the shaded region and the 1st line) The strings between each consecutive pair of+s correspond to the elements 119894 isin [119896] If the fiber over 119894 is empty the sequence is ldquo+1+rdquo If the fibercontains a single element the sequence is ldquo+ minus 1 minus +rdquo and if the fiber contains 119904 gt 1 elements thesequence is ldquo+ minus (1minus)119904minus1+rdquo

Note that the planar orientation of a strictly undulating squiggle diagram makes the numerical la-bels for the lines gaps and the sequence of undulation points redundant Going forward we typicallyomit them

8111 Example (adjunction data in 119964119889119895) For later use we name some of the low-dimensional non-degenerate atomic arrows in119964119889119895 There exist just two atomic 0-arrows which we call

119891 ≔ and 119906 ≔

Since119964119889119895 is a simplicial computad all of its other 0-arrows may be obtained as a unique alternatingcomposite of these two for example

119891119906119891119906119891 ≔

There are exactly two non-degenerate atomic 1-arrows in119964119889119895 these being

120578 ≔ and 120598 ≔

Again since 119964119889119895 is a simplicial computad all of its other 1-arrows are uniquely expressible as hori-zontal composites of the 1-arrows 120578 and 120598 with degenerated 1-arrows obtained from 119891 and 119906 such asfor example

119906119891120578 ≔ and 120598120598 ≔

There exist two non-degenerate atomic 2-arrows with minimal width whose faces are easily com-puted

120572 ≔ 120572 sdot 1205752 ≔ 119891120578 ≔ 120572 sdot 1205751 ≔ 119891 ≔ 120572 sdot 1205750 ≔ 120598119891 ≔

120573 ≔ 120573 sdot 1205752 ≔ 120578119906 ≔ 120573 sdot 1205751 ≔ 119906 ≔ 120573 sdot 1205750 ≔ 119906120598 ≔

For each width119908 ge 3 there exist exactly two non-degenerate atomic 2-arrows the description ofwhich we leave to Exercise 81v There are countably many atomic non-degenerate 119899-arrows in eachdimension 119899 ge 2 which we shall partially enumerate in the proof of Proposition 8211

In the ldquohomotopy 2-categoryrdquo of the simplicial category119964119889119895 which is isomorphic to the 2-category119964119889119895 the atomic 2-arrows 120572 and 120573witness the triangle equalities proving that this 2-category containsan adjunction (119891 119906 120578 ∶ id+ rArr 119906119891 120598 ∶ 119891119906 rArr idminus)

8112 Lemma The simplicial computad119964119889119895 contains a distinguished adjunction

+ minus

119891

⟂119906

with unit 120578∶ id+ rarr 119906119891 isin 119964119889119895(+ +) and counit 120598 ∶ 119891119906 rarr idminus isin 119964119889119895(minus minus)

Proof By Proposition 8110 this follows from Lemma 812 though the reader may prefer to provethis result directly

We can now prove the universal property of the free adjunction 119964119889119895 claimed by Schanuel andStreet

8113 Proposition For any 2-category119966 2-functors119964119889119895 rarr 119966 correspond to adjunctions in the 2-category119966

Proof Lemma 812 identifies an adjunction in 119964119889119895 that we denote in the notation of Lemma8112 by (119891 119906 120578 ∶ id+ rArr 119906119891 120598 ∶ 119891119906 rArr idminus) A 2-functor119964119889119895 rarr 119966 carries this data to an adjunctionin the 2-category 119966

Conversely we suppose that119966 is equipped with an adjunction (119891 119906 120578 ∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860)To construct a 2-functor 119964119889119895 rarr 119966 it suffices because the embedding 2-119966119886119905 119982119982119890119905-119966119886119905 is fullyfaithful to consider both 2-categories as simplicial categories via Lemma 813 and instead define asimplicial functor119964119889119895 rarr 119966 see Exercise 81i Now since119964119889119895 is a simplicial computad and the homsof 119966 are 2-coskeletal it suffices to define a simplicial functor sk2119964119889119895 rarr 119966 from the subcomputad of119964119889119895 generated by its atomic 0- 1- and 2-arrows This functor is given by the mappingbull + ↦ 119861 and minus ↦ 119860 on objectsbull 119891 ↦ 119891 and 119906 ↦ 119906 on atomic 0-arrowsbull 120578 ↦ 120578 and 120598 ↦ 120598 on atomic non-degenerate 1-arrows

and for each of the atomic non-degenerate 2-arrows of Exercise 81v we must by Lemma 813 verifythat the 2-cells in 119966 defined by their boundaries compose vertically as indicated

For 120572 and 120573 the unique non-degenerate atomic 2-arrows of minimal width 3 the required com-position relations are

120598119891 sdot 119891120578 = id119891 and 119906120598 sdot 120578119906 = id119906which hold by the triangle equalities for the adjunction in 119966 For the atomic 2-arrows of odd width2119903 + 1 the required composition relations are the ldquohigher order triangle equalitiesrdquo

120598119903119891 sdot 119891120578119903 = id119891 and 119906120598119903 sdot 120578119903119906 = id119906which can easily be seen to hold by depicting the left-hand expressions as pasting diagramssup2 For theatomic 2-arrows of even with 2119903 the required composites are closely related ldquohigher order triangleequaltiesrdquo

120598119903+1 sdot 119891120578119903119906 = 120598 and 119906120598119903 sdot 120578119903+1 = 120578which again can easily be seen to hold by depicting the left-hand expressions as pasting diagrams

sup2Here 120578119903 refers to the horizontal composite of 119903 copies of the unit and the ldquosdotrdquo expresses a vertical composite of awhiskered copy of this with a whiskered copy of 120598119903 the horizontal composite of 119903-copies of the counit The 1-cell codomainof the former and 1-cell domain of the latter fit together like a cartoon depiction of a closed mouth of pointy teeth

Motivated by this result and the fact that the simplicial category119964119889119895 is a simplicial computad weuse it to define a notion of homotopy coherent adjunction in any quasi-categorically enriched categoryand in particular in anyinfin-cosmos

8114 Definition A homotopy coherent adjunction in a quasi-categorically enriched category119974 isa simplicial functor119964119889119895 rarr 119974 Explicitly it picks outbull a pair of objects 119860119861 isin 119974bull together with four homotopy coherent diagrams

120491+ rarr Fun(119861 119861) 120491op+ rarr Fun(119860119860) 120491⊤ rarr Fun(119860 119861) 120491perp rarr Fun(119861119860) (8115)

that are functorial with respect to the composition action of119964119889119895The 0- and 1-dimensional data of the these diagrams has the form displayed in (801) (802) and(803) We interpret the homotopy coherent diagrams (8115) as defining homotopy coherent versionsof the bar and cobar resolutions of the adjunction (119891 ⊣ 119906 120578 120598)

Exercises

81i Exercise Prove that the embedding 2-119966119886119905 119982119982119890119905-119966119886119905 defined by Lemma 813 is fully faith-ful prove that simplicial functors119964rarr ℬ between 2-categories define 2-functors

81ii Exercise Use Lemma 813 to prove that the homotopy coherent 120596-simplex ℭΔ[120596] is a 2-cat-egorysup3

81iii Exercise Describe the action of a general simplicial operator 120572∶ [119898] rarr [119899] on a strictlyundulating squiggle on 119899 + 1 lines represented by a sequence 119886 = (1198860 1198861 hellip 119886119903minus1 119886119903) of ldquogapsrdquo 119886119894 isinminus 1 hellip 119899 +

81iv Exercise Give a graphical interpretation of the dualities

119964119889119895(minus minus) cong 119964119889119895(+ +)op and 119964119889119895(minus +) cong 119964119889119895(+ minus)op

of Definition 814

81v Exercise(i) Describe the non-degenerate atomic 2-arrows of 119964119889119895 and compute their faces and compare

your results with the composition relations appearing in the last paragraph of the proof ofProposition 8113

(ii) Describe the degenerate atomic 2-arrows of119964119889119895 and ldquocomputerdquo their faces

81vi Exercise Use the graphical calculus presented in Definition 819 to verify the following ob-servation of Karol Szumiło the simplicial category 119964119889119895 is isomorphic to the Dwyer-Kan hammocklocalization [17] of the category consisting of two objects + and minus and a single non-identity arrow+ ⥲ minus that is a weak equivalence

81vii Exercise For any homotopy coherent adjunction as in Definition 8114 define internal ver-sions of monad resolution the comonad resolution the bar resolution and the comonad resolution

119861(119906119891)bullminusminusminusminusrarr 119861120491+ 119860

(119891119906)bullminusminusminusminusrarr 119860120491op+ 119860 barminusminusrarr 119861120491⊤ 119861 cobarminusminusminusminusrarr 119860120491perp

sup3In light of Lemma 813 Theorem 64 of [46] proves more generally that the simplicial computads defined as freeresolutions of strict 1-categories are always 2-categories

and explore the relationships between these functors⁴

82 Homotopy coherent adjunction data

Any homotopy coherent adjunction in an infin-cosmos or general quasi-categorically enriched cat-egory has an underlying adjunction in its homotopy 2-category Remarkably this low-dimensionaladjunction data may always be extended to give a full homotopy coherent adjunction by repeated in-voking the universal property of the unit as expressed in Proposition 832 In this section we filterthe free homotopy coherent adjunction 119964119889119895 by a sequence of ldquoparentalrdquo subcomputads which mustcontain for each atomic 119899-arrow with codomain ldquominusrdquo its ldquofillable parentrdquo the atomic 119899+1-arrow withcodomain ldquo+rdquo obtained by whiskering with 119906 and the ldquoprecomposingrdquo with 120578

In sect83 we then use this filtration to prove that any adjunction in aninfin-cosmos mdash or more preciselyany diagram indexed by a parental subcomputad mdash extends to a homotopy coherent adjunction Ourproof is essentially constructive enumerating the choices necessary tomake each stage of the extensionIn sect84 we give precise characterizations of the homotopical uniqueness of such extensions provingthat the appropriate spaces of extensions are contractible Kan complexes Via the 2-categorical self-duality of 119964119889119895 described in Remark 8212 there is a dual proof that instead exploits the universalproperty of the counit the main steps of which are alluded to in the exercises

Our proof that any adjunction extends to a homotopy coherent adjunction inductively specifiesthe data in the image of a homotopy coherent adjunction by choosing fillers for horns correspondingto ldquofillablerdquo arrows

821 Definition An arrow 119887 of119964119889119895 is fillable ifbull it is non-degenerate and atomicbull its codomain 1198870 = + andbull 119887119894 ne 1198871 for 119894 gt 1

Write Fill119899 sub Atom119899 for the subset of fillable 119899-arrows of the subset of atomic and non-degenerate119899-arrows

822 Definition (distinguished faces of fillable arrows) On account of the graphical calculus werefer to ℎ(119887) ≔ 1198871 minus 1 an integer in 0 hellip 119899 minus 1 that labels the line immediately above the positionof the left-most turn-around as the height of the fillable arrow 119887⁵ The fillability of 119887 implies that noldquotauteningrdquo is required in computing the distinguished codimension-one face 119887 sdot 120575ℎ(119887) which then isnon-degenerate and has the same width as 119887 Further analysis of this face differs by case

⁴For instance there is a commutative diagram119860 119861120491⊤

119861 119861120491+

119906 res

(119906119891)bull

⁵The unique fillable 0-arrow 119906 behaves somewhat differently but nonetheless it is linguistically convenient to includeit among the fillable arrows

bull case ℎ(119887) gt 0 The face 119887 sdot 120575ℎ(119887) is non-degenerate and atomic but it is not fillable since non-degeneracy implies that there is some 119894 gt 1 with 119887119894 = ℎ(119887) whence the entries of 119887 sdot 120575ℎ(119887) at 1 and119894 both equal ℎ(119887)

bull case ℎ(119887) = 0 The face 119887 sdot 120575ℎ(119887) decomposes as 119906 sdot 119886 where 119886 is non-degenerate atomic has widthone less than the width of 119887 and has codomain minus

We write

119887⋄ ≔ 1114108119887 sdot 120575ℎ(119887) ℎ(119887) gt 0119886 ℎ(119887) = 0

for the non-degenerate atomic ℎ(119887)th face in the positive height case and for the non-degenerateatomic factor of the 0th face 119906119886 in the height 0 case and refer to the atomic 119899 minus 1-arrow definedin either case as the distinguished face of the fillable arrow

We now argue that any non-degenerate and atomic 119899-arrow of119964119889119895 that is not fillable arises as thedistinguished face of a unique fillable 119899+ 1-arrow in the form of the first case just described when itscodomain is + and in the form of the second case just described when its codomain is minus

823 Lemma (identifying fillable parents)(i) If 119887 is a non-degenerate and atomic 119899-arrow of119964119889119895 with codomain + that is not fillable then it is the

codimension-one face of exactly two fillable (119899 + 1)-arrows with the same width both of which has 119887as its 1198871th face one which has height 1198871 that we refer to as its fillable parent and denote by 119887

dagger and theother which has height 1198871 minus 1

(ii) If 119886 is a non-degenerate and atomic 119899-arrow of119964119889119895 with codomain minus then the composite arrow 119906119886 isa codimension-one face of exactly one fillable (119899 + 1)-arrow which we call the fillable parent of 119886 anddenote by 119886dagger The fillable parent 119886dagger has width one greater than the width of 119886 has height 0 and has119906119886 as its 0th face

Together these cases define a ldquofillable parentdistinguished facerdquo bijection

Atom119899Fill119899 Atom119899+1

(minus)dagger

cong(minus)⋄

between fillable 119899 + 1-arrows and non-degenerate atomic 119899-arrows which are not fillable

Proof For 82 if 119887 is a non-degenerate atomic 119899-arrow with domain + that is not fillable thenit must be the case that 119887119894 = 1198871 for some 119894 gt 1 This arrow is then a codimension-one face of thetwo atomic 119899 + 1-arrows that are formed by inserting an extra line into the gap labeled 1198871 separatethe entry 1198871 from the other turn-arounds that occur in the same gap the 119899 + 1-arrows obtained inthis way will then clearly have 119887 as their 1198871th face There are exactly two ways to do this as illustratedbelow

119887 ≔012

119887dagger ≔0123

For (ii) given a non-degenerate atomic 119899-arrow 119886 with codomain minus there is a unique way to make119906119886 into the face of an atomic 119899 + 1-arrow with domain + whose width is only one greater by adding

an extra line in the upper-most space above the squiggle diagram

119886 ≔012

119906119886 ≔012

119886dagger ≔0123

824 Definition A subcomputad119964 of119964119889119895 is parental if it contains at least one non-identity arrowand satisfies the conditionbull if 119888 is a non-degenerate atomic arrow in119964 then either it is fillable or its fillable parent 119888dagger is also

in119964

The condition implies that any parental subcomputad 119964 sub 119964119889119895 contains at least one fillablearrow The last vertex of any fillable arrow has the form 119906119886 for some 0-arrow 119886 so it follows that the0-arrow 119906 is contained in any parental subcomputad

Recall from Definition 618 that any collection of arrows 119878 in a computad generates a minimalsubcomputad 119878

825 Example (notable parental subcomputads)

bull The unique fillable 0-arrow119906 generates theminimal parental subcomputad 119906 sub 119964119889119895 containingonly 119906 and its degenerate copies

bull The unit 1-arrow 120578 is fillable and the subcomputad 120578 sub 119964119889119895 it generates has 119906 119891 and 120578 as itsonly non-degenerate atomic arrows Of these 119906 and 120578 are parental and 120578 is the fillable parent of119891 so this subcomputad is parental

bull The triangle identity witness 120573 of Example 8111 is fillable and the subcomputad 120573 sub 119964119889119895 itgenerates has 119906 119891 120578 120598 and 120573 as its only non-degenerate atomic arrows Since 120573 is the fillableparent of 120598 this subcomputad is parental

bull The pair of fillable 3-arrows

120596 ≔ and 120591 ≔

generate a subcomputad 120596 120591 sub 119964119889119895 that has 119906 119891 120578 120598 both triangle identity witnesses 120573 and

120572 of Example 8111 and 120584 ≔ 120596 sdot 1205751 = 120591 sdot 1205751 as its only non-degenerate atomic arrows Since 120596is the fillable parent of 120572 and 120591 is the fillable parent of

120584 ≔

this subcomputad is parentalbull 119964119889119895 is trivially a parental subcomputad of itself

826 Non-Examplebull The subcomputad 120598 has 119906 119891 and 120598 as its only non-degenerate atomic arrows Since both 119891 and120598 are missing their fillable parents this subcomputad is not parental

bull The subcomputad 120578 120598 that has 119906 119891 120578 and 120598 as its only non-degenerate atomic arrows still failsto be parental since the fillable parent of 120598 is missing

bull The subcomputad 120573 120572 that has 119906 119891 120578 120598 and both triangle identity witnesses 120573 and 120572 as itsatomic non-degenerate arrows is not parental since the fillable parent 120596 of 120572 is missing

These examples establish a chain of parental subcomputad inclusions

119906 sub 120578 sub 120573 sub 120596 120591 sub 119964119889119895

Our aim in the remainder of this section is to filter a general parental subcomputad inclusion 119964 sub119964prime as a countable tower of parental subcomputad inclusions with each sequential inclusion pre-sented as the pushout of an explicit simplicial subcomputad inclusion The subcomputad inclusions120794[Λ119896[119899]] 120794[Δ[119899]] from Example 615 will feature where Λ119896[119899] is an inner horn A second familyof inclusions will also be needed which we now introduce

827 Definition For any simplicial set119880 let 120795[119880] denote the simplicial category whose three non-trivial homs are displayed in the following cartoon

120793119880

120793⋆119880

That is 120795[119880] has objects ldquo⊤rdquo ldquominusrdquo and ldquo+rdquo and non-trivial hom-sets

120795[119880](⊤ minus) ≔ 119880 120795[119880](minus +) ≔ 120793 120795[119880](⊤ +) ≔ 120793 ⋆ 119880and whose only non-trivial composition operation is defined by the canonical inclusion

120795[119880](minus +) times 120795[119880](⊤ minus) 120795[119880](⊤ +)

119880 120793 ⋆ 119880

cong cong

Here we define the endo hom-spaces to contain only the respective identities and the remaining hom-spaces to be empty

828 Lemma A simplicial functor 120795[119880] rarr 119974 is uniquely determined by the databull a pair of 0-arrows 119906∶ 119860 rarr 119861 and 119887 ∶ 119883 rarr 119861bull a cone with summit 119880 over the cospan

119880 119887Fun(119883 119861)

Fun(119883119860) Fun(119883 119861)

120587

119906lowast

or equivalently a map 119880 rarr 119887119906lowast whose codomain is its pullback

Proof The objects119883119860 and 119861 are the images of⊤ minus and+ respectively The map 119906∶ 119860 rarr 119861 isthe image of the unique 0-arrow from minus to + Simplicial functoriality then demands the specification

of the vertical maps below making the square commute

119880 120793 ⋆ 119880

Fun(119883119860) Fun(119883 119861)119906lowast

By the join ⊣ slice adjunction of Proposition 425 the simplicial map 120793 ⋆ 119880 rarr Fun(119883 119861) may bedefined by specifying a 0-arrow 119887 ∶ 119883 rarr 119861 the image of the cone point of 120793⋆119880 together with a map119880 rarr 119887Fun(119883 119861) Now the above commutative square transposes to the one of the statement

Similarly a simplicial functor120794[119880] rarr 119974 is uniquely determined by the data119880 rarr Fun(119883119860) of asimplicial map valued in one of the functor spaces of119974 In particular simplicial functors 120794[Δ[119899]] rarr119974 correspond to 119899-arrows in119974

829 Notationbull For each fillable 119899-arrow 119887 of positive height let 119865119887 ∶ 120794[Δ[119899]] rarr 119964119889119895 be the simplicial functor

classified by the 119899-arrow 119887 of119964119889119895bull For each fillable 119899-arrow 119887 of height 0 let 119865119887 ∶ 120795[Δ[119899 minus 1]] rarr 119964119889119895 be the simplicial functor

defined on objects by minus ↦ minus + ↦ + and ⊤ ↦ 119887119908(119887) the domain of 119887 and on the threenon-trivial homs by

120795[Δ[119899 minus 1]](⊤ minus) cong Δ[119899 minus 1] 119964119889119895(119887119908(119887) minus) 120795[Δ[119899 minus 1]](minus +) cong Δ[0] 119964119889119895(minus +)119887⋄ 119906

120795[Δ[119899 minus 1]](⊤ +) cong Δ[119899] 119964119889119895(119887119908(119887) +)119887

8210 Lemma (extending parental subcomputads) Suppose119964 sub 119964119889119895 is a parental subcomputad and 119887is a fillable 119899-arrow of height 119896 that is not a member of119964 but whose faces 119887 sdot 120575119894 are in119964 for all 119894 ne 119896 Thenthe subcomputad119964prime ≔ 119964 119887 generated by119964 and 119887 is defined by the pushout on the left below in the case119896 gt 0 and on the right below in the case 119896 = 0

120794[Λ119896[119899]] 119964 120795[120597Δ[119899 minus 1]] 119964

120794[Δ[119899]] 119964prime 120795[Δ[119899 minus 1]] 119964prime

119865119887

119865119887 119865119887

and in both cases119964prime is again a parental subcomputad

Proof Because 119887 is the fillable parent of its distinguished face 119887⋄ this non-degenerate atomicnon-fillable (119899 minus 1)-arrow cannot be a member of the parental subcomputad119964 Since the other facesof 119887 are assumed to belong to 119964 119887 and 119887⋄ are the only two atomic arrows that are in 119964prime but notin 119964 Since the first is fillable and the second has the first as its fillable parent it is clear that thesubcomputad119964prime is again parental

To verify the claimed pushouts we considerwhat is required to extend a simplicial functor119865∶ 119964 rarr119974 to a simplicial functor 119865∶ 119964prime rarr119974 In the positive height case all that is needed for such an exten-sion is an 119899-arrow 119891 in119974 with the property that 119891 sdot 120575119894 = 119865(119887 sdot 120575119894) for all 119894 ne 119896 which may be specified

by a simplicial functor 119891∶ 120794[Δ[119899]] rarr 119974 that makes the following square commute

120794[Λ119896[119899]] 119964

120794[Δ[119899]] 119974

119865119887

119865

119891

If the height of 119887 is zero then both its 0th face 119887 sdot 1205750 = 119906 sdot119886 and its atomic non-degenerate factor 119886are missing from119964 So to extend a simplicial functor 119865∶ 119964 rarr 119974 to a simplicial functor 119865∶ 119964prime rarr119974requires an (119899 minus 1) arrow 119892 and an 119899-arrow 119891 in 119974 so that 119892 sdot 120575119894 = 119865(119886 sdot 120575119894) for all 119894 isin [119899 minus 1] and119891 sdot 120575119894 = 119865(119887 sdot 120575119894) for all 119894 ne 0 isin [119899] so that 119891 sdot 1205750 = 119865119906 sdot 119892 By Lemma 828 this data may be specifiedby a simplicial functor 119891∶ 120795[Δ[119899 minus 1]] rarr 119974 that makes the following square commute

120795[120597Δ[119899 minus 1]] 119964

120795[Δ[119899 minus 1]] 119974

119865119887

119865

119891

8211 Proposition Any inclusion119964119964prime of parental subcomputads of119964119889119895 may be filtered as a count-able tower of parental subcomputad inclusions

119964 = 1199640 1199641 ⋯1113996119894ge0

119964119894 = 119964prime

in such a way that for each 119894 ge 1 there is a finite non-empty set 119878119894 of fillable arrows that are not themselvescontained in but which have all faces except the distinguished one contained in 119964119894minus1 so that the parentalsubcomputad119964119894 is generated by119964119894minus1 cup 119878119894 Hence the inclusion119964 119964prime may be expressed as a countablecomposite of inclusions which are constructed by pushouts of the form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)gt0

120794[Λℎ(119887)[dim(119887)]]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⊔

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)=0

120795[120597Δ[dim(119887)]]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

119964119894minus1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)gt0

120794[Δ[dim(119887)]]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⊔

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)=0

120795[Δ[dim(119887)]]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

119964119894

⟨119865119887⟩119887isin119878119894

Proof Let 119878 denote the set of fillable arrows in119964prime which are not in119964 and let 119878119908119899119896 denote thesubset of arrows with width 119908 dimension 119899 and height 119896 Note that any non-degenerate arrow of119964119889119895must have dimension strictly less than its width and there are only finitely many non-degeneratearrows of any given width The height is strictly less than the dimension so each set 119878119908119899119896 is finiteand if non-empty we must have 119896 lt 119899 lt 119908 Now we order the triples (119908 119899 119896) that index non-emptysubsets of fillable arrows lexicographically by increasing width increasing dimension and decreasingheight and let 119878119894 ≔ 119878119908119894119899119894119896119894 denote subset of fillable arrows in the 119894th triple in this ordering for 119894 ge 1

Let 119964119894 be the subcomputad of 119964119889119895 generated by 119964 cup (⋃119894119895=1 119878119894) By construction this family filters

the inclusion of119964 into119964prime = 119964 cup (⋃119894 119878119894)We complete the proof by induction on the index starting from the parental subcomputad1199640 =

119964 For the inductive step we must verify that all but the distinguished face of each 119887 isin 119878119894 ≔ 119878119908119894119899119894119896119894lie in119964119894minus1 By iterating Lemma 8210 it will follow that119964119894 is again parental

By construction119964119894minus1 is the smallest subcomputad of119964119889119895 which contains119964 and all of the fillablearrows of119964prime which havebull width less that 119908119894 orbull width 119908119894 and dimension less than 119899119894 orbull width 119908119894 dimension 119899119894 and height greater than 119896119894

Each fillable arrow 119887 isin 119878119894 has width119908119894 dimension 119899119894 and height 119896119894 We must show that its faces 119887 sdot 120575119895lie in119964119894minus1 for each 119895 ne 119896119894 which we verify by a tedious but straightforward case analysis

case 119895 ne 119896119894 + 1 If 119895 ne 119896119894 + 1 then since 119895 ne 119896119894 the line numbered 119895 is not one of the onesseparating the gap 1198871 from the other entries of 119887 which means that this entry will not be eliminatedwhen computing the 119895th face Consider the unique factorization

119887 sdot 120575119895 = (1198871 sdot 1205721) ∘ ⋯ ∘ (119887119903 sdot 120572119903)of the face 119887 sdot120575119895 into non-degenerate and atomic arrows of119964prime acted upon by degeneracy operators Bythe analysis just given 1198871 is fillable with width at most 119908119894 height 119896119894 or 119896119894 minus 1 depending on whether119895 gt 119896119894 +1 or 119895 lt 119896119894 and dimension less than 119899119894 Thus 1198871 it is contained in119964119894minus1 by the hypothesis thatthis subcomputad contains all fillable arrows of width at most 119908119894 and dimension less than 119899119894 Since1198871 has width at least 2 the other atomic factors have width less than or equal to 119908119894 minus 2 Thus each ofthese is either a fillable arrow with width at most 119908119894 minus 2 which means that it is in119964119894minus1 or its fillableparent in119964prime has width at most 119908119894 minus 1 which means that this fillable parent is also in119964119894minus1 As 119887 sdot 120575119895is a composite of degenerate images of arrows in119964119894minus1 it too lies in119964119894minus1

case 119895 = 119896119894+1 Since 119887 notin 119964119894minus1 and all parental subcomputads contain the unique fillable 0-arrow119906 we know that 119887 ne 119906 and so has width at least 2 and dimension at least 1 The only fillable arrowof width 2 is 120578 which has depth 0 and face 120578 sdot 1205751 = id+ which certainly lies in119964119894minus1 so we can safelyassume that the width of 119887 is at least 3 which forces the height 119896119894 to be at most 119899119894 minus 2 otherwise 119887would not be atomic Thus 119895 is at most 119899119894 minus 1 which tells us that 119887 sdot 120575119895 is again atomic Now we havetwo subcasesbull case 1198872 = 1198871 + 1 In this case the line 119895 = 1198961 + 1 = 1198871 separates the gaps 1198871 and 1198872 so the face119887 sdot 120575119895 will have width at least two less than the width of 119887 This face may also be degenerate but inany case the fillable parent of the non-degenerate arrow that it represents has width less than 119908119894and so this face is present in119964119894minus1

bull case 1198872 gt 1198871 + 1 In this case the line 119895 = 1198961 + 1 = 1198871 separates the gap 1198871 from the gapimmediately below it which contains neither 1198870 nor 1198872 So the face 119887 sdot 120575119895 has width 119908119894 and isnon-degenerate Because 119887 is non-degenerate there is some 119904 gt 2 so that 119887119904 = 1198871 + 1 so 119887 sdot 120575119895is not fillable Now Lemma 823 tells us that its fillable parent has width 119908119894 dimension 119899119894 andheight 119896119894 + 1 Thus this fillable parent lies in119964119894minus1 by our hypothesis that that119964119894minus1 contains allfillable arrows of119964prime of width 119908119894 dimension 119899119894 and height greater than 119896119894 so 119887 sdot 120575119895 must also bein there as desired

8212 Remark (a dual form) In the 2-category 119964119889119895co the 0-arrow 119906 is left adjoint to the 0-arrow119891 with unit 120598 and counit 120578 The 2-functor 119964119889119895 rarr 119964119889119895co that classifies this adjunction is an iso-morphism Via this duality we could have introduced a variant notion of ldquofillable 119899-arrowrdquo (a subsetof those 119899-arrows with codomain minus) and ldquoparental subcomputadrdquo of 119964119889119895 so that every parental sub-computad contained the 0-arrow 119891 Dualizing the argument of Example 825 the subcomputads

119891 sub 120598 sub 120572 would all be parental but the subcomputads of Example 825 would no longer be so

Exercises

82i Exercise Give a graphical description of the dual fillable 119899-arrows discussed in Remark 8212

83 Building homotopy coherent adjunctions

We now employ Proposition 8211 to prove that every adjunction in aninfin-cosmos119974 extends to ahomotopy coherent adjunction119964119889119895 rarr 119974which carries the canonical adjunction in119964119889119895 to the chosenadjunction in119974 The data used to present an adjunction in a quasi-categorically enriched category119974in the sense of Definition 211 mdash a pair of infin-categories 119860 and 119861 a pair of 0-arrows 119906 isin Fun(119860 119861)and 119891 isin Fun(119861119860) a pair of 1-arrows 120578∶ id119861 rarr 119906119891 isin Fun(119861 119861) and 120598 ∶ 119891119906 rarr id119860 isin Fun(119860119860)and witnesses to the triangle equalities given by a pair of 2-arrows

119906119891119906 119891119906119891

119906 119906 119891 119891119906120598 isin Fun(119860 119861)

120598119891isin Fun(119861119860)

120578119906120573

119891120578120572

mdash determines a simplicial functor119879∶ 120573 120572 rarr 119974whose domain is the subcomputad of119964119889119895 generatedby the triangle identity 2-arrows This functor then is defined by mapping minus to 119860 and + to 119861 andacting on the non-degenerate and atomic arrows in the way suggested by their syntax 119879(119906) ≔ 119906119879(119891) ≔ 119891 119879(120578) ≔ 120578 119879(120598) ≔ 120598 119879(120573) ≔ 120573 and 119879(120572) ≔ 120572 In Theorem 834 we prove that it isalways possible to extend the data (119891 ⊣ 119906 120578 120598) to a homotopy coherent adjunction 119964119889119895 rarr 119974 butunless 120573 and 120572 satisfy a coherence condition that will be described doing so might require making adifferent choice for one of these triangle equality witnesses

By Proposition 8211 extensions along parental subcomputad inclusions may be built inductivelyfrom two types of extension problem one of which involves attaching fillable arrows with positiveheight and the other of which involves attaching fillable arrows of height zero We prove that bothsimplicial subcomputad extension problems can be solved relatively against any simplicial functor119875∶ 119974 rarr ℒ between quasi-categorically enriched categories that defines a ldquolocal isofibrationrdquo mean-ing that the action of 119875 on functor spaces is by isofibrations This relative lifting result will be usedin the proof of the homotopical uniqueness of such extensions in sect84

831 Lemma Let 119875∶ 119974 rarr ℒ be a simplicial functor that defines a local isofibration of quasi-categoriesThen any lifting problem against the directed suspension of an inner horn inclusion

120794[Λ119896[119899]] 119974

120794[Δ[119899]] ℒ

119875

has a solution

Proof The lifting problem of the statement is equivalent to asking for a lift of the inner horninclusion

Λ119896[119899] Fun(119860 119861)

Δ[119899] Fun(119875119860 119875119861)against the action of 119875 on some functor spaces of119974 andℒ By hypothesis this action is an isofibrationof quasi-categories and so the claimed lift exists

832 Proposition (the relative universal property of the unit) If 119974 has an adjunction (119891 ⊣ 119906 120578)119875∶ 119974 rarr ℒ is a local isofibration of quasi-categorically enriched categories and if 119879∶ 120795[120597Δ[119899]] rarr 119974 isdefined to carry the unique 0-arrow minus rarr + to 119906∶ 119860 rarr 119861 the cone point of 120793 ⋆ 120597Δ[119899] cong Λ0[119899] to119887 ∶ 119883 rarr 119861 and the 1-arrow from 0 to 1 in Λ0[119899] to 120578119887∶ 119887 rarr 119906119891119887 for some 119899 ge 1 then any lifting problem

120795[120597Δ[119899]] 119974

120795[Δ[119899]] ℒ

119879

119875

has a solution

Proof By Lemma 828 the lifting problem of the statement translates to a lifting problem ofthe following form

120597Δ[119899] 119887119906lowast

Δ[119899] 119875119887119875119906lowast

119905

where the upper horizontal map carries the initial vertex 0 isin 120597Δ[119899] to the object 120578119887 isin 119887119906lowast which isinitial in there by Proposition 414 applied to the adjunction between functor spaces

Fun(119883 119861) Fun(119883119860)

119891lowast

119906lowast

and Appendix C which proves that Joyalrsquos slices are fibered equivalent to comma quasi-categoriesThe right-hand vertical is an isofibration that preserves this initial element since it carries it to acomponent of the unit for the adjunction (119875119891 ⊣ 119875119906 119875120578) inℒ Lemma proven in Appendix F nowproves that the desired lift exists

Our theorems about extensions to homotopy coherent adjunctions will follow as special cases ofthe following relative extension and lifting result for parental subcomputads containing the unit ofan adjunction

833 Theorem Suppose 120578 sub 119964 sub 119964prime sub 119964119889119895 are parental subcomputads Then if 119875∶ 119974 rarr ℒ is alocal isofibration between quasi-categorically enriched categories and if119974 has an adjunction (119891 ⊣ 119906 120578) then

we may solve any lifting problem

119964 119974

119964prime ℒ

119879

119875

so long as 119879(119891) = 119891 119879(119906) = 119906 and 119879(120578) = 120578

Proof By Proposition 8211 the inclusion119964 119964prime can be fillered as a sequential composite ofpushouts of coproducts of maps of two basic forms so it suffices to demonstrate that we may solvelifting problems of the following two forms

120794[Λ119896[119899]] 119964 119974 120795[120597Δ[119899 minus 1]] 119964 119974

120794[Δ[119899]] 119964prime ℒ 120795[Δ[119899 minus 1]] 119964prime ℒ

119865119887

119879

119875

119865119887

119879

119875

119865119887 119865119887

for some fillable 119899-arrow 119887 isin 119964119889119895 of positive height in the left-hand case and of height zero in theright-hand case In the case of the left-hand lifting problem such lifts exist immediately from Lemma831

In the case of the right-hand lifting problem we must verify that the hypotheses of Proposition832 are satisfied which amounts to verifying that for any fillable 119899-arrow 119887 of height zero the functor119865119887 ∶ 120795[Δ[119899 minus 1]] rarr 119964119889119895 defined in Notation 829 in relavant part by the map

120795[Δ[119899 minus 1]](⊤ +) cong Δ[119899] 119964119889119895(119887119908(119887) +)119887

carries the 1-arrow from 0 to 1 in Δ[119899] to a component of the unit in 119964 sub 119964119889119895 This is easilyverified using the graphical calculus The image of the 1-arrow from 0 to 1 under the map 119887 ∶ Δ[119899] rarr119964119889119895(119887119908(119887) +) is simply the initial edge of 119887 which may be computed by removing all of the other linesfrom the strictly undulating squiggle diagram By the specifications of a fillable arrow with height 0there are only two possibilities depending on whether the domain of 119887 is + or minus as illustrated by thefollowing diagrams

119887 ≔

11988712057501 ≔ 01 = 0

119887 ≔

11988712057501 ≔ 01 = 0

Thus we see that 119887 sdot 12057501 = 120578119906 or 119887 sdot 12057501 = 120578 both of which satisfy the conditions required forProposition 832 to provide the desired lift

834 Theorem (homotopy coherent adjunctions) If 119974 is a quasi-categorically enriched category con-taining an adjunction (119891 ⊣ 119906 120578 120598 120573 120572)

(i) There exists a simplicial functor120120∶ 119964119889119895 rarr 119974 for which120120(119906) = 119906120120(119891) = 119891 and120120(120578) = 120578(ii) There exists a simplicial functor 120120∶ 119964119889119895 rarr 119974 for which 120120(119906) = 119906 120120(119891) = 119891 120120(120578) = 120578

120120(120598) = 120598 and120120(120573) = 120573(iii) If there exist a pair of 3-arrows 120596 and 120591 in119974

120596 ≔

119906119891

id119861 119906119891

119906119891119906119891

120578

120578sdot120578 119906120598119891119906119891120578

120591 ≔

119906119891

id119861 119906119891

119906119891119906119891

120578

120578sdot120578 119906120598119891120578119906119891

witnessing the coherence relations 1205961205750 = 119906120572 1205911205750 = 120573119891 1205961205751 = 1205911205751 1205961205752 = 1205911205752 = 1205781205901 with the3rd faces of these simplices defined by the pair of 2-arrows given by the horizontal composite

Δ[1] times Δ[1] Fun(119861 119861) times Fun(119861 119861) Fun(119861 119861)120578times120578 ∘

then there exists a simplicial functor120120∶ 119964119889119895 rarr 119974 for which120120(119906) = 119906 120120(119891) = 119891 120120(120578) = 120578120120(120598) = 120598120120(120573) = 120573120120(120572) = 120572120120(120596) = 120596 and120120(120591) = 120591

Coherence conditions of the form stated in ((iii)) appear in the definition of a biadjoint pair in astrongly bicategorically enriched category in [62 138]

Exercises

83i Exercise Unpack Proposition 832 in the case 119899 = 1 and 119899 = 2

83ii Exercise Use Remark 8212 to formulate the dual of Theorem 834 describing homotopycoherent adjunctions generated by a given counit

84 Homotopical uniqueness of homotopy coherent adjunctions

The homotopy 2-category of an infin-cosmos can be regarded as a simplicial category of the formdescribed in Lemma 813 in which context it is equipped with a canonical quotient functor119876∶ 119974 rarr120101119974 of simplicial categories Formally 119876 is a component of the counit of the adjunction described inDigression 142 It follows easily from the characterization of 2-categories in Lemma 813 that119876 is alocal isofibration

By Proposition 8113 any adjunction in the homotopy 2-category of aninfin-cosmos is representedby a unique simplicial functor119879∶ 119964119889119895 rarr 120101119974 To say that a homotopy coherent adjunction120120∶ 119964119889119895 rarr119974 lifts the adjunction in the homotopy 2-category means that120120 is a lift of 119879 along119876 Theorem 834proves that a lift of any adjunction in the homotopy 2-category 120101119974 can be constructed by specifyinga lift 119879∶ 120573 rarr 119974 of 119879∶ 119964119889119895 rarr 120101119974mdash this amounting to a choice of 1-arrows representing the unit

and counit and a 2-arrow witnessing one of the triangle equalities mdash and then extending along theparental subcomputad inclusion to define the homotopy coherent adjunction120120

120573 119974

119964119889119895 120101119974

119879

119876120120

119879

Note that the commutativity of the top left triangle implies the commutativity of the bottom rightone so instead of thinking of 120120 as a lift of 119879∶ 119964119889119895 rarr 120101119974 to a homotopy coherent adjunction wecan equally think of120120 as an extension of 119879∶ 120573 rarr 119974 to a homotopy coherent adjunction

In this section we will define the space of extensions of given adjunction data to a homotopycoherent adjunction Our main theorem in this section is that if the base diagram to be extended isindexed by a parental subcomputad then the space of lifts is a contractible Kan complex

The first step is to construct a (possibly large) simplicial hom-space between two simplicial cate-gories

841 Definition Define the cotensor ℒ119880 of a simplicial category ℒ with a simplicial set 119880 to bethe simplicial category with objℒ119880 ≔ objℒ and hom-spaces

ℒ119880(119883 119884) ≔ ℒ(119883 119884)119880Any simplicial set119880 defines a comonoid (119880 ∶ 119880 rarr 120793Δ∶ 119880 rarr 119880times119880)with respect to the cartesianproduct and diagonal maps so the endofunctor (minus)119880 ∶ 119982119982119890119905-119966119886119905 rarr 119982119982119890119905-119966119886119905 is a monad whose unitand multiplication are defined by restricting along these maps

For simplicial categories 119974 and ℒ let Icon(119974ℒ) denote the large simplicial sets defined by thenatural isomorphism

Icon(119974ℒ)119880 cong Icon(119974ℒ119880) ie Icon(119974ℒ)119899 ≔ Icon(119974ℒΔ[119899])The composition map

Icon(ℒℳ) times Icon(119974ℒ) Icon(119974ℳ)∘

is given by defining the composite of a pair of 119899-arrows 119865∶ 119974 rarr ℒΔ[119899] and 119866∶ ℒ rarr ℳΔ[119899] to bethe Kleisli composite 119899-arrow

119974 ℒΔ[119899] (ℳΔ[119899])Δ[119899] cong ℳΔ[119899]timesΔ[119899] ℳΔ[119899]119865 119866Δ[119899] Δlowast

842 Remark The vertices of Icon(119974ℒ) are simplicial functors119974rarr ℒ The name ldquoiconrdquo is chosenbecause the 1-simplices are analogous to the ldquoidentity component oplax natural transformationsrdquo in2-category theory as defined by Lack [34] In particular each 1-simplex 119974 rarr ℒΔ[1] or 119899-simplex119974rarr ℒΔ[119899] spans simplicial functors119974rarr ℒ that agree on objects

843 Lemma If 119875∶ 119974 ↠ ℒ is a local isofibration between quasi-categorically enriched categories and119868 ∶ 119964 ℬ is a simplicial subcomputad inclusion then if either 119875 is bijective on objects or 119868 is injective onobjects then the Leibniz map

1114024Icon(119868 119875) ∶ Icon(ℬ119974) ↠ Icon(119964119974) timesIcon(119964ℒ)

Icon(ℬℒ)

is an isofibration between quasi-categories

Proof A lifting problem of simplicial sets as below-left transposes into a lifting problem of sim-plicial categories below-right

119880 Icon(ℬ119974) 119964 119974119881

119881 Icon(119964119974) timesIcon(119964ℒ)

Icon(ℬℒ) ℬ 119974119880 timesℒ119880ℒ119881

119894 1114024Icon(119868119875) 119868 1114024119894119875

When 119875 is surjective on objects so is the simplicial functor 1114024119894 119875 and in general the action on homs isgiven by the Leibniz map

1114024119894 119875 ∶ Fun(119883 119884)119881 ↠ Fun(119883 119884)119880 timesFun(119875119883119875119884)119880

Fun(119875119883 119875119884)119881

which is a trivial fibration whenever 119880 119881 is an inner horn inclusion or is the inclusion 120793 120128By Definition 6111 to solve the right-hand lifting problem it suffices to solve the two lifting

problems

empty 119974119881 120794[120597Δ[119899]] 119974119881

120793 119974119880 timesℒ119880ℒ119881 120794[Δ[119899]] 119974119880 times

ℒ119880ℒ119881

1114024119894119875 1114024119894119875

where the left-hand lifting problem is not needed if 119868 is bijective on objects The right-hand liftingproblem can be solved because 1114024119894 119875 is a local trivial fibration and the left-hand lifting problem canbe solved whenever 119875 is surjective on objects

844 Corollary If 119974 is a quasi-categorically enriched category if 119964 is a simplicial computad and if119868 ∶ 119964 ℬ is a simplicial subcomputad inclusion then

Icon(119868119974) ∶ Icon(ℬ119974) ↠ Icon(119964119974)is an isofibration between quasi-categories

845 Lemma Suppose119875∶ 119974 rarr ℒ is a simplicial functor between quasi-categories that is locally conservativein the sense that each Fun(119860 119861) rarr Fun(119875119860 119875119861) reflects isomorphisms Then if119964 is a simplicial computadand119964 ℬ is a bijective-on-objects simplicial subcomputad inclusion then the Leibniz map

Icon(ℬℒ)

is a conservative functor of quasi-categories

Proof In the language of marked simplicial sets we must show that any lifting problem below-left has a solution

120794 Icon(ℬ119974) 119964 119974120794♯

120794♯ Icon(119964119974) timesIcon(119964ℒ)

Icon(ℬℒ) ℬ 119974120794 timesℒ120794ℒ120794♯

1114024Icon(119868119875)

where 120794♯ represents invertible 1-arrows By adjunction this transposes to the lifting problem above-right As in the proof of Lemma 843 it suffices to consider the case where119964 ℬ is 120794[120597[Δ[119899]]] 120794[Δ[119899]] for some 119899 ge 0 in which case the lifting problem of simplicial categories reduces to one ofsimplicial sets

120597Δ[119899] Fun(119860 119861)120794♯ (120597Δ[119899] times 120794♯) cup (Δ[119899] times 120794) Fun(119860 119861)

Δ[119899] Fun(119860 119861)120794 timesFun(119875119860119875119861)120794

Fun(119875119860 119875119861)120794♯ Δ[119899] times 120794♯ Fun(119875119860 119875119861)

The marked simplicial sets on the left have the same underlying sets differing only in their markingsso by the hypothesis that Fun(119860 119861) rarr Fun(119875119860 119875119861) is conservative the result follows

846 Definition The space of homotopy coherent adjunctions in a quasi-categorically enrichedcategory119974 is

cohadj(119974) ≔ Icon(119964119889119895119974)

847 Proposition The space of homotopy coherent adjunctions in119974 is a Kan complex possibly a large one

Proof Since 119964119889119895 is a simplicial computad Corollary 844 implies that cohadj(119974) is a quasi-category By Corollary 1115 we need only show that all of its arrows are isomorphism Since a 1-arrowin a functor space of 119974 is an isomorphism if and only if it represents an invertible 2-cell in 120101119974 thequotient simplicial functor 119876∶ 119974 rarr 120101119974 is locally conservative and by Lemma 845 it suffices toshow that cohadj(120101119974) is a Kan complex

From Proposition 8113 we can extract an explicit description of cohadj(120101119974) that reveals that itis actually isomorphic the nerve of a 1-category itsbull objects are adjunctions (119891 ⊣ 119906 120578 120598) in 120101119974bull arrows (120601 120595) ∶ (119891 ⊣ 119906 120578 120598) rarr (119891prime ⊣ 119906prime 120578prime 120598prime) consist of a pair of 2-cells 120601∶ 119891 rArr 119891prime and120595∶ 119906 rArr 119906prime so that 120598prime sdot (120601120595) = 120598 and 120578prime = (120595120601) sdot 120578 and

bull identities and composition are given componentwiseThe isomorphisms in cohadj(120101119974) are those pairs (120601 120595) whose components 120601 and 120595 are both

invertible From the defining equations 120598prime sdot (120601120595) = 120598 and 120578prime = (120595120601) sdot 120578 it follows that the mate of 120595is an inverse to 120601 and the mate of 120601 is an inverse to 120595 so every arrow is in fact an isomorphism

848 Proposition (homotopical uniqueness of parental subcomputad extensions) Suppose 120578 sub119964 sub 119964prime sub 119964119889119895 are parental subcomputads Suppose 119879∶ 119964 rarr 119974 is a simplicial functor so that 119879(119891) = 119891119879(119906) = 119906 and 119879(120578) = 120578 define an adjunction in119974 Then the fiber of the isofibration

Icon(119964prime 119974) ↠ Icon(119964119974)over 119879 is a contractible Kan complex

Proof The fibers over 119879 are those simplicial functors 119879∶ 119964prime rarr 119974 extending 119879∶ 119964 rarr 119974 ByTheorem 833 this fiber is non-empty We must show for any inclusion 119880 119881 of simplicial sets

the lifting problem119880 119864119879 Icon(119964prime 119974)

119881 120793 Icon(119964119974)

119894

119879

has a solution Transposing we obtain a lifting problem of simplicial categories

119964 119974 119974119881

119964prime 119974119880

119879 119974

against the local isofibration 119974119881 ↠ 119974119880 The simplicial functor 119974 ∶ 119974 rarr 119974119881 like all simplicialfunctors preserves the adjunction (119891 ⊣ 119906 120578) in119974 so Theorem 833 applies to provide a solution

Taking 119964prime = 119964119889119895 Proposition 848 tells us that the space of homotopy coherent adjunctionsextending a simplicial functor119964rarr 119974 indexed by a parental subcomputad whose image specifies theunit of an adjunction is a contractible Kan complex This proves that such extensions are ldquohomotopi-cally uniquerdquo We conclude this section with more refined presentations of this kind of result for twoinstances of basic adjunction data of interest

849 Definition (the space of units) Simplicial functors 120578 rarr 119974 correspond bijectively to thechoice of a pair of 0-arrows 119891 isin Fun(119861119860) 119906 isin Fun(119860 119861) together with a 1-arrow 120578∶ id119861 rarr 119906119891 isinFun(119861 119861) We refer to the simplicial subset unit(119974) sub Icon(120578119974) of those triples (119891 119906 120578) thatspecify the unit of an adjunction and those 1-simplices that define isomorphisms between them as thespace of units and denote its objects by (119891 ⊣ 119906 120578) Since Icon(120578119974) is a quasi-category unit(119974) isa Kan complex

8410 Lemma There is an isomorphism of quasi-categories

Icon(120578119974) cong 1114018119860119861isin119974

HomFun(119861119861)(id119861 ∘)

Fun(119860 119861) times Fun(119861119860) 120793

Fun(119861 119861)

1199011 1199010

120601lArr

∘ ∘

Proof Exercise 84ii

8411 Theorem (uniqueness of homotopy coherent extensions of a unit)(i) The space 119864120578 of homotopy coherent adjunctions extending the counit 120578 is a contractible Kan complex(ii) The forgetful functor 119901119880 ∶ cohadj(119974) ⥲rarr unit(119974) is a trivial fibration of Kan complexes

Proof Both statements follow from specializing Proposition 848 to the parental subcomputad120578 sub 119964119889119895 If 120120∶ 119964119889119895 rarr 119974 is a homotopy coherent adjunction its restriction to 120578 rarr 119974 defines

an object of unit(119974) sub Icon(120578119974) The fiber of the isofibration 119901119880 ∶ cohadj(119974) ⥲rarr unit(119974) over(119891 ⊣ 119906 120578) coincides with the fiber considered in Proposition 848 and is thus a contractible Kancomplex

By Corollay 844 the map 119901119880 ∶ cohadj(119974) ⥲rarr unit(119974) is an isofibration between Kan complexesand is thus a Kan fibration By Proposition any Kan fibration between Kan complexes with con-tractible fibers is a trivial fibration proving (ii)

8412 Definition (the space of right adjoints) Simplicial functors 119906 rarr 119974 correspond bijectivelyto the choice of a 0-arrow 119906 in119974 Indeed

Icon(119906119974) cong 1114018119860119861isin119974

Fun(119860 119861)

We refer to the simplicial subset rightadj(119974) sub Icon(119906119974) of those 0-arrows that possess a leftadjoint and the isomorphisms between them as the space of right adjoints As a quasi-category whose1-arrows are all isomorphisms rightadj(119974) is a Kan complex

The isofibration Icon(120578119974) ↠ Icon(119906119974) restricts to define an isofibration 119902119877 ∶ unit(119974) ↠rightadj(119974) of Kan complexes

8413 Proposition The isofibration 119902119877 ∶ unit(119974) ↠ rightadj(119974) is a trivial fibration of Kan complexes

Proof Since both unit(119974) and rightadj(119974) are Kan complexes automatically 119902119877 is a Kan fibra-tion By Proposition we need only show that its fibers are contractible The fiber of 119902119877 over119906∶ 119860 rarr 119861 is isomorphic to the sub-quasi-category of the fiber of the isofibration Icon(120578119974) ↠

Icon(119906119974) over 119906 whose objects are pairs (119891 120578) which have the property that 119891 is a left adjoint tothe fixed 0-arrow 119906 with unit represented by 120578 and whose 1-simplices are all invertible By Lemma8410 this isofibration is isomorphic to the coproduct of the family of projections

1114018119860119861isin119974

HomFun(119861119861)(id119861 ∘)1199011minusminusrarrrarr Fun(119860 119861) times Fun(119861119860) 120587minusrarrrarr Fun(119860 119861)

whose fiber over 119906 is isomorphic to HomFun(119861119861)(id119861 119906lowast)Applying Proposition 414 to the adjunction between functor spaces

Fun(119861 119861) Fun(119861119860)

119891lowast

119906lowast

and the element id119861 isin Fun(119861 119861) reveals that (119891 120578) is a initial in HomFun(119861119861)(id119861 119906lowast) So the fiberof 119902119877 over 119906 is isomorphic to the sub-quasi-category of HomFun(119861119861)(id119861 119906lowast) spanned by its initialelements and as such is a contractible Kan complex⁶

8414 Theorem (uniqueness of homotopy coherent extensions of a right adjoint)

⁶In any quasi-category in which every element is initial all 1-arrows are isomorphisms since the initial elements arealso initial in its homotopy category which must then be a groupoid This proves that the quasi-category spanned by initialelements is in fact a Kan complex Any adjunction between Kan complexes is an adjoint equivalence so in particular theright adjoint ∶ 119860 ⥲rarr 120793 defines an equivalence and hence a trivial fibration

(i) The space 119864119906 of homotopy coherent adjunctions with right adjoint 119906 is a contractible Kan complex(ii) The forgetful functor 119901119877 ∶ cohadj(119974) ⥲rarr rightadj(119974) is a trivial fibration of Kan complexes

Proof The space 119864119906 is defined as the fiber of the composite fibration

119901119877 ∶ cohadj(119974) unit(119974) rightadj(119974)sim

119901119880 sim

119902119877

By Theorem 8411 and Proposition 8413 both maps are trivial fibrations of Kan complexes hence sois the composite and thus its fiber is a contractible Kan complex

Exercises

84i Exercise State and prove a relative version of Proposition 848 establishing the homotopicaluniqueness of solutions to lifting problems of parental subcomputad inclusions against local isofibra-tions of quasi-categorically enriched categories

CHAPTER 9

The formal theory of homotopy coherent monads

91 Homotopy coherent monads

The free homotopy coherent monad is defined as a full subcategory ℳ119899119889 119964119889119895 on the object+ Via this definition it inherits a graphical calculus from the graphical calculus established for119964119889119895in sect81

911 Definition (the free homotopy coherent monad) The free homotopy coherent monad ℳ119899119889is the full subcategory of the free homotopy coherent adjunction 119964119889119895 on the object + Proposition8110 gives us two definitions ofℳ119899119889bull It is the 2-category regarded as a simplicial category with one object with the hom-category 120491+

and with horizontal composition given by ordinal sum oplus∶ 120491+ times 120491+ rarr 120491+bull It is the simplicial category with one object whose 119899-arrows are strictly undulating squiggles on(119899 + 1)-lines that start and end in the gap labeled +

912 Lemma The simplicial categoryℳ119899119889 is a simplicial computad thoughℳ119899119889 119964119889119895 is not a simpli-cial subcomputad inclusion

Proof Horizontal composition inℳ119899119889 is given by horizontal juxtaposition of squiggle diagramsthat start and end at+ Thus an 119899-arrow is atomic if and only if it has no instances of+ in its interiorThis proves thatℳ119899119889 is a simplicial computad but note thatℳ119899119889 includes atomic arrows such as

119905 ≔ 119906119891 ≔minus+ and 120583 ≔ 119906120598119891 ≔ 1

+(913)

that do have minus in their interiors and thus fail to be atomic in119964119889119895

Employing the graphical calculus we discover another characterization of the atomic 119899-arrows ofℳ119899119889 in reference to the atomic 0-arrow 119905 defined in (913)

914 Lemma An 119899-arrow inℳ119899119889 is atomic if and only if its final vertex is 119905

In particular 119905 is the unique atomic 0-arrow

Proof The 119899-arrows are strictly undulating squiggles on (119899 + 1)-lines that start and end at thespace labeled + these are atomic if and only if there are no instances of + in their interiors

This condition implies that if all the lines are removed except the bottom one a process that computesthe final vertex of the 119899-simplex the resulting squiggle looks like a single hump over one line whichis the graphical representation of the 0-arrow 119905

915 Definition A monad in a 2-category is given bybull an object 119861bull an endofunctor 119905 ∶ 119861 rarr 119861bull and a pair of 2-cells 120578∶ id119861 rArr 119905 and 120583∶ 1199052 rArr 119905

so that the ldquounitrdquo and ldquoassociativityrdquo pasting equalities hold

119861 119861 119861 119861 119861 119861 119861

119861 119861 119861 119861 119861 119861 119861 119861 119861

119905dArr120583

= 119905119905 = =119905dArr120578

dArr120583

119905

119905dArr120583

119905dArr120583 =

119905

dArr120583 119905dArr120583119905

dArr120578

119905 119905

119905

119905119905

When these conditions are satisfied we say that (119905 120578 120583) defines a monad on 119861

916 Lemma The atomic arrows

119905 ≔ 120578 ≔ and 120583 ≔ 119906120598119891 ≔

define a monad (119905 120578 120583) on + in the 2-categoryℳ119899119889

Proof The unit pasting identities of Definition 915 are witnessed by the atomic 2-arrows

119906120572 ≔ and 120573119891 ≔

The pair of atomic 2-arrows

demonstrate that the left hand side and right-hand side of the associativity pasting equality have acommon composite namely the common first face

As a 2-categoryℳ119899119889 has a familiar universal property Lawverersquos characterization of (120491+ oplus [minus1])as the free strict monoidal category containing a monoid ([0] ∶ [minus1] rarr [0] 1205900 ∶ [1] rarr [0]) tells usthatℳ119899119889 is the free 2-category containing a monad [35]

917 Proposition For any 2-category 119966 2-functorsℳ119899119889 rarr 119966 correspond to monads in 119966

These considerations motivate the following definition

918 Definition (homotopy coherent monad) A homotopy coherentmonad in a quasi-categoricallyenriched category 119974 is a simplicial functor 120139∶ ℳ119899119889 rarr 119974 Explicitly a homotopy coherent monadconsists of

bull an object 119861 isin 119974 andbull a homotopy coherent diagram 120139∶ 120491+ rarr Fun(119861 119861) that we refer to as the monad resolution of120139

so that the diagram

120491+ times 120491+ Fun(119861 119861) times Fun(119861 119861)

120491+ Fun(119861 119861)

120139times120139

120139commutes This simplicial functoriality condition implies that the generating 0- and 1-arrows of themonad resolution have the following form

id119861 119905 119905119905 119905119905119905 ⋯ isin Fun(119861 119861)120578120578119905

119905120578

120583 119905120578119905

120578119905119905

119905119905120578

119905120583

120583119905(919)

where (119905 120578 120583) denotes the image of the monad (119905 120578 120583) of Lemma 916 under 120139∶ ℳ119899119889 rarr 119974 Werefer to the 0-arrow 119905 ∶ 119861 rarr 119861 as the functor part of the homotopy coherent monad 120139 and refer tothe 1-arrows 120578 and 120583 as the unit and associativity maps

Note that for any generalized element 119887 ∶ 119883 rarr 119861 the monad resolution (919) restricts to definea monad resolution

119887 119905119887 119905119905119887 119905119905119905119887 ⋯ isin Fun(119883 119861)120578119887

120578119905119887

119905120578119887

120583119887 119905120578119905119887

120578119905119905119887

119905119905120578119887

119905120583119887

120583119905119887(9110)

9111 Example (free monoid monad) Letℳ be a Kan-complex (or topologically) enriched categoryequipped with an enriched monoidal structure minus otimes minus∶ ℳtimesℳrarrℳ that admits countable conicalcoproducts that are preserved by the monoidal product separately in each variable Then there existsa simplicially enriched endofunctor 119879∶ ℳ rarrℳ defined on objects by

119879(119883) ≔1114018119899ge0

119883otimes119899

equipped with simplicial natural transformations 120578∶ idℳ rArr 119879 and 120583∶ 1198792 rArr 119879 defined by includingat the degree-one component and ldquodistributingrdquo the coproduct

1198792(119883) cong1114018119899ge0

11141021114018119898ge0

119883otimes1198981114105otimes119899

The monad resolution of this simplicially enriched monad defines a simplicial functor120491+timesℳrarrℳ regarding120491+ as a simplicial category whose hom-spaces are sets Applying the homotopy coherentnerve120081∶ 119974119886119899-119982119982119890119905 rarr 119980119966119886119905 this simplicially enriched monad defines the left action120491+times120081ℳrarr120081ℳ of a homotopy coherent monad in 119980119966119886119905 on 120081ℳ

More generally any topologically enriched monad on a topologically enriched category defines ahomotopy coherent monad on its homotopy coherent nerve

Any homotopy coherentmonad120139∶ ℳ119899119889 rarr 119974 defines amonad in the homotopy 2-category sim-ply by composing with the canonical quotient functor discussed at the beginning of sect84 and applyingProposition 917

ℳ119899119889 119974 120101119974120139 119876

However ldquomonads up to homotopyrdquo mdash that is monads in the homotopy 2-category of an infin-cosmosmdash cannot necessarily be made homotopy coherent

9112 Non-Example (a monad in the homotopy 2-category that is not homotopy coherent) Stasheffidentifies homotopy associative 119867-spaces that do not extend to 119860infin-spaces that is monoids up tohomotopy cannot necessarily be rectified into homotopy coherent monoids Let

(119872 120578 ∶ lowast rarr 119872 120578∶ 119872 times119872 rarr119872)denote such an up-to-homotopy monoid This structure defines a monad up to homotopy on the(large) quasi-category of spaces by applying the homotopy coherent nerve to the endofunctor 119872 timesminus∶ 119974119886119899 rarr 119974119886119899 and natural transformations induced by 120578 and 120598 This monad in 120101119980119966119886119905 cannot bemade homotopy coherent

Exercises

91i Exercise(i) Show thatℳ119899119889 contains a unique atomic 1-arrow 120583

119899∶ 119905119899 rarr 119905 for each 119899 ge 0 120583

1= id119905 being

degenerate but each of the other 120583119899

being non-degenerate

(ii) Identify the images of these atomic 1-arrows in the monad resolution (919)(iii) Given an interpretation for the 1-arrow 120583

119899that acknowledges the role played by 120583

3in the

proof of Lemma 916

92 Homotopy coherent algebras and the monadic adjunction

Homotopy coherent monads can be defined in any quasi-categorically (or merely simplically) en-riched category but we are particularly interested in homotopy coherent monads valued ininfin-cosmoibecause the flexible weighted limits guaranteed by Corollary 733 permit us to construct the monadicadjunction which relates theinfin-category on which the monad acts to theinfin-category of algebras

The universal property of the monadic adjunction associated to a homotopy coherent monad120139∶ ℳ119899119889 rarr 119974 is very easy to describe though some more work will be required to demonstratethat any infin-cosmos 119974 admits such a construction The monadic adjunction is the terminal adjunc-tion extending the homotopy coherent monad which means that it is given by the right Kan extensionalong the inclusion

119964119889119895

ℳ119899119889 119974

120120120139dArrcong

120139Sinceℳ119899119889 119964119889119895 is fully faithful the value of the right Kan extension at + isin ℳ119899119889 is isomorphicto 119861 ≔ 120139(+) By Example 7119 the value of the right Kan extension at minus isin ℳ119899119889 is computed as the

limit of 120139∶ ℳ119899119889 rarr 119974 weighted by the restriction of the covariant representable of119964119889119895 at minus alongℳ119899119889 119964119889119895 which is how we will now define the weight119882minus for theinfin-category of algebras

921 Observation (weights onℳ119899119889) A weight onℳ119899119889 is a simplicial functor119882∶ ℳ119899119889 rarr 119982119982119890119905Explicitly to specify a weight onℳ119899119889 is equivalent to specifyingbull a simplicial set119882 ≔119882(+)bull equipped with a left action of the simplicial monoidsup1 (120491+ oplus [minus1])

120491+ times119882 119882sdot so that120491+ times 120491+ times119882 120491+ times119882 119882 120491+ times119882

120491+ times119882 119882 119882

idtimessdot

oplustimesid sdot

[minus1]timesid

Frequently the simplicial set119882 happens to be a quasi-category in which case the weight119882 onℳ119899119889is precisely a homotopy coherent monad on the quasi-category119882

Relative to the encoding of weights onℳ119899119889 as simplicial sets with a left120491+-action amap119891∶ 119881 rarr119882 isin 119982119982119890119905ℳ119899119889 is given by a map 119891∶ 119881 rarr 119882 of simplicial sets that is 120491+-equivariant in the sensethat the diagram

119881 119882

120491+ times 119881 120491+ times119882

119881 119882

119891

[minus1]timesid [minus1]timesid

idtimes119891

119891commutes

922 Lemma Let 119882∶ ℳ119899119889 rarr 119982119982119890119905 be a weight on ℳ119899119889 and let 119879∶ ℳ119899119889 rarr 119974 be a homotopycoherent monad on 119861 isin 119974 Then a 119882-shaped cone over 120139 with summit 119883 is specified by a simplicial map120574∶ 119882 rarr Fun(119883 119861) which makes the square

120491+ times119882 Fun(119861 119861) times Fun(119883 119861)

119882 Fun(119883 119861)

120139times120574

120574

Proof By Observation 921 a simplicial natural transformation 120574∶ 119882 rarr Fun(119883120139minus) is givenby its unique component 120574∶ 119882 rarr Fun(119883 119861) subject to the equivariance condition

120491+ times119882 120491+ times Fun(119883 119861)

119882 Fun(119883 119861)

idtimes120574

120574

sup1The strict monoidal category (120491+ oplus [minus1]) is a monoid in (119982119982119890119905 times 120793) Applying the nerve functor (120491+ oplus [minus1])also defines a monoid in (119982119982119890119905 times 120793)

The right-hand action map is the transpose of the action map of the composite simplicial functorFun(119883120139minus) ∶ ℳ119899119889 rarr 119980119966119886119905 this being

120491+ Fun(119861 119861) Fun(Fun(119883 119861)Fun(119883 119861))120139 Fun(119883minus)

which transposes to

120491+ times Fun(119883 119861) Fun(119861 119861) times Fun(119883 119861) Fun(119883 119861)120139timesid ∘

Now the equivariance square coincides with the commutative square of the statement

923 Example (notable weights onℳ119899119889) We fix notation for a few notable weights onℳ119899119889(i) Write119882+ ∶ ℳ119899119889 rarr 119982119982119890119905 for the unique represented functor onℳ119899119889 which is given by the

quasi-category 120491+ = ℳ119899119889(+ +) = 119964119889119895(+ +) acted upon the left by itself via the ordinalsum map

120491+ times 120491+ 120491+oplus

(ii) Write119882minus ∶ ℳ119899119889 rarr 119982119982119890119905 for the restriction of the covariant representable functor119964119889119895 rarr119982119982119890119905 on minus along ℳ119899119889 119964119889119895 This weight is presented by the quasi-category 120491⊤ =119964119889119895(minus +) acted upon the left by 120491+ by the ordinal sum map

120491+ times 120491⊤ 120491⊤oplus

which defines the horizontal composition in119964119889119895 in Definition 811There is a natural inclusion

120491+ 120491⊤

119964119889119895(+ +) 119964119889119895(minus +)

minusoplus[0]

cong cong

minus∘119906

that ldquofreely adjoins a top element in each ordinalrdquo or in the graphical calculus of Definition 819ldquoprecomposes a strictly undulating squiggle with 119906 ≔ (+ minus)rdquo This commutes with the left120491+-actionsand so defines an inclusion of weights119882+ 119882minus by Observation 921sup2

Since119882+ is the representable weight it is automatically flexible By the first axiom of Definition713 the119882+-weighted limit of a homotopy coherent monad120139 recovers theinfin-category lim119882+ 120139 cong 119861on which 120139 acts

924 Lemma The inclusion119882+ 119882minus is a projective cell complex in119982119982119890119905ℳ119899119889 built by attaching projective119899-cells 120597Δ[119899] times119882+ Δ[119899] times119882+ in dimensions 119899 gt 0 In particular119882minus is a flexible weight

Proof We apply Theorem 7212 and prove that119882+ 119882minus is a projective cell complex by verify-ing that coll(119882+) coll(119882minus) is a relative simplicial computad Since119882+ is flexible it follows thenthat119882minus is too The collage coll(119882minus) can be identified with the non-full simplicial subcategory of119964119889119895containing two objects minus (aka ⊤) and + and the hom-spaces 119964119889119895(minus +) and 119964119889119895(+ +) but with thehom-space from minus to minus trivial and the hom-space from + to minus empty Via the graphical calculus ofDefinition 819 we see that coll(119882minus) is a simplicial category whose 119899-arrows are strictly undulatingsquiggles from minus to + or from + to + with composition defined by concatenation at + Its atomic119899-arrows are then those that have no instances of + in their interiors

sup2A ldquoright adjointrdquo to the inclusion119882+ 119882minus will be described in the proof of Proposition 9211

Similarly coll(119882+) is the simplicial category with two objects ⊤ and + and with the hom-spacesfrom ⊤ to + and from + to + both defined to be119964119889119895(+ +) with the hom space from ⊤ to ⊤ trivialand the hom-space from + to ⊤ empty This is also a simplicial computad in which the only atomicarrow from ⊤ to + is the identity 0-arrow corresponding to [minus1] isin 119964119889119895(+ +) = 120491+ as beforethe atomic arrows from + to + are the strictly undulating squiggles which have no instances of + intheir interiors To provide intuition for this simplicial computad structure on coll(119882+) recall thatsince the representable 119882+ defines a projective cell complex empty 119882+ built by attaching a singleprojective 0-cell at + the proof of Theorem 7212 tells us that the simplicial subcomputad inclusion120793 +ℳ119899119889 coll(119882+) is defined by adjoining a single atomic 0-arrow from ⊤ to + to the simplicialcomputad 120793 +ℳ119899119889

The inclusion coll(119882+) coll(119882minus) is bijective on the common subcategory120793+ℳ119899119889 and definedby sending each119899-arrow from⊤ to+ in coll(119882+) represented as a squiggle from+ to+ to the squigglefrom minus to + defined by precomposing with 119906 This function carries the unique atomic 0-arrow from⊤ to + in coll(119882+) to 119906 which is the unique atomic 0-arrow in coll(119882minus) from⊤ to + Now Theorem7212 proves that119882+ 119882minus is a projective cell complex Furthermore since coll(119882+) coll(119882minus)is surjective on atomic 0-arrows only projective cells of positive dimension are needed to present119882+ 119882minus as a sequential composite of pushouts of projective 119899-cells120597Δ[119899]times119882+ Δ[119899]times119882+

Now let119974 be aninfin-cosmos

925 Definition The infin-category of 120139-algebras for a homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974in aninfin-cosmos119974 is the flexible weighted limit lim119882minus 120139 When 120139 acts on theinfin-category 119861 via themonad resolution (919) with functor part 119905 ∶ 119861 rarr 119861 we write

Alg120139(119861) ≔ lim119882minus 120139for theinfin-category of algebras By Proposition 731(ii) the projective cell complex119882+ 119882minus inducesan isofibration

lim119882minus 120139 ↠ lim119882+ 120139upon taking weighted limits defining a map that we denote by 119906119905 ∶ Alg120139(119861) ↠ 119861 and refer to as themonadic forgetful functor This map is the leg of the 119882minus-shaped limit cone indexed by the uniqueobject + isin ℳ119899119889

By Corollary 733 and Lemma 924

926 Proposition Any homotopy coherent monad in aninfin-cosmos admits aninfin-category of algebras

Wenow introduce the generic bar resolution120491⊤ rarr Fun(Alg120139(119861) 119861) associated to theinfin-categoryof 120139-algebras for a homotopy coherent monad acting on 119861

927 Definition (generic bar resolution) The limit cone 120573∶ 119882minus rArr Fun(Alg120139(119861) 120139(minus)) defines thegeneric bar resolution of a homotopy coherent monad120139 acting on aninfin-category 119861 By Lemma 922and Example 923 a 119882minus-cone with summit Alg120139(119861) ≔ lim119882minus 120139 over a homotopy coherent monadacting on 120139 is given by a simplicial map 120573∶ 120491⊤ rarr Fun(Alg120139(119861) 119861) so that the square

120491+ times 120491⊤ Fun(119861 119861) times Fun(Alg120139(119861) 119861)

120491⊤ Fun(Alg120139(119861) 119861)

120139times120573

120573

commutes Under the identification 120491⊤ cong 119964119889119895(minus +) we write 119906119905 and 120573119905 ∶ 119905119906119905 rarr 119906119905 for the 0- and1-arrows of Fun(Alg120139(119861) 119861) defined to be the images of 119906 and 119906120598 under 120573∶ 120491⊤ rarr Fun(Alg120139(119861) 119861)respectively This 0-arrow 119906119905 is the monadic forgetful functor of Definition 925 Then in the notationof (919) the generic bar resolution has the form of a homotopy coherent diagram

119906119905 119905119906119905 119905119905119906119905 119905119905119905119906119905 ⋯ isin Fun(Alg120139(119861) 119861)120578119906119905

120573119905 119905120578119906119905

120578119905119906119905

120583119906119905

119905120573119905

(928)that restricts along the embedding 120491+ 120491⊤ that freely adjoins the top element in each ordinal tothe monad resolution (9110) applied to 119906119905

For any generalized element119883 rarr Alg120139(119861) of theinfin-category of120139-algebras associated to a homo-topy coherent monad acting on 119861 an 119883-family of 120139-algebras in 119861 the generic bar resolution (928)restricts to define a bar resolution

119887 119905119887 119905119905119887 119905119905119905119887 ⋯120578119887

120573 119905120578119887

120578119905119887

120583119887

119905120573

9210 Proposition The monadic forgetful functor 119906119905 ∶ Alg120139(119861) ↠ 119861 is conservative for any 2-cell 120574with codomain Alg120139(119861) if 119906

119905120574 is invertible then so is 120574

Proof Conservativity of the functor 119906119905 asserts that for all 119883 the isofibration of quasi-categories119906119905lowast ∶ Fun(119883Alg120139(119861)) ↠ Fun(119883 119861) reflects invertible 1-cells Working with marked simplicial sets

this is the case just when this map has the right lifting property with respect to the inclusion 120794 120794♯of the walking arrow into the walking marked arrow

By Definition 925 the monadic forgetful functor is defined by applying limminus120139 to the projectivecell complex 119882+ 119882minus of Lemma 924 By Proposition 731 the isofibration 119906119905 ∶ Alg120139(119861) ↠ 119861then factors as the inverse limit of a tower of isofibrations each layer of which is constructed asthe pullback of products of projective cells 119861Δ[119899] ↠ 119861120597Δ[119899] for 119899 ge 1 The cosmological functorFun(119883 minus) ∶ 119974 rarr 119980119966119886119905 preserves this limit so 119906119905lowast ∶ Fun(119883Alg120139(119861)) ↠ Fun(119883 119861) is similarly the

inverse limit of pullbacks of products of maps Fun(119883 119861)Δ[119899] ↠ Fun(119883 119861)120597Δ[119899] for 119899 ge 1 Sinceconservativity of a functor between quasi-categories may be captured by a lifting property it sufficesto show that the maps Fun(119883 119861)Δ[119899] ↠ Fun(119883 119861)120597Δ[119899] reflective invertibility of 1-simplices Sincefor 119899 ge 1 the inclusion 120597Δ[119899] Δ[119899] is bijective on vertices this is immediate from Lemma which says that invertibility in exponentiated quasi-categories is detected pointwise

We now show that any homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 on an infin-category 119861 in aninfin-cosmos extends to a homotopy coherent adjunction120120120139 ∶ 119964119889119895 rarr 119974 whose right adjoint is 119906119905

9211 Proposition (the monadic adjunction) For any homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 on119861 the monadic forgetful functor 119906119905 ∶ Alg120139(119861) rarr 119861 is the right adjoint of a homotopy coherent adjunction

120120120139 ∶ 119964119889119895 rarr 119974

119861 Alg120139(119861)

119891119905

119906119905

120578119905 ∶ id119861 rArr 119906119905119891119905 120598119905 ∶ 119891119905119906119905 rArr idAlg120139(119861)

whose underlying homotopy coherent monad is 120139 This constructs the monadic adjunction of the homotopycoherent monad

In particular the triple (119906119905119891119905 120578119905 119906119905120598119905119891119905) recovers the monad (119905 120578 120583) on 119861 defined in 918

Proof Recall the weights 119882+ and 119882minus are defined in Example 923 to be restrictions of therepresentable functors on119964119889119895 in the case of119882+ this restriction defines the representable functor forℳ119899119889 since the inclusionℳ119899119889 119964119889119895 is full on + The weight for the monadic homotopy coherentadjunction is defined to be the composite of the Yoneda embedding with the restriction functor

119964119889119895op 119982119982119890119905119964119889119895 119982119982119890119905ℳ119899119889

+ 119882+

minus 119882minus

119988 res

119891 ↦⊣ minus∘119906 minus∘119891 ⊢

which can be interpreted as defining an adjunction of weights whose left and right adjoints in theencoding of Observation 921 are given by the maps

120491+ 120491⊤

minus∘119906

perpminus∘119891

that act on strictly undulating 119899-arrows by precomposing with 119906 = (+ minus) or 119891 = (minus +) as appro-priate these maps commute with the left 120491+-actions by postcomposition with a strictly undulatingsquiggle from + to +

Composing with the weighted limit functor limminus120139 defines a simplicial functor

120120120139 ≔ limres119988(minus)120139∶ 119964119889119895 rarr 119974ie a homotopy coherent adjunction between lim119882+ 120139 cong 119861 and lim119882minus 120139 cong Alg120139(119861) whose rightadjoint is given by the action of the 0-arrow 119906 which is the monadic forgetful functor 119906119905 ∶ Alg120139(119861) rarr119861 is defined in 925

Finally the underlying homotopy coherent monad of the homotopy coherent adjunction just con-structed is defined to be the limit of 120139∶ ℳ119899119889 rarr 119974 weighted by

ℳ119899119889op 119964119889119895op 119982119982119890119905119964119889119895 119982119982119890119905ℳ119899119889119988 res

which is just the Yoneda embedding for ℳ119899119889 By the first axiom of Definition 713 this functor isisomorphic to 120139

In sect94 we give a characterization of the monadic adjunction of a homotopy coherent monad Tobuild towards this result we spend the next section establishing important special properties of the

monadic forgetful functor 119906119905 ∶ Alg120139(119861) ↠ 119861 and its left adjoint 119891119905 ∶ 119861 rarr Alg120139(119861) whose essentialimage identifies the free 120139-algebras

Exercises

92i Exercise Prove that the infin-category of algebras associated to the homotopy coherent monad119882+ ∶ ℳ119899119889 rarr 119980119966119886119905 on 120491+ is 120491perp What is the monadic adjunction

93 Limits and colimits in theinfin-category of algebras

The key technical insight enabling Beckrsquos proof of the monadicity theorem [1] is the observationthat any algebra is canonically a colimit of a particular diagram of free algebras In the case of a monad(119905 120578 120583) acting on a 1-category 119861 the data of a 119905-algebra in 119861 is given by a 119906119905-split coequalizer diagram

1199052119887 119905119887 119887120583119887

119905120573

119905120578119887

120578119905119887

120573

120578119887

Here the solid arrows are maps which respect the 119905-algebra structure where the dotted splittings donot Split coequalizers are examples of absolute colimits which are preserved by any functor and inparticular by 119905 ∶ 119861 rarr 119861 a fact we may exploit to show that the underlying fork of (931) defines areflexive coequalizer diagram in the category of 119905-algebras

In the infin-categorical context we require a higher-dimensional version of the diagram (931)namely the bar resolution constructed in (929) for any generalized element 119883 rarr Alg120139(119861) of theinfin-category of 120139-algebras for a homotopy coherent monad acting on 119861 This replaces the 119906119905-splitcoequalizer by a canonically-defined 119906119905-split augmented simplicial object

Before defining this special class of colimits we establish a more general result

932 Proposition Let 120139∶ ℳ119899119889 rarr 119974 be a homotopy coherent monad on aninfin-category 119861 with functorpart 119905 ∶ 119861 rarr 119861 Then if 119861 admits and 119905 preserves colimits of shape 119869 then the monadic forgetful functor119906119905 ∶ Alg120139(119861) rarr 119861 creates colimits of shape 119869

Proof The 0-arrows in the image of a homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 are given bythe identity functor at 119861 the ldquofunctor partrdquo 119905 ∶ 119861 rarr 119861 defined as the image of the unique atomic0-arrow ofℳ119899119889 and finite composites 119905119899 ∶ 119861 rarr 119861 for each 119899 ge 1 If 119905 preserves colimits of shape 119869 in119861 then so does 119905119899 Thus in the case where 119861 admits and 119905 preserves 119869-shaped colimits the homotopycoherent monad lifts to homotopy coherent monad120139∶ ℳ119899119889 rarr 119974perp119869 in theinfin-cosmos of Proposition7414 Since the inclusion119974perp119869 119974 creates flexible weighted limits such as those weighted by119882minusit follows that the limit cone 119906119905 ∶ Alg120139(119861) ↠ 119861 lifts to 119974perp119861 This monadic forgetful functor is theunique 0-arrow component of the limit cone so by Proposition 7414 this tells us that Alg120139(119861) admitsand 119906119905 ∶ Alg120139(119861) ↠ 119861 creates 119869-shaped colimits

A dual argument employing the infin-cosmos 119974⊤119869 of infin-categories that admit and functors thatpreserve 119869-indexed limits proves that if 119861 admits and 119905 preserves limits of shape 119869 then the monadicforgetful functor 119906119905 ∶ Alg120139(119861) ↠ 119861 creates limits of shape 119869 We donrsquot explicitly consider this dual ver-sion here however because we will prove a stronger result in Theorem 939 that drops the hypotheseson 119905

933 Definition (119906-split simplicial objects) The image of the embedding

120491op+ 120491⊤

119964119889119895(minus minus) 119964119889119895(minus +)

[0]oplusminus

cong cong

119906∘minus

is the subcategory of 120491⊤ generated by all of its elementary operators except for the face operators1205750 ∶ [119899 minus 1] rarr [119899] for each 119899 ge 1 We refer to these extra face maps as splitting operators By Defini-tion 239 a simplicial object 119883 rarr 119861120491op

in 119861 admits an augmentation if it lifts along the restrictionfunctor 119861120491

op+ ↠ 119861120491op

and an augmented simplicial object 119883 rarr 119861120491op+ in 119861 admits a splitting if it

lifts along the restriction functor 119861120491⊤ ↠ 119861120491op+ Thus given any functor 119906∶ 119860 rarr 119861 ofinfin-categories

the infin-categories 119878120491op(119906) and 119878120491op+ (119906) of 119906-split simplicial objects and 119906-split augmented simplicial

objects in 119860 are defined by the pullbacks

119878120491op(119906) 119861120491⊤ 119878120491op+ (119906) 119861120491⊤

119860120491op 119861120491op 119860120491op+ 119861120491

119906120491op

119906120491op+

and there exists a forgetful functor

119860120491op+ 119878120491

op+ (119906)

119861120491⊤

119860120491op 119878120491op(119906)

Our interest in these notions is explained by the following example if 119906∶ 119860 rarr 119861 is a right adjointfunctor between infin-categories any homotopy coherent adjunction extending 119906 defines a canonical119906-split augmented simplicial object

934 Lemma Let120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint 119906∶ 119860 rarr 119861 Thenthe comonad resolution and bar resolution

119860 119861120491⊤

119860120491op+ 119861120491

119896bull

119906120491op+

jointly define a 119906-split augmented simplicial object 119860 rarr 119878120491op+ (119906)

Proof Functoriality of120120 supplies a commutative diagram below-left

120491op+ cong 119964119889119895(minus minus) Fun(119860119860)

120491⊤ cong 119964119889119895(minus +) Fun(119860 119861)

120120

119906∘minus 119906∘minus

120120

which internalizes to the commutative diagram of the statement By the definition of theinfin-categoryof 119906-split augmented simplicial objects in 933 this induces the claimed functor 119860 rarr 119878120491

op+ (119906) Thus

the comonad resolution 119896bull ∶ 119860 rarr 119860120491op+ defines an augmented simplicial object in119860 that is 119906-split by

the bar resolution for120120

935 Proposition The monadic forgetful functor 119906119905 ∶ Alg120139(119861) rarr 119861 creates colimits of 119906119905-split simplicialobjects Moreover for any 119906119905-split augmented simplicial object the augmentation defines the colimit cone forthe underlying simplicial object in Alg120139(119861)

Proof Theinfin-category of 119906119905-split simplicial objects is defined by the pullback

119878120491op(119906119905) 119861120491⊤

Alg120139(119861)120491op 119861120491op

(119906119905)120491op

By Proposition 2311 the canonical 2-cell (2310) defined by the initial object in120491+ defines an absoluteleft lifting diagram

119861

119878120491op(119906119905) 119861120491⊤ 119861120491op+ 119861120491op

uArr119861120584op Δ

res res

evminus1 (936)

that is also an absolute colimit in 119861 preserved by all functors and in particular by 119905 ∶ 119861 rarr 119861Now Proposition 932 tells us that 119906119905 ∶ Alg120139(119861) ↠ 119861 creates this colimit which means that there

exists an absolute left lifting diagram as below-left

Alg120139(119861)

119878120491op(119906119905) Alg120139(119861)120491op

uArr120582Δcolim

Alg120139(119861) 119861 119861

119878120491op(119906119905) Alg120139(119861)120491op 119861120491op 119878120491op(119906119905) 119861120491

op+ 119861120491op

uArr120582Δ

119906119905

Δ =uArr119861120584

opΔcolim

(119906119905)120491op res

evminus1

that when postcomposed with (119906119905)120491op ∶ Alg120139(119861)120491op ↠ 119861120491op

recovers the absolute left lifting diagram(936) in the sense expressed by the pasting equality above-right Thus the monadic forgetful functorcreates colimits of 119906119905-split simplicial objects in Alg120139(119861)

Upon precomposing with the 119878120491op+ (119906119905) rarr 119878120491op(119906119905) the fact that Proposition 932 tells us that

119906119905 ∶ Alg120139(119861) ↠ 119861 creates the colimit (936) also means that whenever there exists a pasting equality

Alg120139(119861) 119861 119861

119878120491op+ (119906119905) Alg120139(119861)

120491op+ Alg120139(119861)

120491op 119861120491op 119878120491op+ (119906119905) 119861120491⊤ 119861120491

op+ 119861120491op

uArrAlg120139(119861)120584op Δ

119906119905

Δ =uArr119861120584

evminus1

(119906119905)120491op res res

evminus1

such as arises here by 2-functoriality of the simplicial cotensor construction the 2-cell

Alg120139(119861)

119878120491op+ (119906119905) Alg120139(119861)

120491op+ Alg120139(119861)

120491opuArrAlg120139(119861)

120584op Δ

evminus1

is an absolute left lifting diagram This proves that 119906119905 ∶ Alg120139(119861) ↠ 119861 creates colimits of 119906119905-splitsimplicial objects

Now as displayed by the bar resolution (928) any 120139-algebra in 119861 canonically gives rise to a119906119905-split simplicial object towhich Proposition 935 applies the bar resolution120491⊤ rarr Fun(Alg120139(119861) 119861)internalizes to a diagrambar ∶ Alg120139(119861) rarr 119861120491⊤ The colimit cone inAlg120139(119861) is given by the120491op

+ -shapedsubdiagram of the bar resolution that omits the dashed maps

119906119905 119905119906119905 119905119905119906119905 119905119905119905119906119905 ⋯120578119906119905

120573119905 119905120578119906119905

120578119905119906119905

120583119906119905

119905120573119905

This subdiagram admits a concise description it is the comonad resolution for the comonad inducedby the monadic adjunction 119891119905 ⊣ 119906119905 on Alg120139(119861) this being a functor 120491op

+ rarr Fun(Alg120139(119861)Alg120139(119861))that internalizes to a functor 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)

120491op+

937 Theorem (canonical colimit representation of algebras) For any homotopy coherent monad 120139on 119861 the induced comonad resolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)

120491op+ on the infin-category of 120139-algebras in 119861

encodes an absolute left lifting diagram

Alg120139(119861) Alg120139(119861)

Alg120139(119861) Alg120139(119861)120491op

Alg120139(119861) Alg120139(119861)120491op+ Alg120139(119861)

120491opuArr120573

Δ ≔uArrAlg120139(119861)

120584opΔ

119896119905bull 119896119905bullres

evminus1 (938)

created from the 119906119905-split simplicial object in 119861

Alg120139(119861) 119861 119861

Alg120139(119861) Alg120139(119861)120491op+ Alg120139(119861)

120491op 119861120491opAlg120139(119861) 119861120491⊤ 119861120491

op+ 119861120491op

uArrAlg120139(119861)120584op Δ

119906119905

Δ =uArr119861120584

119896119905bullres

evminus1

(119906119905)120491op bar res res

evminus1

Thus (938) exists the algebras for a homotopy coherent monad as colimits of canonical simplicial objects of freealgebras

Proof Applying Lemma 934 to the monadic adjunction of Proposition 9211 we see that thecomonad resolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)

120491op+ on Alg120139(119861) and the bar resolution bar ∶ Alg120139(119861) rarr

119861120491⊤ defined in (928) together define a canonical 119906119905-split augmented cosimplicial object

Alg120139(119861)

119878120491op+ (119906119905) 119861120491⊤

Alg120139(119861)120491op+ 119861120491

119896119905bull

(119906119905)120491op+

Now the claimed result follows immediately from Proposition 935

Our final task for this section is to generalize the dual of Proposition 932 proving that themonadic forgetful functor creates all limits that 119861 admits whether or not 119905 preserves them

939 Theorem Let120139∶ ℳ119899119889 rarr 119974 be a homotopy coherent monad on aninfin-category119861 Then the monadicforgetful functor 119906119905 ∶ Alg120139(119861) rarr 119861 creates all limits that 119861 admits

Proof See [50 sect5] for now

Exercises

94 The monadicity theorem

Consider an adjunction

119860 119861119906perp119891

120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860

between infin-categories that is in the homotopy 2-category of an infin-cosmos Theorem 834 provesthat this data lifts to a homotopy coherent adjunction 120120∶ 119964119889119895 rarr 119974 which then restricts to definea homotopy coherent monad 120139∶ 119964119889119895 rarr 119974 on 119861 Proposition 9211 then constructs a new homo-topy coherent adjunction with 120139 as its underlying homotopy coherent monad namely the monadicadjunction 119891119905 ⊣ 119906119905 between 119861 and the infin-category of 120139-algebras Alg120139(119861) Immediately from itsconstruction as a right Kan extension mdash there is a simplicial natural transformation from the firsthomotopy coherent adjunction to the second whose component at + is the identity and whose com-ponent at minus defines a functor that we call 119903 ∶ 119860 rarr Alg120139(119861) commuting strictly with all of the data ofeach homotopy coherent adjunction

119860 Alg120139(119861)

119861

119903

119906⊤

119906119905

119891

119891119905

This monadicity theorem originally proven for 1-categories by Beck [1] and first proven for quasi-categories by Lurie [37] supplies conditions under which this comparison functor 119903 is an equivalenceso that theinfin-category 119860 can be identified with theinfin-category of 120139-algebras

To construct this simplicial natural transformation we re-express theinfin-category of algebras as aweighted limit of the full homotopy coherent adjunction diagram not merely as a weighted limit ofits underlying homotopy coherent monad

941 Proposition The infin-category of algebras associated to the homotopy coherent monad underlying ahomotopy coherent adjunction120120∶ 119964119889119895 rarr 119974 is the limit weighted by the weight lan119882minus defined by the leftKan extension

119964119889119895

ℳ119899119889 119982119982119890119905

lan119882minusuArrcong

119882minus

Explicitly lan119882minus ∶ 119964119889119895 rarr 119982119982119890119905 is the homotopy coherent adjunction displayed on the top below

120491op 120491⊤ 119964119889119895 119982119982119890119905

119964119889119895(minus minus) 119964119889119895(minus +) 119964119889119895 119982119982119890119905

119906∘minus

119891∘minus

lan119882minus

119906∘minus

119891∘minus

119964119889119895minus

defined by restricting the domain of the right adjoint and codomain of the left adjoint of the representableadjunction119964119889119895minus along the canonical inclusion 120491op 120491op

Proof Recall Lemma 7120 which says that the weighted limit of a restricted diagram can becomputed as the limit of the original diagram weighted by the left Kan extension of the weight Thus

lim119964119889119895lan119882minus

120120 cong limℳ119899119889119882minus res120120

recovers theinfin-category of algebras for the homotopy coherent monad underlying120120All that remains is to compute the functor lan119882minus ∶ 119964119889119895 rarr 119982119982119890119905 explicitly Because the inclusion

ℳ119899119889 119964119889119895 is full on + lan119882minus(+) cong 119882minus(+) cong 119964119889119895(minus +) since119882minus was defined as the restrictionof the covariant representable functor 119964119889119895minus ∶ 119964119889119895 rarr 119982119982119890119905 along ℳ119899119889 119964119889119895 By the standardformula for left Kan extensions reviewed in Example 7119 presented in the form of (716) the valueof lan119882minus at the object minus is computed by

lan119882minus(minus) cong 1114009ℳ119899119889

119964119889119895(+ minus) times 119964119889119895(minus +)

cong coeq 1114102 119964119889119895(+ minus) times 119964119889119895(+ +) times 119964119889119895(minus +) 119964119889119895(+ minus) times 119964119889119895(minus +)∘timesid

idtimes∘1114105

By associativity of composition in119964119889119895 the composition map

119964119889119895(+ minus) times 119964119889119895(minus +) 119964119889119895(minus minus)∘

defines a cone under the coequalizer diagram By the graphical calculus and Proposition 8110 theimage of this map in119964119889119895(minus minus) cong 120491op

+ is comprised of those strictly undulating squiggles from minus to minus

that pass through + This is the subcategory 120491op 120491op+ In fact we claim that

119964119889119895(+ minus) times 119964119889119895(+ +) times 119964119889119895(minus +) 119964119889119895(+ minus) times 119964119889119895(minus +) 120491op∘timesid

idtimes∘∘ (942)

is a coequalizer diagram The map from the coequalizer to 120491op is surjective for the reason just de-scribed a strictly undulating squiggle from minus to minus that passes through + can be decomposed as ahorizontal composite of a squiggle in119964119889119895(minus +) followed by a squiggle in119964119889119895(+ minus) To see that themap from the coequalizer to 120491op is injective consider two distinct subdivisions of a squiggle from minusto minus into a pair of squiggles from minus to + and from + to minus The subsquiggle between the two chosen+ symbols in this an element of119964119889119895(+ +) and thus this pair of elements of119964119889119895(+ minus) times 119964119889119895(minus +)are identified in the coequalizer diagram

943 Lemma For any homotopy coherent adjunction120120∶ 119964119889119895 rarr 119974 there exists a simplicial natural trans-formation from120120 to the monadic adjunction lim119964119889119895

lan119882minus120120∶ 119964119889119895 rarr 119974 whose components at + and minus defined

on weights by the counit of the adjunction

119982119982119890119905119964119889119895 119982119982119890119905ℳ119899119889

lanlan119882minus = lan res119964119889119895minus 119964119889119895minus

119964119889119895+ cong lan res119964119889119895+ 119964119889119895+

120598minus

120598+cong

Proof Consider the diagram of weights in 119982119982119890119905119964119889119895

119964119889119895minus lan res119964119889119895minus

119964119889119895+ cong lan res119964119889119895+

minus∘119891

120598

perplan res(minus∘119891)minus∘119906

lan res(minus∘119906)

Applying theseweights to a homotopy coherent adjunction120120∶ 119964119889119895 rarr 119974with underlying adjunction119891 ⊣ 119906∶ 119860 rarr 119861 yields

119860 Alg120139(119861)

119861

119903

119906⊤

119906119905

119891

119891119905

with the component 120598minus inducing the non-identity component of the canonical comparison functorwith the monadic adjunction

944 Example Returning to Example 9111 there is a Kan-complex enriched categoryℳ119900119899(ℳ) ofmonoids inℳ equipped with a simplicially enriched adjunction

ℳ119900119899(ℳ) ℳ119880

perp119865

Applying the homotopy coherent nerve this defines a homotopy coherent adjunction between thequasi-categories 120081ℳ119900119899(ℳ) and 120081ℳ By Lemma 943 there is a canonical comparison map to themonadic homotopy coherent adjunction

120081ℳ119900119899(ℳ) Alg120139(120081ℳ)

120081ℳ

119903

119906

⊤119906119905

119891

119891119905

that is not an equivalence Elements of 120081ℳ119900119899(ℳ) are strict monoids in ℳ while elements ofAlg120139(120081ℳ) are homotopy coherent monoids objects 119883 isin ℳ equipped with 119899-ary multiplicationmaps 120583119899 ∶ 119883otimes119899 rarr 119883 for all 119899 that are coherently associative up to higher homotopy

945 Lemma Let 120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint 119906∶ 119860 rarr 119861 andlet120120120139 ∶ 119964119889119895 rarr 119974 be the associated monadic adjunction Then there is a canonical functor 119871∶ Alg120139(119861) rarr119860120491op

that(i) is 119906-split by the bar resolution bar ∶ Alg120139(119861) rarr 119861120491op

(ii) is so that the composite 119871 ∘ 119903 ∶ 119860 rarr 119860120491op

is the simplicial object underlying the comonad resolution119896bull ∶ 119860 rarr 119860120491

op+ and

(iii) is so that the composite 119903120491op ∘ 119871 ∶ Alg120139(119861) rarr Alg120139(119861)120491op

is the simplicial object underlying the

comonad resolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)120491op+

Proof The first two statements ask for a functor 119871 that fits into a commutative diagram below-left

119860 119860120491op+ 119964119889119895minus 119964119889119895minus times 120491

Alg120139(119861) 119860120491op lan res119964119889119895minus 119964119889119895minus times 120491op

119861120491⊤ 119861120491op 119964119889119895+ times 120491⊤ 119964119889119895+ times 120491op

119896bull

119903 res

119871

bar 119906120491op

120598

idtimes(119906∘minus)

(minus∘119906)timesid

in which each of the objects and all but the map 119871 have been described as maps induced by takingweighted limits of the homotopy coherent adjunction diagram with the weights in119982119982119890119905119964119889119895 displayedabove-right By the Yoneda lemma each of the three maps of weights labeled ldquo∘rdquo are defined by asingle map of simplicial sets In the case of ∘ ∶ 119964119889119895minus times120491

op+ rarr119964119889119895minus the Yoneda lemma says it suffices

to define a map 120491op+ rarr 119964119889119895minus(minus) = 120491op

+ we take this map to be the identity which implies that∘ ∶ 119964119889119895minustimes120491

op+ rarr119964119889119895minus acts in both components by composing over minus in119964119889119895 In light of the explicit

description of the adjunction lan res119964119889119895minus given in Proposition 941 the other two maps labelled ldquo∘rdquomay be defined similarly by identity maps Since the dashed map makes the right-hand diagram ofweights commute the induced functor on weighted limits has the desired properties (i) and (ii)

The final statements demands commutativity of the diagram below left which again follows fromthe commutativity of the corresponding diagram of weights below-right

Alg120139(119861) lan res119964119889119895minus

119860120491opAlg120139(119861)

120491op 119964119889119895minus times 120491op lan res119964119889119895minus times 120491

119896119905bull119871

119903120491op

∘ ∘

120598timesid

this just amounting to the simple observation that the counit component 120598 ∶ lan res119964119889119895minus 119964119889119895minus isjust given by the natural inclusion 120491op 120491op

946 Lemma Given any homotopy coherent adjunction with left adjoint 119891∶ 119861 rarr 119860 the diagram definedby restricting the canonical cone (2310) built from the interalized comonad resolution 119896bull ∶ 119860 rarr 119860120491

op+ along

119891∶ 119861 rarr 119860119860 119860

119861 119860120491op 119861 119860 119860120491op+ 119860120491op

uArr120577Δ ≔

uArr119860120584op Δ119891

119896bull119891 119891 119896bull res

evminus1

displays 119891 as an absolute colimit of the family of diagrams 119896bull119891∶ 119861 rarr 119860120491op

Proof The homotopy coherent adjunction provides a commutative diagram below-left

120491op+ Fun(119860119860) 119861 119860

120491perp Fun(119861119860) 119860120491⊤ 119860120491op+

120120

119891∘minus 119891∘minus cobar

119891

119896bull

120120 res

which transposes to the commutative diagram above-right which tells us that when the internalizedcomonad resolution 119896bull ∶ 119860 rarr 119860120491

op+ is restricted along 119891 it extends to a split augmented simplicial

object with the splittings on the opposite side as usual this is no matter since120491op+ considered as a full

sub 2-category of 119982119982119890119905 spanned by finite ordinals is isomorphic to its co-dual via an isomorphismthat commutes with 120491⊤co cong 120491perp This tells us that the colimit cone of the statement is the one ofProposition 2311

There are many versions of the monadicity theorem For expediencyrsquos sake we prove just onefor now We break its statement into two parts first constructing a left adjoint to the canonicalcomparison functor which under additional hypotheses we prove defines an adjoint equivalence

947 Theorem Let120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint 119906∶ 119860 rarr 119861 withunderlying homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 If 119860 admits colimits of 119906-split simplicial objects thenthe canonical comparison functor admits a left adjoint

119860 Alg120139(119861)119903

perpℓ

Proof If 119860 admits colimits of 119906-split simplicial objects then there exists an absolute left liftingof the 119906-split simplicial object 119871∶ Alg120139(119861) rarr 119860120491op

defined in Lemma 945

119860

Alg120139(119861) 119860120491opuArr120582

119871

ℓ (948)

whose functor part we take to be the definition of the left adjoint ℓ ∶ Alg120139(119861) rarr 119860 By Lemma945(ii) the diagram defined by restricting along 119903 agrees with the cosimplicial object underlying thecomonad resolution which has a canonical cone (2310) as displayed below-left

119860 119860

119860 119860120491op+ 119860120491op 119860 Alg120139(119861) 119860120491op

uArr119860120584op Δ = existuArr120598

uArr120582Δ

119896bull

evminus1

res 119903

119871

By the universal property of the absolute left lifting diagram (ℓ119903 120582119903) this induces a unique 2-cell120598 ∶ ℓ119903 rArr id119860

The unit is induced from the absolute left lifting diagram (938) By Lemma 945(iii) the comonadresolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)

120491opfactors as 119903120491op sdot 119871 so the pasted composite below left factors

through the absolute left lifting diagram as below right

119860 Alg120139(119861) 119860 Alg120139(119861)

Alg120139(119861) 119860120491opAlg120139(119861)

120491opAlg120139(119861) Alg120139(119861)

120491opuArr120582

119903

existuArr120578

uArr120573Δ

119871

119903120491op 119896119905bull

ℓ (9410)

To verify the triangle equalities note that by construction

119860 Alg120139(119861) 119860 Alg120139(119861)

119860 Alg120139(119861) Alg120139(119861)120491op 119860 Alg120139(119861) 119860120491op

Alg120139(119861)120491op

uArr120598

119903

uArr120578

uArr120573Δ = uArr120598

uArr120582Δ

119903

119903 119896119905bull

119903 119871

119903120491op

119860 Alg120139(119861) Alg120139(119861)

119860 119860120491opAlg120139(119861)

120491op 119860 Alg120139(119861) Alg120139(119861)120491op

=uArr119860120584

119903

Δ = uArr120573Δ

119896bull 119903120491op 119903 119896119905bull

the last equality following from simplicial naturality of 119903 and the definition of 120573 as Alg120139(119861)120584op in

Theorem 937 Thus the triangle equality composite 119903120598 sdot 120578119903 = id119903It follows that the other triangle equality composite 120601 ≔ 120598ℓ sdot ℓ120578 is an idempotent

120601 sdot 120601 ≔ (120598ℓ sdot ℓ120578) sdot (120598ℓ sdot ℓ120578) = 120598ℓ sdot 120598ℓ119903ℓ sdot ℓ119903ℓ120578 sdot ℓ120578 = 120598ℓ sdot ℓ119903120598ℓ sdot ℓ120578119903ℓ sdot ℓ120578 = 120598ℓ sdot ℓ120578 ≕ 120601

so to prove that 120598ℓ sdot ℓ120578 = idℓ it suffices to show that 120601 is an isomorphism To demonstrate this wewill show

(i) that 120601119891119905 is invertible ie that 120601 is an isomorphism when restricted to free 120139-algebras(ii) and that the putative left adjoint ℓ preserves the canonical colimit (938) that expresses every

120139-algebra as a colimit of free 120139-algebrasWe then combine (i) and (ii) to argue that 120601 is invertible

To this end we first observe by Lemma 946 and the definition of ℓ above that we have a pair ofabsolute left lifting diagrams

119860 119860

119861 119860120491op 119861 Alg120139(119861) 119860120491opuArr

Δ conguArr120582

Δ119891

119896bull119891 119891119905 119871

By simplicial naturality of the canonical comparison map 119903119891 = 119891119905 and by Lemma 945(ii) 119871119891119905 =119871119903119891 = 119896bull119891 Thus the absolute left lifting problems coincide and we obtain a canonical natural iso-morphism 120574∶ ℓ119891119905 cong 119891

Now to prove the claim of (i) that 120601119891119905 is an isomorphism it suffices to prove that 120578119891119905 is an iso-morphism and that 120598ℓ119891119905 is an isomorphism mdash and by naturality of whiskering and the isomorphism120574∶ ℓ119891119905 cong 119891 just constructed 120598ℓ119891119905 is an isomorphism if and only if 120598119891 is an isomorphism

By (9410) the construction of 120574 and simplicial naturality of 119903 which implies that 119903120491op119896bull = 119896119905bull119903

119860 Alg120139(119861) 119860 Alg120139(119861)

119861 Alg120139(119861) Alg120139(119861)120491op 119861 Alg120139(119861) 119860120491op

Alg120139(119861)120491op

conguArr120574

119903

uArr120578

uArr120573Δ = conguArr120574

uArr120582Δ

119903

119891119905

119891

119896119905bull

119891119905

119891

119871

119903120491op

119860 Alg120139(119861) Alg120139(119861)

119861 119860120491opAlg120139(119861)

120491op 119861 Alg120139(119861) Alg120139(119861)120491op

uArrΔ

119903

Δ = uArr120573 Δ119891

119896bull119891 119903120491op 119891119905 119896119905bull

So 120578119891119905 is the inverse of the isomorphism 119903120574 and is consequently invertibleSimilarly by (949)

119860 119860 119860

119861 119860 Alg120139(119861) 119860120491op 119861 119860120491op 119861 Alg120139(119861) 119860120491op

uArr120598uArr120582

Δ = uArr120577Δ =

uArr120582uArrcong120574 Δ

119891 119903

119871 119896bull119891

119891

119891119905

119891

119871

so 120598119891 = 120574 is also an isomorphism Thus we conclude that 120601119891119905 is invertible as claimed in (i)

To prove (ii) we must show that

Alg120139(119861) 119860

Alg120139(119861) Alg120139(119861)120491op 119860120491op

uArr120573 Δ

119896119905bull ℓ120491op

(9411)

is an absolute left lifting diagram Of course we expect this to be true because left adjoints preservecolimits by Theorem but as we have not yet shown that ℓ is a left adjoint this requires a directargument

By the equational characterization of Theorem the cosmological functor (minus)120491op ∶ 119974 rarr 119974 pre-serves the absolute left lifting diagram (948) thus

119860120491op

Alg120139(119861)120491op (119860120491op)120491op

uArr120582120491op Δ120491

119871120491op

ℓ120491op

is absolute left lifting Since 120582120491op sdot Δ = Δ120491op sdot 120582 there are two equivalent ways to compute thehorizontal composite of this 2-cell with 120573 displayed below-left and below-right

119860 119860

Alg120139(119861) 119860120491opAlg120139(119861) 119860120491op

Alg120139(119861) Alg120139(119861)120491op (119860120491op)120491op

ΔuArr120582

uArr120573

ΔuArr120582120491

op Δ120491op

uArr120573

119871

Δ Δ120491op

119896119905bull 119871120491op

ℓ120491op

119896119905bull 119871120491op

By Lemma 241 to show that (9411) is absolute left lifting ie that ℓ preserves the absolute left liftingdiagram 120573 it suffices to prove that 119871 preserves the absolute left lifting diagram 120573 By Proposition4315 to show that this diagram is absolute left lifting it suffices to show that for each [119899] isin 120491 thatthe diagram

Alg120139(119861) 119860120491op 119860

Alg120139(119861) Alg120139(119861)120491op (119860120491op)120491op 119860120491op

uArr120573

119871

Δ Δ120491op

ev119899

119896119905bull 119871120491op ev119899

is absolute left liftingBy the construction of 119871 in Lemma 945 ev119899 119871∶ Alg120139(119861) rarr 119860 is the map induced by taking

the weighted limits of the homotopy coherent adjunction 120120∶ 119964119889119895 rarr 119974 by the map of weightsminus ∘ (119891119906)119899+1 ∶ 119964119889119895minus rarr lan res119964119889119895minus Thus we see that ev119899 119871 is the map 119891119905119899119906119905 ∶ Alg120139(119861) rarr 119860 and in

particular factors through 119906119905 ∶ Alg120139(119861) rarr 119861 Since the canonical colimit of Theorem 937 is 119906119905-split(119906119905 119906119905120573) is an absolute left lifting diagram preserved under postcomposition by all functors and in

particular by 119891(119906119891)119899 Thus the above diagram is absolute left lifting as claimed which tells us that 119871and thus ℓ preserves the colimit (938)

It remains only to combine (i) and (ii) to argue that 120601 is invertible For this we consider thepasting equality

Alg120139(119861) 119860 Alg120139(119861) 119860

Alg120139(119861) Alg120139(119861)120491op 119860120491op

uArr120573 Δ

Δ =uArr120573 Δ

uArr120601

119896119905bull

ℓ120491op

uArr120601120491op

ℓ120491op

119896119905bull ℓ120491op

By the definition of 119896119905bull the components of the whiskered natural transformation120601120491op sdot119896119905bull at [119899] isin 120491op

is 120601(119891119905119906119905)119899+1 which is an isomorphism by (i) By Lemma this proves that 120601120491op sdot 119896119905bull is invertibleThus by (ii) the left hand diagram is isomorphic to an absolute left lifting diagram and thus is absoluteleft lifting The pasting equality describes a factorization of the left hand absolute left lifting diagramthrough the absolute left lifting diagram of (ii) via 120601 so by the uniqueness in the universal property ofabsolute left lifting diagrams we conclude that 120601 is invertible as desired This proves that (ℓ ⊣ 119903 120578 120598)defines an adjunction as claimed

We now describe conditions under which the adjunction just constructed defines an adjoint equiv-alence As the proof will reveal condition (ii) implies that the unit is an isomorphism while conditions(ii) and (iii) together imply that the counit is an isomorphism

9412 Theorem (monadicity) Let120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint119906∶ 119860 rarr 119861 with underlying homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 If

(i) 119860 admits colimits of 119906-split simplicial objects(ii) 119906∶ 119860 rarr 119861 preserves them and(iii) 119906∶ 119860 rarr 119861 is conservative

then the canonical comparison functor 119903 ∶ 119860 rarr Alg120139(119861) admits a left adjoint ℓ ∶ Alg120139(119861) rarr 119860 defining anadjoint equivalence

Note that Theorem 937 and Proposition 9210 establish these properties for the monadic adjunc-tion

Proof Theorem 947 constructs an adjunction (ℓ ⊣ 119903 120578 120598) under the hypothesis (i) with the leftadjoint ℓ ∶ Alg120139(119861) rarr 119860 defined as the colimit of the 119906-split family of diagrams 119871∶ Alg120139(119861) rarr 119860120491op

with colimit cone 120582∶ Δℓ rArr 119871 It remains to show that this defines an adjoint equivalenceBy hypothesis (ii) 119906 preserves the 119906-split colimit diagram that defines ℓ By Lemma 945(i)

119906120491op119871∶ Alg120139(119861) rarr 119861120491opis the monadic bar resolution so the absolute left lifting diagrams below-left

and below-center are isomorphic

119860 119861 119861 Alg120139(119861) 119861

Alg120139(119861) 119860120491op 119861120491opAlg120139(119861) 119861120491⊤ 119861120491op

Alg120139(119861) Alg120139(119861)120491op 119861120491op

uArr120582Δ

119906

Δ conguArr119861120584

opΔ =

uArr120573

119906119905

119871

119906120491op

119906119905

bar res

119896119905bull (119906119905)120491op

By the construction of the bar resolution in Lemma 934 the absolute left lifting diagram above-leftcoincides with the one above-right Now consulting (9410) we see that the left-hand diagram abovefactors through the right-hand diagram above via 119906119905120578 Thus 119906119905120578 is invertible and by Proposition9210 120578∶ idAlg120139(119861)

cong 119903ℓ is also an isomorphism

By the same line of reasoning the diagrams 119871119903 ∶ 119860 rarr 119860120491opand 119896bull ∶ 119860 rarr 119860120491op

are both 119906-splitby the bar resolution by Lemmas 945 and 934 respectively Again by the hypothesis that 119906∶ 119860 rarr 119861preserves 119906-split colimits the absolute left lifting diagram below-left must be isomorphic to the onebelow-center which equals the absolute left lifting diagram below-right

119860 119861 119861 119860 119861

119860 119860120491op 119861120491op 119860 119861120491⊤ 119861120491op 119860 119860120491op 119861120491opuArr120582119903

119906

op Δ =uArr119860120584

119906

119871119903

ℓ119903

119906120491op

119906

bar res

119896bull 119906120491op

By the uniqueness of factorization through absolute lifting diagrams it follows that 119906120598 must be anisomorphism and since 119906 is conservative this means that 120598 ∶ ℓ119903 cong id119860 is also invertible

This completes the proof but in fact we can sidestep the most difficult part of the proof of Theo-rem 947 mdash the proof of the triangle equality 120598ℓ sdot ℓ120578 = idℓ mdash under the additional hypotheses (ii) and(iii) that imply invertibility of the unit and counit constructed there Since 120578 and 120598 are both invertiblethe triangle equality composite 120598ℓ sdot ℓ120578 is an isomorphism Since the other triangle equality compositeis an identity as in the proof of Proposition 2111 this composite is also an idempotent and hence bycancelation this idempotent isomorphism is also an identity

Exercises

94i Exercise Dualize the work of this section to define and characterize the comonadic adjunctionbetween 119860 and theinfin-category of coalgebras for the homotopy coherent comonad acting on 119860

95 Monadic descent

Consider once more the canonical comparison functor

119860 Alg120139(119861)

119861

119903

119906⊤

119906119905

119891

119891119905

defining the component at minus isin 119964119889119895 of a simplicial natural transformation 120120 rarr 120120120139 from a homo-topy coherent adjunction to the monadic homotopy coherent adjunction built from its underlyinghomotopy coherent monad The monadicity theorem of the previous section characterizes when thecomparison functor 119903 ∶ 119860 rarr Alg120139(119861) is an equivalence Our aim in this section is to present a theo-rem proven by Sulyma [60] which characterizes when the functor 119903 ∶ 119860 rarr Alg120139(119861) is fully faithfulwhich is the case just when the canonical map

id119903 ∶ 119860120794 rarr HomAlg120139(119861)(119903 119903)

is an equivalence over 119860 times 119860

To analyze this situation we consider the homotopy coherent comonad120130∶ 119966119898119889 rarr 119974 underlyingthe homotopy coherent adjunction 120120∶ 119964119889119895 rarr 119974 We write 119896 ≔ 119891119906 for the functor part of thehomotopy coherent comonad an endomorphism 119896 ∶ 119860 rarr 119860 defined as the image of the unique atomic0-arrow in 119966119898119889 Our first partial result true for formal reasons under no additional hypothesesobserves that the canonical comparison functor 119903 is always fully faithful on maps out of the comonad119896 ≔ 119891119906∶ 119860 rarr 119860 of the homotopy coherent adjunction

951 Lemma ([60 35]) The canonical comparison map pulls back along the substitution of the comonad119896 ∶ 119860 rarr 119860 into its domain variable to define a fibered equivalence

Hom119860(119896 119860) 119860120794

HomAlg120139(119861)(119903119896 119903) HomAlg120139(119861)

(119903 119903)

119860 times 119860 119860 times 119860

sim id119903

idtimes119896

Proof By Proposition 411 applied to the adjunctions 119891 ⊣ 119906 and 119891119905 ⊣ 119906119905 and the relations119896 = 119891119906 119906 = 119906119905119903 and 119903119891 = 119891119905

Hom119860(119896 119860) ≔ Hom119860(119891119906119860) ≃119860times119860 Hom119860(119906 119906) = Hom119860(119906119905119903 119906119905119903) ≃119860times119860 Hom119860(119891119905119906119905119903 119903)= Hom119860(119903119891119906 119903) ≕ Hom119860(119903119896 119903)

and this equivalence is the one induced by id119903 the first equivalence is induced by pasting with 120578 = 120578119905while the second equivalence is induced by pasting with 120598119905119903 = 119903120598 By the triangle equality 120598119891sdot119891120598 = id119891the composite map is the one induced by pasting with id119903

The statement of the main theorem requires the following definition recall that the homotopycoherent comonad120130 internalizes to define a functor 119896bull ∶ 119860 rarr 119860120491

op+ that we call the comonad reso-

lution which restricts to a canonical cosimplicial object together with a cone under it

952 Definition A generalized element 119886 ∶ 119883 rarr 119860 is120130-cocomplete if the restriction of the canon-ical diagram

119860

119883 119860 119860120491op+ 119860120491op

uArr119860120584op Δ

119886 119896bull res

evminus1

is an absolute left lifting diagram

The terminology is adapted fromHess [22] who refers to such elements as ldquostrongly120130-cocompleterdquo

953 Example (algebras are cocomplete) Theorem 937 proves that in a monadic homotopy coher-ent adjunction the identity functor at the infin-category of algebras is 120130120139-cocomplete By restrict-ing the absolute left lifting diagram all generalized elements in the infin-category of 120139-algebras are120130120139-cocomplete

We now argue that the canonical comparison functor associated to a homotopy coherent adjunc-tion is full and faithful if and only if all generalized elements of 119860 are120130-cocomplete

954 Theorem ([60 314]) The canonical comparison functor 119903 ∶ 119860 rarr Alg120139(119861) for a homotopy coherentadjunction 120120 with underlying homotopy coherent comonad 120130 is full and faithful if and only if the identityid119860 ∶ 119860 rarr 119860 is120130-cocomplete ie

119860 119860

119860 119860Δop 119860 119860120491op+ 119860120491op

uArr120572Δ ≔

uArr119860120584op Δ

119896bull 119896bull res

evminus1 (955)

In the special case where 119860 admits colimits of 119906-split simplicial objects the diagram (949) inthe proof of Theorem 9412 establishes the result the composite ℓ119903 ∶ 119860 rarr 119860 defines the colimitof the comonad resolution 119896bull ∶ 119860 rarr 119860120491op

and this functor is related to the identity via the counit120598 ∶ ℓ119903 rArr id119860 To say that id119860 is 120130-cocomplete is to say that id119860 ∶ 119860 rarr 119860 is the colimit of thisdiagram which is the case if and only if 120598 is fully faithful Since a right adjoint 119903 is fully faithful justwhen the counit is an isomorphism (see Proposition 1235 or Exercise 95i) the result follows

In the absence of this hypothesis we cannot construct the left adjoint ℓ ∶ Alg120139(119861) rarr 119860 In itsplace we make use of the functor 119871∶ Alg120139(119861) rarr 119860120491op

of Lemma 945 In the terminology of sect34the following lemma proves that theinfin-category of cones under this diagram is right-represented bythe canonical comparison functor 119903 ∶ 119860 rarr Alg120139(119861)

956 Lemma There is a fibered equivalenceHom119860120491op(119871 Δ) ≃ HomAlg120139(119861)(Alg120139(119861) 119903) over119860timesAlg120139(119861)

Proof The inverse equivalences are defined by 1-cell induction

Hom119860120491op(119871 Δ)

119860 Alg120139(119861)

Alg120139(119861) 119860120491op

Alg120139(119861)120491op

1199011 1199010

120601lArr

Δ119903119871

119896119905bullΔ

119903120491op

Hom119860120491op(119871 Δ)

119860 Alg120139(119861)

Alg120139(119861)

Alg120139(119861)120491op

1199011 1199010

exist120577lArr

119903

120573lArr 119896119905bull

and then

Hom119860120491op(119871 Δ)

119860 Alg120139(119861)

1199011 1199010120577lArr

119903

Hom119860120491op(119871 Δ)

HomAlg120139(119861)(Alg120139(119861) 119903)

119860 Alg120139(119861)

1199011 1199010120577

1199011 1199010120601lArr

119903

And conversely

HomAlg120139(119861)(Alg120139(119861) 119903)

119860 Alg120139(119861)

119860120491op

1199011 1199010120601lArr

119903119896bull

120572119871

HomAlg120139(119861)(Alg120139(119861) 119903)

Hom119860120491op(119871 Δ)

119860 Alg120139(119861)

119860120491op

1199011 1199010

120601lArr

Δ 119871

We leave the verification that these maps define inverse images to the reader

Proof of Theorem 954 By Lemma 945(ii) the fibered equivalence of Lemma 956 pulls backalong 119860 times 119903∶ 119860 times 119860 rarr 119860 times Alg120139(119861) to a fibered equivalence

Hom119860120491op(119896bull Δ) ≃ HomAlg120139(119861)(119903 119903)

over 119860 times 119860 Moreover this equivalence commutes with the canonical functors up to natural isomor-phism

119860120794

congid119903120572

Under the universal property of Proposition 337 the functor id119903 is classified by the 2-cell

119860120794 119860 Alg120139(119861)1199011

1199010

uArr120581 119903

defined as the whiskered composite of the generic arrow over119860 The equivalence converts this into a

2-cell under119860120794 and over the cospan 119860 119860120491op 119860Δ 119896bull by forming the horizontal compositeof 120581 with 120572

119860

119860120794 119860 Alg120139(119861) 119860120491op

uArr120572 Δ1199011

1199010

uArr120581 119903

119896bull

119871

and this is the 2-cell that classifies the map labelled 120572By Theorem 343 the left-diagonal map is an equivalence if and only if (955) is an absolute left

lifting diagram By Corollary 346 the right-diagonal map is an equivalence if and only if 119903 is fullyfaithful By the 2-of-3 property either map is an equivalence if and only if both are which is what wewished to show

957 Remark Sulyma observes that the same argument shows that the canonical comparison functor119903 ∶ 119860 rarr Alg120139(119861) is fully faithful on maps out of 119886 ∶ 119883 rarr 119860 if and only if 119886 ∶ 119883 rarr 119860 is120130-cocomplete[60 314]

The motivation for the result of Theorem 954 arises from the theory of monadic descent Toexplain we first require some definitions

958 Definition Theinfin-category of descent data Dsc120139(119861) of a homotopy coherent monad120139 actingon aninfin-category 119861 is theinfin-category of120130120139-coalgebras in theinfin-category of 120139-algebras

Dsc120139(119861) ≔ Coalg120130120139(Alg120139(119861))Elements are called descent data

Unpacking Definition 958 we can clarify the meaning of ldquodescent datardquo

959 Proposition Let120139 be a homotopy coherent monad acting on aninfin-category 119861 isin 119974 Theinfin-categoryof descent data is the limit of 120139∶ ℳ119899119889 rarr 119974 weighted by the weight119882dsc ∶ ℳ119899119889 rarr 119982119982119890119905 given by thecategory 120491 with the left 120491+-action by ordinal sum

Proof Recall from Proposition 9211 that the monadic adjunction associated to a homotopycoherent monad is defined as the limit of the homotopy coherent monad diagram 120139 weighted bythe restriction of the Yoneda embedding 119988 isin 119982119982119890119905119964119889119895

optimes119964119889119895 along the inclusion ℳ119899119889 119964119889119895in the codomain variable a weight we denote by resℳ119899119889

119964119889119895 119988 From the monadic homotopy coherentadjunction

120120120139 ≔ limℳ119899119889resℳ119899119889119964119889119895 119988

120139∶ 119964119889119895 rarr 119974

the induced homotopy coherent comonad on Alg120139(119861) is defined by restricting along the inclusion119966119898119889 119964119889119895 producing a diagram

120139120130 ≔ res119966119898119889119964119889119895 (limℳ119899119889resℳ119899119889119964119889119895 119988

120139)∶ 119966119898119889 rarr 119974

Now dualizing Definition 925 the descent object is defined to be the infin-category of coalgebras forthis homotopy coherent comonad which is define to be the lifted weighted by the restriction of therepresentable functor119964119889119895+ ∶ 119964119889119895 rarr 119982119982119890119905 along the inclusion119966119898119889 119964119889119895 in its codomain variable

Dsc120139(119861) ≔ lim119966119898119889res119966119898119889119964119889119895 119964119889119895+

res119966119898119889119964119889119895 (limℳ119899119889resℳ119899119889119964119889119895 119988

120139)

Our aim is now to express this iterated weighted limit as a single weighted limit of the diagram120139By Lemma 7120 the weighted limit of the restricted diagram res119966119898119889119964119889119895 (lim

ℳ119899119889resℳ119899119889119964119889119895 119988

120139)∶ 119966119898119889 rarr 119974 is

isomorphic to the limit of the diagram limℳ119899119889resℳ119899119889119964119889119895 119988

120139)∶ 119964119889119895 rarr 119974 weighted by the left Kan extension

of the weight along 119966119898119889 119964119889119895 Thus

Dsc120139(119861) ≔ lim119966119898119889res119966119898119889119964119889119895 119964119889119895+

res119966119898119889119964119889119895 (limℳ119899119889resℳ119899119889119964119889119895 119988

120139)

cong lim119964119889119895lan

119964119889119895119966119898119889 res

119966119898119889119964119889119895 119964119889119895+

(limℳ119899119889resℳ119899119889119964119889119895 119988

120139)

By Definition 713(ii) the weighted limit of a weighted limit of a diagram is the limit of that diagramweighted by the weighted colimit of the weights

cong limℳ119899119889colim

119964119889119895

lan119964119889119895119966119898119889 res

119966119898119889119964119889119895 119964119889119895+

resℳ119899119889119964119889119895 119988

120139

By the symmetry of Definition 714 this weighted colimit of the weights is isomorphic to the weightedcolimit of weights with the weight and diagram swapped yielding the first isomorphism below Nowby Definition 713(i) this reduces to the restricting of the diagram alongℳ119899119889 119964119889119895 yielding thesecond isomorphism below

colim119964119889119895lan

119964119889119895119966119898119889 res

119966119898119889119964119889119895 119964119889119895+

resℳ119899119889119964119889119895 119988 cong colim119964119889119895

resℳ119899119889119964119889119895 119988

lan119964119889119895119966119898119889 res119966119898119889119964119889119895 119964119889119895+ cong resℳ119899119889

119964119889119895 lan119964119889119895119966119898119889 res119966119898119889119964119889119895 119964119889119895+

Upon substituting into our formula for Dsc120139(119861) we conclude that

cong limℳ119899119889resℳ119899119889119964119889119895 lan

119964119889119895119966119898119889 res

119966119898119889119964119889119895 119964119889119895+

120139

as desiredIt remains only to simplify the description of the weight

119882dsc ≔ resℳ119899119889119964119889119895 lan119964119889119895119966119898119889 res

119966119898119889119964119889119895 119964119889119895+ isin 119982119982119890119905

ℳ119899119889ByObservation 921 this diagramdefines a category119882dsc(+)mdashwhichwewill shortly identify definedby evaluating at+ isin ℳ119899119889mdash and a homotopy coherent monad on it This is the dual of the calculationof Proposition 941 By the formula for pointwise left Kan extensions the value of lan119964119889119895119966119898119889 res

119966119898119889119964119889119895 119964119889119895+

at the object + is computed by

119882dsc(+) ∶= (lan119964119889119895119966119898119889 res

119966119898119889119964119889119895 119964119889119895+)(+)

cong 1114009119966119898119889

119964119889119895(minus +) times 119964119889119895(+ minus)

cong coeq 1114102 119964119889119895(minus +) times 119964119889119895(minus minus) times 119964119889119895(+ minus) 119964119889119895(minus +) times 119964119889119895(+ minus)∘timesid

idtimes∘1114105

cong coeq 1114102 120491⊤ times 120491op+ times 120491perp 120491⊤ times 120491perp

∘timesid

idtimes∘1114105

cong 120491The monad structure is given by a ldquoleft action of 120491+rdquo in this case by the functor

oplus∶ 120491+ times 120491 rarr 120491obtained by restricting the ordinal sum composition in119964119889119895

9510 Remark The119882dsc-weighted limit cone on Dsc120139(119861) takes the form of a120491+-invariant diagram120491 rarr Fun(Dsc120139(119861) 119861) that we refer to as the generic descent datum Writing 119887 ∶ Dsc120139(119861) rarr 119861 for

the 0-arrow in the image of the terminal element of 120491 this diagram has the form

119887 119905119887 1199052119887120578

120579120573

120578

119905120579

119905120578119905120573

120583⋯

The generic descent datum internalizes to a functor Dsc120139(119861) rarr 119861120491

9511 Observation The weights119882minus119882dsc119882+ isin 119982119982119890119905ℳ119899119889 are given by the categories 120491 120491⊤ and

120491+ respectively with the left 120491+-action in each case given by ordinal sum Thus the inclusions

120491 120491⊤ 120491+ 119882minus rarr119882dsc rarr119882+ isin 119982119982119890119905ℳ119899119889

are 120491+-equivariant defining maps of weights On weighted limits we thus get canonical maps thatfit into a commutative diagram

lim119882+ 120139 lim119882dsc120139 lim119882minus 120139

119861 Dsc120139(119861) Alg120139(119861)

cong cong cong119888

119891119905

The canonical functor 119888 ∶ 119861 rarr Dsc120139(119861) can also be described as the non-identity component ofthe simplicial natural transformation from the monadic homotopy coherent adjunction 120120120139 to thecomonadic homotopy coherent adjunction associated to its homotopy coherent comonad120130120139

119861 Dsc120139(119861)

Alg120139(119861)

119888

119891119905

perp119906119896119905

perp119906119905

119888119896119905

9512 Definition The homotopy coherent monad 120139 satisfies descent if 119888 ∶ 119861 rarr Dsc120139(119861) is fullyfaithful and satisfies effective descent if 119888 ∶ 119861 rarr Dsc120139(119861) is an equivalence The homotopy coherentmonad 120139 satisfies effective descent just when the monadic homotopy coherent adjunction is alsocomonadic

Specializing the dual of Theorem 954 we have

9513 Corollary A homotopy coherent monad 120139 acting on 119861 satisfies descent if and only if id119861 ∶ 119861 rarr 119861is 120139-complete finish

Dually for a homotopy coherent comonad120130 acting on aninfin-category119860 theinfin-category of code-scent data is

Codesc120130(119860) ≔ Alg120139120130(Coalg120130(119860))The homotopy coherent comonad 120130 satisfies codescent if the canonical comparison map 119888 ∶ 119860 rarrCodesc120130(119860) is fully faithful and satisfies effective codescent if 119888 ∶ 119860 rarr Codesc120130(119860) is an equivalencewhich is the case just when the comonadic homotopy coherent adjunction is also monadic The formerholds just when id119860 ∶ 119860 rarr 119860 is120130-cocomplete

Exercises

95i Exercise UseCorollary 3410 and Proposition 411 to anticipate the proof of Proposition 1235show that a right adjoint is fully faithful if and only if the counit of the adjunction is an isomorphism

95ii Exercise Prove the claim made in Remark 957

96 Homotopy coherent monad maps

Two adjunctions are equivalent just when there exists a pair of equivalences as displayed horizon-tally below that commute up to isomorphism with the right adjoints

119860 119860prime

119861 119861prime

sim119886

cong119906 119906prime

sim119887

119891 ⊣119891prime

and so that the mate of the isomorphism 119887119906 cong 119906prime119886 defines an isomorphism 119891prime119887 cong 119886119891 in the squareformed with the left adjoints The pair 119891 ⊣ 119906 extends to a homotopy coherent adjunction whichdefines a homotopy coherent monad on 119861 and similarly 119891prime ⊣ 119906prime defines a homotopy coherent monadon 119861prime But are the monadic adjunctions induced by these homotopy coherent monads similarly equiv-alent

A simpler question also requires some argument Consider just a single adjunction 119891 ⊣ 119906 inthe homotopy 2-category 120101119974 of aninfin-cosmos and two extensions to homotopy coherent adjunctions120120120120prime ∶ 119964119889119895 rarr 119974 By Proposition 848 120120 and 120120prime are isomorphic as vertices of the Kan complexcohadj(119974) But what does this actually mean

Unpacking Definition 841 we are given a homotopy coherent adjunction ∶ 119964119889119895 rarr 119974120128 whosetarget is the quasi-categorically enriched category whose objects are the same as in 119974 and whosefunctor-spaces are defined as the cotensor of the functor spaces of119974 with 120128 so that when is evalu-ated at the endpoints of 120128 this returns the homotopy coherent adjunctions120120 and120120prime Note this datadoes not define a simplicial natural transformation120120rarr 120120prime in particular there would be no obviouschoices for its components other than the identity functors at 119860 and 119861 so we cannot directly applyProposition 731 to construct an equivalence between the infin-categories of algebras The followingresult explains how to construct an equivalence between two homotopy coherent adjunctions that areisomorphic as vertices of cohadj(119974) in slightly greater generality than described here

961 Proposition Suppose119906 119906prime ∶ 119860 rarr 119861 are naturally isomorphic right adjoints in the homotopy 2-category120101119974 of aninfin-cosmos 119974 Then any homotopy coherent adjunctions120120∶ 119964119889119895 rarr 119974 and120120prime ∶ 119964119889119895 rarr 119974 ex-tending 119906 and 119906prime are connected by a zig zag of simplicial natural equivalences and hence the monadic homotopycoherent adjunctions for120120 and120120prime are naturally equivalent

Proof By Theorem 8414 the forgetful functor 119901119877 ∶ cohadj(119974) ⥲rarr rightadj(119974) defines a trivialfibration of Kan complexes where rightadj(119974) is defined as a sub-quasi-category of∐119860119861isin119974 Fun(119860 119861)

containing the right adjoint functors and isomorphisms between them The postulated natural iso-morphism 120572∶ 119906 cong 119906prime defines a lifting problem

120597120128 cohadj(119974)

120128 rightadj(119974)

120120+120120prime

sim 119901119877

120572

whose solution defines a simplicial functor ∶ 119964119889119895 rarr 119974120128Consulting the definition of the cotensor of a quasi-categorically enriched category with a sim-

plicial set 119880 in Definition 841 we see that when 119974 is an infin-cosmos (and in particular admits sim-plicial cotensors of objects inside 119974) 119974119880 is concisely defined as the Kleisli category for the monad(minus)119880 ∶ 119974 rarr 119974 with the unit defined by the constant map Δ∶ 119860 rarr 119860119880 and the multiplicationdefined by the fold map nabla∶ (119860119880)119880 cong 119860119880times119880 rarr 119860119880 both arising from the comonoid (119880 ∶ 119880 rarr120793Δ∶ 119880 rarr 119880 times 119880) in 119982119982119890119905 As such the quasi-categorically enriched categories 119974 and 119974119880 areconnected by the Kleisli adjunction

119974 119974119880Δ

perp119870

whose left adjoint is identity on objects and acts on homs by post-composing with the constant map

Fun(119883119860) Fun(119883119860119880) cong Fun(119883119860)119880Δlowast

while the right adjoint sends 119860 to 119860119880 and acts on homs by

Fun(119883119860)119880 cong Fun(119883119860119880) Fun(119883119880 (119860119880)119880)) Fun(119883119880 119860119880)(minus)119880 nablalowast

Now for each vertex 119906∶ 120793 rarr 119880 there is a simplicial natural transformation 119870 rArr ev119906 whose compo-nents are given by ev119906 ∶ 119860119880 ↠ 119860

Specializing to 119880 = 120128 we obtain a zig-zag of simplicial natural transformations

119974120128 119974119870

ev0uArr≀

ev1dArr≀

whose components are the trivial fibrations ev0 ev1 ∶ 119860120128 ⥲rarr 119860 and in particular define equivalencesPrecomposing with prime this defines a zig-zag of simplicial natural equivalences between the homotopycoherent adjunction120120 and the homotopy coherent adjunction120120prime The homotopy coherent monadicadjunction is computed as a flexible weighted limit of these diagrams so Proposition 731 implies thatthe homotopy coherent monadic adjunctions are equivalent as claimed

Proposition 961 can be understood as presenting an affirmative answer to the question posedat the start of this section The equivalence 119886 and 119887 can be promoted to adjoint equivalences byProposition 2111 then composed with the adjunctions 119891 ⊣ 119906 and 119891prime ⊣ 119906prime to produce a pair ofnaturally isomorphic adjunctions between the sameinfin-categories to which Proposition 961 appliesThe details are left as Exercise 96i But even with this question resolved such considerations inspire

a more general avenue of inquiry that is worth pursuing which we do following Zaganidis work in hisPhD thesis [65]

To state the question we first review a bit of classical 2-category theory962 Definition (monad morphism in a 2-category) Let (119905 120578 120583) be a monad on 119861 and let (119904 120580 120584)be a monad on 119860 in a 2-category A (lax) monad morphism (119891 120594) ∶ (119905 120578 120583) rarr (119904 120580 120584) is given bybull a functor 119891∶ 119861 rarr 119860bull a 2-cell 120594∶ 119904119891 rArr 119891119905

so that the following pasting equalities hold

119860 119860

119860 119860 119860 119861 119860

119861 119861 119861 119861

119904dArr120584

119904119904

119904

dArr120594

119904

dArr120594

119905dArr120583

119891

dArr120594

119891

119905

119891 119891 119905

119905

119891

119860 119860 119860 119860

119861 119861 119861 119861

dArr120580119904

dArr120594 =119891

119905

119891 119891

119905dArr120578

119891

If (119892 120595) ∶ (119905 120578 120583) rarr (119904 120580 120584) is a second monad morphism a monad transformation 120572∶ (119891 120594) rArr(119892 120595) is given by a 2-cell 120572∶ 119891 rArr 119892 so that

119861 119860 119861 119860

119905

119891

dArr120594119904 = 119905

119891

dArr120572

119892dArr120595

119904119891

dArr120572

119892119892

This defines the 2-category of monads and monad morphisms in a fixed 2-category 119966 [56] If119966 admits Eilenberg-Moore objects which are to represent the ldquocategory of algebras functorrdquosup3 then this2-category defines a reflective subcategory of a particular 2-category of adjunctions that we now in-troduce963 Definition A right adjunction morphism is a commutative square between the right adjoints

119860 119860prime

119861 119861prime119906

119886

119906prime

119887

119891 ⊣ ⊢119891prime

and a right adjunction transformation is a pair of natural transformations

119860 119860prime

119861 119861prime

119906

119886

119886prime

dArr120572

119906prime119887

119887prime

dArr120573

119891 ⊣ ⊢119891prime

sup3A 2-category 119966 admits Eilenberg-Moore objects whenever it admits the PIE limits alluded to in Digression 726 [30]

so that 119906prime120572 = 120573119906

964 Proposition If 119966 is a 2-category admitting the construction of Eilenberg-Moore objects the forgetful2-functor from the 2-category of adjunctions right adjunction morphisms and right adjunction transformationsto the 2-category of monads admits a fully faithful right 2-adjoint

Proof Exercise 96ii or see [12]

965 Remark A lax morphism of monads (119891 ∶ 119861 rarr 119860 120594∶ 119904119891 rArr 119891119905) as in Definition 962 corre-sponds to a lift of 119891 along the right adjoints of the monadic adjunctions for 119905 and 119904 but this liftedfunctor does not commute with the left adjoints In general the mate of the identity functor is non-invertible

The question is what is a homotopy coherent monad morphism

966 Digression Consider homotopy coherent monads 120139∶ ℳ119899119889 rarr 119974 and 120138∶ ℳ119899119889 rarr 119974 on 119861and 119860 A simplicial natural transformation 119891∶ 120139 rArr 120138 is given by its unique component 119891∶ 119861 rarr 119860satisfying a strict naturality condition relative to the bar resolutions

120491+ Fun(119861 119861)

Fun(119860119860) Fun(119861119860)

120139

120138 119891lowast

119891lowast

Such data defines a monad morphism in the homotopy 2-category whose component 120594∶ 119904119891 rArr 119891119905 isthe identity 2-cell So in general this definition is too strict

By Remark 842 if the vertices 120139 and 120138 are in the same connected component of Icon(ℳ119899119889119974)then necessarily 119861 = 119860 which is also too rigid a notion of monad morphism to consider in general

We present the data of a homotopy coherent monad morphism by means of a simplicial computadℳ119899119889120794 introduced by Zaganidis Generalizing the relationship between the simplicial computadℳ119899119889and the simplicial computad 119964119889119895 ℳ119899119889120794 defines a simplicial subcategory of a simplicial category119964119889119895120794 that we introduce first via a graphical calculus developed in Zaganidis [65 ] This graphicalcalculus extends to a sequence of composable adjunction morphisms so we might as well introduce119964119889119895120159 in its full generality

The simplicial category119964119889119895120159 is a 2-category whose hom-wise nerve can be presented via a graphi-cal calculus exactly as was the case for119964119889119895 A 2-categorical description of119964119889119895120159 is given in [65 sect311]The graphical calculus that presents 119964119889119895120159 as a simplicial category is described in more detail in [65sect312]

967 Definition The objects of 119964119889119895120159 are pairs whose first component is either + or minus and whosesecond component 119894 isin 0 1 hellip 119899 minus 1 is best thought of as a color drawn from a linearly ordered setwhere the color 0 is the ldquolightestrdquo and the color 119899minus1 is the ldquodarkestrdquo Note the objects of119964119889119895120159 are theobjects of the product simplicial category119964119889119895 times 120159 we write +119894 or minus119894 for the two objects with color 119894or either plusmn119894 or ∓119894 to denote a generic object of119964119889119895 times 120159

The 119899-arrows in 119964119889119895120159 are strictly undulating colored squiggles on 119899-lines In more detail an119899-arrow plusmn119896 rarr ∓119895 is permitted only if 119896 ge 119895 that is the color of the domain appearing on the rightmust be ldquodarkerrdquo than the color of the codomain appearing on the left The data of a morphism

plusmn119896 rarr ∓119895 is given by a strictly undulating squiggle of 119964119889119895 from plusmn to ∓ as appropriate with all fourchoices of + or minus possible together with a coloring with colors drawn from the interval [119895 119896] of theconnected components of each of the strips between the lines 119894 and 119894minus1 for some 119894 isin [119899] of the shadedregion under the squiggle diagram This coloring must satisfy the axiomsbull In the strip between the 119894th and 119894 minus 1th lines for 119894 isin [119899] the coloring function is monotone

becoming darker as you move from left to rightbull If a connected component of a strip above the line 119894 shares a boundary with a connected compo-

nent of a strip below the line 119894 then color of the top strip is lighter than the color of the bottomstrip ie the coloring function is monotone becoming darker as you move from top to bottomin a single vertical section of the squiggle diagram

bull Finally in the case of a morphism minus119896 rarr plusmn119895 each of the strips that touches the right-hand bound-ary must be colored the maximal color 119896

For a fully formal definition together with a description of the composition face and degeneracyactions are described in [65 sect312]

The monochromatic strictly undulating squiggles in119964119889119895120159 define the data found in just one of its119899 homotopy coherent adjunctions

968 Example The atomic 0-arrows of 119964119889119895120159 define 119899 adjunctions that we denote by 119891119894⊣ 119906

119894and

119899 minus 1 adjunction morphisms as depicted below

minus0 minus1 ⋯ minus119899minus2 minus119899minus1

+0 +1 ⋯ +119899minus2 +119899minus1

1199060⊣ 1199061⊣

1198861

119906119899minus2⊣

119886119899minus1

119906119899minus1⊣1198910

1198911

1198871

119891119899minus2

119891119899minus1

119887119899minus1

We now state a few of the key results from Zaganidisrsquo thesis whose proofs are too long to reproducehere Note that119964119889119895120159 is not a simplicial computad since the diagram of right adjoints and adjunctionmorphisms in Example 968 commutes strictly at the level of 0-arrows However this is the onlyobstruction in a sense made precise by the following result

969 Proposition ([65 325]) Consider the inclusion 120794 times 120159 119964119889119895120159 whose image is the subcategorycomprised of the right adjoints 119906

119894and the 0-arrow components 119886

119894and 119887

119894of the adjunction morphisms Then

120794 times 120159 119964119889119895120159 is a relative simplicial computad

9610 Remark Diagrams of morphisms between adjunctions of more general 2-categorical past-ing diagram shapes are also of interest Zaganidis constructs a generalization 119964119889119895119878120159 of 119964119889119895120159 for any

ldquoshaperdquo 119878 sub ℭΔ[119899 minus 1] with the universal property that for a 2-category 119966 2-functors 119964119889119895119878120159 rarr 119966correspond to 2-functors from 119878 into the 2-category of adjunctions right adjunction morphisms andright adjunction transformations of Definition 963 Zaganidisrsquo construction requires 119878 to be a sim-plicial computad although 119878 ℭΔ[119899 minus 1] need not be a simplicial subcomputad inclusion Insteadany factorization inℭΔ[119899minus1] of an atomic arrow of 119878must be through an object that is not containedin the subcategory 119878

An argument along the lines of that given in sect82 and sect83 proves that homotopy coherent adjunc-tion morphisms are generated by what we term a diagram of 119899-right adjoints this being given by acommutative diagram in119974

1198600 1198601 ⋯ 119860119899minus2 119860119899minus1

1198610 1198611 ⋯ 119861119899minus2 119861119899minus1

1199060

1198861

1199061 119906119899minus2

119886119899minus1119906119899minus1

1198871 119887119899minus1

together with a choice of left adjoint and unit for each 119906119894 We state just one of the many extensiontheorems proven as Theorems A and 512-13 in [65]

9611 Theorem Any diagram of 119899-right adjoints in a quasi-categorically enriched category extends to asimplicial functor119964119889119895120159 rarr 119974 And moreover the space of extensions over a given diagram of right adjointsis a trivial Kan complex

9612 Definition Let ℳ119899119889120159 be the full subcategory of 119964119889119895120159 on the objects +0 hellip +119899minus1 By [65519] it is a simplicial computad

9613 Theorem ([65 520524-5]) For any homotopy coherent diagram of monads120139∶ ℳ119899119889120159 rarr119974 thesimplicially enriched right Kan extension 120120120139 ∶ 119964119889119895120159 rarr 119974 exists For each 119894 isin 0 hellip 119899 minus 1 the object120120120139(minus119894) is equivalent to theinfin-category of algebras for the 119894th underlying homotopy coherent monad

Note that its not the case that the value of the right Kan extension along ℳ119899119889120159 119964119889119895120159 at minus119894recovers theinfin-category of algebras for the 119894th homotopy coherent monad on the nose However byapplying Theorem 9412 thisinfin-category can be shown to be equivalent to theinfin-category of algebrasSee sect53 of [65] for more details

Exercises

96i Exercise Use Proposition 961 to prove that the infin-categories of algebras associated to anyhomotopy coherent adjunctions extending equivalent adjunctions in the sense described at the startof this section are equivalent

96ii Exercise Prove Proposition 964 perhaps just in the 2-category of categories

Part III

The calculus of modules

CHAPTER 10

Two-sided fibrations

Recall fromProposition 7415 that for anyinfin-category119861 in aninfin-cosmos119974 the quasi-categoricallyenriched categories 119888119900119966119886119903119905(119974)119861 and 119966119886119903119905(119974)119861 define sub infin-cosmoi of 119974119861 In this section we in-troduce another subinfin-cosmos 119860ℱ119894119887(119974)119861 sub 119974119860times119861 whose objects are two-sided fibrations from 119860 to119861 Several equivalent definitions of this notion are given in sect101 Iterating Proposion 7415 revealsthat 119860ℱ119894119887(119974)119861 is again an infin-cosmos which we study in sect102 Importantly two-sided fibrationfrom 119861 to 1 is simply a cocartesian fibration over 119861 while a two-sided fibration from 1 to 119861 is a carte-sian fibration over 119861 so results about two-sided fibrations simultaneously generalize these one-sidednotions For instance in sect103 we introduce two-sided representables and prove the Yoneda lemmageneralizing Theorem 553 for representable (co)cartesian fibrations

Another reason for our interest in two-sided fibrations is the fact that the discrete objects in 119860ℱ119894119887119861are precisely the modules from 119860 to 119861 which we define and study in sect104 The calculus of modulesdeveloped in Chapter 11 is the main site of the formal category theory ofinfin-categories which is thesubject of Chapter 12

101 Four equivalent definitions of two-sided fibrations

By factoring any span in119974 from 119860 to 119861 may be replaced up to equivalence by a two-sided isofi-bration a span 119860

119902larrminuslarr119864

119901minusrarrrarr 119861 for which the functor (119902 119901) ∶ 119864 ↠ 119860 times 119861 is an isofibration Two-sided

isofibrations from 119860 to 119861 are the objects of the infin-cosmos 119974119860times119861 In this section we describe whatit means for a two-sided isofibration to be cocartesian on the left or cartesian on the right and thenintroduce two-sided fibrations which integrate these notions

1011 Lemma (cocartesian on the left) For a two-sided isofibration119860119902larrminuslarr119864

119901minusrarrrarr 119861 in aninfin-cosmos119974 the

following are equivalent(i) The functor

119864 119860 times 119861

119861119901

(119902119901)

120587

is a cocartesian fibration in the sliceinfin-cosmos119974119861(ii) The functor

119864 119860 times 119861

119860119902

(119902119901)

120587

in119974119860 lies in the subinfin-cosmos 119888119900119966119886119903119905(119974)119860

(iii) The functor induced by id119902

119864 Hom119860(119902 119860)

119860 times 119861

119894perp

(119902119901) (11990111199011199010)

admits a left adjoint in119974119860times119861(iv) The isofibration 119902 ∶ 119864 ↠ 119860 is a cocartesian fibration in119974 and for every 119902-cocartesian transformation

119883 119864119890

119890primedArr120594 the composite 119883 119864 119861

119890

119890primedArr120594

119901is an isomorphism

If any of these equivalent conditions are satisfied we say that 119860119902larrminuslarr119864

119901minusrarrrarr 119861 is cocartesian on the

Proof We first prove that the equivalence (i)hArr(iii) is exactly the interpretation of the equiva-lence Theorem 5111(i)hArr(ii) applied to the isofibration (119902 119901) ∶ 119864 ↠ 119860times119861 in theinfin-cosmos119974119861 Thislatter result asserts that the isofibration (119902 119901) ∶ 119864 ↠ 119860 times 119861 is a cocartesian fibration in 119974119861 if andonly if a certain canonical functor from 119864 to the left representation of the functor (119902 119901) ∶ 119864 ↠ 119860times119861admits a left adjoint over the codomain 120587∶ 119860 times 119861 ↠ 119861 since (119974119861)120587 ∶ 119860times119861↠119861 cong 119974119860times119861 this is thesame as asserting this adjunction over 119860 times 119861

The only subtlety in interpreting Theorem 5111 in119974119861 has to do with the correct interpretationof the left representable commainfin-category in119974119861 for the functor (119902 119901) ∶ 119864 ↠ 119860 times 119861 This commainfin-category is constructed as a subobject of the 120794-cotensor of the object 120587∶ 119860times119861 rarr 119861 in119974119861 whichProposition 1217 tells us is formed as the left-hand vertical of the pullback diagram

119860120794 times 119861 (119860 times 119861)120794

119861 119861120794120587

120587120794

By (332) the commainfin-category is constructed by the pullback in119974119861

Hom119860(119902 119860) 119860120794 times 119861 119860 times 119861

119864 119860 times 119861

119861

1199010 1199011199010 120587

1199011timesid

120587

(119902119901)

119901 120587

1199010timesid

(1012)

which is created by the forgetful functor 119974119861 rarr 119974 and its codomain-projection functor is thetop composite (1199011 1199011199010) ∶ Hom119860(119902 119860) ↠ 119860 times 119861 Now we see that the interpretation of Theorem5111(i)hArr(ii) is exactly the equivalence (i)hArr(iii)

It remains to demonstrate the equivalence with (ii) and 104 Assuming (iii) and composing with120587∶ 119860times119861 ↠ 119860 we are left with a fibered adjunction that demonstrates that 119902 is a cocartesian fibration

The unit of both fibered adjunctions is the same and by Theorem 5111(v) the composite

Hom119860(119902 119860) Hom119860(119902 119860) 119864

119864ℓ

dArr120578 1199010

119894

is the generic 119902-cocartesian cell Since 120578 is fibered over119860times119861 when we postcompose with 119901 we get anidentity which tells us that 119901∶ 119864 ↠ 119861 carries 119902-cocartesian cells to isomorphisms This proves 104

In fact 104 can easily be seen to be equivalent to (ii) By Example 525 the cocartesian cells forthe cocartesian fibration 120587∶ 119860times119861 ↠ 119860 are precisely those 2-cells whose component with codomain119861 is an isomorphism so 104 says exactly that (119902 119901) ∶ 119864 ↠ 119860 times 119861 defines a cartesian functor from 119902to 120587 Thus 104 implies (ii)

For the converse assume (ii) and consider a 2-cell in119974119861

119883 119864

119860 times 119861

119890

(119886119887)

dArr(120572id) (119902119901)

Because 119902 is a cocartesian fibration 120572∶ 119902119890 rArr 119886 has a 119902-cocartesian lift 120594∶ 119890 rArr 119890prime ∶ 119883 rarr 119864 andsince (119902 119901) is a cartesian functor the whiskered 2-cell 119901120594∶ 119887 rArr 119901119890prime is an isomorphism Because(119902 119901) ∶ 119864 ↠ 119860 times 119861 is an isofibration we may lift the 2-cell (id 119901120594minus1) ∶ 119890prime rArr (119886 119887) to an invertible2-cell 120574∶ 119890prime rArr 119890Prime with 119902120574 = id119886 The composite 120574 sdot 120594∶ 119890 rArr 119890Prime is a lift of (120572 id) along (119902 119901) over 119861which is easily verified to define a (119902 119901)-cocartesian lift of (120572 id) in119974119861

If (119902 119901) ∶ 119864 ↠ 119860times119861 is a discrete cocartesian fibration in119974119861 then the converse to the last statementof 104 holds any 2-cell 120594 for which 119901120594 is an isomorphism is 119902-cocartesian See Exercise 101i

By Lemma 1011 and its dual

1013 Corollary A two-sided isofibration 119860119902larrminuslarr119864

119901minusrarrrarr 119861 is cocartesian on the left and cartesian on the

right if the following equivalent conditions are satisfied(i) The functor

119864 119860 times 119861

119860119902

(119902119901)

120587

lies in 119888119900119966119886119903119905(119974)119860 and defines a cartesian fibration in119974119860(ii) The functor

119864 119860 times 119861

119861119901

(119902119901)

120587

lies in 119966119886119903119905(119974)119861 and defines a cocartesian fibration in119974119861

A two-sided fibration is a span that is cocartesian on the left cartesian on the right and satisfiesa further compatibility condition that can be stated in a number of equivalent ways which boil downto the assertion that the processes of taking 119902-cocartesian and 119901-cartesian lifts commute

1014 Theorem For a two-sided isofibration 119860119902larrminuslarr119864

119901minusrarrrarr 119861 in119974 the following are equivalent

(i) The functor

119864 119860 times 119861

119860119902

(119902119901)

120587

defines a cartesian fibration in 119888119900119966119886119903119905(119974)119860(ii) The functor

119864 119860 times 119861

119861119901

(119902119901)

120587

defines a cocartesian fibration in 119966119886119903119905(119974)119861(iii) The canonical functors admit the displayed adjoints in119974119860times119861

119864 Hom119861(119861 119901)

Hom119860(119902 119860) Hom119860(119902 119860) times119864 Hom119861(119861 119901)

119894⊤

119894⊣ (1198941199011id) ⊢

119903

(id1198941199010)perp

119903

and themate of the identity 2-cell in this displayed commutative square defines an isomorphism ℓ119903 cong 119903ℓ(iv) 119860

119901minusrarrrarr 119861 is cocartesian on the left cartesian on the right and for any 119886 ∶ 119883 rarr 119860 119887 ∶ 119883 rarr 119861

and 119890 ∶ 119883 rarr 119864 and pair of 2-cells 120572∶ 119902119890 rArr 119886 and 120573∶ 119887 rArr 119901119890 the map 120573lowast120572lowast(119890) ∶ 119883 rarr 119864 obtainedas the domain of a 119901-cartesian lift of the composite of 120573 with a 119902-cocartesian lift 120594120572 ∶ 119890 rArr 120572lowast(119890) of120572 over 119861 is isomorphic over 119860 times 119861 to the map 120572lowast120573lowast(119890) ∶ 119883 rarr 119864 obtained as the codomain of a119902-cocartesian lift of the composite of 120572 with a 119901-cartesian lift 120594120573 ∶ 120573lowast(119890) rArr 119890 of 120573 over 119860

A span 119860119902larrminuslarr119864

119901minusrarrrarr 119861 defines a two-sided fibration from 119860 to 119861 if any of these equivalent condi-

tions are satisfied

Proof Our strategy will be to show that condition (i) is equivalent to (iii) an equationally wit-nessed condition in the slice infin-cosmos 119974119860times119861 A dual argument will show that condition (ii) isequivalent to (iii) We then unpack this condition to prove its equivalence with (iv)

We use Theorem 5111(ii) which provides a condition that characterizes the cartesian fibrationsin any infin-cosmos via the presence of a fibered adjunction to re-express (i) in 119974119860times119861 To apply thischaracterization to the map displayed in (i) we must first compute the right representable commaobject in 119888119900119966119886119903119905(119974)119860 sub 119974119860 associated to the functor (119902 119901) ∶ 119864 rarr 119860 times 119861 over 119860 By Proposition7415 it suffices to compute this comma object in119974119860 By the dual of the calculation (1012) we gave inthe proof of Lemma 1011 the comma object covariantly representing the functor (119902 119901) ∶ 119864 rarr 119860times119861in 119888119900119966119886119903119905(119974)119860 sub 119974119860 is the cocartesian fibration 1199021199011 ∶ Hom119861(119861 119901) ↠ 119860

Now Theorem 5111(ii) applied in 119888119900119966119886119903119905(119974)119860 tells us that condition (i) holds if and only if thecanonical functor 119894 admits a right 119903 over 119860 times 119861

119864 Hom119861(119861 119901)

119860 times 119861(119902119901)

119894perp

(11990211990111199010)119903 (1015)

where the three displayed objects are all cocartesian fibrations over 119860 and each of the four displayedmaps are cartesian functors between these cocartesian fibrations

By Lemma 1011 to say that (119902 119901) defines a cartesian functor between cocartesian fibrations is to

say that the span 119860119902larrminuslarr119864

119901minusrarrrarr 119861 is cocartesian on the left which is the case if and only if the canonical

functor 119894 admits a left adjoint

119864 Hom119860(119902 119860)

119860 times 119861

119894perp

(119902119901) (11990111199011199010)

in 119974119860times119861 Similarly to say that (1199021199011 1199010) defines a cartesian functor between cocartesian fibrations

is to say that the span 1198601199021199011larrminusminusminuslarrHom119861(119861 119901)

1199010minusminusrarrrarr 119861 is cocartesian on the left By Lemma 1011 againthis is equivalent to the hypothesis that the canonical functor from Hom119861(119861 119901) to the comma objectcontravariantly representing 1199021199011 admits a left adjoint

Hom119861(119861 119901) Hom119860(119902 119860) times119864 Hom119861(119861 119901)

119860 times 119861

(1198941199011id)perp

(11990211990111199010) (11990111199010)

in119974119860times119861Finally by Theorem 5119 to say that the functor 119894 of (1015) is cartesian is to say that in the

commutative solid-arrow diagram in119974119860times119861

119864 Hom119861(119861 119901) 119864 Hom119861(119861 119901)

Hom119860(119902 119860) Hom119860(119902 119860) times119864 Hom119861(119861 119901) Hom119860(119902 119860) Hom119860(119902 119860) times119864 Hom119861(119861 119901)

119894

119894⊣ (1198941199011id) ⊢ 119894119894⊤

(1198941199011id)

119903

(id1198941199010)

ℓ(id1198941199010)perp

119903(1016)

the mate of this identity 2-cell involving the left adjoints ℓ is an isomorphism This ldquoBeck-Chevalleyrdquocondition is equivalent to saying that the other mate displayed above right associated to the functor119903 of (1015) is an isomorphism Finally to say that 119903 is also a cartesian functor between cocartesian

fibrations is to say that the further mate

119864 Hom119861(119861 119901)

Hom119860(119902 119860) Hom119860(119902 119860) times119864 Hom119861(119861 119901)

119903

ℓ uArrcong

119903

ℓ (1017)

is an isomorphismThus we have shown that condition (i) is equivalent to (iii) positing the existence of adjunctions

(1016) in119974119860times119861 so that all of the mates of the solid-arrow diagram are isomorphisms Dualizing andreversing this argument we see that this is equivalent to condition (ii)

Finally (iii) and (iv) are equivalent since the existence of the left adjoints in (1016) is equivalentto the span being cocartesian on the left the existence of the right adjoints is equivalent to beingcartesian on the right and the compatibility condition for the cartesian and cocartesian lifts is themeaning of the isomorphism (1017)

1018 Corollary Any two-sided isofibration (119886 119887) ∶ 119883 ↠ 119860 times 119861 that is equivalent over 119860 times 119861 to atwo-sided fibration (119902 119901) ∶ 119864 ↠ 119860 times 119861 is a two-sided fibration

Proof The assertion of Theorem 1014(iii) is invariant under fibered equivalence

Theorem 1014 will help us establish an important family of examples

1019 Proposition For anyinfin-category119860 and any 119899 ge 2 the two-sided isofibration119860119901119899minus1larrminusminusminusminuslarr119860120159

1199010minusminusrarrrarr 119860defines a two-sided fibration

This result is a generalization of Proposition 5123 and its dual and the proof uses similar ideas

Proof We use Theorem 1014(iii) The right representable comma infin-category associated to1199010 ∶ 119860120159 ↠ 119860 is constructed by the pullback

119860120794or120159 119860120794

119860120159 119860

1199011

1199010

which is equivalent to (119901119899 1199010) ∶ 119860120159+120793 ↠ 119860 times 119860 over the endpoint evaluation maps The canonicalmap

119860120159 119860120159+120793 120159 120159 + 120793

119860 times 119860 120793 + 120793(119901119899minus11199010)

1198601205900⊤

(1199011198991199010)

1198601205751 1205751

1205900perp

(1198990)(119899minus10)

that tests whether (119901119899minus1 1199010) ∶ 119860120159 ↠ 119860times119860 is cartesian on the right is given by restriction along theepimorphism 1205900 ∶ 120159 + 120793 rarr 120159 that sends the objects 0 1 isin 120159 + 120793 to 0 isin 120159 This functor admits aleft adjoint under the endpoint inclusions displayed above-right which provides the desired fiberedright adjoint displayed above left

Adual argument shows that (119901119899minus1 1199010) ∶ 119860120159 ↠ 119860times119860 is cocartesian on the left The final conditionasks that the mate of the commutative square defined by the degeneracy maps

120159 120159 + 120793

120159 + 120793 120159 + 120794

1205751

1205751198991205900perp

120575119899+1120590119899⊢

1205751

120590119899+1 ⊣

1205900⊤

is an isomorphism as is easily verified The square in Theorem 1014(iii) is obtained by applying119860(minus)

Theorem 1014 has a relative analogue whose proof is left to the reader which characterizes whatwe call a cartesian functor between two-sided fibrations from 119860 to 119861

10110 Lemma A map of spans between a pair of two-sided fibrations from 119860 to 119861

119864

119860 119861

119865

119902 119901

119892

119904 119903

(10111)

defines a cartesian functor between the cartesian fibrations in 119888119900119966119886119903119905(119974)119860 if and only if it defines a cartesianfunctor between the cocartesian fibrations in 119966119886119903119905(119974)119861

Proof A similar argument to that given in Theorem 1014 shows that the map 119892∶ 119864 rarr 119865 of(10111) is a cartesian functor between cartesian fibrations in 119888119900119966119886119903119905(119974)119860 if and only if in two com-mutative diagrams over119860times119861 the mates are isomorphisms This condition also characterizes cartesianfunctors between cocartesian fibrations in 119966119886119903119905(119974)119861 The details are left as Exercise 101ii

A cartesian functor is just a map of spans (10111) that defines a cartesian functor between thecocartesian fibrations 119902 and 119904 and also a cartesian functor between the cartesian fibrations 119901 and 119903 Itfollows from the internal characterization (iii) of Theorem 1014 and a similar internal characteriza-tion of cartesian functors derived from Theorem 5119 that

10112 Corollary Any cosmological functor preserves two-sided fibrations and cartesian functors betweenthem

Exercises

101i Exercise Suppose (119902 119901) ∶ 119864 ↠ 119860 times 119861 is a discrete cocartesian fibration in119974119861 Prove for any2-cell 120594 with codomain 119864 that if 119901120594 is an isomorphism then 120594 is 119902-cocartesian

101iii Exercise Prove that(i) A two-sided isofibration 1 larrminuslarr119864

119901minusrarrrarr 119861 defines a two-sided fibration from 1 to 119861 if and only if

119901∶ 119864 ↠ 119861 is a cartesian fibration

(ii) A map of spans119864

1 119861

119865

119901

119892

119902

defines a cartesian functor of two-sided fibrations if and only if 119892∶ 119864 rarr 119865 defines a cartesianfunctor from 119901 to 119902

102 Theinfin-cosmos of two-sided fibrations

The equivalent conditions (i) and (ii) of Theorem 1014 provide two equivalent ways to define theinfin-cosmos of two-sided fibrations

1021 Definition (the infin-cosmos of two-sided fibrations) By Theorem 1014 and Lemma 10110the pair of quasi-categorically enriched subcategories

119966119886119903119905(119888119900119966119886119903119905(119974)119860)120587 ∶ 119860times119861↠119860 and 119888119900119966119886119903119905(119966119886119903119905(119974)119861)120587 ∶ 119860times119861↠119861

of 119974119860times119861 coincide By Proposition 7415 applied twice this subcategory is an infin-cosmos which wecall theinfin-cosmos of two-sided fibrations from 119860 to 119861 and denote by

119860ℱ119894119887(119974)119861 sub 119974119860times119861By definition

Fun119860ℱ119894119887(119974)119861(119864 119865) ≔ Func119860times119861(119864 119865) sub Fun119860times119861(119864 119865)

is the quasi-category of maps of spans that define cartesian functors from 119864 to 119865 in the sense of Lemma10110

1022 Proposition The simplicial subcategory 119860ℱ119894119887(119974)119861 119974119860times119861 creates aninfin-cosmos structure ontheinfin-cosmos of two-sided fibrations from theinfin-cosmos of two-sided isofibrations

Proof This inclusion factors as

119860ℱ119894119887(119974)119861 cong 119888119900119966119886119903119905(119966119886119903119905(119974)119861)119860times119861↠119861 (119966119886119903119905(119974)119861)119860times119861↠119861 (119974119861)119860times119861↠119861 cong 119974119860times119861

Applying Proposition 7415 we see that both inclusions createinfin-cosmos structures

1023 Observation (two-sided fibrations generalize (co)cartesian fibrations) By Exercise 101iii atwo-sided fibration from 119861 to 1 is a cocartesian fibration over 119861 while a two-sided fibration from 1to 119861 is a cartesian fibration over 119861 Indeed

119888119900119966119886119903119905(119974)119861 cong 119861119865119894119887(119974)1 and 119966119886119903119905(119974)119861 cong 1119865119894119887(119974)119861In this sense statements about two-sided fibrations simultaneously generalize statements about carte-sian and cocartesian fibrations

1024 Proposition For any pair of functors 119886 ∶ 119860prime rarr 119860 and 119887 ∶ 119861prime rarr 119861 the cosmological pullbackfunctor

(119886 119887)lowast ∶ 119974119860times119861 ⟶119974119860primetimes119861prime

restricts to define a cosmological functor

(119886 119887)lowast ∶ 119860ℱ119894119887(119974)119861 ⟶ 119860primeℱ119894119887(119974)119861primeIn particular the pullback of a two-sided fibration is again a two-sided fibration

Proof By factoring (119886 119887) as119860prime times119861prime idtimes119887minusminusminusminusrarr 119860prime times119861 119886timesidminusminusminusrarr 119860times119861 we see that it suffices to considerpullback along one side at a time Proposition 5120 proves that pullback along 119887 ∶ 119861prime rarr 119861 preservescartesian fibrations and cartesian functors defining a restricted functor

119966119886119903119905(119974)119861 119974119861

119966119886119903119905(119974)119861prime 119974119861prime

Since limits and isofibrations in119966119886119903119905(119974)119861 are created in119974119861 this restricted functor is a cosmologicalfunctor Applying this result to the map

119860 times 119861prime 119860 times 119861

119860

idtimes119887

120587 120587

in theinfin-cosmos 119888119900119966119886119903119905(119974)119860 we conclude that pullback restricts to define a cosmological functor

119860ℱ119894119887(119974)119861 cong 119966119886119903119905(119888119900119966119886119903119905(119974)119860)120587 ∶ 119860times119861↠119860 (119888119900119966119886119903119905(119974)119860)120587 ∶ 119860times119861↠119860 119974119860times119861

119860ℱ119894119887(119974)119861prime cong 119966119886119903119905(119888119900119966119886119903119905(119974)119860)120587 ∶ 119860times119861prime↠119860 (119888119900119966119886119903119905(119974)119860)120587 ∶ 119860times119861prime↠119860 119974119860times119861prime

(idtimes119887)lowast (idtimes119887)lowast (idtimes119887)lowast

1025 Lemma If 119860119902larrminuslarr119864

119901minusrarrrarr 119861 is a two-sided fibration from 119860 to 119861 119907∶ 119860 ↠ 119862 is a cocartesian fibration

and 119906∶ 119861 ↠ 119863 is a cartesian fibration then the composite span

119862 119860 119864 119861 119863119907 119902 119901 119906

defines a two-sided fibration from 119862 to119863 Moreover a cartesian functor between two-sided fibrations from 119860to 119861 induces a cartesian functor

119864

119862 119860 119861 119863

119865

119892

119901119902

119907 119906

119904 119903

between two-sided fibrations from 119862 to 119863

Note composition in the sense being described here does not define a cosmological functor from119860ℱ119894119887(119974)119861 to 119862ℱ119894119887(119974)119863 because it does not preserve flexible limits

Proof By Theorem 1014 it suffices to consider composition on one side at a time say witha cocartesian fibration 119907∶ 119860 ↠ 119862 Working in the infin-cosmos 119966119886119903119905(119974)119861 we are given cocartesianfibrations

119864 119860 times 119861 119860 times 119861 119862 times 119861

119861 119861119901

(119902119901)

120587 120587

119907timesid

120587

These compose vertically to define a cocartesian fibration

119864 119862 times 119861

119861119901

(119907119902119901)

120587

and hence a two-sided fibration from 119862 to 119861 as desiredNow the 119907119902-cocartesian cells are the 119902-cocartesian lifts of the 119907-cocartesian cells If 119892 is a cartesian

functor from 119902 to 119904 then these are clearly preserved proving that 119892 also defines a cartesian functorfrom 119907119902 to 119907119904

Proposition 1024 and Lemma 1025 combine to prove that two-sided fibrations can be composedldquohorizontallyrdquo

1026 Proposition The pullback of a two-sided fibration from 119860 to 119861 along a two-sided fibration from 119861to 119862

119864 times119861119865

119864 119865

119860 119861 119862

1205871 1205870

119902 119901 119904 119903

defines a two-sided fibration from 119860 to 119862 as displayed

Proof The composite two-sided fibration is constructed in two stages first by pulling back

119864 times119861119865 119865

119864 times 119862 119861 times 119862

(12058711199031205870)

(119904119903)

119901times119862

and then by composing the left leg with the cocartesian fibration 119902 ∶ 119864 ↠ 119860 By Proposition 1024and Lemma 1025 this results is a two-sided fibration from 119860 to 119862 Alternatively the composite canbe constructed by pulling back along 119860 times 119904 and composing with the cartesian fibration 119903 ∶ 119865 ↠ 119862resulting in another two-sided fibration from 119860 to 119862 that is canonically isomorphic to the first

1027 Example If 119901∶ 119864 ↠ 119861 is a cartesian fibration and 119902 ∶ 119865 ↠ 119860 is a cocartesian fibration thenthe span

119860119902larrminuslarr119865 120587larrminuslarr119865 times 119864 120587minusrarrrarr 119864

defines a two-sided fibration from 119860 to 119861

1028 Example By Proposition 1019 and Proposition 1024 a general comma span

119862 Hom119860(119891 119892) 1198611199011 1199010

is a two-sided fibration as a pullback of 1198601199011larrminusminuslarr119860120794

1199010minusminusrarrrarr 119860 By Proposition 1026 ldquohorizontal com-positesrdquo of comma spans are also two-sided fibrations In certain cases a horizontal composite again

defines a span that is equivalent to a comma span as is the case for

Hom119860(119860 119892) times119860 Hom119860(119891 119860)

Hom119860(119860 119892) Hom119860(119891 119860)

119862 119860 119861

1205871 1205870

1199011 1199010 1199011 1199010

which is equivalent to (1199011 1199010) ∶ Hom119860(119891 119892) ↠ 119862times119861 over119862times119861 Certain other horizontal compositesare not equivalent to comma spans but nonetheless define two-sided fibrations as is the case for

Hom119860(119886 119860) times119883 Hom119861(119861 119887)

Hom119860(119886 119860) Hom119861(119861 119887)

119860 119883 119861

1205871 1205870

1199011 1199010 1199011 1199010

Exercises

102i Exercise Consider a diagram

119864 119865

119860 119861 119862

119902 119901

119892 ℎ

119904 119903

119902prime 119901prime 119904prime 119903prime

in which 119892 and ℎ define cartesian functors between two-sided fibrations from 119860 to 119861 and from 119861 to119862 respectively Prove that 119892 and ℎ pullback to define a cartesian functor (119892 ℎ) ∶ 119864 times

119861119865 rarr 119864prime times

119861119865prime

102ii Exercise Prove the assertion made in Example 1027

103 Representable two-sided fibrations and the Yoneda lemma

Wenow introduce representable two-sided fibrations and prove a two-sided version of the externalYoneda lemma

1031 Definition Specializing Example 1027 for any pair of elements 119886 ∶ 1 rarr 119860 and 119887 ∶ 1 rarr 119861the span

1198601199011120587larrminusminusminuslarrHom119860(119886 119860) times Hom119861(119861 119887)

1199010120587minusminusminusrarrrarr 119861defines a two-sided fibration from119860 to 119861 that we refer to as the two-sided fibration represented by 119886and 119887 Note there is a canonical element (id119886 id119887) ∶ 1 rarr Hom119860(119886 119860) times Hom119861(119861 119887) in the fiber over(119886 119887) ∶ 1 rarr 119860 times 119861

The terminology of Definition 1031 is justified by the Yoneda lemma for two-sided fibrations

1032 Theorem (Yoneda lemma) For any elements 119886 ∶ 1 rarr 119860 and 119887 ∶ 1 rarr 119861 and any two-sided fibration

119901minusrarrrarr 119861 restriction along (id119886 id119887) ∶ 1 rarr Hom119860(119886 119860) times Hom119861(119861 119887) defines an equivalence of

quasi-categories

Func119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119886 119860) times Hom119861(119861 119887)

119860 times 119861

119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119860 times 119861

(119886119887) 119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Interpreting this result in a slice infin-cosmos will enable us to replace the elements 119886 and 119887 with apair of generalized elements 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119883 rarr 119861 see Corollary 1035

Proof The two-sided fibration represented by 119886 and 119887 is defined by a pullback

Hom119860(119886 119860) times Hom119861(119861 119887) Hom119860(119886 119860) times 119861

119860 times Hom119861(119861 119887) 119860 times 119861

idtimes1199010

1199011timesid

idtimes1199010

We will argue by applying the Yoneda lemma of Theorem 553 twice first for cocartesian fibrationsover the object 120587∶ 119860 times Hom119861(119861 119887) ↠ Hom119861(119861 119887) the infin-cosmos 119966119886119903119905(119974)Hom119861(119861119887) and then forcartesian fibrations over the object 119861 in the119974

To set up the first use of the Yoneda lemma we begin by pulling back the two-sided fibration

119901minusrarrrarr 119861 along 1199010 ∶ Hom119861(119861 119887) rarr 119861 to obtain a two-sided fibration

Hom119861(119901 119887)

119864 Hom119861(119861 119887)

119860 119861

119901prime1199010

119902 119901

1199010

from 119860 to Hom119861(119861 119887) By the adjoint correspondence between cartesian functors established byLemma 1033(iii) below there is an isomorphism

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119886 119860) times Hom119861(119861 119887)

119860 times 119861

1199011times1199010 119864

119860 times 119861

(119902119901)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

cong Func119860timesHom119861(119861119887)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119886 119860) times Hom119861(119861 119887)

119860 times Hom119861(119861 119887)

1199011timesid Hom119861(119901 119887)

119860 times Hom119861(119861 119887)

(1199021199010119901prime)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Consider the object 120587∶ 119860 times Hom119861(119861 119887) ↠ Hom119861(119861 119887) in the infin-cosmos 119966119886119903119905(119974)Hom119861(119861119887) Theelement

Hom119861(119861 119887) 119860 times Hom119861(119861 119887)

Hom119861(119861 119887)

(119886id)

120587

has 1199011 times id ∶ Hom119860(119886 119860) times Hom119861(119861 119887) rarr 119860 times Hom119861(119861 119887) as its representing cocartesian fibra-tion in 119966119886119903119905(119974)Hom119861(119861119887) Applying the Yoneda lemma to this representable cocartesian fibration in119966119886119903119905(119974)Hom119861(119861119887) we see that restricting along this map defines an equivalence whose codomain

Func119860timesHom119861(119861119887)(Hom119861(119861 119887) rarr 119860 times Hom119861(119861 119887)Hom119861(119901 119887) ↠ 119860 times Hom119861(119861 119887))

is the mapping quasi-category defined by the pullback

bull FuncHom119861(119861119887)(idHom119861(119861119887)Hom119861(119901 119887) ↠ Hom119861(119861 119887))

1 FuncHom119861(119861119887)(idHom119861(119861119887) 119860 times Hom119861(119861 119887)

120587minusrarrrarr Hom119861(119861 119887))

(1199021199010119901prime)∘minus

(119886id)

By Lemma 1033(iii) applied to the adjunction (1199010 ∘minus) ⊣ 119901lowast0 the right-hand vertical map is isomorphicto the left-hand vertical map displayed below

Func119861(Hom119861(119861 119887)

1199010minusminusrarrrarr 119861119864119901minusrarrrarr 119861) Fun119861(1

119887minusrarr 119861 119864119901minusrarrrarr 119861)

Func119861(Hom119861(119861 119887)

1199010minusminusrarrrarr Hom119861(119861 119887) 119860 times 119861120587minusrarrrarr 119861) Fun119861(1

119887minusrarr 119861119860 times 119861 120587minusrarrrarr 119861)

(119902119901)∘minus

which is equivalent to the vertical map on the right by the Yoneda lemma applied to the cartesian fibra-tion 1199010 ∶ Hom119861(119861 119887) ↠ 119861 corresponding to the element 119887 ∶ 1 rarr 119861 Combining these observationswe obtain an equivalence to the pullback

Fun119860times119861(1(119886119887)minusminusminusrarr 119860 times 119861 119864

(119902119901)minusminusminusminusrarrrarr 119860times 119861) Fun119861(1

1 Fun119861(1119887minusrarr 119861119860 times 119861 120587minusrarrrarr 119861)

(119902119901)∘minus

(119886119887)

This composite equivalence is the map defined by precomposition with the canonical element

(id119886 id119887) ∶ 1 rarr Hom119860(119886 119860) times Hom119861(119861 119887)completing the proof

1033 Lemma(i) Suppose 119904 ∶ 119861 rarr 119860 is a discrete cartesian fibration and 119901∶ 119864 rarr 119861 is an isofibration Then 119901∶ 119864 rarr 119861

is a cocartesian fibration if and only if 119904119901 ∶ 119864 ↠ 119860 is a cocartesian fibration

(ii) Suppose that 119904 is a discrete cocartesian fibration and 119901 and 119902 are cocartesian fibrations Then (119892 119904)defines a cartesian functor from 119901 to 119902 if and only if 119892 defines a cartesian functor from 119904119901 to 119903

119864 119865 119864 119865

119861 119860 119860

119892

119901 119902

119892

119904119901 119902

119904

(1034)

(iii) If 119904 ∶ 119861 ↠ 119860 is a discrete cocartesian fibration then the adjunction 119904 ∘ minus ⊣ 119904lowast

Fun119860(119864119901minusrarrrarr 119861 119904minusrarrrarr 119860 119865

119902minusrarrrarr 119860) cong Fun119861(119864

119901minusrarrrarr 119861 119904lowast119865

119904lowast119902minusminusminusrarrrarr 119861)

between composition with and pullback along 119904 restricts to cartesian functors

Func119860(119864

119901minusrarrrarr 119861 119904minusrarrrarr 119860 119865

119902minusrarrrarr 119860) cong Func

119861(119864119901minusrarrrarr 119861 119904lowast119865

Proof For (i) recall that Lemma 517 proves that cocartesian fibrations compose with 119904119901-cocart-esian cells being the 119901-cocartesian lifts of 119904-cocartesian cells This proves that 119904119901 is cocartesian if 119901 isand the converse follows as well by the proof of that result if 119904 is a discrete cocartesian fibration thenany 2-cell with codomain 119861 is 119904-cocartesian so we may take the 119901-cocartesian cells to be precisely the119904119901-cocartesian cells

For (ii) note that in proving (i) we have just argued that 119904119901-cocartesian cells are precisely the sameas 119901-cocartesian cells when 119901 is a cocartesian fibration and 119904 is a discrete cocartesian fibration Thisproves that the two notions of cartesian functor coincidesup1

Finally (iii) follows immediately from (ii) and Proposition 5120 which tells us that a map to thepullback of a cocartesian fibration along 119904 ∶ 119861 rarr 119860 defines a cartesian functor over 119861 if and only ifthe square as displayed on the left of (1034) defines a cartesian functor

1035 Corollary The inclusion119860ℱ119894119887119861 119974119860times119861

admits a left biadjoint defined by sending a two-sided isofibration 119860 119886larrminus 119883 119887minusrarr 119861 to the composite two-sidedfibration

Hom119860(119886 119860) times119883 Hom119861(119861 119887)

Hom119860(119886 119860) Hom119861(119861 119887)

119860 119883 119861

1199011 1199010 1199011 1199010

and equipped with a natural equivalence

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119886 119860) times119883 Hom119861(119861 119887)

119860 times 119861

119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119883

119860 times 119861

(119886119887) 119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

sup1If 119904 is cocartesian but not discrete then if the left-hand diagram defines a cartesian functor then so does the rightone but the converse no longer holds

of functor spaces

Proof Let (119902 119901) ∶ 119864 ↠ 119860 times 119861 be a two-sided fibration in119974 Then since minus times 119883∶ 119974 rarr 119974119883 is acosmological functor the span

119860 times 119883119902timesidlarrminusminusminusminuslarr119864 times 119883

119901timesidminusminusminusminusrarrrarr 119861 times 119883

is a two-sided fibration in 119974119883 The maps 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119883 rarr 119861 induce a pair of elements(119886 id) ∶ 119883 rarr 119860 times 119883 and (119887 id) ∶ 119883 rarr 119861 times 119883 in 119974119883 which are respectively covariantly and con-travariantly represented by

(1199011 1199010) ∶ Hom119860(119886 119860) ↠ 119860 times 119883 and (1199011 1199010) ∶ Hom119861(119861 119887) ↠ 119861 times 119883Applying Theorem 1032 in 119974119883 to this two-sided isofibration and pair of elements we find thatrestriction along the canonical map

119894 ∶ 119883 rarr Hom119860(119886 119860) times119883 Hom119861(119861 119887)

defines an equivalence of quasi-categories

Func119860times119861times119883

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119886 119860) times119883 Hom119861(119861 119887)

119860 times 119861 times 119883

119864 times 119883

119860 times 119861 times 119883

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861times119883

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119883

119860 times 119861 times 119883

(119886119887id) 119864 times 119883

119860 times 119861 times 119883

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Transposing the domain and codomain across the adjunction

119974 119974119883minustimes119883

sum119883

yields the equivalence of functor spaces

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119886 119860) times119883 Hom119861(119861 119887)

119860 times 119861

119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119883

119860 times 119861

(119886119887) 119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

of the statement

The argument used to establish the equivalence of functor spaces in Corollary 1035 works for anypair of generalized elements 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119883 rarr 119861 whether or not this span defines a two-sidedisofibration In the case of spans represented by a single non-identity functor a ldquoone-sidedrdquo version ofCorollary 1035 which is much more simply established may be preferred

1036 Proposition (one-sided Yoneda for two-sided fibrations) For any two-sided isofibration 119860119902larrminuslarr

119864119901minusrarrrarr 119861 and functor 119891∶ 119860 rarr 119861 restriction along the canonical functor

119860

119860 119861

Hom119861(119861 119891)

id119891

119891

1199011 1199010

induce equivalences of functor spaces

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119861(119861 119891)

119860 times 119861

(11990111199010) 119864

119860 times 119861

(119902119901)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119860

119860 times 119861

(id119860119891) 119864

119860 times 119861

(119902119901)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Proof This follows immediately from the external Yoneda lemma of Theorem 553 applied tothe element (id119860 119891) ∶ 119860 rarr 119860 times 119861 and the cartesian fibration (119902 119901) ∶ 119864 ↠ 119860 times 119861 in the infin-cosmos119888119900119966119886119903119905(119974)119860

Exercises

103i Exercise State and prove the other one-sided Yoneda lemma for two-sided fibrations estab-lishing an equivalence of functor spaces below-left induced by restricting along the functor below-right

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119891 119860)

119860 times 119861

(11990111199010) 119864

119860 times 119861

(119902119901)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119861

119860 times 119861

(119891id119861) 119864

119860 times 119861

(119902119901)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

119861

119860 119861

Hom119860(119891 119860)

119891

119894

1199011 1199010

104 Modules as discrete two-sided fibrations

We are not so much interested in two-sided fibrations but the special case of those two-sidedfibrations that are discrete as objects in119974119860times119861

1041 Definition A module from 119860 to 119861 is a two-sided fibration 119860119902larrminuslarr119864

119901minusrarrrarr 119861 that is a discrete

object in 119860ℱ119894119887(119974)119861

Note that an object in the replete subcosmos 119860ℱ119894119887(119974)119861 119974119860times119861 is discrete in there if and onlyif it is discrete as an object of119974119860times119861 see Exercise 74v Our work in this chapter enables us to give adirect characterization of modules

1042 Lemma A two-sided isofibration 119860119902larrminuslarr119864

119901minusrarrrarr 119861 defines a module from 119860 to 119861 if and only if it is

(i) cocartesian on the left(ii) cartesian on the right(iii) and discrete as an object of119974119860times119861

Proof This follows immediately from the characterization of two-sided fibrations given as The-orem 1014(iv)

Unpacking the definition we easily establish the following properties of modules

119901minusrarrrarr 119861 defines a module from 119860 to 119861 then

(i) The functors

119864 119860 times 119861 119864 119860 times 119861

119861 119860119901

(119902119901)

120587 119902

(119902119901)

120587

define a discrete cocartesian fibration and a discrete cartesian fibration respectively in the sliceinfin-cosmoi119974119861 and119974119860

(ii) The functors 119902 ∶ 119864 ↠ 119860 and 119901∶ 119864 ↠ 119861 respectively define a cocartesian fibration and a cartesianfibration in119974

(iii) For any 2-cell 120594with codomain 119864 120594 is 119901-cartesian if and only if 119902120594 is invertible and 120594 is 119902-cocartesianif and only if 119901120594 is invertible

(iv) In particular any 2-cell that is fibered over 119860 times 119861 is both 119901- and 119902-cocartesian and any map of spansfrom a two-sided fibration

119865

119860 119861

119865

119904 119903

119892

119902 119901

to a module defines a cartesian functor of two-sided fibrations in the sense of Lemma 10110 and alsoa cartesian functor from 119904 to 119902 and from 119903 to 119901

Proof By Lemma 1011 conditions (i) and (iii) together assert exactly that (119902 119901) ∶ 119864 ↠ 119860 times 119861defines a discrete cocartesian fibration in 119974119861 proving (i) Statement (ii) holds for any two-sidedfibration the point of reasserting it here is that it is not necessarily the case that the legs of a moduleare themselves discrete fibrations (see Exercise 104i)

One direction of statement (iii) is proven as Lemma 1011 while the converse is proven in Exercise101i Statement (iv) follows

An important property of modules is that they are stable under pullback

1044 Proposition If 119860 119864 ↠ 119861 is a module from 119860 to 119861 and 119886 ∶ 119860prime rarr 119860 and 119887 ∶ 119861prime rarr 119861 are anypair of functors then the pullback defines a module 119860prime 119864(119887 119886) ↠ 119861prime from 119860prime to 119861prime

Proof The cosmological functor

(119886 119887)lowast ∶ 119860ℱ119894119887(119974)119861 ⟶ 119860primeℱ119894119887(119974)119861primeof Proposition 1024 preserves discrete objects in theseinfin-cosmoi

Applying Proposition 1044 to a pair of elements 119886 ∶ 1 rarr 119860 and 119887 ∶ 1 rarr 119861 we see that a modulefrom 119860 to 119861 is a two-sided fibration whose fibers 119864(119887 119886) are discrete objects The converse does notgenerally hold being discrete as an object of the slicedinfin-cosmos119974119860times119861 is a stronger condition thanmerely having discrete fibers However in theinfin-cosmos119980119966119886119905 the point 1 is a generator in a suitablesense and we have

1045 Lemma If 119864 ↠ 119860 times 119861 is a two-sided fibration of quasi-categories whose fibers are Kan complexesthen 119864 is a module

Proof By definition 119864 ↠ 119860 times 119861 defines a discrete object in 119980119966119886119905119860times119861 if any 1-simplex in 119864119883that lies over a degenerate 1-simplex in (119860 times 119861)119883 is an isomorphism As isomorphisms in functorquasi-categories are determined pointwise (see Lemma ) it suffices to consider the case 119883 = 1 andnow this reduces to the hypothesis that the fibers 119864(119887 119886) over any pair of points (119886 119887) isin 119860 times 119861 areKan complexes

The prototypical example of a module is given by the arrowinfin-category construction

1046 Proposition For anyinfin-category 119860 the arrowinfin-category 119860120794 defines a module from 119860 to 119860

The fact that119860120794 ↠ 119860times119860 is discrete is related to but stronger than the fact that each fiber over apair of elements in 119860 the internal hom-space between those elements of 119860 is a discreteinfin-categoryproven in Proposition 3310

Proof Proposition 1019 proves that (1199011 1199010) ∶ 119860120794 ↠ 119860times119860 is a two-sided fibration so it remainsonly to verify the discreteness condition which we can do in119974119860times119860 Discreteness of 119860120794 ↠ 119860times119860 asan object of 119974119860times119860 is an immediate consequence of 2-cell conservativity of Proposition 325 if 120574 isany 2-cell with codomain 119860120794 so that 1199010120574 and 1199011120574 are invertible then 120574 is itself invertible

By the construction of (332) and Proposition 1044

1047 Corollary The commainfin-category Hom119860(119891 119892) ↠ 119862times119861 associated to a pair of functors 119891∶ 119861 rarr119860 and 119892∶ 119862 rarr 119860 defines a module from 119862 to 119861

Two special cases of these comma modules those studied in sect34 deserve a special name

1048 Definition To any functor 119891∶ 119860 rarr 119861 betweeninfin-categories(i) the module Hom119861(119861 119891) from 119860 to 119861 is right or covariantly represented by 119891 while(ii) the module Hom119861(119891 119861) from 119861 to 119860 is left or contravariantly represented by 119891

As we saw in sect55 the Yoneda lemma for two-sided fibrations simplifies when mapping into amodule on account of the observation in Lemma 1043(iv) that any map of spans from a two-sidedfibration to a module defines a cartesian functor

1049 Theorem (Yoneda lemma for modules) For any elements 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119883 rarr 119861 and any

module 119860119902larrminuslarr 119864

119901minusrarrrarr 119861 restriction along (id119886 id119887) ∶ 119883 rarr Hom119860(119886 119860) times119883 Hom119861(119861 119887) defines an

equivalence of Kan complexes

Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119886 119860) times119883 Hom119861(119861 119887)

119860 times 119861

119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119883

119860 times 119861

(119886119887) 119864

119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Similarly for any functors 119891∶ 119860 rarr 119861 or 119892∶ 119861 rarr 119860 restriction along id119891 ∶ 119860 rarr Hom119861(119861 119891) orid119892 ∶ 119861 rarr Hom119860(119892 119860) define equivalences of Kan complexes

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119861(119861 119891) 119864

119860 times 119861 119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119860 119864

119860 times 119861 119860 times 119861

(id119891)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hom119860(119892 119860) 119864

119860 times 119861 119860 times 119861

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⥲ Fun119860times119861

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

119861 119864

119860 times 119861 119860 times 119861

(119892id)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Exercises

104i Exercise Demonstrate by means of an example that if 119860119902larrminuslarr119864

119901minusrarrrarr 119861 defines a module from

119860 to 119861 then it is not necessarily the case that 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration or 119902 ∶ 119864 ↠ 119860is a discrete cocartesian fibration

104ii Exercise(i) Explain why the two-sided fibration (119901119899minus1 119901119899) ∶ 119860120159 ↠ 119860times119860 of Proposition 1019 does not

define a module for 119899 gt 2(ii) Conclude that the horizontal composite of modules as defined in Proposition 1026 is not

necessarily a module

CHAPTER 11

The calculus of modules between infin-categories bears a strong resemblance to the calculus of(bi)modules between unital rings Here infin-categories take the place of rings with functors betweeninfin-categories playing the role of ring homomorphisms which we display vertically on the table belowA module 119864 from119860 to 119861 like the two-sided fibrations considered in Chapter 10 is aninfin-category onwhich 119860 ldquoacts on the leftrdquo and 119861 ldquoacts on the rightrdquo and these actions commute this is analogous tothe situation for bimodules in ring theory and explains our choice of terminologysup1 Modules will now

be depicted as 119860 119864⇸119861 whenever explicit names for the legs of the constituent span are not neededunital rings 119860 infin-categories

ring homomorphisms119860prime

119860

119886 infin-functors

bimodules between rings 119860 119864⇸119861 modules betweeninfin-categories

module maps119860prime 119861prime

119860 119861

119886 dArr

119864prime

119887

119864

module maps

Finally there is a notion of module map that we shall introduce below whose boundary in the mostgeneral case is a square comprised of two modules and two functors as above In ring theory a modulemap with this boundary is given by an119860primendash119861prime module homomorphism 119864prime rarr 119864(119887 119886) whose codomainis the119860primendash119861prime bimodule defined by restricting the scalar multiplication in the119860ndash119861module 119864 along thering homomorphisms 119886 and 119887

The analogy extends deeper than this unital rings ring homomorphisms bimodules and modulemaps define a proarrow equipment in the sense of Wood [64]sup2 Our main result in this chapter is The-orem 1126 which asserts that infin-categories functors modules and module maps in any infin-cosmosdefine a virtual equipment in the sense of Cruttwell and Shulman [15]

As a first step in sect111 we introduce the double category of two-sided fibrations which restricts todefine a virtual double category of modules A double category is a sort of 2-dimensional category with ob-jects two varieties of 1-morphisms the ldquohorizontalrdquo and the ldquoverticalrdquo and 2-dimensional cells fitting

sup1In the 1- and infin-categorical literature the names ldquoprofunctorrdquo ldquocorrespondencerdquo and ldquodistributorrdquo are all used assynonyms for ldquomodulerdquo

sup2This can be seen as a special case of the prototypical equipment comprised of119985-categories119985-functors119985-modulesand 119985-natural transformations between then for any closed symmetric monoidal category 119985 The equipment for ringsis obtained from the case where 119985 is the category of abelian groups by restricting to abelian group enriched categorieswith a single object

into ldquosquaresrdquo whose boundaries consist of horizontal and vertical 1-morphisms with compatible do-mains and codomains A motivating example from abstract algebra is the double category of modulesobjects are rings vertical morphisms are ring homomorphisms horizontal morphisms are bimodulesand whose squares are bimodule homomorphisms In the literature this sort of structure is sometimescalled a pseudo double category mdash morphisms and squares compose strictly in the ldquoverticalrdquo directionbut only up to isomorphism in the ldquohorizontalrdquo direction mdash but wersquoll refer to this simply as a ldquodoublecategoryrdquo as it is the only variety that we will consider

Our aim in sect111 is to describe a similar structure whose objects and vertical morphisms are theinfin-categories and functors in any fixed infin-cosmos whose horizontal morphisms are modules andwhose squares are module maps as will be defined in 1116 If the horizontal morphisms are replacedby the larger class of two-sided fibrations this does define a double category with the horizontalcomposition operation defined by Proposition 1026 However on account of Exercise 104ii thehorizontal composition of two-sided fibrations does not preserve the class of modules the arrowinfin-category119860120794 defines a module from119860 to119860 whose horizontal composite with itself is equivalent tothe two-sided fibration (1199012 1199010) ∶ 119860120795 ↠ 119860times119860 of Proposition 1019 which is not discrete in119974119860times119860To define a genuine ldquotensor product for modulesrdquo operation would require a two-stage constructionfirst forming the pullback that defines a composite two-sided fibration as in Proposition 1026 andthen reflecting this into a two-sided discrete fibration by means of some sort of ldquohomotopy coinverterrdquoconstruction As colimits that are not within the purview of the axioms of aninfin-cosmos this presentssomewhat of an obstacle

Rather than leave the comfort of our axiomatic framework in pursuit of a double category of mod-ules we instead describe the structure that naturally arises within the axiomatization it turns out tobe familiar to category theorists and robust enough for our desired applications We first demon-strate thatinfin-categories functors modules and module maps assemble into a virtual double categorya weaker structure than a double category in which cells are permitted to have a multi horizontalsource as a ldquovirtualrdquo replacement for horizontal composition of modules

Once the definition of a virtual equipment is given in sect112 these axioms are very easily checkedThe final two sections are devoted to exploring the consequences of this structure which will be putto full use in Chapter 12 which develops the formal category theory of infin-categories by introducingKan extensions in the virtual equipment of modules In sect113 we explain how certain horizontal com-posites of modules can be recognized in the virtual equipment even if the general construction of thetensor product of an arbitrary composable pair of modules is not known The final sect114 collects to-

gether many special properties of the modules119860Hom119861(119861119891)⇸ 119861 and 119861

Hom119861(119891119861)⇸ 119860 represented by a functor119891∶ 119860 rarr 119861 ofinfin-categories revisiting some of the properties first established in sect34

111 The double category of two-sided fibrations

Our first task is to define the 2-dimensional morphisms in the double categories that we willintroduce

1111 Definition Let 119860119902larrminuslarr119864

119901minusrarrrarr 119861 and 119860 119904larrminuslarr119865 119903minusrarrrarr 119861 be two-sided isofibrations A map of spans

from 119860119902larrminuslarr119864

119901minusrarrrarr 119861 to 119860 119904larrminuslarr119865 119903minusrarrrarr 119861 is a fibered isomorphism class of strictly commuting functors

119864

119860 119861

119865

119892

119901119902

119903119904

where two such functors 119892 and 119892prime are considered equivalent if there exists a natural isomorphism120572∶ 119892 cong 119892prime so that 119903120572 = id119902 and 119904120572 = id119901

119901minusrarrrarr 119861 and 119860 119904larrminuslarr119865 119903minusrarrrarr 119861 be two-sided fibrations A map of two-sided

fibrations from 119860119902larrminuslarr 119864

119901minusrarrrarr 119861 to 119860 119904larrminuslarr 119865 119903minusrarrrarr 119861 is a map of spans in which any and hence every

representing map119864

119860 119861

119865

119892

119901119902

119903119904

defines a cartesian functor of two-sided fibrations defined in Lemma 10110

119901minusrarrrarr 119861 and 119860 119904larrminuslarr119865 119903minusrarrrarr 119861 be modules A map of modules from 119864 to

119865 is just a map of spans from 119860119902larrminuslarr119864

119901minusrarrrarr 119861 to 119860 119904larrminuslarr119865 119903minusrarrrarr 119861 that is a fibered isomorphism class of

strictly commuting functors119864

119860 119861

119865

119892

119901119902

119903119904

1114 Observation (the 1-categories of spans and maps) The 1-category of two-sided isofibrationsfrom 119860 to 119861 and maps of spans may be obtained as a quotient of the quasi-categorically enrichedcategory119974119860times119861 or of its homotopy 2-category 120101(119974119860times119861) or of the slice homotopy 2-category 120101119974119860times119861it is the 1-category with the same set of objects and in which the morphisms are isomorphism classesof 0-arrows

Similarly the 1-category of two-sided fibrations from119860 to119861 and maps of two-sided fibrations maybe obtained as a quotient of the quasi-categorically enriched category 119860ℱ119894119887(119974)119861 or of its homotopy2-category 120101(119860ℱ119894119887(119974)119861) it is the 1-category with the same set of objects and in which the morphismsare isomorphism classes of 0-arrows

By Lemma 1043 the 1-category of modules from 119860 to 119861 and modules maps is a full subcategoryof either of the two 1-categories just considered

The 1-categories of Observation 1114 are of interest because they precisely capture the correct no-tion of equivalence between two-sided isofibrations two-sided fibrations or modules first introducedin Definition 327

1115 Lemma(i) A pair of two-sided isofibrations are equivalent in 119974119860times119861 if and only if they are isomorphic in the

1-category of spans from 119860 to 119861(ii) A pair of two-sided fibrations are equivalent in 119860ℱ119894119887(119974)119861 if and only if they are isomorphic in the

1-category of two-sided fibrations from 119860 to 119861(iii) A pair of modules are equivalent over119860 times 119861 if and only if they are isomorphic in the 1-category of

modules from 119860 to 119861

Each of the Definitions 1113 admits a common generalization which defines the 2-dimensionalmaps inhabiting squares

1116 Definition (maps in squares) A map of modules or map of two-sided fibrations or map of

two-sided isofibration from 119860prime119902primelarrminusminuslarr119864prime

119901primeminusminusrarrrarr 119861prime to 119860

119901minusrarrrarr 119861 over 119886 ∶ 119860prime rarr 119860 and 119887 ∶ 119861prime rarr 119861 is

119860prime 119861prime 119864prime 119864

119860 119861 119860prime times 119861prime 119860 times 119861

119886 dArr

119864prime

119887 (119902prime119901prime)

119892

(119902119901)

119864119886times119887

is a fibered isomorphism class of strictly commuting functors 119892 as displayed above-right which in thecase of two-sided fibrations must preserve the cartesian and cocartesian transformations where twosuch functors 119892 and 119892prime are considered equivalent if there exists a natural isomorphism 120572∶ 119892 cong 119892prime sothat 119902120572 = id119886119902prime and 119901120572 = id119887119901prime

In the case of modules or two-sided isofibrations the functor-space Fun119886times119887(119864prime 119864) of maps from119864prime to 119864 over 119886 times 119887 is defined by the pullback

Fun119886times119887(119864prime 119864) Fun(119864prime 119864)

120793 Fun(119864prime 119860 times 119861)

(119902119901)

(119886119902prime119887119901prime)

In the case of two-sided fibrations the functor space is taken to be the full sub quasi-category

Func119886times119887(119864prime 119864) sub Fun119886times119887(119864prime 119864)

of all 119899-simplices whose vertices define cartesian functors

We occasionally extend the notion of map to allow the domain to be an identity span 119862id119862larrminusminus

119862id119862minusminusrarr 119862 but unless the domain is the identity span we always require the codomain to be at least a

two-sided isofibration

We now introduce the double categories of isofibrations and of two-sided fibrations These struc-tures can be viewed either as a collection of data present in the homotopy 2-category 120101119974 of aninfin-cosmos or as a quotient of quasi-categorically enriched structures presented by a non-unital in-ternal category defined up to natural isomorphism in the category of infin-cosmoi and cosmologicalfunctors see Exercise 111i For economy of language we adopt the former approach

1117 Proposition (the double category of two-sided (iso)fibrations) The homotopy 2-category of aninfin-cosmos supports a double categorysup3 of two-sided isofibrations whosebull objects areinfin-categoriesbull vertical arrows are functors

bull horizontal arrows 119860 119864⇸119861 are two-sided isofibrations 119860119902larrminuslarr119864

119901minusrarrrarr 119861 and

bull 2-cells with boundary

119860 119861

119886 dArr

119864prime

119887

119864are maps of two-sided isofibrations as defined in 1116 or equivalently are isomorphism classes of objectsin the quasi-category Fun119886times119887(119864prime 119864)

Vertical composition of arrows and 2-cells is by composition in119974 while horizontal composition of arrows and2-cells is by pullback which is well-defined and associative up to isomorphism The double category of two-sidedfibrations is the sub double category that has the same objects and vertical arrows but whose

bull horizontal arrows 119860 119864⇸119861 are two-sided fibrations 119860119902larrminuslarr119864

119860 119861

119886 dArr

119864prime

119887

119864are maps of two-sided fibrations as defined in 1116 or equivalently are isomorphism classes of objects inFunc

119886times119887(119864prime 119864)

Proof The composition of horizontal arrows is defined in Proposition 1026 while the hori-zontal composition of 2-cells is defined in Exercise 102i By simplicial functoriality of pullback andcomposition in119974 both constructions are associative up to canonical natural isomorphism

1118 Remark (why we left out the horizontal unit) We could have formally added the identity spanΔ∶ 119860 rarr 119860times119860 to serve as a horizontal unit in the double categories of Proposition 1117 but we findit less confusing to leave them out because when we restrict to the structure of greatest interest the

virtual equipment category of modules we will see that the arrowinfin-category119860Hom119860⇸ 119860 plays the role

of the horizontal unit for composition in a sense to be described in Proposition 1124

1119 Definition (virtual double category) A virtual double category consists ofbull a category of objects and vertical arrowsbull for any pair of objects 119860119861 a class of horizontal arrows 119860 ⇸ 119861

sup3As discussed previously our double categories support vertical composition laws that are strictly unital and associa-tive but horizontal composition laws that are only associative and unital up to isomorphism For reasons to be explainedin Remarks 1118 and 11113 we choose not to require a horizontal unit arrow or 2-cell

bull cells with boundary depicted as follows

1198600 1198601 ⋯ 119860119899

1198610 119861119899

119891

1198641

1198642 119864119899

119892

119865

(11110)

including those whose horizontal source has length zero in the case 1198600 = 119860119899bull a composite cell as below-right for any configuration as below-left

1198600 1198601 ⋯ 119860119899

1198610 1198611 ⋯ 119861119899

1198620 119862119899

1198910

11986411hellip11986411198961

11986421hellip11986421198962

1198911 dArr

1198641198991hellip119864119899119896119899

⋯ dArr 119891119899

119892

1198651 1198652

119865119899

119866

≕1198600 sdot sdot 119860119899

1198620 119862119899

1198921198910

11986411 11986412hellip119864119899119896119899minus1

119864119899119896119899

ℎ119891119899

119866

bull an identity cell for every horizontal arrow

119860 119861

119864

dArrid119864

119864

so that composition of cells is associative and unital in the usual multi-categorical sense

11111 Lemma The double category of two-sided isofibrations and the double category of two-sided fibrationsextend to a virtual double category in which thebull objects areinfin-categoriesbull vertical arrows are functorsbull horizontal arrows are two-sided isofibrations or two-sided fibrations as appropriate andbull 119899-ary cells (11110) are 2-cells

1198600 119860119899

1198610 119861119899

119891

1198641 times1198601⋯ times119860119899minus1

119864119899

dArr 119892

119865

(11112)

whose single vertical source is the (119899 minus 1)-fold pullback of the sequence of spans comprising the verticalsource in (11110)

Proof The only thing perhaps worth commenting on is the nullary cells which have an emptysequence as their vertical domain

119860 119860

119861 119862

119891 dArr 119892

119865which we interpret as a 0-fold pullback this being the identity span from119860 to119860 So the nullary cellsdisplayed above are fibered isomorphism classes of maps

119860

119861 119862

119865

119891 119892

119904 119903

where ℎ and ℎprime lie in the same equivalence class if there exists a natural isomorphism 120572∶ ℎ cong ℎprime so that119904120572 = id119891 and 119903120572 = id119892

11113 Remark For instance the map

119860

119860 119860

119860120128

1199021 1199020

119860 119860

119860 119860dArr

119860120128

defines a nullary morphism with codomain 119860 119860120128⇸119860 in the virtual double category of two-sided isofi-brations Note however that despite the fact that Δ∶ 119860 ⥲ 119860120128 defines an equivalence in119974 this celldoes not define an isomorphism in the virtual double category of any kind It is for this sort of reasonthat we left out the identity horizontal arrows in Proposition 1117

Our main example of interest is a full sub virtual double category defined by restricting the class ofhorizontal arrows and taking all cells between them Since the only operations given in the structure ofa virtual double category are vertical sources and targets vertical identities and vertical compositionit is clear that this substructure is closed under all of these operations and thus inherits the structureof a virtual double category

11114 Proposition For anyinfin-cosmos119974 there is a virtual double category of modulesℳ119900119889(120101119974) definedas a full subcategory of either the virtual double categories of isofibrations or the virtual double category of two-sided fibrations whosebull objects areinfin-categoriesbull vertical arrows are functors

bull horizontal arrows 119860 119864⇸119861 are modules 119864 from 119860 to 119861

bull 119899-ary cells are fibered isomorphism classes of maps of spans

1198600 1198601 ⋯ 119860119899

1198610 119861119899

119891

1198641

1198642 119864119899

119892

119865

1198641 times1198601⋯ times

119860119899minus1119864119899

1198600 119860119899

1198610 119861119899

119865

119891 119892 (11115)

These maps introduced in Definition 1116 can be thought of as special cases of the 119899-ary cells of Lemma11111 where 1198641 hellip 119864119899 119865 are all required to be modules the single horizontal source in the diagram(11112) is the two-sided fibration defined by the (119899minus1)-fold pullback of the sequence of modules comprisingthe horizontal source in the left-hand diagram

We refer to (11115) as an 119899-ary module map Note a 1-ary module map is just a module map asin Definition 1116 We refer to the finite sequence of modules occurring as the horizontal domain ofan 119899-ary module map as a composable sequence of modules which just means that their horizontalsources and targets are compatible in the evident way

A hint at the relevance of this notion of 119899-ary module map is given by the following special case

11116 Lemma There is a bijection between 119899-ary module maps whose codomain module 119861Hom119863(ℎ119896)⇸ 119862 is a

comma as displayed below-left and 2-cells in the homotopy 2-category whose boundary is displayed above-right

1198600 1198601 ⋯ 119860119899

119861 119862

119891

1198641

1198642 119864119899

119892

Hom119863(ℎ119896)

1198641 times1198601⋯ times

119860119899minus1119864119899

1198600 119860119899

119861 119862

119863

lArr119891 119892

119896 ℎ

Proof Combine Definition 1116 with Proposition 337

For any pair of objects 119860 and 119861 in the virtual double category of modules there is a vertical1-category of modules from119860 to 119861 and module maps over a pair of identity functors which coincideswith the 1-category of modules from 119860 to 119861 introduced in Observation 1114

11117 Lemma A parallel pair of modules119860 119864⇸119861 and119860 119865⇸119861 are isomorphic as objects of vertical 1-categoryof modules in the virtual double category of modules if and only if the modules 119864 and 119865 are equivalent as spansfrom 119860 to 119861

Proof This is a restatement of Lemma 1115(iii)

For consistency with the rest of the text we write 119864 ≃ 119865 or 119864 ≃119860times119861 119865 whenever the modules

119860 119864⇸119861 and 119860 119865⇸119861 are isomorphic as objects of the vertical 1-category of modules from 119860 to 119861 Forinstance Proposition 411 proves that a functor 119891∶ 119861 rarr 119860 is left adjoint to a functor 119906∶ 119860 rarr 119861 if

and only if Hom119860(119891 119860) ≃ Hom119861(119861 119906) over 119860 times 119861 that is if and only if the modules 119861Hom119860(119891119860)⇸ 119860

and 119860Hom119861(119861119906)⇸ 119861 are isomorphic as objects of the vertical 1-category of modules from 119860 to 119861

Exercises

111i Exercise Generalize the proof of Proposition 741 to prove that for anyinfin-cosmos119974 there isaninfin-cosmos119974⤩ whose objects are two-sided isofibrations between an arbitrary pair ofinfin-categoriesProve that the domain and codomain define cosmological functors119974⤩ 119974 Use this to give a seconddescription of the double category of two-sided isofibrations as as quotient of a structure defined atthe level of quasi-categorically enriched categories

111ii Exercise Prove that any double category defines a virtual double category⁴

112 The virtual equipment of modules

The virtual double category of modules ℳ119900119889(120101119974) in aninfin-cosmos119974 has two special propertiesthat characterize what Cruttwell and Shulman term a virtual equipment Before stating the definitionwe explore each of these in turn

1121 Proposition (restriction) Any diagram inℳ119900119889(120101119974) as below-left can be completed to a cartesiancell as below-right

119860 119861 119860 119861

119886 119887 119886

119864(119887119886)

dArr120588 119887

119864 119864characterized by the universal property that any cell as displayed below-left factors uniquely through 120588 asbelow-right

1198830 1198831 ⋯ 119883119899

119860 119861

119886119891

1198641

1198642 119864119899

119887119892

119864

1198830 1198831 ⋯ 119883119899

119860 119861

119891

1198641

existdArr

1198642 119864119899

119892

119886

119864(119887119886)

dArr120588 119887

119864

Proof The horizontal source of the cartesian cell is defined by restricting the module 119860 119864⇸119861along the functors 119886 and 119887

119864(119887 119886) 119864

119860prime times 119861prime 119860 times 119861

120588

(119886119887)

(1122)

⁴If the double category lacks horizontal identity morphisms the corresponding virtual double category may lacknullary morphisms mdash unless these can be defined in some other way as we did in the proof of Lemma 11111 Note that ifwe added identity spans to our double category of two-sided isofibrations then the corresponding virtual double categorywould be the correct one which contains the virtual double category of modules See Remark 11113 however

By Proposition 1044 this left-hand isofibration defines a module from 119860prime to 119861prime while by Defini-tion 1116 the top horizontal functor represents a module map inhabiting the desired square As inDefinition 1116 the simplicial pullback in119974 induces an equivalence⁵ of functor spaces

Fun119891times119892(1198641 times1198601⋯ times

119860119899minus1119864119899 119864(119887 119886)) Fun119886119891times119887119892(1198641 times1198601

⋯ times119860119899minus1

119864119899 119864)sim

120588∘minus

which descends to a bijection on isomorphism classes of objects This defines the unique factorizationof cells as displayed above left through the cartesian restriction cell 120588

We refer to the module 119860prime119864(119887119886)⇸ 119861prime as the restriction of 119860 119864⇸119861 along the functors 119886 and 119887 be-

cause the pullback (1122) is analogous to the restriction of scalars of a bimodule along a pair of ringhomomorphisms

1123 Example The module 119862Hom119860(119891119892)⇸ 119861 is the restriction of the module 119860119860120794⇸119860 along 119892∶ 119862 rarr 119860

and 119891∶ 119861 rarr 119860 To make this restriction relationship more transparent we typically write 119860Hom119860⇸ 119860

when regarding the arrow infin-category as a module Since the common notation for ldquohomsrdquo placesthe contravariant variable on the left and the covariant variable on the right wersquove adopted a similarnotation convention for restrictions in Proposition 1121

1124 Proposition (units) Any object 119860 inℳ119900119889(120101119974) is equipped with a canonical nullary cocartesiancell as displayed below

119860 119860

119860 119860dArr120580

Hom119860

characterized by the universal property that any cell inℳ119900119889(120101119974) whose horizontal source includes the object119860 factors uniquely through 120580 as below-right

119883 ⋯ 119860 ⋯ 119884

119861 119862

119891

1198641 119864119899 1198651

119865119898

119892

119866

119883 ⋯ 119860 119860 ⋯ 119884

119861 119862

1198641

dArrid1198641 dArrid119864119899⋯

119864119899

dArr120580

1198651

dArrid1198651

119865119898

⋯ dArrid119865119898

119891

1198641 119864119899 Hom119860

dArrexist

1198651 119865119898

119892

119866

Proof The canonical nullary cell is represented by the map of spans

119860

119860 119860

Hom119860

120580

1199011 1199010

⁵If the pullbacks are defined strictly then in fact pullback induces an isomorphism of functor spaces but even if 119864(119887 119886)is replaced by an equivalent module the functor spaces are still equivalent which enough to induce a bijection on isomor-phism classes of objects

induced by the generic arrow with codomain 119860 (323) recall from Example 1123 that we write

119860Hom119860⇸ 119860 for the module encoded by the arrowinfin-category constructionIn the case where both of the sequences 119864119894 and 119865119895 are empty the one-sided version of the Yoneda

lemma for modules given by Theorem 1049 tells us that restriction along the this map induces anequivalence of functor spaces

Fun119860times119860(Hom119860 119866(119892 119891)) ⥲ Fun119860times119860(119860119866(119892 119891))Taking isomorphism classes of objects gives the bijection of the statement

In the case where one or both of the sequences are non-empty we may form their horizontalcomposite two-sided fibrations and then form either the horizontal composite 119864 times

119860Hom119860 or the

horizontal composite Hom119860 times119860 119865 of the composable triple below

119864 times119860

Hom119860 times119860 119865

Hom119860(119860 119901) Hom119860(119904 119860)

119864 Hom119860 119865

119883 119860 119860 119884

119902 119901 1199011 1199010 119904 119903

In the former case this constructs the two-sided fibration (1199021199011 1199010) ∶ Hom119860(119860 119901) ↠ 119883 times 119860 and inthe latter case this constructs the two-sided fibration (1199011 1199031199010) ∶ Hom119860(119904 119860) ↠ 119860times119884 In both casesby Lemma 1011 the map 120580 pulls back to define a fibered adjoint functor

119864 Hom119860(119860 119901) 119865 Hom119860(119904 119860)

119883 times 119860 119860 times 119884

119864times119860120580

(119902119901) (11990211990111199010)119903

120580times119860119865perp

(119904119903) (11990111199031199010)

We only require one of these adjunctions so without loss of generality we use the former This fiberedadjunction pulls back along 119904 and pushes forward along 119903 to define a fibered adjunction

119864 times119860119865 119864 times

119860Hom119860 times119860 119865

119883 times 119884

119864times119860120580times119860119865

119903

between two-sided fibrations Upon mapping into the discrete object 119866(119892 119891) ↠ 119883 times119884 this adjunc-tion becomes an adjoint equivalence In particular restriction along 120580 induces an equivalence of Kancomplexes

Fun119883times119884(119864 times119860 Hom119860 times119860 119865119866(119892 119891)) Fun119883times119884(119864 times119860 119865119866(119892 119891))

minus∘(119864times119860120580times119860119865)

and once again taking isomorphism classes of objects gives the bijection of the statement

Propositions 1121 and 1124 imply that the virtual double category of modules is a virtual equip-ment in the sense introduced by Cruttwell and Shulman [15 sect7]

1125 Definition A virtual equipment is a virtual double category so that

(i) For any horizontal arrow119860 119864⇸119861 and pair of vertical arrows 119886 ∶ 119860prime rarr 119860 and 119887 ∶ 119861prime rarr 119861 there

exists a horizontal arrow119861prime119864(119887119886)⇸ 119860prime and unary cartesian cell 120588 satisfying the universal property

of Proposition 1121

(ii) Every object 119860 admits a unit horizontal arrow 119860Hom119860⇸ 119860 equipped with a nullary cocartesian

cell 120580 satisfying the universal property of Proposition 1124

Thus Propositions 1121 and 1124 combine to prove

1126 Theorem The virtual double categoryℳ119900119889(120101119974) of modules in aninfin-cosmos119974 is a virtual equip-ment

By abstract nonsense the relatively simple axioms (i) and (ii) established in Theorem 1126 es-tablish a robust ldquocalculus of modulesrdquo In an effort to familiarize the reader with this little-knowncategorical structure and expedite the proofs of the formal category theory ofinfin-categories in Chap-ter 12 we devote the remainder of this chapter to proving a plethora of results that actually followformally from this axiomatization namely Lemmas 1135 and 11311 Proposition 1141 Theorem1144 Corollary 1146 Proposition 1147 Corollary 1148 and the bijection of Proposition 11410between unary cells in the virtual equipment of modules One additional result is left as Exercise114iii for the reader

1127 Notation We adopt the following notational conventions to streamline certain virtual equip-ment diagramsbull We adopt the convention that an unlabeled unary cell whose vertical boundaries are identities

and whose horizontal sources and targets is an identity cell

119860 119861 119860 119861

119864

≔ dArrid119864

119864

119864 119864

bull Cells whose vertical boundary functors are identities and therefore whose source and target spanslie between the same pair ofinfin-categories

120583∶ 1198641⨰ ⋯ ⨰ 119864119899 rArr 119864 ≔

1198600 1198601 ⋯ 119860119899

1198600 119860119899

1198641

dArr120583

1198642 119864119899

119864

may be displayed in line using the notation120583∶ 1198641⨰ ⋯ ⨰ 119864119899 rArr 119864 an expressionwhich implicitly

asserts that the modules appearing in the domain define a composable sequence with the symbolldquo⨰rdquo meant to suggest the pullback appearing as the horizontal domain of (11110) rather than aproduct

Exercises

112i Exercise Prove that unital rings ring homomorphisms bimodules and bimodule maps alsodefine a virtual equipment

113 Composition of modules

In a virtual equipment one cannot define the composite of a generic pair 119860 119864⇸119861 and 119861 119865⇸119862 of

horizontal morphisms but there is a mechanism by which a particular horizontal composite 119860 119866⇸119862that happens to exist can be recognized in which case we write 119864 otimes 119865 ≃ 119866 to reinforce the intuitionprovided by the analogy with bimodules In the virtual equipment of modules a composition relation119864 otimes 119865 ≃ 119866 does not mean that the module 119866 is equivalent to the horizontal composite two-sidedisofibration 119864 ⨰ 119865 Rather in the notation of 1127 a composition relation is witnessed by a modulemap 119864 ⨰ 119865 rArr 119866 that defines a cocartesian cell in a sense analogous to the universal property statedin Proposition 1124⁶

1131 Definition A composable sequence of modules

11986001198641⇸1198601 1198601

1198642⇸1198602 hellip 119860119899minus1119864119899⇸119860119899

admits a composite if there exists a module 1198600119864⇸119860119899 and a cocartesian cell

120583∶ 1198641⨰ ⋯ ⨰ 119864119899 rArr 119864

characterized by the universal property that any cell of the form

119883 ⋯ 1198600 ⋯ 119860119899 ⋯ 119884

119861 119862

119891

1198651 119865119898 1198641

119864119899 1198661 119866119896

119892

119867

factors uniquely through 120583 as follows

119883 ⋯ 1198600 ⋯ 119860119899 ⋯ 119884

119883 ⋯ 1198600 119860119899 ⋯ 119884

119861 119862

1198651 119865119898

1198641

dArr120583

119864119899 1198661 119866119896

119891

1198651 119865119898 119864

dArrexist

1198661 119866119896

119892

119867

A composite 120583∶ 1198641⨰ ⋯ ⨰ 119864119899 rArr 119864 can be used to reduce the domain of a cell by replacing

any occurrence of the composable sequence 1198641⨰ ⋯ ⨰ 119864119899 from 1198600 to 119860119899 by a single module 119864

Particularly in the case of binary composites we write 1198641otimes1198642 to denote the composite of the modules

⁶In nearly all cases where 119864otimes119865 ≃ 119866 precomposition with the cocartesian map 120583∶ 119864times119861 119865 rArr 119866 which by Definition1111 corresponds to a fibered isomorphism class of maps of spans over 119860 times 119862 induces an equivalence of Kan complexesminus ∘ 120583∶ Fun119860times119862(119866119867) ⥲ Fun119860times119862(119864 times119861 119865119867) for all modules 119867 so the module 119866 may be understood as the ldquoreflectionrdquoof the two-sided isofibration 119864 times119861 119865 into the subcategory of modules Corollary 1035 reappearing as Proposition 1137below is one instance of this

1198641 and 1198642 appearing as the codomain of the cocartesian cell 1198641times1198642 rArr 1198641otimes1198642 Lemma 11117 easilyimplies that composites are unique up to vertical isomorphism ie by Lemma 11117 up to equivalenceof modules Moreover

1132 Lemma Suppose the cells 120583119894 ∶ 1198641198941⨰ ⋯ ⨰ 119864119894119896119894 rArr 119864119894 exhibit composites for 119894 = 1hellip 119899

(i) If 120583∶ 1198641⨰ ⋯ ⨰ 119864119899 rArr 119864 exhibits a composite then the composite cell

11986411⨰ ⋯ ⨰ 119864119899119896119899 1198641

⨰ ⋯ ⨰ 119864119899 1198641205831⨰⋯⨰120583119899 120583

exhibits 119864 as a composite of the sequence 11986411⨰ ⋯ ⨰ 119864119899119896119899

(ii) If the composite cell

11986411⨰ ⋯ ⨰ 119864119899119896119899 1198641

⨰ ⋯ ⨰ 119864119899 1198641205831⨰⋯⨰120583119899 120583

exhibits 119864 as a composite of the sequence 11986411⨰ ⋯ ⨰ 119864119899119896119899 then 120583∶ 1198641

⨰ ⋯ ⨰ 119864119899 rArr 119864 exhibits 119864as a composite of the sequence 1198641

⨰ ⋯ ⨰ 119864119899

Proof For (i) the required bijection factors as a composite of 119899 + 1 bijections induced by themaps 1205831 hellip 120583119899 and 120583 For (ii) the required bijection induced by 120583 composes with the bijectionssupplied by the maps 1205831 hellip 120583119899 to a bijection and is thus itself a bijection

The trivial instances of composites are easily verified

1133 Lemma(i) The units 120580 ∶ empty rArr Hom119860 of Proposition 1124 define nullary composites(ii) A unary cell 120583∶ 119864 rArr 119865 is a composite if and only if it is an isomorphism in the vertical category of

modules from119860 to 119861 and module maps over identity functors that is if and only if the modules 119864 and119865 are equivalent as spans

Proof Exercise 113ii

1134 Remark On account of the universal property of restrictions established in Proposition 1121to prove that a cell 120583∶ 1198641

⨰ ⋯ ⨰ 119864119899 rArr 119864 exhibits a composite it suffices to prove the factorizationproperty of Definition 1131 in the case where the vertical functors are identities

As one might hope the unit modules 119860Hom119860⇸ 119860 are units for the composition of Definition 1131

for any module 119860 119864⇸119861 Hom119860 otimes 119864 otimes Hom119861 ≃ 119864

1135 Lemma (composites with units) For any module119860 119864⇸119861 the unique cell ∘ ∶ Hom119860times119864timesHom119861 rArr 119864defined using the universal property of the unit cell

119860 119860 119861 119861

119860 119861

dArr120580

119864

dArr120580Hom119860 119864

dArr∘

Hom119861

119864

≔119860 119861

119860 119861

119864

displays 119864 as the composite Hom119860 otimes 119864 otimes Hom119861

Proof The result is immediate from Lemma 1132(ii) and Lemma 1133

The argument used in the proof of Proposition 1124 to demonstrate that units define nullarycomposites in the virtual equipment of modules applies also to more general composites

1136 Lemma Consider an 119899-ary module morphism 120583∶ 1198641 times1198601⋯ times

119860119899minus1119864119899 rArr 119864 in the virtual equipment of

modules whose codomain is a module from 119860 to 119861 If any representing map of spans

1198641⨰ ⋯ ⨰ 119864119899 119864

119860 times 119861

120583

admits a fibered adjoint over 119860 times 119861 then 120583 exihibits 119864 as the composite 1198641 otimes⋯otimes 119864119899

Proof To verify the universal property of Definition 1131 consider a composable sequence ofmodules 1198651 hellip 119865119898 from 119883 to 119860 and a composable sequence of modules 1198661 hellip 119866119896 from 119861 to 119884 andform the horizontal composite two-sided fibrations

119865

119883 119860

119902 119901 and119866

119861 119884119904 119903

The fibered adjunction of the statement pulls back along 119901 times 119904 ∶ 119865 times 119866 ↠ 119860times119861 and pushes forwardalong 119902 times 119903 ∶ 119865 times 119866 rarr 119883 times 119884 to a fibered adjoint to

119865 times1198601198641 times1198601

119864119899 times119861 119866 119865 times119860119864 times119861119866

119883 times 119884

119865times119860120583times

119861119866

Via Remark 1134 it suffices to verify the universal property of the composite for modules 119883 119867⇸119884Since modules are discrete applying Fun119883times119884(minus119867) transforms this fibered adjunction into an adjointequivalence

Fun119883times119884(119865 times119860 119864 times119861 119866119867) Fun119883times119884(119865 times119860 1198641 times1198601⋯ times

119860119899minus1119864119899 times119861 119866119867)

minus∘(119865times119860120583times

119861119866)

Passing to isomorphism classes of objects this now gives the universal property of Definition 1131

For instance

1137 Proposition Let 119860 119864⇸119861 be a module encoded by the span 119860119902larrminuslarr119864

119901minusrarrrarr 119861 Then the binary module

119860 119864 119861

119860 119861

Hom119860(119902119860)

dArr120583

Hom119861(119861119901)

119864represented by composite left and right adjoints of Theorem 1014(iii) exhibits119864 ≃ Hom119860(119902 119860)otimesHom119861(119861 119901)as the composite of the modules representing its legs

Proof Immediate from Theorem 1014(iii) and Lemma 1136 applied twice⁷

1138 Remark Nothing in the proof of Proposition 1137 requires that the span 119860119902larrminuslarr 119864

is actually a module except for the interpretation that 120583 is a binary cell in the virtual equipment of

modules For any two-sided fibration119860119902larrminuslarr119864

119901minusrarrrarr 119861 it is still the case that restriction along 120583 defines a

bijection between 119899-ary maps whose source includes the span119864 and whose codomain is a module standin bijection with (119899 + 1)-ary maps whose source includes Hom119860(119902 119860)

⨰ Hom119861(119861 119901) By Theorem1049 this result can be extended further to mere spans 119864 not necessarily two-sided fibrations

1139 Remark Weprovide another description of compositionmap120583∶ Hom119860(119902 119860)⨰ Hom119861(119861 119901) rArr

119864 The comma conesHom119860(119902 119860) times119864 Hom119861(119861 119901)

Hom119860(119902 119860) Hom119861(119861 119901)

119860 119864 119861

1199011 1199010120601lArr

119902 119901

Define a pair of 2-cells to which the premises of Theorem 1014(iv) apply The conclusion asserts thatthere is a well-defined fibered isomorphism class of functors 120583 ∶ Hom119860(119902 119860) times119864 Hom119861(119861 119901) rarr 119864defined by taking the 119902-cocartesian lift of the left comma cone composing with the right commacone and then taking the codomain of a 119901-cartesian lift of this composite cell mdash or by first taking the119901-cartesian lift composing and then taking the domain of a 119902-cartesian lift of this composite mdash this

being the functor ℓ119903 ≃ 119903ℓ in the notation of Theorem 1014(iii) In the case where 119860 119864⇸119861 is itself acomma module the resulting fibered isomorphism class of functors 120583 is the one that classifies thepasted composite above with the comma cone for 119864

Any virtual double category has an identity cell for each horizontal arrow 119860 119864⇸119861 whose verticalboundary arrows are identities In a virtual equipment we also have a unit cell for each vertical

arrow 119891∶ 119860 rarr 119861 whose horizontal boundary is given by the unit modules 119860Hom119860⇸ 119860 and 119861

Hom119861⇸ 119861 ofProposition 1124

11310 Definition Using the unit modules inℳ119900119889(120101119974) for any functor 119891∶ 119860 rarr 119861 we may definea unary unit cell as displayed below-left by appealing to the universal property of the nullary unit cell120580 ∶ empty rArr Hom119860 for 119860 in the equation below-right

119860 119860

119861 119861

119891

Hom119860

dArrHom119891 119891

Hom119861

119860 119860 119860 119860

119860 119860 119861 119861

119861 119861 119861 119861

dArr120580 119891 119891

119891

Hom119860

dArrHom119891 119891

≔dArr120580

Hom119861 Hom119861

⁷Of course the composite of a left and a right adjoint is not an adjoint but here wersquore effectively composing adjointequivalences in which case the direction does not matter

In the characterization of Lemma 11116 both sides of the pasting equality defining the unary unit

cell correspond to the identity 2-cell 119860 119861119891

119891

dArrid119891

As one might hope the unit cells are units for the vertical composition of cells in the virtualequipment of modules

11311 Lemma (composition with unit cells) Any cell 120572 as below-right equals the pasted composite below-left

1198600 1198600 1198601 ⋯ 119860119899 119860119899

119861 119861 119862 119862

119861 119862

dArr120580

1198641 1198642

119864119899

dArr120580

119891

Hom1198600

dArrHom119891 119891

1198641

dArr120572

1198642 119864119899

119892

Hom119860119899

dArrHom119892 119892

dArr∘Hom119861 119864 Hom119862

119864

=1198600 1198601 ⋯ 119860119899

119861 119862

119891

1198641

dArr120572

1198642 119864119899

119892

119864

Proof By Definition 11310 and the laws for composition with identity cells in a virtual doublecategory stated in Definition 1119 the left-hand composite of the statement equals the left-hand com-posite cell displayed below and the right-hand side of the statement equals the right-hand compositecell displayed below

1198600 1198600 1198601 ⋯ 119860119899 119860119899

119861 119861 119862 119862

119861 119862

119891 119891

1198641

dArr120572

1198642 119864119899

119892 119892

dArr120580119864

dArr120580

dArr∘Hom119861 119864 Hom119862

119864

1198600 1198601 ⋯ 119860119899

119861 119862

119891

1198641

dArr120572

1198642 119864119899

119892

119864

By Lemma 1135 the left-hand side equals the right-hand side

11312 Definition (horizontal composition of cells) If given a horizontally composable sequence ofunary cells

1198600 1198601 ⋯ 119860119899

1198610 1198611 ⋯ 119861119899

1198910

1198641

dArr1205721 1198911

1198642

dArr1205722 ⋯

119864119899

dArr120572119899 119891119899

1198651 1198652 119865119899

for which the composable sequences 1198641⨰ ⋯ ⨰ 119864119899 and 1198651

⨰ ⋯ ⨰ 119865119899 both admit compositesthen there exists a horizontal composite unary cell 1205721 lowast ⋯ lowast 120572119899 that is uniquely determined up tothe specification of the composites ∘ ∶ 1198641

⨰ ⋯ ⨰ 119864119899 rArr 119864 and 1198651⨰ ⋯ ⨰ 119865119899 rArr 119865 by the pasting

identity

1198600 1198601 ⋯ 119860119899 1198600 1198601 ⋯ 119860119899

1198600 119860119899 = 1198610 1198611 ⋯ 119861119899

1198610 119861119899 1198610 119861119899

1198641

dArr∘

1198642 119864119899

1198910

1198641

dArr1205721 1198911

1198642

dArr1205722 ⋯

119864119899

dArr120572119899 119891119899

1198910 dArr1205721lowast⋯lowast120572119899

119864

119891119899 dArr∘1198651 1198652 119865119899

119865 119865

By an argument very similar to the proof of Lemma 11311 using the composites of Lemma 1135the horizontal composite Hom119891 lowast 120572 lowast Hom119892 of a unary cell 120572 with the unit cells Hom119891 and Hom119892 atits vertical boundary functors recovers 120572 The upshot of Definition 11312 is that ℳ119900119889(120101119974) can beunderstood to contain various ldquoverticalrdquo and ldquohorizontalrdquo bicategories

11313 Proposition (the vertical 2-category in the virtual equipment) Any virtual equipment containsa vertical 2-categorywhose objects are the objects of the virtual equipment whose arrows are the vertical arrowsand whose 2-cells are those unary cells

119860 119860

119861 119861

119892

Hom119860

dArr 119891

Hom119861

whose horizontal boundary arrows are given by the unit modules

Proof To prove that these structures define a 2-category we must define ldquohorizontalrdquo composi-tion of 2-cells (composing along a boundary 0-cell) and ldquoverticalrdquo composition of 2-cells (composingalong a boundary 1-cell) The ldquohorizontalrdquo composition in the 2-category is defined via the verticalcomposition in the virtual double category described in Definition 1119 The ldquoverticalrdquo compositionin the 2-category is defined by Definition 11312 To see that this yields a 2-category and not a bicat-egory note that any bicategory in which the composition of 1-cells is strictly associative and unital isa 2-category in this case the 1-cells are the vertical arrows of the virtual double category which doindeed compose strictly

Proposition 11410 will prove that the vertical 2-category in the virtual equipment ℳ119900119889(120101119974) isisomorphic to the homotopy 2-category 120101119974

11314 Remark (horizontal bicategories in the virtual equipment) Via Definition 11312 a virtualequipment can also be understood to contain various ldquohorizontalrdquo bicategories defined by taking the1-cells to be composable modules and the 2-cells to be unary module maps whose vertical boundaryfunctors are identities Particular horizontal bicategories of interest are described inDefinition 11411

Exercises

113i Exercise Extending Exercise 111ii prove that in a virtual double category arising from anactual double category that every composable sequence of horizontal morphisms admits a compositein the sense of Definition 1131

114 Representable modules

Any vertical arrow 119891∶ 119860 rarr 119861 in a virtual equipment has a pair of associated horizontal arrows

119861Hom119861(119891119861)⇸ 119860 and 119860

Hom119861(119861119891)⇸ 119861 mdash defined as restrictions of the horizontal unit arrows mdash that haveuniversal properties similar to companions and conjoints in an ordinary double category In the virtualequipment of modules these are sensible referred to as the left and right representations of a functoras a module and coincide exactly with the left and right representable first introduced in sect34 andreappearing in Definition 1048

1141 Proposition (companion and conjoint relations for representables) To any functor 119891∶ 119860 rarr 119861in the virtual equipment of modules there exist canonical restriction cells displayed below-left and applicationcells displayed below-right

119861 119860 119860 119861 119860 119860 119860 119860

119861 119861 119861 119861 119861 119860 119860 119861

Hom119861(119891119861)

dArr120588 119891 119891 dArr120588

Hom119861(119861119891)

119891 dArr120581

Hom119860

dArr120581

Hom119860

119891

Hom119861 Hom119861 Hom119861(119891119861) Hom119861(119861119891)

defining unary module maps between the unit modules 119860Hom119860⇸ 119860 and 119861

Hom119861⇸ 119861 and the left and right repre-

sentable modules 119861Hom119861(119891119861)⇸ 119860 and 119860

Hom119861(119861119891)⇸ 119861 These satisfy the identities

119860 119860

119861 119860

119861 119861

119891 dArr120581

Hom119860

dArr120588

Hom119861(119891119861)

119891

Hom119861

=119860 119860

119861 119861

119891

Hom119860

dArrHom119891 119891

Hom119861

119860 119860

119860 119861

119861 119861

dArr120581

Hom119860

119891

119891 dArr120588

Hom119861(119861119891)

Hom119861

(1142)

119861 119860 119860

119861 119861 119860

119861 119860

Hom119861(119891119861)

dArr120588 119891

Hom119860

dArr120581Hom119861 Hom119861(119891119861)

dArr∘

Hom119861(119891119861)

=119861 119860 119860

119861 119860

Hom119861(119891119861)

dArr∘

Hom119860

Hom119861(119891119861)

119860 119860 119861

119860 119861

Hom119860

dArr∘

Hom119861(119861119891)

119860 119860 119861

119860 119861 119860

119860 119861

Hom119860

dArr120581 119891

Hom119861(119861119891)

dArr120588Hom119861(119861119891) Hom119861

dArr∘

Hom119861(119861119891)(1143)

Proof The unary module maps 120581 are defined by the equations (1142) by appealing to the uni-versal property of the restriction cells in Proposition 1121 The relations (1143) could also be verifieddirectly from the axioms of Theorem 1126 via Propositions 1124 and Lemmas 1135 and 1135 In-stead we appeal to Lemma 11116 to characterize each of the cells in the virtual equipment as 2-cellsin the homotopy 2-category

We prove this for the right representables the co-dual then proves this for the left representablesThe binary module morphism on the left-hand side of the equality of (1143) represents the 2-cellbelow-left while the right-hand composite is below-right

119860120794 times119860

Hom119861(119861 119891)

119860120794 Hom119861(119861 119891)

119860 119860 119861

1199011 1199010120581lArr

1199011 1199010120601lArr

119891

119860120794 times119860

Hom119861(119861 119891)

119860120794 Hom119861(119861 119891)

Hom119861(119861 119891) 119860 119861120794

119860 119861 119861

11989611990111199010 1205881199011

1199010

1199011 1199010120601lArr

119891 1199011 1199010120581lArr

119891

By the definition of the induced functor 119896 ∶ 119860120794 rarr Hom119861(119861 119891) 120601119896 = 119891120581 and by the definition of thecomma cone 120601 for Hom119861(119861 119891) 120601 = 120581120588 Thus the left-hand side equals the right-hand side

1144 Theorem In the virtual equipment of modules there are bijections between cells of the following fourforms

119862 119860 119861 119860 119861 119860 119861 119863

119862 119863 119862 119863 119862 119863

119862 119860 119861 119863

119862 119863

Hom119862(119891119862) 119864

dArr 119892 119891

119864

dArr 119892 119891

119864 Hom119863(119863119892)

119865

119865 119865

Hom119862(119891119862)

119864 Hom119863(119863119892)

119865implemented by composing with the canonical cells 120581 and 120588 of Proposition 1141 and with the composition andnullary cells associated to the units

Proof The composite bijection carries cells 120572 and 120573 to the cells displayed below-left and below-right respectively

119862 119860 119861 119863

119862 119862 119863 119863

119862 119863

Hom119862(119891119862)

dArr120588 dArr120572119891

119864

119892

Hom119863(119863119892)

dArr120588Hom119862

dArr∘

119865 Hom119863

119865

and ≔

119860 119860 119861 119861

119862 119860 119861 119863

119862 119863

dArr120580

119864

dArr120580

119891 dArr120581

Hom119860 119864

dArr120581

Hom119861

119892Hom119862(119891119862)

dArr120573

119864 Hom119863(119863119892)

119865

By Proposition 1141 and Lemma 11311

119860 119860 119861 119861

119862 119860 119861 119863

119862 119862 119863 119863

119862 119863

dArr120580

119864

dArr120580

119891 dArr120581

Hom119860 119864

dArr120581

Hom119861

119892Hom119862(119891119862)

dArr120588 dArr120572119891

119864

119892

Hom119863(119863119892)

dArr120588Hom119862

dArr∘

119865 Hom119863

119865

119860 119860 119861 119861

119862 119862 119863 119863

119862 119863

dArr120580

119864

dArr120580

119891 dArrHom119891 dArr120572

Hom119860 119864

119891 dArrHom119892119892

Hom119861

119892Hom119862

dArr∘

119865 Hom119863

119865

=119860 119861

119862 119863

119891

119864

dArr120572 119892

119865

The other composite is

119862 119860 119860 119861 119861 119863

119862 119862 119860 119861 119863 119863

119862 119862 119863 119863

119862 119863

Hom119862(119891119862)

dArr120580

119864

dArr120580

Hom119863(119863119892)

Hom119862(119891119862)

dArr120588 119891 dArr120581

Hom119860 119864

dArr120581

Hom119861

119892

Hom119863(119863119892)

dArr120588Hom119862 Hom119862(119891119862)

dArr120573

119864 Hom119863(119863119892) Hom119863

Hom119862

dArr∘119865

Hom119863

119865

(1145)

By Lemma 1135 we have

119862 119862 119860 119861 119863 119863

119862 119862 119863 119863

119862 119863

dArr120580

Hom119862(119891119862) 119864 Hom119863(119863119892)

dArr120580Hom119862 Hom119862(119891119862)

dArr120573

119864 Hom119863(119863119892) Hom119863

Hom119862

dArr∘119865

Hom119863

119865

119862 119862 119860 119861 119863 119863

119862 119860 119861 119863

119862 119863

dArr120580

Hom119862(119891119862) 119864 Hom119863(119863119892)

dArr120580Hom119862

dArr∘

Hom119862(119891119862) 119864 Hom119863(119863119892)

dArr∘

Hom119863

Hom119862(119891119862)

dArr120573

119864 Hom119863(119863119892)

119865

since both composites equal 120573 By the universal property of the unit cells in Proposition 1124 thebottom two rows of these diagrams are equal so we may substitute the bottom two rows of the right-hand diagram for the bottom two rows of (1145) to obtain

119862 119860 119860 119861 119861 119863

119862 119862 119860 119861 119863 119863

119862 119860 119861 119863

119862 119863

Hom119862(119891119862)

dArr120580

119864

dArr120580

Hom119863(119863119892)

Hom119862(119891119862)

dArr120588 119891 dArr120581

Hom119860 119864

dArr120581

Hom119861

119892

Hom119863(119863119892)

dArr120588Hom119862

dArr∘

Hom119862(119891119862) 119864 Hom119863(119863119892)

dArr∘

Hom119863

Hom119862(119891119862)dArr120573

119864

Hom119863(119863119892)

119865

By Proposition 1141 and Lemma 1135 this reduces to 120573

119862 119860 119860 119861 119861 119863

119862 119860 119861 119863

119862 119863

Hom119862(119891119862)

dArr120580

119864

dArr120580

Hom119863(119863119892)

Hom119862(119891119862)

dArr∘

Hom119860 119864 Hom119861

dArr∘

Hom119863(119863119892)

Hom119862(119891119862)dArr120573

119864

Hom119863(119863119892)

119865

=119862 119860 119861 119863

119862 119863

Hom119862(119891119862)

dArr120573

119864 Hom119863(119863119892)

119865

By vertically bisecting this construction one obtains the one-sided bijections of the statement

We frequently apply this result in the following form

1146 Corollary Given modules 119860 119864⇸119861 and 119862 119865⇸119863 and functor 119891∶ 119860 rarr 119862 and 119892∶ 119861 rarr 119863 there isa bijection between ternary module maps Hom119862(119891 119862)

⨰ 119864 ⨰ Hom119863(119863 119892) rArr 119865 and unary module maps119864 rArr 119865(119892 119891) ie between cells

119862 119860 119861 119863 119860 119861

119862 119863 119860 119861

Hom119862(119891119862)

119864 Hom119863(119863119892)

119864

119865 119865(119892119891)

Proof Combine Theorem 1144 with Proposition 1121

1147 Proposition For any module119860 119864⇸119861 and pair of functors 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119884 rarr 119861 the compositeHom119860(119860 119886)otimes119864otimesHom119861(119887 119861) exists and is given by the restriction 119864(119887 119886) with the ternary composite map120583∶ Hom119860(119860 119886)

⨰ 119864 ⨰ Hom119861(119887 119861) rArr 119864(119887 119886) defined by the universal property of the restriction by thepasting diagram

119883 119860 119861 119884

119883 119884

119860 119861

Hom119860(119860119886)

dArr120583

119864 Hom119861(119887119861)

119886119864(119887119886)dArr120588 119887

119864

119883 119860 119861 119884

119860 119860 119861 119861

119860 119861

119886 dArr120588

Hom119860(119860119886) 119864

dArr120588

Hom119861(119887119861)

119887

Hom119860dArr∘

119864

Hom119861

119864

Proof The horizontal composite two-sided fibration of the composable sequence is

Hom119860(119860 119886) times119860 119864 times119861 Hom119861(119887 119861)

Hom119860(119902 119886) Hom119861(119887 119901)

Hom119860(119860 119886) 119864 Hom119861(119887 119861)

119883 119860 119861 119884

1199011 1199010 119902 119901 1199011 1199010

from which we see that the binary composite cell

Hom119860(119902 119860) times119864 Hom119861(119861 119901) 119864

119860 times 119861(119902119901)

of Proposition 1137 pulls back along 119886 times 119887∶ 119883 times 119884 rarr 119860times 119861 to define the map 120583 By Lemma 1136we conclude that 120583∶ Hom119860(119860 119886)

⨰ 119864 ⨰ Hom119861(119887 119861) rArr 119864(119887 119886) is a composite

As a special case right representable modules can always be composed with each other and duallyleft representable modules can always be composed with each other

1148 Corollary Any composable pair of functors 119860119891minusrarr 119861

119892minusrarr 119862 defines a composable pair of right-

represented modules and a composable pair of left-represented modules

119860 119861 119862 119862 119861 119860Hom119861(119861119891) Hom119862(119862119892) Hom119862(119892119862) Hom119861(119891119861)

and moreover

Hom119861(119861 119891)otimesHom119862(119862 119892) ≃ Hom119862(119862 119892119891) and Hom119862(119892 119862)otimesHom119861(119891 119861) ≃ Hom119862(119892119891 119862)

Combining this result with Proposition 1137 provides a generalization of Theorem 3411 whichallows us to detect representable modules

1149 Proposition Let 119860119902larrminuslarr119864

119901minusrarrrarr 119861 encode a module

(i) The module 119860 119864⇸119861 is right representable if and only if its left leg 119902 ∶ 119864 ↠ 119860 has a right adjoint119903 ∶ 119860 rarr 119864 in which case 119864 ≃ Hom119861(119861 119901119903)

(ii) The module119860 119864⇸119861 is left representable if and only if its right leg 119901∶ 119864 ↠ 119861 has a left adjoint ℓ ∶ 119861 rarr119864 in which case 119864 ≃ Hom119860(119902ℓ 119860)

Proof Lemma 348 proves that the claimed adjoints to the legs 1199011 ∶ Hom119861(119861 119891) ↠ 119860 and1199010 ∶ Hom119861(119891 119861) ↠ 119860 of left or right representable modules exist so it remains only to prove the con-

verse By Proposition 1137 any module 119860 119864⇸119861 can be expressed as a composite 119864 ≃ Hom119860(119902 119860) otimesHom119861(119861 119901) of the left representation of its left leg followed by the right right representation of itsright leg If 119902 ⊣ 119903 then by Proposition 411 Hom119860(119902 119860) ≃ Hom119864(119864 119903) so 119864 ≃ Hom119864(119864 119903) otimesHom119861(119861 119901) By Corollary 1148 Hom119864(119864 119903) otimes Hom119861(119861 119901) ≃ Hom119861(119861 119901119903) so it follows that 119864 ≃Hom119861(119861 119901119903) The case of left representability is dual

Finally we revisit the ldquocheaprdquo version of the Yoneda lemma presented in Corollary 3410 whichencodes natural transformations in the homotopy 2-category as maps of represented modules

11410 Proposition For any parallel pair of functors there are natural bijections between 2-cells in thehomotopy 2-category

119860 119861119891

119892dArr

and cells in the virtual equipment of modules

119860 119861 119860 119860 119861 119860

119860 119861 119861 119861 119861 119860dArr

Hom119861(119861119891)

dArr119892

Hom119860

119891

Hom119861(119892119861)

Hom119861(119861119892) Hom119861 Hom119861(119891119861)

Proof The bijection between 2-cells in the homotopy 2-category and unary module maps be-tween left or right representables is simply a restatement of Corollary 3410 Alternatively the bi-jection between 2-cells in the homotopy 2-category and the cells displayed in the center follows fromLemma 11116 and Proposition 1124 The bijections inℳ119900119889(120101119974) can then be derived from Theorem1144 and Corollary 1146

11411 Definition (the covariant and contravariant embeddings) Proposition 11410 defines theaction on 2-cells of two identity-on-objects locally fully faithful homomorphisms

120101119974 ℳ119900119889(120101119974) 120101119974coop ℳ119900119889(120101119974)

119860119891minusrarr 119861 119860

Hom119861(119861119891)⇸ 119861 119860119891minusrarr 119861 119861

Hom119861(119891119861)⇸ 119860that embed the homotopy 2-category fully faithfully into the sub ldquobicategoryrdquo ofℳ119900119889(120101119974) containingonly those unary cells whose vertical boundaries are identities

This substructure of ℳ119900119889(120101119974) isnrsquot quite a bicategory because not all horizontally composablemodules can be composed but if we restrict only to the right representable modules or only to the leftrepresentable modules then by Corollary 1148 the composites do exist and moreover the embeddings

are horizontally as well as vertically pseudofunctorial given 119860119891minusrarr 119861

119892minusrarr 119862 we have Hom119861(119861 119891) otimes

Hom119862(119862 119892) ≃ Hom119862(119862 119892119891) and Hom119862(119892 119862) otimes Hom119861(119891 119861) ≃ Hom119862(119892119891 119862) We refer to these asthe covariant and contravariant embeddings respectively

As described in Remark 11314 ℳ119900119889(120101119974) can be understood to contain genuine bicategorieswhose 2-cells are unary cells between right-represented modules (or between left-represented mod-ules) whose vertical boundaries are identities In this way the covariant and contravariant embeddingscan be understood to define genuine bicategorical homomorphisms

There is a third locally fully faithful embedding of the homotopy 2-category 120101119974 into ℳ119900119889(120101119974)that is identity on objects sends 119891∶ 119860 rarr 119861 to the corresponding vertical 1-cell and uses the thirdbijection of Proposition 11410 to define the action on 2-cells Since Hom119860 otimes Hom119860 ≃ Hom119860 theunary cells in this image of this embedding can be composed horizontally as well as vertically and thisembedding is functorial in both directions vertical composites of natural transformations in 120101119974 co-incide with horizontal composites of unary cells and horizontal composites of natural transformationsin 120101119974 coincide with vertical composites of unary cells The image is precisely the vertical 2-categoryof Proposition 11313 We will make much greater use of the covariant and contravariant embeddingsof Definition 11411 however

Exercises

114i Exercise Prove Proposition 1141 in any virtual equipment without appealing to Lemma11116

114ii Exercise Prove Proposition 1147 in any virtual equipment without appealing to Lemma1136

114iii Exercise For any functor 119891∶ 119860 rarr 119861 define a unary cell 120578∶ Hom119860 rArr Hom119861(119891 119891) congHom119861(119861 119891) otimes Hom119861(119891 119861) and a binary cell 120598 ∶ Hom119861(119891 119861)

⨰ Hom119861(119861 119891) rArr Hom119861 Use this data

to demonstrate that the modules 119860Hom119861(119861119891)⇸ 119861 and 119861

Hom119861(119891119861)⇸ 119860 are ldquoadjointrdquo in a suitable sense

CHAPTER 12

Formal category theory in a virtual equipment

Mac Lane famously asserted that ldquoall concepts are Kan extensionsrdquo [38 sectX7] mdash at least in categorytheory Kelly later amended this to assert that the pointwise Kan extensions which he calls simply ldquoKanextensionsrdquo are the important ones writing ldquoOur present choice of nomenclature is based on ourfailure to find a single instance where a [non-pointwise] Kan extension plays any mathematical rolewhatsoeverrdquo [31 sect4] Using the calculus of modules we can now add the theory of pointwise Kanextensions of functors betweeninfin-categories to the basicinfin-category theory developed in Part I

Right and left extensions of a functor 119891∶ 119860 rarr 119862 along a functor 119896 ∶ 119860 rarr 119861 can be definedinternally to any 2-category mdash at this level of generality the eponym ldquoKanrdquo is typically dropped Wereview this notion in Definition 1221 However in the homotopy 2-category of an infin-cosmos theuniversal property defining left and right extensions is not strong enough and indeed the correctuniversal property is associated to the stronger notion of a pointwise extension for which the valuesof a right or left extension at an element of 119861 can be computed as limits or colimits indexed by theappropriate commainfin-category see Corollary 1243 for a precise statement

Our aim in this chapter is to define and study pointwise extensions for functors betweeninfin-cate-gories In fact we givemultiple definitions of pointwise extension One is fundamentally 2-categoricala pointwise extension is an ordinary 2-categorical extension in the homotopy 2-category that is stableunder pasting with comma squares Another definition is that a 2-cell defines a pointwise right ex-tension if and only if its image under the covariant embedding into the virtual equipment of modulesdefines a right extension there Theorem 1227 proves that these two notions coincide

Before turning our attention to pointwise extensions we first introduce exact squares in sect121 aclass of squares in the homotopy 2-category that include comma squares and which will be used tocharacterize the pointwise extensions internally to the homotopy 2-category Pointwise extensionsare introduced in a variety of equivalent ways in sect122 and applied in sect123 to develop a few aspects ofthe formal theory ofinfin-categories In sect124 we conclude with a discussion of pointwise extensions ina cartesian closedinfin-cosmos in which context these relate to the absolute lifting diagrams and limitsand colimits studied in sect23-24

121 Exact squares

By Proposition 11410 natural transformations in the homotopy 2-category of aninfin-cosmos cor-respond bijectively to unary squares in the virtual equipment of modules of various forms By thisresult Theorem 1144 Proposition 1124 and Proposition 1121 we have bijections

119863

119862 119861

119860

ℎ119896120572lArr

119892 119891

119863 119863 119862 119863 119861

119860 119860 119862 119861

119892119896

Hom119863

dArr 119891ℎ

Hom119862(119896119862)

Hom119861(119861ℎ)

Hom119860 Hom119860(119891119892)

1211 Definition (exact squares) A square in the homotopy 2-category of aninfin-cosmos

119863

119862 119861

119860

ℎ119896120572lArr

119892 119891

is exact if and only if the corresponding cell below-left which under the bijection of Lemma 11116encodes the below-right pasted composite

119862 119863 119861

119862 119861

Hom119862(119896119862)

Hom119861(119861ℎ)

Hom119860(119891119892)

Hom119862(119896 119862) times119863 Hom119861(119861 ℎ)

Hom119862(119896 119862) Hom119861(119861 ℎ)

119863

119862 119861

119860

1199010

1199011120601lArr

1199011

1199010120601lArrℎ119896

120572lArr

119892 119891

(1212)

displays Hom119860(119891 119892) as the composite Hom119862(119896 119862) otimes Hom119861(119861 ℎ) as defined in Definition 1131

When the boundary square is clear from context for economy of language we may write thatldquo120572∶ 119891ℎ rArr 119892119896 is an exact squarerdquo but note that the definition of exactness requires the specification ofthe four boundary components of the square inhabited by the 2-cell 120572sup1

1213 Remark (exactness as a Beck-Chevalley condition) By Proposition 1147 the canonical cell120583Hom119860(119860 119892)timesHom119860(119891 119860) rArr Hom119860(119891 119892) encoded by the map of spans defined by 1-cell induction

Hom119860(119860 119892) times119860 Hom119860(119891 119860)

Hom119860(119860 119892) Hom119860(119891 119860)

119862 119860 119861

1199011 1199010120601lArr

119892 119891

Hom119860(119860 119892) times119860 Hom119860(119891 119860)

Hom119860(119891 119892)

119862 119861

119860

120583

1199011 1199010120601lArr

119892 119891

is a composite Exactness says that 120572 induces an isomorphism

∶ Hom119862(119896 119862) otimes Hom119861(119861 ℎ) ≃ Hom119860(119860 119892) otimes Hom119860(119891 119860)of modules from 119862 to 119861

The remainder of this section is devoted to examples of exact squares

sup1For instance compare the statements of Exercise 121i and Lemma 1233

1214 Lemma (comma squares are exact) For any cospan 119862119892minusrarr 119860

119891larrminus 119861 the comma cone defines an exact

squareHom119860(119891 119892)

119862 119861

119860

1199011 1199010120601lArr

119892 119891

Proof By Proposition 1137 the module 119862Hom119860(119891119892)⇸ 119861 is the composite of the left representation

of its left leg followed by the right representation of its right leg Hom119862(1199011 119862) otimes Hom119861(119861 1199010) ≃Hom119860(119891 119892) By Remark 1139 the composition map Hom119862(1199011 119862)

⨰ Hom119861(119861 1199010) rArr Hom119860(119891 119892)classifies the pasted composite

Hom119862(1199011 119862)⨰ Hom119861(119861 1199010)

Hom119862(1199011 119862) Hom119861(119861 1199010)

Hom119860(119891 119892)

119862 119861

119860

1199010

1199011

120601lArr

1199011

1199010

120601lArr

11990101199011

120601lArr

119892 119891

which recovers the cell defined by (1212) that tests the exactness of comma square 120601∶ 1198911199010 rArr1198921199011

1215 Lemma If 119892∶ 119862 rarr 119860 is a cartesian fibration or 119891∶ 119861 rarr 119860 is a cocartesian fibration then thepullback square

119875

119862 119861

119860

1205871 1205870

119892 119891

is exact

Proof The two statements are dual though the positions of the cocartesian and cartesian fibra-tions cannot be interchanged as the proof will reveal If 119891∶ 119861 ↠ 119860 is a cocartesian fibration observethat the functor 119894 ∶ 119875 rarr Hom119860(119891 119892) induced by the identity 2-cell 1198911205870 = 1198921205871 is a pullback of the

functor 119894 ∶ 119861 rarr Hom119860(119891 119860) induced by the identity 2-cell id119891

119875 119861

Hom119860(119891 119892) Hom119860(119891 119860)

119862 119860

1205870

119894

1205871119891

119894perp

1199011

119892

Since 119891 is a cocartesian fibration Theorem 5111(ii) tells us that 119894 ∶ 119861 rarr Hom119860(119891 119860) has a fiberedleft adjoint over 119860 This fibered adjunction pulls back via the cosmological functor 119892lowast ∶ 119974119860 rarr 119974119862to define a fibered left adjoint to 119894 ∶ 119875 rarr Hom119860(119891 119892)

Since 1205870 = 1199010119894 Corollary 1148 implies that the canonical cell

HomHom119860(119891119892)(Hom119860(119891 119892) 119894)⨰ Hom119861(119861 1199010) rArr Hom119861(119861 1205870)

is a composite Since ℓ ⊣ 119894 Hom119875(ℓ 119875) ≃ HomHom119860(119891119892)(Hom119860(119891 119892) 119894) by Proposition 411 Andsince 1199011 = 1205871ℓ Corollary 1148 again implies that the canonical cell

Hom119862(1205871 119862)⨰ Hom119875(ℓ 119875) rArr Hom119862(1199011 119862)

is a composite Composing these bijections we see that cells with domain Hom119862(1205871 119862)⨰ Hom119861(119861 1205870)

correspond bijectively to cells with domain Hom119862(1199011 119862)⨰ Hom119861(119861 1199010)

The equation id = 120601119894 ∶ 1198911205870 = 1198911199010119894 rArr 1198921199011119894 = 1198921205871 asserts that the identity is the transpose alongℓ ⊣ 119894 of the cell 120601∶ 1198911199010 rArr 1198921199011 = 1198921205871ℓ which tells us that the cells

id ∶ Hom119862(1205871 119862)⨰ Hom119861(119861 1205870) rArr Hom119860(119891 119892) and

∶ Hom119862(1199011 119862)⨰ Hom119861(119861 1199010) rArr Hom119860(119891 119892)

correspond under the bijection just described Since Lemma 1214 proves that is a composite byLemma 1132 so is id

1216 Lemma For any pair of functors 119891∶ 119860 rarr 119861 and 119892∶ 119862 rarr 119863 the square

119860 times 119862 119861 times 119862

119860 times 119863 119861 times 119863

119891times119862

119860times119892 119861times119892

119891times119863

is exact

1217 Lemma The product of a comma square 120601∶ 1198911199010 rArr 1198921199011 with any infin-category 119870 defines an exactsquare

Hom119860(119891 119892) times 119870

119862 times 119870 119861 times 119870

119860 times 119870

1199011times119870 1199010times119870

120601times119870lArr

119892times119870 119891times119870

Finally the plethora of examples of exact squares just established can be composed to yield furthercomma squares

1218 Lemma (composites of exact squares) Given a diagram of squares in the homotopy 2-category

119867 119866

119865 119863 119864

119862 119861

119860

119903119904120574lArr

119901119902

120573lArr

119888 ℎ119896120572lArr

119887

119892 119891

if 120572∶ 119891ℎ rArr ℎ119896 120573∶ 119887119901 rArr ℎ119902 and 120574∶ 119896119903 rArr 119888119904 are all exact squares then so are the composite rectangles120572119902 sdot 119891120573 and 119892120574 sdot 120572119903 Consequently arbitrary ldquodouble categoricalrdquo composites of exact squares define exactsquares

Proof The two cases are co-duals so it suffices to prove that the rectangle120572119902sdot119891120573 ∶ 119892(119896119902) rArr (119891119887)119901is exact The corresponding cell 1114025120572119902 sdot 119891120573 displayed below-left factors as below-right

119862 119863 119866 119864

119862 119864

Hom119862(119896119862)

dArr1114025120572119902sdot119891120573

Hom119863(119902119863) Hom119864(119864119901)

Hom119860(119891119887119892)

119862 119863 119866 119864

119862 119863 119864

119862 119864

Hom119862(119896119862) Hom119863(119902119863)

Hom119864(119864119901)

Hom119862(119896119862)dArr

Hom119861(119887ℎ)

Hom119860(119891119887119892)

through a cell defined by the universal property of the composite Hom119861(119861 ℎ) otimes Hom119861(119887 119861) ≃Hom119861(119887 ℎ) of Proposition 1147 by the pasting equality

119862 119863 119861 119864 119862 119863 119861 119864

119862 119863 119864 ≔ 119862 119861 119864

119862 119864 119862 119864

Hom119862(119896119862) Hom119861(119861ℎ)

dArr∘

Hom119861(119887119861) Hom119862(119896119862)

Hom119861(119861ℎ) Hom119861(119887119861)

Hom119862(119896119862)dArr

Hom119861(119887ℎ) Hom119860(119891119892)dArr∘

Hom119861(119887119861)

Hom119860(119891119887119892) Hom119860(119891119887119892)

By exactness of 120572 and both cells named ∘ are composites so by Lemma 1132 is a composite aswell By exactness of 120573 and Lemma 1132 again it now follows that 1114025120572119902 sdot 119891120573 is also a composite provingexactness of the rectangle 120572119902 sdot 119891120573 ∶ 119892(119896119902) rArr (119891119887)119901

Exercises

121i Exercise Prove that the identity cells

119860 119860

119860 119861 119861 119860

119861 119861

119896 119896

define exact squares

122 Pointwise right and left extensions

In this section we give four definitions of pointwise extension and prove they are equivalent Webegin with a general categorical notion that on its own is too weak

1221 Definition A right extension of a 1-cell 119891∶ 119860 rarr 119862 along a 1-cell 119896 ∶ 119860 rarr 119861 is given by a pair(119903 ∶ 119861 rarr 119862 120584 ∶ 119903119896 rArr 119891) as below-left

119860 119862 119860 119862 119860 119862

119861 119861 119861119896

119891

119896

119891

= 119896

119891

119903uArr120584

119892uArr120574

119892

119903uArr120584uArrexist

so that any similar pair as above-center factors uniquely through 120584 as above right The co-dual definesa left extension of a 1-cell 119891∶ 119860 rarr 119862 along a 1-cell 119896 ∶ 119860 rarr 119861

The op-dual of Definition 1221 defines a notion of right lifting in any 2-category Analogousnotions of right extension and right lifting can be defined for horizontal morphisms in a virtual doubleequipment where in the presence of restrictions of modules it suffices to consider cells whose verticalfunctors are identities We specialize our language to the virtual equipment of modules as this will bethe one case of interest

1222 Definition A right extension of a module 119860 119865⇸119862 along a module 119860 119870⇸119861 consists of a pair

given by a module 119861 119877⇸119862 together with a binary cell

119860 119861 119862

119860 119862

119870 119877

dArr120584

119865

with the property that every 119899 + 1-ary cell of the form displayed below-left factors uniquely through120584∶ 119870 ⨰ 119877 rArr 119865 as below-right

119860 119861 ⋯ 119862

119860 119862

119870

1198641 119864119899

119865

119860 119861 ⋯ 119862

119860 119861 119862

119860 119862

119870

dArrexist

1198641 119864119899

119870

dArr120584

119877

119865

Dually a right lifting of119860 119865⇸119862 through119861 119867⇸119862 consists of a pair given by a module119860 119871⇸119861 togetherwith a binary cell

119860 119861 119862

119860 119862

119871 119867

dArr120582

119865with the property that every 119899 + 1-ary cell of the form displayed below-left factors uniquely through120582∶ 119871 ⨰ 119867 rArr 119865 as below-right

119860 ⋯ 119861 119862

119860 119862

1198641

119864119899 119867

119865

119860 ⋯ 119861 119862

119860 119861 119862

119860 119862

dArrexist

1198641 119864119899 119867

dArr120582

119871 119867

119865

Because of the asymmetry in Definition 1119 there is no corresponding notion of left extensionor left lifting

1223 Lemma For any functor 119891∶ 119860 rarr 119861 the binary cell

119861 119860 119861

119861 119861

Hom119861(119891119861)

dArr120598

Hom119861(119861119891)

Hom119861

defined in Exercise 114iii defines a right extension of 119861Hom119861⇸ 119861 through 119861

Hom119861(119891119861)⇸ 119860 and a right lifting of

119861Hom119861⇸ 119861 through 119860

Hom119861(119861119891)⇸ 119861

Proof Exercise 122i

We now explain how Definition 1222 relates to Definition 1221 via the covariant and contravari-ant embeddings of Definition 11411

1224 Lemma If

119860 119861 119862

119860 119862

Hom119861(119861119896) Hom119862(119862119903)

dArr120584

Hom119862(119862119891)

defines a right extension in the virtual equipment of modules then 120584∶ 119903119896 rArr 119891 defines a right extension in thehomotopy 2-category Dually if

119860 119861 119862

119860 119862

Hom119860(ℓ119860) Hom119861(ℎ119861)

dArr120582

Hom119860(119892119860)

defines a right lifting in the virtual equipment of modules then 120582∶ 119892 rArr ℓℎ is a left extension in the homotopy2-category

Proof By Corollary 1148 binary cells 120584∶ Hom119861(119861 119896)⨰ Hom119862(119862 119903) rArr Hom119862(119862 119891) corre-

spond to unary cells 120584∶ Hom119862(119862 119903119896) rArr Hom119862(119862 119891) which by Proposition 11410 correspond tonatural transformations 120584∶ 119903119896 rArr 119891 in the homotopy 2-category Via this correspondence the univer-sal property of Definition 1222 clearly subsumes that of Definition 1221 The left extension case issimilar via the contravariant embedding of Definition 11411

1225 Definition (stability of extensions under pasting) A right extension diagram 120584∶ 119903119896 rArr 119891 in a2-category is said to be stable under pasting with a square 120572

119860 119862

119861119896

119891

119903uArr120584

119863

119860 119864

119861

ℎ119892

120572lArr

119896 119887

if the pasted composite

119863 119860 119862

119864 119861

119892

ℎ 120572 119896

119891

119887

119903uArr120584 (1226)

defines a right extension 119903119887 ∶ 119864 rarr 119862 of 119891119892∶ 119863 rarr 119862 along ℎ∶ 119863 rarr 119864 The co-dual of this statementdefines what it means for a left extension diagram to be stable under pasting with a square

1227 Theorem (pointwise right extensions) For a diagram

119860 119862

119861119896

119891

119903uArr120584

in the homotopy 2-category of aninfin-cosmos119974 the following are equivalent

(i) 120584∶ 119903119896 rArr 119891 defines a right extension in 120101119974 that is stable under pasting with exact squares(ii) 120584∶ 119903119896 rArr 119891 defines a right extension in 120101119974 that is stable under pasting with comma squares(iii) 120584∶ Hom119861(119861 119896)

⨰ Hom119862(119862 119903) rArr Hom119862(119862 119891) defines a right extension inℳ119900119889(120101119974)(iv) For any exact square 120572∶ 119887ℎ rArr 119896119892 120584119892sdot119903120572 ∶ Hom119864(119864 ℎ)

⨰ Hom119862(119862 119903119887) rArr Hom119862(119862 119891119892) definesa right extension inℳ119900119889(120101119974)

When these conditions hold we say 119903 ∶ 119861 rarr 119862 defines a pointwise right extension of 119891∶ 119860 rarr 119862 along119896 ∶ 119860 rarr 119861

Proof Lemma 1214 proves (i)rArr(ii)To show (ii)rArr(iii) suppose 120584∶ 119903119896 rArr 119891 defines a right extension in 120101119974 that is stable under pasting

with comma squares and consider a cell inℳ119900119889(120101119974)

119860 119861 ⋯ 119862

119860 119862

Hom119861(119861119896)

1198641 119864119899

Hom119862(119862119891)

Let 119861119902larrminuslarr 119864

119901minusrarrrarr 119862 denote the composite two-sided fibration 1198641

⨰ ⋯ ⨰ 119864119899 By Remark 1138

applied to 119861119902larrminuslarr 119864

119901minusrarrrarr 119862 module maps Hom119861(119861 119896)

⨰ 1198641⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 119891) stand in

bijection with module maps Hom119861(119861 119896)⨰ Hom119861(119902 119861)

⨰ Hom119862(119862 119901) rArr Hom119862(119862 119891) By Corollary1146 and Proposition 1147 such module maps stand in bijection with module maps Hom119861(119902 119896) rArrHom119862(119901 119891) By Lemma 11116 these module maps correspond bijectively to 2-cells

Hom119861(119902 119896)

119860 119864

119862

1199011 1199010120572lArr

119891 119901

in the homotopy 2-category By the hypothesis (ii)

Hom119861(119902 119896) 119860 119862

119864 119861

1199011

1199010 120601 119896

119891

119902

119903uArr120584

defines a right extension in 120101119974 so 120572 factors uniquely through this pasted composite via a map 120574∶ 119901 rArr119903119902 By Proposition 11410 this defines a cell 120574∶ Hom119862(119862 119901) rArr Hom119862(119862 119903119902) which by Corollary1146 gives rise to a canonical cell ∶ Hom119861(119902 119861)

⨰ Hom119862(119862 119901) rArr Hom119862(119862 119903) By Remark 1138

again this produces the desired unique factorization

119860 119861 ⋯ 119862

119860 119862

Hom119861(119861119896)

1198641 119864119899

Hom119862(119862119891)

119860 119861 ⋯ 119862

119860 119861 119862

119860 119862

Hom119861(119861119896)

dArrexist

1198641 119864119899

Hom119861(119861119896)

dArr120584

Hom119862(119862119903)

Hom119862(119862119891)

To show (iii)rArr(iv) consider a diagram (1226) in which120572∶ 119887ℎ rArr 119896119892 is exact and 120584∶ Hom119861(119861 119896)⨰

Hom119862(119862 119903) rArr Hom119862(119862 119891) defines a right extension diagram inℳ119900119889(120101119974) Now by Corollary 1146a cell

Hom119864(119864 ℎ)⨰ 1198641

⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 119891119892)corresponds to a cell

Hom119860(119892 119860)⨰ Hom119864(119864 ℎ)

⨰ 1198641⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 119891)

By exactness of 120572 this corresponds to a cell

Hom119861(119887 119896)⨰ 1198641

⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 119891)or equivalently upon restricting along the composite map Hom119861(119861 119896)

⨰ Hom119861(119887 119861) rArr Hom119861(119887 119896)to a cell

Hom119861(119861 119896)⨰ Hom119861(119887 119861)

⨰ 1198641⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 119891)

Since 119861Hom119862(119862119903)⇸ 119862 is the right extension of119860

Hom119862(119862119891)⇸ 119862 along119860Hom119861(119861119896)⇸ 119861 this corresponds to a cell

Hom119861(119887 119861)⨰ 1198641

⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 119903)which transposes via Corollary 1146 to a cell

1198641⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 119903119887)

which gives the factorization required to prove that119864Hom119862(119862119903119887)⇸ 119862 is the right extension of119863

Hom119862(119862119891119892)⇸ 119862along 119863

Hom119864(119864ℎ)⇸ 119864 A slightly more delicate argument is required to see that this bijection is imple-mented by composing with the map of modules Hom119864(119864 ℎ)

⨰ Hom119862(119862 119903119887) rArr Hom119862(119862 119891119892) corre-sponding to the pasted composite (1226) but for this it suffices by the Yoneda lemma to start withthe identity cell idHom119862(119862119903119887) and trace back up through the bijection just described By Lemma 11116this is straightforward

Finally Lemma 1224 and the trivial example of Exercise 121i prove that (iv)rArr(i)

1228 Corollary The pasted composite (1226) of a pointwise right extension with an exact square is apointwise right extension

Proof Lemma 1218 the pasted composite of two exact squares remains an exact square so byTheorem 1227(i) the pasted composite of a pointwise right extension remains stable under pastingwith exact squares

Exercises

122ii Exercise Using Lemma 1224 as a hint state and prove a dual version of Theorem 1227defining pointwise left extensions in the homotopy 2-category of aninfin-cosmos

123 Formal category theory in a virtual equipment

1231 Proposition For any pair of functors 119906∶ 119860 rarr 119861 the following are equivalent(i) The natural transformation 120598 ∶ 119891119906 rArr id119860 is the counit of an adjunction 119891 ⊣ 119906(ii) There is a pointwise right extension diagram in 120101119974

119860 119860

119861

119906119891

uArr120598 (1232)

that is absolute preserved by any functor ℎ∶ 119860 rarr 119862(iii) The square

119860

119860 119861

119860

120598lArr

119906

119891

is exact

Proof Without the adjective ldquopointwiserdquo the equivalence (i)hArr(ii) is the op-dual of Lemma 236whose proof works in any 2-category and in particular in 120101119974op It remains only to demonstrate thatthe right extension produced by (i)rArr(ii) is pointwise To see this recall that for any 119891∶ 119861 rarr 119860Lemma 1223 provides a right lifting inℳ119900119889(120101119974) as below left

119860 119861 119860

119860 119860

Hom119860(119891119860)

dArr120598

Hom119860(119860119891)

Hom119860

cong119860 119861 119860

119860 119860

Hom119861(119861119906)

dArr120598

Hom119860(119860119891)

Hom119860

If 119891 ⊣ 119906 then by Proposition 411 the counit induces an equivalence Hom119861(119861 119906) ≃ Hom119860(119891 119860) sowe have an isomorphic right lifting diagram above right By Proposition 11410 and Theorem 1227this supplies a natural transformation 120598 ∶ 119891119906 rArr id119860 that defines a pointwise right extension (1232)

Proposition 411 which demonstrates that 120598 ∶ 119891119906 rArr id119860 is a counit if and only if it induces anequivalence Hom119861(119861 119906) ≃ Hom119860(119891 119860) also proves (i)hArr(iii) via Lemma 1135 which tells us thatHom119860 otimes Hom119861(119861 119906) ≃ Hom119861(119861 119906)

1233 Lemma A functor 119896 ∶ 119860 rarr 119861 is fully faithful when any of the following equivalent conditions hold

(i) The square119860

119860 119860

119861119896 119896

is exact(ii) The unary cell id119896 ∶ Hom119860 rArr Hom119861(119896 119896) is a composite(iii) The module map id119896 defines an equivalence of modules Hom119860 ≃ Hom119861(119896 119896) from 119860 to 119860

Proof The equivalence follows from Definition 1211 and Lemma 1133

1234 Proposition (extensions along fully faithful functors) If

119860 119862

119861119896

119891

119903uArr120584

is a pointwise right extension and 119896 is fully faithful then 120584 is an isomorphism

Proof Pasting the pointwise right extension with the exact square of Lemma 1233 yields a point-wise right extension diagram

119860 119862

119860

119891

119903119896uArr120584

By Lemma 1135 and Theorem 1227(iii) 119891∶ 119860 rarr 119862 also defines a pointwise right extension of itselfalong the identity The universal property of these extension diagrams in 120101119974 described in Definition1221 now suffices to construct an inverse isomorphism to 120584

The dual result that the 2-cell in a pointwise left extension along a fully faithful functor is invert-ible is left to Exercise 123i

1235 Proposition A right adjoint 119906∶ 119860 rarr 119861 is fully faithful if and only if the counit 120598 ∶ 119891119906 rArr id119860 isan isomorphism

Proof If 119891 ⊣ 119906 with counit 120598 ∶ 119891119906 rArr id119860 then Proposition 411 reveals that pasting with 120598induces an equivalence of modules Hom119861(119861 119906) ≃ Hom119860(119891 119860) Equivalently by Proposition 1231the counit 120598 ∶ 119891119906 rArr id119860 is exact If 119906 is fully faithful then by Lemma 1218 the composite rectangleis also exact

119860

119860 119860

119860 119861

119860

120598lArr

119906119906

119891

which by Lemma 1133 is to say that 120598 induces an equivalence ofmodules Hom119860(119860119860) ≃ Hom119860(119891119906119860)under the contravariant embedding of Definition 11411 By Proposition 11410 which says this em-bedding is fully faithful it follows that 120598 ∶ 119891119906 rArr id119860 is an isomorphism

Conversely assume 119891 ⊣ 119906 with invertible counit 120598 ∶ 119891119906 cong id119860 By Proposition 1147 we have acomposite Hom119861(119861 119906) otimes Hom119861(119906 119861) ≃ Hom119861(119906 119906) Substituting the equivalence Hom119860(119891 119860) ≃Hom119861(119861 119906) gives another composite Hom119860(119891 119860) otimes Hom119861(119906 119861) ≃ Hom119861(119906 119906) but now Corollary1148 gives a third composite Hom119860(119891 119860) otimes Hom119861(119906 119861) ≃ Hom119860(119891119906119860) Factoring one compos-ite through the other we obtain an equivalence Hom119860(119891119906119860) ≃ Hom119861(119906 119906) which composes withthe equivalence Hom119860 ≃ Hom119860(119891119906119860) induced by the invertible counit cell to define a compositeHom119860 ≃ Hom119860(119891119906119860) ≃ Hom119861(119906 119906) proving by Lemma 1233 that 119906∶ 119860 rarr 119861 is fully faith-ful

Exercises

123i Exercise Prove that if

119860 119862

119861119896

119891

ℓdArr120582

is a pointwise left extension and 119896 is fully faithful then 120582 is an isomorphism

124 Limits and colimits in cartesian closedinfin-cosmoi

In this section we work in aninfin-cosmos119974 that is cartesian closed satisfying the extra assumption ofDefinition 1218 In a cartesian closedinfin-cosmos a functor 119891∶ 119860 rarr 119862 transposes to define an element119891∶ 1 rarr 119862119860 of the infin-category of functors from 119860 to 119862 and a family of diagrams 119891∶ 119860 times 119863 rarr 119862 ofshape119860 parametrized by aninfin-category119863 transposes to a generalized element 119891∶ 119863 rarr 119860119862 The aimis to reinterpret the limits and colimits of sect23 as pointwise right and left extensions

1241 Proposition Suppose the diagram below-left is a pointwise right extension diagram in a cartesianclosedinfin-cosmos

119860 times119863 119862 119862119861

119861 times 119863 119863 119862119860119896times119863

119891

dArr120584119862119896119903

uArr120584 119903

119891

Then its transpose above-right defines an absolute right lifting diagram in the homotopy 2-category which more-over is stable under pasting with the square 119862120601 induced from any exact square 120601

Proof The universal property that characterizes the absolute right lifting diagram

119883 119862119861 119883 119862119861

119863 119862119860 119863 119862119860119889

119890

dArr120594 119862119896 = 119889

119890

dArrexist120577

dArr120584119862119896

119891 119891

119903

transposes to

119860 times 119883 119860 times 119863 119862 119860 times 119883 119860 times 119863 119862

119861 times 119883 119861 times 119883 119861 times 119863119896times119883

119860times119889 119891

= 119896times119883

119860times119889

119896times119863

119891

119890

uArr120594

119890

119861times119889

119903uArr120584

uArrexist120577

Similarly the pasted composite of the absolute right lifting diagram with an exponentiated exactsquare transposes below-left to the diagram below-right

119862119861 119862119864

119863 119862119860 119862119865dArr120584

119862119896

119862ℎ

dArr119862120601 119862119902119903

119891 119862119901

119865 times119863 119860 times119863 119862

119864 times 119863 119861 times 119863

119901times119863

119902times119863 dArr120601times119863 119896times119863

119891

ℎtimes119863

119903uArr120584

Lemma 1216 and Lemma 1217 prove that the square defining the product 119896times119889 and the square120601times119863are exact

Now if 120584 is a pointwise right extension then by Lemma 1216 and Corollary 1228 so is 120584 sdot (119860times119889)The transposed universal property of this right extension diagram proves that 120584 defines an absoluteright lifting Since by Corollary 1228 the pasted composite of 120584 with the exact square 120601 times 119863 givesanother pointwise right extension the same argument shows that the pasted composite of 120584 with 119864120601is again an absolute right lifting diagram

1242 Proposition In a cartesian closedinfin-cosmos any limit as encoded by the absolute right lifting dia-gram below-right transposes to define a pointwise right extension diagram as below-left

119860 119862 119862

1 1 119862119860

119891

dArr120584Δ

ℓuArr120584 ℓ

119891

Conversely any pointwise right extension diagram of this form transposes to define a limit in 119862

Proof The converse is a special case of Proposition 1241 so it remains only to demonstratethat absolute right lifting diagrams of the form displayed above-right transpose to pointwise rightextension diagrams as above-left Comma squares over ∶ 119860 rarr 1 have the form

119860 times 119883

119860 119883

120587120587

Wecan show that the pasted composite of 120584∶ ℓ rArr 119891with this square defines a right extension diagramin 120101119974 by proving that the transposed diagram

119862 119862119883

1 119862119860 119862119860times119883dArr120584

Δ 119862120587ℓ

119891 119862120587

defines a right lifting diagram In fact by transposing across the 2-adjunction (minus) times119883 ⊣ (minus)119883 on 120101119974of Proposition 145 this is easily seen to define an absolute right lifting diagram since 120584∶ Δℓ rArr 119891 isan absolute right lifting diagram

1243 Corollary For any pointwise right extension in a cartesian closedinfin-cosmos

119860 119862

119861119896

119891

119903uArr120584

and any element 119887 ∶ 1 rarr 119861 the element 119903119887 ∶ 1 rarr 119862 is the limit of the diagram

Hom119861(119887 119896) 119860 1198621199011 119891

Proof By Theorem 1227(ii) the composite

Hom119861(119887 119896) 119860 119862

1 119861

1199011

120601 119896

119891

119887

119903uArr120584

is a pointwise right extension By Proposition 1242 this can be interpreted as saying that 119903119887 ∶ 1 rarr 119861defines the limit of the restriction of the diagram 119891∶ 119860 rarr 119862 along 1199011 ∶ Hom119861(119887 119896) ↠ 119860

Recall from Definition 246 that a functor 119896 ∶ 119868 rarr 119869 is final if and only if for any infin-category 119860the square

119860 119860

119860119869 119860119868Δ Δ

119860119896

preserves and reflects all absolute left lifting diagrams Dually a functor 119896 ∶ 119868 rarr 119869 is initial if this squarepreserves and reflects all absolute right lifting diagrams We can now give a more concise formulationof these notions

1244 Definition A functor 119896 ∶ 119868 rarr 119869 is final if and only if the square below-left is exact and initialif and only if the square below-right is exact

119868 119868

1 119869 119869 1

119896 119896

Note that the functor ∶ 119869 rarr 1 is represented on the right and on the left by the modules 119869119869⇸1 and

1119869⇸119869 So we see that 119896 ∶ 119868 rarr 119869 is final if and only if 1199010 ∶ Hom119869(119869 119896) ⥲rarr 119869 is a trivial fibration and119896 ∶ 119868 rarr 119869 is initial if and only if 1199011 ∶ Hom119869(119896 119869) ⥲rarr 119869 is a trivial fibration

To reconcile Definition 1244 with Definition 246 we must prove

1245 Proposition In a cartesian closedinfin-cosmos if 119896 ∶ 119868 rarr 119869 is initial and 119891∶ 119869 rarr 119862 is any diagramthen a limit of 119891 also defines a limit of 119891119896∶ 119868 rarr 119862 and conversely if the limit of 119891119896∶ 119868 rarr 119862 exists then italso defines a limit of 119891∶ 119869 rarr 119862

Proof By Proposition 1242 a limit of 119891 defines a pointwise right extension as below left whichby Corollary 1228 and Definition 1244 gives rise to another pointwise right extension below-right

119869 119862 119868 119869 119862

119891

119896

119891

uArr120584

By Proposition 1242 again this tells us that ℓ is the limit of 119891119896For the converse suppose we are given a pointwise right extension diagram

119868 119862

119891119896

uArr120583

By Theorem 1227(iii) this universal property tells us that for any composable sequence of modules1198641⨰ ⋯ ⨰ 119864119899 from 1 to 119862 composing with 120583∶ Hom(1 )

⨰ Hom119862(119862 ℓ) rArr Hom119862(119862 119891119896) defines abijection between cells 1198641

⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 ℓ) and the cells below-left

119868 1 ⋯ 119862 119869 119868 1 ⋯ 119862

119868 119862 119862

Hom1(1)

1198641 119864119899 Hom119869(119896119869) Hom1(1)

1198641 119864119899

Hom119862(119862119891119896) Hom119862(119862119891)

By Corollary 1146 composing with 120583∶ Hom(1 )⨰ Hom119862(119862 ℓ) rArr Hom119862(119862 119891119896) also defines a

bijection between cells 1198641⨰ ⋯ ⨰ 119864119899 rArr Hom119862(119862 ℓ) and the cells above-right

To say that 119896 ∶ 119868 rarr 119869 is initialmeans byDefinition 1244 that themap Hom119869(119896 119869)⨰ Hom1(1 ) rArr

Hom1(1 ) of modules from 119869 to 1 induced by the identity is a composite Thus composing with

120583∶ Hom(1 )⨰ Hom119862(119862 ℓ) rArr Hom119862(119862 119891119896) also defines a bijection between cells 1198641

⨰ ⋯ ⨰

119864119899 rArr Hom119862(119862 ℓ) and the cells

119869 1 ⋯ 119862

119869 119862

Hom1(1)

1198641 119864119899

Hom119862(119862119891)

But this says exactly that the cell 120584∶ Hom1(1 )⨰ Hom119862(119862 ℓ) rArr Hom119862(119862 119891) that corresponds

to 120583 under this series of bijections displays 1Hom119862(119862ℓ)⇸ 119862 as the right extension of 119869

Hom119862(119862119891)⇸ 119862 along

119869Hom1(1)⇸ 1 unpacking this definition we see that 120584119896 = 120583 Thus 120584∶ ℓ rArr 119891 is a pointwise right

extension and by Proposition 1242 we conclude that ℓ ∶ 1 rarr 119862 also defines a limit of 119891∶ 119869 rarr 119862 asclaimed

1246 Definition In a cartesian closedinfin-cosmos119974 aninfin-category 119864 admits functorial pointwiseright extensions along 119896 ∶ 119860 rarr 119861 if there is a pointwise right extension

119860 times 119864119860 119864

119861 times 119864119860119896times119864119860

ran119896uArr120598

of the evaluation functor along 119896 times 119864119860

By Proposition 1241 if 119864 admits functorial pointwise right extensions along 119896 ∶ 119860 rarr 119861 there isan absolute right lifting diagram

119864119861

119864119860 119864119860dArr120598

resran119896

that is stable under pasting with 119864120601 for any exact square 120601

1247 Proposition (Beck-Chevalley condition) For any exact square

119863

119862 119861

119860

119902 119901

120601lArr

119892 119891

in a cartesian closedinfin-cosmos and anyinfin-category 119864 whenever 119864 admits functorial pointwise left and rightextensions the Beck-Chevalley condition is satisfied for the induced 2-cell 120601lowast below-left the mates of 120601lowast below

center and below right are both isomorphisms

119864119860 119864119860 119864119860

119864119862 119864119861 119864119862 119864119861 119864119862 119864119861

119864119863 119864119863 119864119863

120601lowastlArr

119891lowast

dArr120601

119892lowast

uArr120601

ran119892

lan119891

ran119901 lan119902

Proof By Proposition 1241 the pointwise right extensions defining ran119892 and ran119901 transpose todefine absolute right lifting diagrams

119864119860 119864119861 119864119861

119864119862 119864119862 119864119863 119864119862 119864119863 119864119863dArr120598

119892lowast

119891lowast

dArr120601lowast 119901lowastdArr120598

119901lowastran119892

ran119901

Moreover the mate 120601 of 120601lowast defines a factorization of the left-hand diagram through the right-handdiagram

119864119860 119864119861 119864119860 119864119861

119864119862 119864119862 119864119863 119864119862 119864119863 119864119863dArr120598

119892lowast

119891lowast

dArr120601lowast 119901lowast = dArr120601

119891lowast

dArr120598119901lowast

ran119892

ran119892 ran119901

Immediately from the universal property of the absolute right liftings of 119902lowast through 119901lowast we have that120601 is an isomorphism The proof for 120601 is similar using the absolute left lifting diagrams arising fromthe pointwise left extensions defining lan119891 and lan119902

Part IV

Change of model and model independence

CHAPTER 13

Cosmological biequivalences

In this chapter we study a certain class of cosmological functors between infin-cosmoi that do notmerely preserve infin-categorical structure but also reflect and create it We refer to these functors asbiequivalences because the 2-functors they induce between homotopy 2-categories are biequivalencessurjective on objects up to equivalence and defining a local equivalence of hom-categories Infor-mally we refer to cosmological biequivalences as ldquochange-of-model functorsrdquo For example the fourinfin-cosmoi introduced in Example 1219 whose objects model (infin 1)-categories are connected by thefollowing biequivalences briefly described in Example 137

119966119982119982 119982119890119892119886119897

119980119966119886119905

1-119966119900119898119901

discretization

prismcylinder

119880

(1301)

In sect131 we collect together a number of results about cosmological functors that are scatteredthroughout the text In sect132 we introduce the special class of biequivalences and discuss severalexamples Then in sect133 we establish the basic 2-categorical properties of biequivalences which willprovide the essential ingredient in the proof of the model-independence results We preview this lineof reasoning in sect134 before developing a more systematic approach to model-invariance in Chapter14

Recall fromDefinition 131 that a cosmological functor is a simplicial functor119865∶ 119974 rarr ℒ betweeninfin-cosmoi that preservesbull the specified classes of isofibrations andbull all of the simplicial limits enumerated in 121(i)

By Lemma 132 it follows that cosmological functors also preserves the equivalences and the trivialfibrations By Corollary 733 cosmological functors preserve all flexible weighted limits since Propo-sition 731 reveals that these can be constructed out of simplicial limits of diagrams of isofibrations ofthe form listed in 121(i)

Our aim in this section is to show that cosmological functors preserve all of the infin-categoricalstructures we have introduced In many cases this will not be evident from the original 2-categoricaldefinitions (eg of cartesian fibrations in Definition 516) but can be deduced rather easier from the ac-companying ldquointernal characterizationrdquo of each categorical notion (such as given inTheorem 5111(iii))

First we orient ourselves to the main examples Recall Proposition 133

1311 Example (cosmological functors)(i) The nerve embedding defines a cosmological functor 119966119886119905 119980119966119886119905(ii) For any object 119860 in aninfin-cosmos119974

Fun(119860 minus) ∶ 119974 rarr 119980119966119886119905defines a cosmological functor

(iii) In particular eachinfin-cosmos has an underlying quasi-category functor

(minus)0 ≔ Fun(1 minus) ∶ 119974 rarr 119980119966119886119905defined bymapping out of the terminalinfin-category We use the notation ldquo(minus)0rdquo for this functorbecause the underlying quasi-categories of a complete Segal space or Segal category displayedin (1301) are defined by evaluating these simplicial objects at 0 and because this notation foran ldquounderlying categoryrdquo is commonly used in enriched category theory (see Definition A15)

(iv) For anyinfin-cosmos119974 and any simplicial set 119880 the simplicial cotensor defines a cosmologicalfunctor

(minus)119880 ∶ 119974 rarr 119974(v) For any object 119860 in a cartesian closed infin-cosmos 119974 exponentiation defines a cosmological

functor(minus)119860 ∶ 119974 rarr 119974

(vi) For any map 119891∶ 119860 rarr 119861 in aninfin-cosmos119974 pullback defines a cosmological functor

119891lowast ∶ 119974119861 rarr119974119860In the special case of the map ∶ 119860 rarr 1 the pullback cosmological functor has the form

minus times 119860∶ 119974 rarr 119974119860(vii) For any cosmological functor 119865∶ 119974 rarr ℒ and any 119860 isin 119974 the induced map on slices

119865∶ 119974119860 rarr ℒ119865119860

1312 Example The inclusion of a replete subinfin-cosmos described in Proposition 746 is a cosmo-logical functor Examples include

(i) The full subcategory of discrete objects

119967119894119904119888(119974) 119974defines a replete subinfin-cosmos see Proposition 1222

(ii) The inclusions119974⊤ 119974 119974perp

of theinfin-cosmoi ofinfin-categories possessing and functors preserving a terminal or initial ele-ment More generally the inclusions

119974⊤119869 119974 119974perp119869

of theinfin-cosmoi ofinfin-categories possessing and functors preserving limits or colimits of shape119869 see Propositions 747 and 7414

(iii) The inclusionsℛ119886119903119894(119974)119861 119974119861 ℒ119886119903119894(119974)119861

orℛ119886119903119894(119974) 119974120794 ℒ119886119903119894(119974)

ofinfin-cosmoi whose objects are isofibrations admitting a right or left adjoint right inverse seeCorollary 7410 and Proposition 7412

(iv) The inclusions119966119886119903119905(119974)119861 119974119861 119888119900119966119886119903119905(119974)119861

or119966119886119903119905(119974) 119974120794 119888119900119966119886119903119905(119974)

of theinfin-cosmoi of cartesian or cartesian fibrations and cartesian functors between them seeProposition 7415

1313 Observation Inclusions119974prime 119974 andℒprime ℒ of replete subinfin-cosmoi create isofibrationsand the simplicial limits of axiom 121(i) As these inclusions are full on positive-dimensional arrowsa simplicial functor

119974 ℒ

119974prime ℒprime

119865

restricts to a simplicial functor on the subcategories if and only if it carries the objects and 0-arrowsof the subcategory 119974prime into the objects and 0-arrows of ℒprime as will be the case whenever the repletesubinfin-cosmoi are ldquonaturally definedrdquo by the sameinfin-categorical property Whenever this is the casethe restricted functor is cosmological

For example by Proposition 5120 and Exercise 51iv pullback 119891lowast ∶ 119974119861 rarr 119974119860 preserves carte-sian fibrations and cartesian functors Thus pullback along any functor 119891∶ 119860 rarr 119861 in119974 restricts todefine a cosmological functor

119974119861 119974119860

119966119886119903119905(119974)119861 119966119886119903119905(119974)119860

119891lowast

1314 Example There are also cosmological functors connecting the infin-cosmoi of ldquoReedy fibrantdiagramsrdquo

(i) By Proposition 741 the domain codomain and identity functors

119974120794 119974dom

are all cosmological(ii) By Exercise 111i the domain and codomain functors

119974⤩ 119974dom

from theinfin-cosmos of two-sided isofibrations are cosmological

Cosmological functors may fail dramatically to be surjective on objects mdash id ∶ 119974 rarr 119974120794 comes tomind Thus a 2-categorical property involving a universal quantifier is not obviously preserved Forinstance by Definition 1221 an object 119864 isin 119974 is discrete just when for all 119883 isin 119974 Fun(119883 119864) is a Kancomplex However by Exercise 12iv which we belatedly solve discrete objects admit an internalcharacterization given in (iv) below

1315 Proposition An object 119864 in an infin-cosmos 119974 is discrete if the following equivalent conditions aresatisfied

(i) 119864 is a discrete object in the homotopy 2-category 120101119974 that is every 2-cell with codomain 119864 is invertible(ii) For each 119883 isin 119974 the hom-category hFun(119883 119864) is a groupoid(iii) For each 119883 isin 119974 the mapping quasi-category Fun(119883 119864) is a Kan complex(iv) The isofibration 119864120128 ↠ 119864120794 induced by the inclusion of simplicial sets 120794 120128 is a trivial fibration

Proof Here (ii) is an unpacking of (i) The equivalence of (ii) and (iii) is a well-known resultof Joyal [24 14] reproduced in Corollary 1115 Condition (iv) is equivalent to the assertion thatFun(119883 119864)120128 ↠ Fun(119883 119864)120794 is a trivial fibration between quasi-categories for all 119883 If this is a trivialfibration then surjectivity on vertices implies that every 1-simplex in Fun(119883 119864) is an isomorphismproving (iii) By Exercise 11iv 120794 120128 is in the class of maps cellularly generated by the outer horninclusions so (iii) implies (iv)

Note that if 119864 is discrete then any equivalent object 119864prime is also discrete Now the result we want iseasy to establish

1316 Corollary Cosmological functors preserve discrete objects

Proof If 119864 isin 119974 is discrete and 119865∶ 119974 rarr ℒ is cosmological then

119865(119864120128 ⥲rarr 119864120794) cong (119865119864)120128 ⥲rarr (119865119864)120794

is a trivial fibration inℒ proving that 119865119864 is discrete by Proposition 1315(iv)

Importantly

1317 Proposition Cosmological functors preserve commainfin-categories if 119865∶ 119974 rarr ℒ is a cosmologicalfunctor and the diagram below-left

119864 119865119864

119862 119861 119865 119865119862 119865119861

119860 119865119860

1198901 1198900120598lArr

1198651198901 1198651198900

119865120598lArr

119892 119891 119865119892 119865119891

(1318)

is a comma cone in119974 then the diagram above-right is a comma cone inℒ

Proof The simplicial pullback (332) that constructs the comma cone is preserved by any cosmo-logical functor By Proposition 3311 any comma cone as above left arises from a fibered equivalence119890 ∶ 119864 ⥲ Hom119860(119891 119892) where 120598 = 120601119890 and any fibered equivalence of this form defines a comma coneSince 119865119890 ∶ 119865119864 ⥲ 119865(Hom119860(119891 119892)) cong Hom119865119860(119865119891 119865119892) we conclude that the right-hand data againdefines a comma cone

Using Proposition 1317 we can quickly establish the following preservation properties of cosmo-logical functors by arguing along similar lines to Corollary 1316 For ease of reference we include onthis list the preservation properties that have been previously established

1319 Proposition Cosmological functors preserve(i) Equivalences betweeninfin-categories(ii) Adjunctions between infin-categories including right adjoint right inverse adjunctions and left adjoint

right inverse adjunctions(iii) Invertible 2-cells and mates(iv) Homotopy coherent adjunctions and monads(v) Trivial fibrations and discrete objects(vi) Flexible weighted limits(vii) Comma spans and comma cones(viii) Absolute right and left lifting diagrams(ix) Limits or colimits of diagrams indexed by a simplicial set 119869 inside aninfin-category(x) Cartesian and cocartesian fibrations and cartesian functors between them(xi) Discrete cartesian fibrations and discrete cocartesian fibrations(xii) Two-sided fibrations and cartesian functors between them and also modules(xiii) Modules and represented modules

Proof Lemma 132 proves that cosmological functors preserve equivalences betweeninfin-categoriessince these can be characterized as 2-categorical equivalences which are preserved by any 2-functorSimilarly as observed in Lemma 213 adjunctions are preserved by any 2-functor as are invertible2-cells and mates In the same way homotopy coherent adjunctions are preserved by any simplicialfunctor So the preservation properties (ii) (iii) and (iv) hold under much weaker hypotheses

The preservation of trivial fibrations was also established in Lemma 132 and Corollary 1316uses this to prove that discrete objects are also preserved proving (v) Corollary 733 proves thatcosmological functors preserve all flexible weighted limits as stated in (vi) Proposition 1317 provesthat cosmological functors preserve comma spans in theinfin-cosmos and comma cones in the homotopy2-category as stated in (vii)

The preservation property (viii) follows from Theorem 343 which characterizes absolute liftingdiagrams as fibered equivalences of comma infin-categories Proposition 1317 which says that cosmo-logical functors preserve commas and the fact that cosmological functors preserve equivalences Now(ix) follows from this by Definition 237 and the fact that cosmological functors preserve simplicialtensors

The preservation properties (x) and (xi) also follow from Proposition 1317 and the fact that cos-mological functors preserve right or left adjoint right inverse adjunctions mates and trivial fibrationsvia the characterizations of Theorem 5111(iii) Theorem 5119(iii) and Proposition 546 More detailsare given in the proof of Corollary 5116 which also observes that cartesian natural transformationsare preserved by cosmological functors

Directly from the internal characterization of Theorem 1014 and the preservation of adjunc-tions and invertible 2-cells cartesian functors preserve two-sided fibrations preservation of cartesianfunctors between them is left to Exercise 131i This establishes (xii) By Proposition 133(vi) a cos-mological functor induces a direct image cosmological functor between slicedinfin-cosmoi which thenpreserves discrete objects by Corollary 1316 Thus modules are also preserved By specializing Propo-sition 1317 to cospans involving identities it becomes clear that left and right representable commas

are preserved Since a module is representable if and only if it is fibered equivalent to one of theserepresentable modules are preserved as well completing the proof of (xiii)

13110 Non-Example The composite of the underlying quasi-category functor with the homotopycategory functor

119974 119980119966119886119905 119966119886119905(minus)0 h

defines a simplicial functor that carries aninfin-category119860 to its homotopy category h119860 first introducedin Definition 1412 This functor does preserve isofibrations but does not preserve simplicial cotensorsand so is not cosmological This is a good thing because if it were Proposition 1319(ix) would implythat limits in aninfin-category would also define limits in its homotopy category which is not generallytrue

Exercises

131i Exercise Prove that any cosmological functor preserves cartesian functors between two-sidedfibrations Conclude that any cosmological functor 119865∶ 119974 rarr ℒ induces a cosmological functor119865∶ 119860ℱ119894119887(119974)119861 rarr 119865119860ℱ119894119887(ℒ)119865119861

132 Biequivalences ofinfin-cosmoi as change-of-model functors

Special cosmological functors called biequivalences preserve and also reflect the category theorydeveloped in this text Recall Definition 135

1321 Definition (cosmological biequivalences) A cosmological functor 119865∶ 119974 rarr ℒ is a biequiva-lence when it is

(i) surjective on objects up to equivalence ie if for all 119862 isin ℒ there exists 119860 isin 119974 so that119865119860 ≃ 119862 and

(ii) a local equivalence of quasi-categories ie if for every pair 119860119861 isin 119974 the map

Fun(119860 119861) Fun(119865119860 119865119861)sim

is an equivalence of quasi-categoriesMore generally two infin-cosmoi are biequivalent if there exists a finite zig-zag of biequivalences con-necting them

1322 Example (biequivalences between infin-cosmoi of (infin 1)-categories) In Appendix E we willshow that the functors (1301) are all biequivalences Except for the functor ∶ 119980119966119886119905 rarr 1-119966119900119898119901each of the functors (1301) arises from a simplicially enriched right Quillen equivalence betweenmodel categories enriched over the Joyal model structure with all objects cofibrant Functors of thisform are easily seen to encode cosmological biequivalences

Recall that anyinfin-cosmos has an underlying quasi-category functor

119974 119980119966119886119905(minus)0≔Fun(1minus)

defined by mapping out of the terminal infin-category We now show that for every infin-cosmos that isbiequivalent to 119980119966119886119905 its underlying quasi-category functor is a biequivalence

1323 Proposition If aninfin-cosmos119974 is biequivalent to119980119966119886119905 then the underlying quasi-category functor(minus)0 ∶ 119974 rarr 119980119966119886119905 is a cosmological biequivalence

Proof We will show that in the presence of cosmological biequivalences

119974 ℒ 119980119966119886119905sim119866 sim119865

the underlying quasi-category functor (minus)0 ∶ 119974 rarr 119980119966119886119905 is a cosmological biequivalence By in-duction perhaps taking one of these functors to be the identity the same conclusion holds for anyinfin-cosmos connected by a finite zig-zag of biequivalences to 119980119966119886119905

As 119865 is a biequivalence for each quasi-category 119876 there exists an infin-category 119861 isin ℒ so that119865119861 ≃ 119876 Because 119865 and 119866 are both local equivalences preserving the terminal infin-category 1 (forwhich we adopt the same notation in each of119974ℒ and119980119966119886119905) there is then a zig-zag of equivalencesof quasi-categories

(119866119861)0 = Fun(1 119866119861) Fun(1 119861) Fun(1 119865119861) cong 119865119861 ≃ 119876sim sim

This proves that there exists an infin-category 119866119861 isin 119974 whose underlying quasi-category is equivalentto 119876

To show that the underlying quasi-category functor (minus)0 ∶ 119974 rarr 119980119966119886119905 is a local equivalence con-sider a pair ofinfin-categories 119860119861 isin 119974 By essential surjectivity of 119866 there existinfin-categories 119883119884 isinℒ so that119866119883 ≃ 119860 and 119861 ≃ 119866119884 By pre- and post-composing with these equivalences Corollary 149implies that Fun(119860 119861) rarr Fun(1198600 1198610) is equivalent to Fun(119866119883119866119884) rarr Fun((119866119883)0 (119866119884)0) so itsuffices to prove that the latter map is an equivalence of quasi-categories

By simplicial functoriality of the local equivalences 119865 and119866 there is a commutative diagram whosevertical maps are equivalences

Fun(119866119883119866119884) Fun(Fun(1 119866119883) Fun(1 119866119884))

Fun(Fun(1 119883)Fun(1 119866119884))

Fun(119883 119884) Fun(Fun(1 119883) Fun(1 119884))

Fun(Fun(1 119883) Fun(1 119865119884))

Fun(119865119883 119865119884) Fun(Fun(1 119865119883)Fun(1 119865119884))

simsim

Any quasi-category is isomorphic to its underlying quasi-category so the bottom horizontal map is anisomorphism By the 2-of-3 property it follows that the top horizontal map is an equivalence whichis what we wanted to show

Recall from Proposition 133(vi) that a cosmological functor 119865∶ 119974 rarr ℒ induces a cosmologicalfunctor 119865∶ 119974119861 rarr ℒ119865119861 for any 119861 isin 119974

1324 Proposition If 119865∶ 119974 rarr ℒ is a cosmological biequivalence then for any 119861 isin 119974 the induced functor119865∶ 119974119861 rarr ℒ119865119861 is also a cosmological biequivalence

Proof We first argue that the 119865 defines a local equivalence of sliced mapping quasi-categoriesas defined in Proposition 1217(ii) Given a pair of isofibration 119901∶ 119864 ↠ 119861 and 119901prime ∶ 119864prime ↠ 119861 in119974 the

induced functor on mapping quasi-categories is defined by

Fun119861(119901 119901prime) Fun(119864 119864prime)

1 Fun(119864 119861)

1 Fun(119865119864 119865119861)

sim sim

As the maps between the cospans in 119980119966119886119905 are equivalences so is the induced map between the pull-backs

For surjectivity up to equivalence consider an isofibration 119902 ∶ 119871 ↠ 119865119861 inℒ As 119865 is surjective onobjects up to equivalence there exists some 119860 isin 119974 together with an equivalence 119894 ∶ 119865119860 ⥲ 119871 isin ℒAs 119865 defines a local equivalence of mapping quasi-categories there is moreover a functor 119891∶ 119860 rarr 119861in ℒ so that 119865119891∶ 119865119860 rarr 119865119861 is isomorphic to 119902119894 in 120101ℒ The map 119891 need not be an isofibration butLemma 1212 allows us to factor 119891 as 119860 ⥲ 119870

119901minusrarrrarr 119861 Choosing an equivalence inverse 119895 ∶ 119870 ⥲ 119860 this

data defines a diagram in 120101ℒ that commutes up to isomorphism

119865119870 119865119860 119871

119865119861119865119901

119865119895

119865119891

sim119894

119902cong cong

Proposition 1410 tells us that isofibrations in infin-cosmoi define isofibrations in the homotopy2-category In particular we may lift the displayed isomorphism along the isofibration 119902 ∶ 119871 ↠ 119865119861 todefine a commutative triangle

119865119870 119865119860 119871 119865119870 119871

119865119861 119865119861119865119901

119865119895

119865119891

sim119894

119902 119865119901

119894sdot119865119895

congsim119890 119902

cong cong=

As noted in the proof of Theorem 147 since 119890 is isomorphic to an equivalence 119894 sdot119865119895 it must also definean equivalence Thus by Proposition 1217(vii) we have shown that the isofibration 119901∶ 119870 ↠ 119861mapsunder 119865 to an isofibration that is equivalent to our chosen 119902 ∶ 119871 ↠ 119865119861

A similar argument proves that a cosmological biequivalence induces a biequivalence between thecorrespondinginfin-cosmoi of isofibrations of Proposition 741

1325 Proposition If 119865∶ 119974 rarr ℒ is a cosmological biequivalence then the induced functor 119865∶ 119974120794 rarr ℒ120794

is a biequivalence

Proof Exercise 132i

To establish another family of biequivalences of slicedinfin-cosmoi we require a general lemma

1326 Lemma In anyinfin-cosmos the pullback of an equivalence along an isofibration is an equivalence

119875 119864

119860 119861

119902

119892

119901

sim119891

Proof The map 119892∶ 119875 rarr 119864 is an equivalence if and only if 119892lowast ∶ Fun(119883 119875) rarr Fun(119883 119864) isan equivalence of quasi-categories for all 119883 By the simplicial universal property of the pullback 119875this map 119892lowast is the pullback of the equivalence 119891lowast ∶ Fun(119883119860) ⥲ Fun(119883 119861) along the isofibration119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861) Thus it suffices to prove this result for quasi-categories where a numberof proofs are available One invokes a standard bit of model category theory in any model categorythe pullback of any weak equivalence between fibrant objects along a fibration is a weak equivalence(see for instance [41 1542]) An alternate approach left to the reader is to use Lemma 315 and theisomorphism lifting property to directly lift the data of an equivalence from 119891 to 119892

1327 Proposition If 119891∶ 119860 ⥲ 119861 is an equivalence in119974 then the pullback functor 119891lowast ∶ 119974119861 rarr119974119860 is acosmological biequivalence

Proof To see that 119891lowast ∶ 119974119861 rarr119974119860 is essentially surjective consider an object 119903 ∶ 119863 ↠ 119860 factorthe composite map 119891119903 ∶ 119863 rarr 119861 as an equivalence followed by an isofibration and pull the result backalong 119891

119863

119875 119864

119860 119861

119903sim

119902

119901

sim119891

By Lemma 1326 the pullback of 119891 is an equivalence so by the 2-of-3 property 119903 is equivalent to theisofibration 119902 ∶ 119875 ↠ 119860 which is in the image of 119891lowast ∶ 119974119861 rarr119974119860

To show that this simplicial functor is a local equivalence consider a pair of isofibrations 119901∶ 119864 ↠119861 and 119902 ∶ 119865 ↠ 119861 We will show that the quasi-category of functors over 119861 is equivalent to the quasi-category of functors over 119860 between their pullbacks

119891lowast119864 119864

119891lowast119865 119865

119860 119861

119903

simℎ

119901

119904

119896sim

119902

119891

To define the comparison map Fun119861(119864 119865) rarr Fun119860(119891lowast119864 119891lowast119865) consider the following commuta-tive prism

Fun119861(119864 119865) Fun(119864 119865)

Fun119860(119891lowast119864 119891lowast119865) Fun(119891lowast119864 119891lowast119865) Fun(119891lowast119864 119865)

120793 Fun(119864 119861)

120793 Fun(119891lowast119864119860) Fun(119891lowast119864 119861)

sim 119902lowast

simℎlowast

sim119896lowast

119901

simℎlowast

119903

119904lowast

sim119891lowast

119902lowast

The front-right square is a pullback by the simplicial universal property of 119891lowast119865 while the front-leftsquare and back face are the pullbacks that define Fun119860(119891lowast119864 119891lowast119865) and Fun119861(119864 119865) The universalproperty of the composite front pullback rectangle induces the map Fun119861(119864 119865) rarr Fun119860(119891lowast119864 119891lowast119865)As this functor is the pullback of the equivalences ℎlowast of Corollary 149 the induced map defines anequivalence of quasi-categories

Exercises

132i Exercise Prove Proposition 1325

132ii Exercise Give a second proof of Lemma 1326 that directly constructs an inverse equivalenceto the pullback of an equivalence in anyinfin-cosmos

132iii Exercise If 119865∶ 119974 ⥲ ℒ is a biequivalence and 119860 isin 119974 and 119861 isin ℒ are so that 119865119860 ≃ 119861 provethat the sliceinfin-cosmoi119974119860 andℒ119861 are biequivalent

133 Properties of change-of-model functors

We refer to biequivalence between infin-cosmoi as change-of-model functors In this section we enu-merate their basic properties First we observe that cosmological biequivalences descend to biequiv-alences between homotopy 2-categories hence the name

1331 Proposition A cosmological biequivalence 119865∶ 119974 rarr ℒ induces a biequivalence 119865∶ 120101119974 rarr 120101ℒ ofhomotopy 2-categories ie the 2-functor 119865 is

(i) surjective on objects up to equivalence and(ii) defines a local equivalence of categories hFun(119860 119861) ⥲ hFun(119865119860 119865119861) for all 119860119861 isin 119974

Proof This is immediate from the definition Theorem 147 observed thatinfin-cosmos-level equiv-alences coincide with 2-categorical equivalences and by Lemma 1215 the homotopy category functorh ∶ 119980119966119886119905 rarr 119966119886119905 carries equivalences of quasi-categories to equivalences of categories

Any biequivalence between 2-categories induces a variety of local and global bijections as enu-merated below which can be put to use to solve Exercise 133i

1332 Corollary Consider any cosmological biequivalence 119865∶ 119974 rarr ℒ(i) The biequivalence 119865 preserves reflects and creates equivalences betweeninfin-categories and induces a

bijection between equivalence classes of objects

(ii) The biequivalence 119865 induces local bijections between isomorphism classes of functors extending thebijection of (i) choosing any pairs of objects 119860119861 isin 119974 and 119860prime 119861prime isin ℒ and equivalences 119865119860 ≃ 119860primeand 119865119861 ≃ 119861prime the map

hFun(119860 119861) ⥲ hFun(119865119860 119865119861) ⥲ hFun(119860prime 119861prime) (1333)

defines a bijection between isomorphism classes of functors 119860 rarr 119861 in 119974 and isomorphism classes offunctors 119860prime rarr 119861prime inℒ

(iii) The biequivalence 119865 induces local bijections between 2-cells with specified boundary extending the bijec-tions of (i) and (ii) choosing any pairs of objects 119860119861 isin 119974 and 119860prime 119861prime isin ℒ equivalences 119865119860 ≃ 119860primeand 119865119861 ≃ 119861prime functors 119891 119892 ∶ 119860 119861 and 119891prime 119892prime ∶ 119860prime 119861prime and natural isomorphisms

119865119860 119865119861 119865119860 119865119861

119865119891sim cong120572

119865119892

sim cong120573

the map (1333) induces a bijection between 2-cells 119891 rArr 119892 in119974 and 2-cells 119891prime rArr 119892prime inℒ

Proof For (i) any 2-functor preserves equivalences so our first take is to show that equivalencesare also reflected If 119891∶ 119860 rarr 119861 is a morphism so that 119865119891∶ 119865119860 ⥲ 119865119861 is an equivalence then by the2-of-3 property and the commutative diagram

Fun(119883119860) Fun(119883 119861)

Fun(119865119883 119865119860) Fun(119865119883 119865119861)

119891lowast

sim sim

sim119865119891lowast

119891lowast is an equivalence for all 119883 proving that 119891∶ 119860 rarr 119861 is an equivalenceTo prove that equivalences are also created mdash meaning that if 119865119860 ≃ 119865119861 then 119860 ≃ 119861mdash note that

the equivalence Fun(119860 119861) ⥲ Fun(119865119860 119865119861) implies that any morphism 119865119860 rarr 119865119861 has a lift 119860 rarr 119861up to isomorphism In this way the morphisms of an equivalence 119891∶ 119865119860 ⥲ 119865119861 and 119892∶ 119865119861 ⥲ 119865119860can be lifted to morphisms 119891 ∶ 119860 rarr 119861 and ∶ 119861 rarr 119860 so that 119891 and id119860 have isomorphic imagesunder the equivalence Fun(119860119860) ⥲ Fun(119865119860 119865119860) Since an equivalence of quasi-categories inducesa bijection on isomorphism classes of vertices we conclude that 119891 cong id119860 isin Fun(119860119860) Similarly 119891and id119861 have isomorphic images under Fun(119861 119861) ⥲ Fun(119865119861 119865119861) and thus 119891 cong id119861 proving that119860 ≃ 119861

Finally the preservation reflection and creation of equivalences together with essential surjec-tivity of a cosmological biequivalence implies that such functors induces a bijection on equivalenceclasses of objects

For (ii) by Corollary 149 the chosen equivalences 119865119860 ≃ 119860prime and 119865119861 ≃ 119861prime induce an equivalenceof quasi-categories

Fun(119860 119861) Fun(119865119860 119865119861) Fun(119860prime 119861prime)sim sim

which descends to the equivalence of homotopy categories (1333) Since equivalences of quasi-categoriesinduce bijections between isomorphism classes of vertices this yields in particular a bijection betweenisomorphism classes of functors

For (iii) the equivalence (1333) is full and faithful inducing a bijection between cells 119891 rArr 119892and 119865119891 rArr 119865119892 This bijection can be transported along any chosen isomorphisms 120572 and 120573 to yield abijection between natural transformations 119891 rArr 119892 in hFun(119860 119861) in 119974 and natural transformations119891prime rArr 119892prime in hFun(119860prime 119861prime) inℒ

As an application ofCorollary 1332 we now show that the internal hom119861119860 betweeninfin-categories119860 and 119861 in a cartesian closed infin-cosmos 119974 that is biequivalent to 119980119966119886119905 is equivalent to the simpli-cial cotensor 1198611198600 of 119861 with the underlying quasi-category of 119860 Even if 119974 is not cartesian closedas an infin-cosmos on account of the biequivalence (minus)0 ∶ 119974 rarr 119980119966119886119905 of Proposition 1323 its homo-topy 2-category 120101119974 is cartesian closed in the bicategorical sense replacing the natural isomorphismsof Proposition 145(ii) with natural equivalences we define 119861119860 isin 119974 to be any infin-category whoseunderlying quasi-category is 11986111986000 By the 2-of-3 property for equivalences

Fun(119883 119861119860) Fun(1198830 11986111986000 )

Fun(119883 times 119860 119861) Fun(1198830 times 1198600 1198610)

simsim cong

we see that Fun(119883 119861119860) ≃ Fun(119883 times 119860 119861) for any 119883 For this reason the statement of Proposition1334 does not require that119974 is cartesian closed as aninfin-cosmos the exponentials can be inferred toexist a posteriori

1334 Proposition Suppose 119974 is aninfin-cosmos that is biequivalent to 119980119966119886119905 Then for anyinfin-categories119860119861 isin 119974 the exponential 119861119860 ≃ 1198611198600 is equivalent to the cotensor of 119861 with the underlying quasi-category of119860

Proof By Corollary 1332(i) cosmological biequivalences reflect equivalences of infin-categoriesThus to prove 119861119860 ≃ 1198611198600 it suffices by Proposition 1323 and Corollary 1332 to prove that 119861119860 and1198611198600 have equivalent underlying quasi-categories The defining universal properties of the exponentialand cotensor provide equivalences

Fun(1 119861119860) ≃ Fun(119860 119861) Fun(1198600 1198610) cong Fun(1 1198611198600)sim

which compose with the local equivalence of the underlying quasi-category functor to provide thedesired equivalence

We now prove that biequivalences reflect and create as well as preserve theinfin-categorical struc-tures considered in Section 131

1335 Proposition A biequivalence 119865∶ 119974 rarr ℒ betweeninfin-cosmoi(i) Preserves and reflects invertibility of 2-cells(ii) Preserves reflects and creates adjunctions betweeninfin-categories including right adjoint right inverse

adjunctions and left adjoint right inverse adjunctions(iii) Preserves and reflects discreteness(iv) Preserves reflects and creates fibered equivalences(v) Preserves and reflects comma and arrowinfin-categories a cell as on the left of (1318) is a comma cone

in119974 if and only if its image is a comma cone inℒ(vi) Preserves and reflects cartesian and cocartesian fibrations and cartesian functors between them

(vii) Preserves and reflects discrete cartesian fibrations and discrete cocartesian fibrations(viii) Preserves and reflects two-sided fibrations and cartesian functors between them(ix) Preserves and reflects modules and represented modules and induces a bijection on equivalence classes

of modules between a fixed pair ofinfin-categories(x) Preserves reflects and creates absolute right and left lifting diagrams over a given cospan(xi) Preserves and reflects limits or colimits of diagrams indexed by a simplicial set 119869 inside aninfin-category

and creates the property of aninfin-category in119974 admitting a limit or colimit of a given diagram

Proof Properties (i) and (ii) hold for any biequivalence between 2-categories such as 119865∶ 120101119974 rarr120101ℒ and are left to Exercise 133i as a useful exercise to familiarize oneself with the 2-categorical notionof biequivalence

Property (iii) follows from (i) and Proposition 1315(i) if 119865119864 is discrete then the image under 119865of any 2-cell in119974 with codomain 119864 is invertible which implies that that 2-cell is invertible in 119864

Property (iv) is a special case of Corollary 1332(i) applied to the induced biequivalence of Proposi-tion 1324 Since both119974 andℒ admit commainfin-categories and Proposition 3311 shows that commaspans are characterized by a fibered equivalence class of two-sided isofibrations (v) follows from (iv)

Proposition (vi) follows from (ii) and (i) and (vii) follows from Corollary 1332(ii) which showsthat invertibility of the morphism considered in Proposition 546 is reflected Property (viii) followssimilarly from Theorem 1014(iii) and (ii) and (i) Preservation and reflection of modules now followsfrom (iii) and the bijection between equivalence classes follows from (iv) we defer full details untilProposition 1411 The final statement of (ix) on representability combines (iv) with (v) as will beelaborated upon in Proposition 1412

The reflection properties of (x) and (xi) follow from Theorem 343 and the creation propertiesfollow from Theorem 3411 and (ix) The precise steps are spelled out further in Theorem 1341

1336 Corollary If 119865∶ 119974 rarr ℒ is a cosmological biequivalence then the following induced cosmologicalfunctors are all biequivalences

(i) 119865∶ 119967119894119904119888(119974) rarr 119967119894119904119888(ℒ)(ii) 119865∶ 119974⊤119869 rarr ℒ⊤119869 and 119865∶ 119974perp119869 rarr ℒperp119869(iii) 119865∶ ℛ 119886119903119894(119974)119861 rarrℛ119886119903119894(ℒ)119865119861 and 119865∶ ℒ119886119903119894(119974)119861 rarr ℒ119886119903119894(ℒ)119865119861(iv) 119865∶ ℛ 119886119903119894(119974) rarr ℛ119886119903119894(ℒ) and 119865∶ ℒ119886119903119894(119974) rarr ℒ119886119903119894(ℒ)(v) 119865∶ 119966119886119903119905(119974)119861 rarr 119966119886119903119905(ℒ)119865119861 and 119865∶ 119888119900119966119886119903119905(119974)119861 rarr 119888119900119966119886119903119905(ℒ)119865119861(vi) 119865∶ 119966119886119903119905(119974) rarr 119966119886119903119905(ℒ) and 119865∶ 119888119900119966119886119903119905(119974) rarr 119888119900119966119886119903119905(ℒ)(vii) 119865∶ 119860ℱ119894119887(119974)119861 rarr 119865119860ℱ119894119887(ℒ)119865119861(viii) 119865∶ 119860ℳ119900119889(119974)119861 rarr 119865119860ℳ119900119889(ℒ)119865119861

Proof In each case we start with a cosmological biequivalence mdash perhaps 119974119861 rarr ℒ119865119861 or119974120794 rarr ℒ120794 mdash and must show that the restricted cosmological functor of Observation 1313 is againa biequivalence between the replete sub infin-cosmoi Each of the arguments is similar we know that119865∶ 119974 rarr ℒ induces a bijection between equivalence classes of objects Since the sub infin-cosmoi arereplete the equivalence classes of objects of the replete sub infin-cosmos are given by a subset of theequivalence classes of objects of119974 Since moreover Proposition 1335 proves that the property thatcharacterizes the objects of the subinfin-cosmos is preserved reflected and created by any biequivalenceit is clear that the bijection between equivalence classes of objects in 119974 and ℒ restricts to define abijection between equivalence classes of objects in the subinfin-cosmoi

Now suppose 119860119861 isin 119974 are objects of its replete subinfin-cosmos We must show the local equiva-lence

Fun(119860 119861) Fun(119865119860 119865119861)sim

restricts to a local equivalence of the functor spaces of the sub infin-cosmoi Since we know that thebiequivalence defines a bijection between isomorphism classes of 0-arrows in the sub infin-cosmoi theproof is completed by the following lemma

1337 Lemma Let 119891∶ 119870 ⥲ 119871 be an equivalence of quasi-categories Let 119870prime sub 119870 and 119871prime sub 119871 be repletesub quasi-categories mdash that is full and replete up to isomorphism on some collection of vertices mdash and supposefurther that 119891 restricts to define a simplicial functor

119870 119871

119891

that is bijective on isomorphism classes of vertices Then 119891∶ 119870prime ⥲ 119871prime is an equivalence of quasi-categories

Proof Choose any inverse equivalence 119892∶ 119871 ⥲ 119870 Since 119870prime sub 119870 is replete and full on somesubset of vertices and 119891∶ 119870 rarr 119871 is bijective on isomorphism classes of vertices 119892 restricts to definea functor 119892∶ 119871prime rarr 119870prime for each 119910 isin 119871prime 119891119892(119910prime) cong 119910prime isin 119871 so 119892(119910prime) must lie in 119870prime By repleteness andfullness again the simplicial natural isomorphisms 119870 rarr 119870120128 and 119871 rarr 119871120128 that witness 119892119891 cong id119870 and119891119892 cong id119871 restrict to 119871prime and 119870prime to define the desired equivalence

Exercise

133i Exercise Consider a 2-functor 119865∶ 119966 rarr 119967 that defines a biequivalence in the sense of Propo-sition 1331 Prove that

(i) A 2-cell 119860 119861119891

119892dArr120572 in 119966 is invertible if and only if 119865120572 is invertible in119967

(ii) 119906∶ 119860 rarr 119861 admits a left adjoint in 119966 if and only if 119865119906∶ 119865119860 rarr 119865119861 admits a left adjoint in119967in which case 119865 preserve the adjunction

133ii Exercise Prove that cosmological biequivalences between cartesian closedinfin-cosmoi preserveexponential objects up to equivalence

134 Ad hoc model invariance

An ad hoc approach to proving themodel-independence of the basic category theory ofinfin-categoriesis given by elaborations of Proposition 1335 as described for instance in the following theorem

1341 Theorem (model independence of basic category theory I) The following notions are preservedreflected and created by any cosmological biequivalence

(i) A left or right adjoint to a functor 119906∶ 119860 rarr 119861(ii) A limit or a colimit for a 119869-indexed diagram 119889∶ 1 rarr 119860119869 in aninfin-category 119860

Proof For (i) if 119906∶ 119860 rarr 119861 admits a left adjoint 119891∶ 119861 rarr 119860 then this adjunction is preserved byany 2-functor and hence by any cosmological functor 119865∶ 119974 rarr ℒ If there merely exists 119891∶ 119861 rarr 119860in 119974 so that 119865119891 ⊣ 119865119906 in ℒ then in the case where 119865 is a biequivalence Corollary 1332(iii) can be

used to lift the unit and counit from ℒ to define 2-cells 120578∶ id119861 rArr 119906119891 and 120598 ∶ 119891119906 rArr id119860 in 119974 Apriori the composites 120598119891 sdot 119891120578 and 119906120598 sdot 120578119906 need not be identities but they will be invertible and thestandard 2-categorical argument appearing in the proof of Proposition 2111 can be used to modifyone of these 2-cells to produce a genuine adjunction 119891 ⊣ 119906 A similar line of reasoning can be usedin the absence of a candidate left adjoint using Corollary 1332(ii) to lift the left adjoint from ℒ to afunctor 119891∶ 119861 rarr 119860 in119974whose image is isomorphic to the left adjoint to 119865119906 The rest of the argumentproceeds as before

For (ii) given an absolute lifting diagram in 120101ℒ as displayed on the right

119865119860 119860

1 119865119860119869 1 119860119869dArr

ΔdArr120582

119865119889

119889

Corollary 1332(ii) and (iii) yield a 1-cell ℓ ∶ 1 rarr 119860 with 119865ℓ cong ℓ and a 2-cell 120582∶ Δℓ rArr 119889 as displayedon the right so that 119865120582 composes with the isomorphism 119865ℓ cong ℓ to Our task is to show that thatthis data defines an absolute right lifting diagram in 119974 which amounts to showing for all objects 119883and for all maps 119886 ∶ 119883 rarr 119860 that the composition operation with 120582 induces a bijection

hFun(119883119860)(119886 ℓ) hFun(119883119860119869)(Δ119886 119889)

hFun(119865119883 119865119860)(119865119886 119865ℓ) hFun(119865119883 119865119860119869)(Δ119865119886 119865119889)

hFun(119865119883 119865119860)(119865119886 ℓ) hFun(119865119883 119865119860119869)(Δ119865119886 119865119889)

120582∘minus

cong cong

119865120582∘minus

cong cong

∘minus

On account of the isomorphisms of Corollary 1332(ii) and (iii) this follows from the correspondingencoding of the universal property of inℒ

This argument also shows that the universal property of absolute lifting diagrams is reflected Tosee that absolute lifting diagrams are also preserved Corollary 1332(i) must be invoked to lift anycone inℒ with summit 119884 over the 2-cell 119865120582 to an equivalent one in119974 with summit 119883 chosen so that119865119883 ≃ 119884 As the local bijections Corollary 1332(ii) and (iii) can also be defined relative to specifiedequivalences of objects the same style of argument goes through

In the next chapter we approach this model-independence question more systematically by prov-ing that a cosmological biequivalence induces a biequivalence of virtual equipments

CHAPTER 14

Proof of model independence

Recall from Chapter 11 that the virtual double category of modules ℳ119900119889(120101119974) in an infin-cosmos119974 consists ofbull a category of objects and vertical arrows here theinfin-categories andinfin-functors

bull for any pair of objects 119860119861 a class of horizontal arrows 119860 119864⇸119861 here the modules (119902 119901) ∶ 119864 ↠119860 times 119861 from 119860 to 119861

1198600 1198601 ⋯ 119860119899

1198610 119861119899

119891

1198641

1198642 119864119899

119892

119865

including those whose horizontal source has length zero in the case 1198600 = 119860119899 Here a cell withthe displayed boundary is an isomorphism class of objects in the functor space

Fun119891119892(1198641 times1198601⋯ times

119860119899minus1119864119899 119865) Fun(1198641 times1198601

119864119899 119865)

120793 Fun(1198641 times1198601⋯ times

119860119899minus1119864119899 1198610 times 119861119899)

ie a fibered isomorphism class of maps of spans over 119891 times 119892bull a composite cell for any configuration

1198600 1198601 ⋯ 119860119899

1198610 1198611 ⋯ 119861119899

1198620 119862119899

1198910

11986411hellip11986411198961

11986421hellip11986421198962

1198911 dArr

1198641198991hellip119864119899119896119899

⋯ dArr 119891119899

119892

1198651 1198652

119865119899

119866

119860 119861

119864

dArrid119864

119864

so that composition of cells is strictly associative and unital in the usual multi-categorical senseThe virtual double category of modules is a virtual equipment in the sense introduced by Cruttwell

and Shulman [15 sect7] which means that it satisfies the two further properties(i) For anymodule and pair of functors as displayed on the left there exists amodule and cartesian

cell as displayed on the right

119860 119861 119860 119861

119886 119887 119886

119864(119887119886)

dArr120588 119887

119864 119864

characterized by the universal property that any cell as displayed below-left factors uniquelythrough 120588 as below-right

1198830 1198831 ⋯ 119883119899

119860 119861

119886119891

1198641

1198642 119864119899

119887119892

119864

1198830 1198831 ⋯ 119883119899

119860 119861

119891

1198641

existdArr

1198642 119864119899

119892

119886

119864(119887119886)

dArr120588 119887

119864

(ii) Every object 119860 admits a unit module equipped with a nullary cocartesian cell

119860 119860

119860 119860dArr120580

Hom119860

satisfying the universal property that any cell in the virtual double category of modules whosehorizontal source includes the object 119860 as displayed on the left

119883 ⋯ 119860 ⋯ 119884

119861 119862

119891

1198641 119864119899 1198651

119865119898

119892

119866

119883 ⋯ 119860 119860 ⋯ 119884

119861 119862

1198641

dArrid1198641 dArrid119864119899⋯

119864119899

dArr120580

1198651

dArrid1198651

119865119898

⋯ dArrid119865119898

119891

1198641 119864119899 Hom119860

dArrexist

1198651 119865119898

119892

119866

factors uniquely through 120580 as displayed on the right

The module in (i) is defined by pulling back a module 119860 119864⇸119861 along functors 119886 ∶ 119860prime rarr 119860 and119887 ∶ 119861prime rarr 119861 The simplicial pullback defining 119864(119887 119886) induces an equivalence of functor spaces

Fun119886119891119887119892(1198641 times⋯times 119864119899 119864) ≃ Fun119891119892(1198641 times⋯times 119864119899 119864(119887 119886))which gives rise to the universal property See Proposition 1121

The unit module is the arrow infin-category given the notation 119860Hom119860⇸ 119860 when considered as a

module from 119860 to 119860 The universal property of (ii) follows from the Yoneda lemma see Proposition1124

The virtual equipment ofmodules in119974 has a lot of pleasant properties which follow formally fromthe axiomatization of a virtual equipment For instance certain sequences of composable modules canbe said to have composites witnessed by cocartesian cells as in (ii) (see 1135 11311 1147 and 1148 and

the non-formal composite given in 1137) Also for any functor 119891∶ 119860 rarr 119861 the modules119860Hom119861(119861119891)⇸ 119861

and119861Hom119861(119891119861)⇸ 119860 behave like adjoints is a sense suitable to a virtual double category more precisely the

module119860Hom119861(119861119891)⇸ 119861 defines a companion and the module 119861

Hom119861(119891119861)⇸ 119860 defines a conjoint to 119891∶ 119860 rarr 119861(see 1141 1144 and 1146) Another consequence of Theorem 1126 is the following for any parallelpair of functors there are natural bijections between 2-cells in the homotopy 2-category

119860 119861119891

119892dArr

119860 119861 119860 119860 119861 119860

119860 119861 119861 119861 119861 119860dArr

Hom119861(119861119891)

dArr119892

Hom119860

119891

Hom119861(119892119861)

Hom119861(119861119892) Hom119861 Hom119861(119891119861)

See Proposition 11410As remarked upon in Definition 11411 as a consequence of these results there are two locally-

fully-faithful homomorphisms 120101119974 ℳ119900119889(120101119974) and 120101119974coop ℳ119900119889(120101119974) mdash referred to as thecovariant and contravariant embeddings respectively mdash embedding the homotopy 2-category into thesubstructuresup1 ofℳ119900119889(120101119974) comprised only of unary cells whose vertical boundaries are identities Themodules in the image of the covariant embedding are the right representables and the modules in theimage of the contravariant embedding are the left representables

The theme of Chapter 12 could be summarized by saying that the virtual equipment of modules inaninfin-cosmos is a robust setting to develop the category theory ofinfin-categories On the one hand itcontains the homotopy 2-category of theinfin-cosmos which was the setting for the results of Part I Itis also a very natural home to study infin-categorical properties that are somewhat awkward to expressin the homotopy 2-category For instance the weak 2-universal property of comma infin-categoriesis now encoded by a bijection in Lemma 11116 cells in the virtual equipment whose codomain is acommamodule correspond bijectively to natural transformations of a particular form in the homotopy2-category Fibered equivalences of modules as used to express the universal properties of adjunctionslimits and colimits in Chapter 4 are now vertical isomorphisms in the virtual equipment betweenparallel modules The virtual equipment also cleanly encodes the universal property of pointwise leftand right Kan extensions

In this chapter we show that a cosmological biequivalence 119865∶ 119974 ⥲ ℒ induces a biequivalence ofvirtual equipments 119865∶ ℳ119900119889(120101119974) ⥲ ℳ119900119889(120101ℒ) in a suitable sense that we introduce to describe what

sup1This substructure is very nearly a bicategory with horizontal composites of unary cells constructed as in Definition11312 except that compatible sequences of modules do not always admit a horizontal composite

is true in our setting We then explain the interpretation of this result that the category theory ofinfin-categories is preserved reflected and created by any ldquochange-of-modelrdquo functor of this form

141 A biequivalence of virtual equipments

We first elaborate on Proposition 1331(ix)

1411 Proposition A cosmological biequivalence 119865∶ 119974 ⥲ ℒ preserves reflects and creates modules(i) An isofibration 119864 ↠ 119860 times 119861 is a module in119974 if and only if 119865119864 ↠ 119865119860 times 119865119861 is a module inℒ(ii) For every module 119866 ↠ 119860prime times 119861prime in ℒ and every pair of infin-categories 119860119861 in 119974 with specified

equivalences 119865119860 ≃ 119860prime and 119865119861 ≃ 119861prime there is a module 119864 ↠ 119860times 119861 in119974 so that 119865119864 is equivalent to119866 over the pair of equivalences

Proof For (i) consider an isofibration 119864 ↠ 119860 times 119861 in 119974119860times119861 and the induced biequivalence119865∶ 119974119860times119861 ⥲ ℒ119865119860times119865119861 of Proposition 1324 By Proposition 1335(iii) 119864 defines a discrete object in119974119860times119861 if and only if 119865119864 defines a discrete object inℒ119865119860times119865119861 By Theorem 1341(i) 119864 rarr Hom119861(119861 119901)admits a right adjoint and 119864 rarr Hom119860(119902 119860) admits a left adjoint in 119974119860times119861 if and only if 119865119864 rarr119865Hom119861(119861 119901) cong Hom119865119861(119865119861 119865119901) admits a right adjoint and 119865119864 rarr 119865Hom119860(119902 119860) cong Hom119865119860(119865119902 119865119860)admits a left adjoint inℒ119865119860times119865119861 By Lemma 1042 these properties characterize modules

For (ii) fix a pair of equivalences 119865119860 ≃ 119860prime and 119865119861 ≃ 119861prime defining an equivalence 119890 ∶ 119860prime times 119861prime ⥲119865119860 times 119865119861 and consider the composite biequivalence

119974119860times119861 ℒ119865119860times119865119861 ℒ119860primetimes119861primesim119865 sim119890lowast

given by Propositions 1324 and 1327 Consider a module 119866 ↠ 119860prime times 119861prime By essential surjectivitythere is an isofibration 119864 ↠ 119860 times 119861 whose image under this cosmological functormdashthe pullback of119865119864 ↠ 119865119860 times 119865119861 along 119890 ∶ 119860prime times 119861prime ⥲ 119865119860 times 119865119861mdashdefines an isofibration (119902 119901) ∶ 119864prime ↠ 119860prime times 119861prime that isequivalent to 119866 inℒ119860primetimes119861prime It remains only to argue that 119864 defines a module from 119860 to 119861 which willfollow essentially as in the proof of (i) from the fact that 119864prime ≃ 119866 defines a module from 119860prime to 119861prime

As the image 119864prime of 119864 is equivalent to a discrete object Proposition 1335(iii) tells us 119864 is discretein 119974119860times119861 The final step is to argue that the desired right adjoint to 119864 rarr Hom119861(119861 119901) is present inthe image of the biequivalence119974119860times119861 ⥲ ℒ119860primetimes119861prime and apply Theorem 1341(i) to deduce its presencein 119974119860times119861 a similar argument of course applies to the functor 119864 rarr Hom119860(119902 119860) To see this notethat 119865∶ 119974119860times119861 ⥲ ℒ119865119860times119865119861 carries Hom119861(119861 119901) ↠ 119860 times 119861 to Hom119865119861(119865119861 119865119901) ↠ 119865119860 times 119865119861 By (i) itsuffices to argue that this functor has a right adjoint over 119865119860times119865119861 Applying Theorem 1341(i) to thebiequivalence 119890lowast ∶ ℒ119865119860times119865119861 ⥲ ℒ119860primetimes119861prime this follows from the the fact that 119864prime rarr Hom119861prime(119861prime 119901prime) has aright adjoint over 119860prime times 119861prime

1412 Proposition Let 119865∶ 119974 ⥲ ℒ be a cosmological biequivalence Then a module 119860 119864⇸119861 in119974 is right

representable if and only if the module 119865119860 119865119864⇸119865119861 is right representable in ℒ in which case 119865 carries therepresenting functor 119891∶ 119860 rarr 119861 in119974 to a representing functor 119865119891∶ 119865119860 rarr 119865119861 inℒ

Proof To say that119860 119864⇸119861 is right representable in119974 is to say that there exists a functor119891∶ 119860 rarr 119861together with an equivalence 119864 ≃119860times119861 Hom119861(119861 119891) of modules over 119861 If this is the case then anycosmological functor 119865∶ 119974 rarr ℒ carries this to a fibered equivalence 119865119864 ≃119865119860times119865119861 Hom119865119861(119865119861 119865119891)and hence the module 119865119860 119865119864⇸119865119861 is right-represented by 119865119891∶ 119865119860 rarr 119865119861 inℒ

Conversely if 119865119860 119865119864⇸119865119861 is right-represented by some functor 119892∶ 119865119860 rarr 119865119861 then by Corollary1332(ii) there exists a functor 119891∶ 119860 rarr 119861 in119974 so that 119865119891 cong 119892 inℒ By Proposition 11410 naturallyisomorphic functors represent equivalent modules that is Hom119865119861(119865119861 119892) ≃119865119860times119865119861 Hom119865119861(119865119861 119865119891)Thus 119865119864 ≃119865119860times119865119861 Hom119865119861(119865119861 119865119891) By Corollary 1332(i) this fibered equivalence lifts along thecosmological functor 119865∶ 119974119860times119861 rarr ℒ119865119860times119865119861 to a fibered equivalence 119864 ≃119860times119861 Hom119861(119861 119891) whichproves that 119864 is right represented by 119891∶ 119860 rarr 119861 in119974

Because cosmological functors preserve modules simplicial pullbacks and the arrow constructionwe see that a functor 119865∶ 119974 rarr ℒ induces a functor of virtual equipments 119865∶ ℳ119900119889(120101119974) rarrℳ119900119889(120101ℒ)preserving all of the structures described in Theorem 1126

1413 Theorem (model independence ofinfin-category theory) If 119865∶ 119974 rarr ℒ is a cosmological biequiv-alence then the induced functor of virtual equipments

119865∶ ℳ119900119889(120101119974) rarrℳ119900119889(120101ℒ)defines a biequivalence of virtual equipments ie it is

(i) bijective on equivalence classes of objects(ii) locally bijective on isomorphism classes of parallel vertical functors extending the bijection of (i)(iii) locally bijective on equivalence classes of parallel modules extending the bijection of (ii)(iv) locally bijective on cells extending the bijections of (i) (ii) and (iii)

Consequently anyinfin-categorical notion that can be encoded as an equivalence-invariant proposition in the vir-tual equipment of modules is model invariant preserved reflected and created by cosmological biequivalences

Note further that if two infin-cosmoi are connected by a finite zig-zag of biequivalences then thebijections described in Theorem 1413 compose

Proof Properties (i) and (ii) are restatements of Corollary 1332(i) and (ii)The local bijection (iii) follows immediately from Proposition 1411 and the fact that for any pair

of equivalences 119890 ∶ 119860prime times 119861prime ⥲ 119865119860 times 119865119861 the composite biequivalence

preserves reflects and creates equivalences between objects (Corollary 1332(i)) Finally (iv) is anapplication of Corollary 1332(ii) to this cosmological biequivalence

For instance the presence of an adjunctions betweeninfin-categories and the existence of limits andcolimits inside an infin-category can both be encoded as an equivalence-invariant proposition in thevirtual equipment of modules Using Theorem 1413 we can reprove Theorem 1341

Module-theoretic proof of Theorem 1341 By Proposition 411 119906∶ 119860 rarr 119861 admits a left

adjoint if and only if themodule119860Hom119861(119861119906)⇸ 119861 is contravariantly represented Any representing functor

inℒ lifts up to isomorphism to define a functor 119891∶ 119861 rarr 119860 in119974 and now by Theorem 1413(iii) thereis an equivalence Hom119860(119891 119860) ≃ Hom119861(119861 119906) of modules from 119860 to 119861 in 119974 if and only if there is anequivalence Hom119865119860(119865119891 119865119860) ≃ Hom119865119861(119865119861 119891119906) of modules from 119865119860 to 119865119861 inℒ

Similarly a diagram 119889∶ 1 rarr 119860119869 admits a limit if and only if the module 1Hom119860119869(Δ119889)⇸ 119860 is covariantly

represented and Proposition 1412 or the argument just given completes the proof in this case aswell

A biequivalence of virtual equipments preserves reflects and creates composites of modules

1414 Lemma Let 119865∶ 119974 ⥲ ℒ be a cosmological biequivalence(i) Then a composable sequence of modules in119974 admits a composite inℳ119900119889(120101119974) if and only if the image

of this sequence admits a composite inℳ119900119889(120101ℒ)(ii) Hence cosmological biequivalences preserve and reflect exact squares

Proof Via Definition 1211 (ii) follows immediately from (i) so it remains only to show thata biequivalence of virtual equipments 119865∶ ℳ119900119889(120101119974) ⥲ ℳ119900119889(120101ℒ) preserves reflects and createscomposites of modules To see that an 119899-ary composite cell 120583∶ 1198641

⨰ ⋯ ⨰ 119864119899 rArr 119864 in 120101119974 is preservednote that by Theorem 1413(iv) any cell in ℳ119900119889(120101ℒ) is isomorphic to a cell in the image of 119865 firstreplace the objects by equivalent ones in the image then replace the vertical functors by naturallyisomorphic ones in the image then replace the modules by equivalent ones in the image over thespecified equivalences between theirinfin-categorical sources and targets and then finally apply the localbijection (iv) to replace the cell inℳ119900119889(120101ℒ) by a unique cell in the image ofℳ119900119889(120101119974) by composingwith this data Now by local full and faithfulness and essential surjectivity the universal propertyof the cocartesian cell 120583∶ 1198641

⨰ ⋯ ⨰ 119864119899 rArr 119864 implies that its image 119865120583∶ 1198651198641⨰ ⋯ ⨰ 119865119864119899 rArr

119865119864 is again a cocartesian cell Thus composites 1198641 otimes ⋯ otimes 119864119899 ≃ 119864 are preserved by cosmologicalbiequivalences

Now if 119865120583∶ 1198651198641⨰ ⋯ ⨰ 119865119864119899 rArr 119865119864 is a composite the fact that 119865∶ ℳ119900119889(120101119974) rarr ℳ119900119889(120101ℒ) is

locally fully faithful implies immediately that 120583∶ 1198641⨰ ⋯ ⨰ 119864119899 rArr 119864 is a composite thus composites

1198651198641 otimes⋯otimes 119865119864119899 ≃ 119865119864 are reflected by cosmological biequivalencesFinally suppose the sequence 1198651198641

⨰ ⋯ ⨰ 119865119864119899 of modules in ℳ119900119889(120101ℒ) admits a composite

1198651198600119866⇸119865119860119899 since the composable sequence of modules is in the image of 119865 the source and target

infin-categories of the composite module 119866 are as well By Theorem 1413(iii) there exists a module

1198600119864⇸119860119899 in 120101119974 so that 119865119864 ≃ 119866 as modules from 1198651198600 to 119865119860119899 The cocartesian cell 1198651198641

⨰ ⋯ ⨰

119865119864119899 rArr 119866 composes with the unary cell of this equivalence to define a cocartesian cell 1198651198641⨰ ⋯ ⨰

119865119864119899 rArr 119865119864 By Theorem 1413(iv) this lifts to an 119899-ary cell 1198641⨰ ⋯ ⨰ 119864119899 rArr 119864 in ℳ119900119889(120101119974) As

wersquove just seen that cocartesianness of cells is reflected by biequivalences 119865∶ ℳ119900119889(120101119974) rarrℳ119900119889(120101ℒ)this completes the proof that composites are created by cosmological biequivalences

Exercises

141i Exercise Prove that cosmological biequivalences between cartesian closedinfin-cosmoi preserveand reflect initial and final functors

Appendix of Abstract Nonsense

APPENDIX A

Basic concepts of enriched category theory

Enriched category theory exists because enriched categories exist in nature To explain considerthe data of a (small) 1-category 119966 given bybull a set of objectsbull for each pair of objects 119909 119910 isin 119966 a set 119966(119909 119910) of arrows in 119966 from 119909 to 119910bull for each 119909 isin 119966 a specified identity element id119909 ∶ 1 rarr 119966(119909 119909) and for each 119909 119910 119911 isin 119966 a specified

composition map ∘ ∶ 119966(119910 119911) times 119966(119909 119910) rarr 119966(119909 119911) satisfying the associativity and unit conditions

119966(119910 119911) times 119966(119908 119910) 119966(119908 119911)idtimes∘

∘timesid

119966(119909 119910) 119966(119909 119910) times 119966(119909 119909)

119966(119910 119910) times 119966(119909 119910) 119966(119909 119910)

idtimes id119909

id119910 times id ∘

∘(A01)

In many mathematical examples of interest the set119966(119909 119910) can be given additional structure in whichcase it would be strange not to take it account when performing further categorical constructions

Perhaps there exists a specified zero arrow 0119909119910 isin 119966(119909 119910) the set of which defines a two-sidedideal for composition 119892 ∘ 0 ∘ 119891 = 0 Or extending this perhaps 119966(119909 119910) admits an abelian groupstructure for which composition is ℤ-bilinear Or in another direction perhaps the set of arrowsfrom 119909 to 119910 in 119966 form the objects of a 1-category In this setting we regard the objects of 119966(119909 119910) asldquo1-dimensionalrdquo morphisms from 119909 to 119910 and the arrows of 119966(119909 119910) as ldquo2-dimensionalrdquo morphisms from119909 to 119910 in 119966 now itrsquos natural natural to require the composition map to define a functor Or perhapsthe set of arrows from 119909 to 119910 in 119966 form the vertices of a simplicial set whose higher simplices nowprovides arrows in each positive dimension in this setting it is natural to ask composition to definea simplicial map In such settings one says that the 1-category 119966 can be enriched over the category119985 in which the objects 119966(119909 119910) and diagrams (A01) livesup1 mdash with119985 equal to the category of pointedsets abelian groups categories or simplicial sets in the examples just described

An alternate point of view of enriched category theory is often emphasized mdash adopted for instancein the classic textbook [31] from which we stole the title of this chapter To borrow a distinction usedby Peter May the term ldquoenrichedrdquo can be used as a compound noun mdash enriched categories mdash or asan adjective mdash enriched categories In the noun form an enriched category 119966 has no pre-existingunderlying ordinary category although we shall see below that the underlying unenriched 1-categorycan always be identified a posteriori When used as an adjective an enriched category 119966 is perhaps

sup1To interpret the diagrams (A01) in 119985 one needs to specify an interpretation for the monoidal product ldquotimesrdquo andits unit object 1 (which is not displayed in the diagram) In the examples we will consider this product is the cartesianproduct and this unit is the terminal object

most naturally an ordinary category whose hom-sets can be given additional structuresup2 While thenoun perspective is arguably more elegant when discussing the general theory of enriched categoriesthe adjective perspective dominates when discussing examples sowe choose to emphasize the adjectiveform and focus on enriching unenriched categories here

Before giving a precise definition of enriched category and the enriched functors between them insectA2 in sectA1 we study that category119985 that defines the base for enrichment in which the hom-objectswill ultimately live Here we are interested primarily in two examples 119985 = 119966119886119905 and 119985 = 119982119982119890119905as well as the unenriched case 119985 = 119982119890119905 each of which has the special property of being cartesianclosed categories Since there are some simplifications in enriching over a cartesian closed categorywe grant ourselves the luxury of working explicitly with this notion see [31] or [47 Chapter 3] for anintroduction to categories enriched over a more general symmetric monoidal category

We continue in sectA3 with an introduction to enriched natural transformations and the enrichedYoneda lemma These notions allow us to correctly state the universal properties that characterizetensors and cotensors in sectA4 and conical limits and colimits in sectA5 both of which are characterized byuniversal properties involving an enriched natural isomorphism We conclude in sectA6 with a generaltheory of change of base mdash the one part of the theory of enriched categories that is not covered inencyclopedic detail in [31] the original reference instead being [20] mdash which allows us to be moreprecise about the procedure by which a 2-category may be regarded as a simplicial category (as usedto great effect in Chapters 8 and 9) or by which a simplicial category may be quotiented to define a2-category (see eg Definition 141) as alluded to in Digression 142

A1 Cartesian closed categories

Throughout this text the base category for enrichment will always be taken to be a cartesian closedcategory

A11 Definition A category119985 is cartesian closed when itbull admits finite products or equivalently a terminal object 1 isin 119985 and binary products andbull for each 119860 isin 119985 the functor 119860 times minus∶ 119985 rarr 119985 admits a right adjoint (minus)119860 ∶ 119985 rarr 119985

A12 Lemma In a cartesian closed category 119985 the product bifunctor is the left adjoint of a two-variableadjunction this being captured by a commutative triangle of natural isomorphisms

119985(119860 times 119861119862)

119985(119860119862119861) 119985(119861 119862119860)congcong

cong(A13)

Proof The family of functors (minus)119860 ∶ 119985 rarr 119985 extend to bifunctors

(minus)minus ∶ 119985op times119985 rarr 119985in a unique way so that the isomorphism defining each adjunction 119860 times minus ⊣ (minus)119860

119985(119860 times 119861119862) cong 119985(119861 119862119860)becomes natural in 119860 (as well as 119861 and 119862) The details are left as Exercise A1i or to [48 436] Thisdefines the natural isomorphism on the right-hand side of (A13) The natural isomorphism on the

sup2To quote [41] ldquoThinking from the two points of view simultaneously it is essential that the constructed ordinary cat-egory be isomorphic to the ordinary category that one started out with Either way there is a conflict of notation betweenthat preferred by category theorists and that in common use by lsquoworking mathematiciansrsquo (to whom [39] is addressed)

left-hand is defined by composing with the symmetry isomorphism119860times119861 cong 119861times119860 The third naturalisomorphism is taken to be the composite of these two

A14 Example (cartesian closed categories)(i) The category of sets is cartesian closed with 119861119860 defined to be the set of functions from 119860 to119861 Transposition across the natural isomorphism (A13) is referred to as ldquocurryingrdquo

(ii) The category 119966119886119905 of small categories is cartesian closed with 119861119860 defined to be the category offunctors and natural transformations from 119860 to 119861sup3 The natural isomorphism

119966119886119905(119860 119862119861) cong 119966119886119905(119860 times 119861119862) cong 119966119886119905(119861 119862119860)identifies natural transformations which are arrows 120794 rarr 119862119860 in the category of functors withldquohomotopiesrdquo 119860 times 120794 rarr 119862

(iii) For any small category 119966 the category 119982119890119905119966op

is cartesian closed By the Yoneda lemma for119865119866 isin 119982119890119905119966

op the value of 119866119865 at 119888 isin 119966 must be defined by

119866119865(119888) cong 119982119890119905119966op(119966(minus 119888) 119866119865) cong 119982119890119905119966

op(119865 times 119966(minus 119888) 119866)

By the proof of Lemma A12 the action of 119866119865 on a morphism 119891∶ 119888 rarr 119888prime isin 119966 is definedby precomposition with the corresponding natural transformation 119891lowast ∶ 119966(minus 119888) rarr 119966(minus 119888prime)This defines the functor 119866119865 Since any functor 119867 isin 119982119890119905119966

opis canonically a colimit of rep-

resentables this definition extends to the required natural isomorphism 119982119890119905119966op(119867119866119865) cong

119982119890119905119966op(119865 times 119867119866)

(iv) In particular taking 119966 = 120491op the category of simplicial sets 119982119982119890119905 ≔ 119982119890119905120491op

is cartesianclosed

The exponential 119861119860 is frequently referred to as an internal hom As this name suggests the internalhom 119861119860 ca be viewed as a lifting of the hom-set119985(119860 119861) along a functor that we now introduce

A15 Definition For any cartesian closed category119985 the underlying set functor is the functor

119985 119982119890119905(minus)0≔119985(1minus)

represented by the terminal object 1 isin 119985

A16 Lemma For any pair of objects 119860119861 isin 119985 in a cartesian closed category the underlying set of theinternal hom 119861119860 is119985(119860 119861) ie

(119861119860)0 cong 119985(119860 119861)

Proof Combining Definition A15 with (A13)

(119861119860)0 ≔ 119985(1 119861119860) cong 119985(1 times 119860 119861) cong 119985(119860 119861)since there is a natural isomorphism 1 times 119860 cong 119860

It makes sense to ask whether an isomorphism of underlying sets can be ldquoenrichedrdquo to lie in 119985that is lifted along the underlying set functor (minus)0 ∶ 119985 rarr 119982119890119905

sup3Size matters here the category of large but locally small categories is not cartesian closed though it is still possibleto define 119861119860 in the case where 119860 is a small category

A17 Lemma The natural isomorphisms (A13) characterizing the defining two-variable adjunction of acartesian closed category lift to119985 for any 119860119861 119862 isin 119985

119862119860times119861

(119862119861)119860 (119862119860)119861congcong

cong(A18)

Proof This follows from the associativity of finite products and the Yoneda lemma To prove(A18) it suffices to show that 119862119860times119861 (119862119861)119860 and (119862119860)119861 represent the same functor We have a se-quence of natural isomorphisms

119985(119883 (119862119861)119860) cong 119985(119883 times 119860119862119861)cong 119985((119883 times 119860) times 119861 119862) cong 119985(119883 times (119860 times 119861) 119862)cong 119985(119883119862119860times119861)cong 119985((119860 times 119861) times 119883119862) cong 119985(119860 times (119861 times 119883) 119862)cong 119985(119861 times 119883119862119860)cong 119985(119883 (119862119860)119861)

and hence119985(119883 (119862119861)119860) cong 119985(119883119862119860times119861) cong 119985(119883 (119862119860)119861)

A19 Remark Note (minus)1 ∶ 119985 rarr 119985 is naturally isomorphic to the identity functor mdash ie 1198611 cong 119861mdash since it is right adjoint to a functor minus times 1∶ 119985 rarr 119985 that is naturally isomorphic to the identity

A bicomplete cartesian closed category is a special case of a complete and cocomplete closed sym-metric monoidal category this being deemed a cosmos by Beacutenabou to signify that such bases are anideal setting for enriched category theory For obvious reasons we wonrsquot use this term here Thereis a competing 2-categorical notion of (fibrational) ldquocosmosrdquo due to Street [57] that is more similarto the notion we consider here which was the direct inspiration for the terminology we introduce inDefinition 121

Exercises

A1i Exercise Prove that in a cartesian closed category 119985 the family of functors (minus)119860 ∶ 119985 rarr 119985extend to bifunctors

119985(119860 times 119861119862) cong 119985(119861 119862119860)becomes natural in 119860 (as well as 119861 and 119862)

A1ii Exercise The data of a closed symmetric monoidal category generalizes Definition A11 byreplacing finite products by an arbitrary bifunctor minusotimesminus∶ 119985times119985 rarr119985 replacing the terminal objectby an object 119868 isin 119985 and requiring the additional specification of natural isomorphisms

119860 otimes (119861 otimes 119862) cong (119860 otimes 119861) otimes 119862 119868 otimes 119860 cong 119860 cong 119860 otimes 119868 119860 otimes 119861 cong 119861 otimes 119860satisfying various coherence axioms [20 27] see also [28] Explain why these isomorphisms do notneed to specified in the special case of a cartesian closed category and why the coherence conditionsare automatic

A2 Enriched categories

We now briefly switch perspectives and explain the meaning of the noun phrase ldquoenriched cate-goryrdquo because discussion what is required to ldquoenrichedrdquo an ordinary 1-category Throughout we fix acartesian closed category (119985times 1) to serve as the base for enrichment

A21 Definition A119985-enriched category 119966 is given bybull a set of objectsbull for each pair of objects 119909 119910 isin 119966 an object of arrows 119966(119909 119910) isin 119985bull for each 119909 isin 119966 a specified identity element encoded by a map id119909 ∶ 1 rarr 119966(119909 119909) isin 119985 and for

each 119909 119910 119911 isin 119966 a specified composition map ∘ ∶ 119966(119910 119911) times 119966(119909 119910) rarr 119966(119909 119911) isin 119985 satisfying theassociativity and unit conditions both commutative diagrams lying in119985⁴

119966(119910 119911) times 119966(119908 119910) 119966(119908 119911)idtimes∘

∘timesid

119966(119909 119910) 119966(119909 119910) times 119966(119909 119909)

119966(119910 119910) times 119966(119909 119910) 119966(119909 119910)

idtimes id119909

Evidently from the diagrams of (A01) a locally-small 1-category defines a category enriched in119982119890119905 The underlying set functor of Definition A15 can be used to define the ldquounderlying categoryrdquo ofan enriched category

A22 Definition If 119966 is a 119985-category its underlying category is the 1-category with the same setof objects and with hom-sets 119966(119909 119910)0 defined by applying the underlying set functor (minus)0 ∶ 119985 rarr 119982119890119905to the hom-objects 119966(119909 119910) isin 119985

Note the identity arrow id119909 ∶ 1 rarr 119966(119909 119909) of the 119985-category is by definition an element of thehom-set 119966(119909 119909)0 ≔ 119985(1 119966(119909 119909)) The composite of two arrows 119891∶ 1 rarr 119966(119909 119910) and 119892∶ 1 rarr 119966(119910 119911)in the underlying category is defined to be the arrow constructed as the composite

1 119966(119910 119911) times 119966(119909 119910) 119966(119909 119911)119892times119891 ∘

In analogy with the discussion around Definition A15 when one speaks of ldquoenrichingrdquo an a prioriunenriched category119966 over119985 the task is to define a119985-enriched category as in Definition A21 whoseunderlying category recovers 119966 When119985 = 119966119886119905 the task is to define a 2-category whose underlying1-category is the one given When119985 = 119982119982119890119905 the task is define simplicial hom-sets of 119899-arrows so thatthe 0-arrows are the ones given When a simplicially enriched category 119966 is encoded as a simplicialobject 119966bull in 119966119886119905 as explained in Digression 122 its underlying category is the category 1199660 furtherjustifying the notion introduced in Definition A22

For example a cartesian closed category 119985 as in Definition A11 can be enriched to define a119985-category

A23 Lemma A cartesian closed category119985 defines a119985-category whosebull objects are the objects of119985bull hom-object in119985 from 119860 to 119861 is given by the internal hom 119861119860 and

⁴These diagrams suppose the associativity and unit natural isomorphisms involving the product bifunctors times and itsunit object 1 In a cartesian closed category these are canonical mdash rather than given by extra data as is the case in the moregeneral closed symmetric monoidal category see Exercise A1ii

bull the identity map id119860 ∶ 1 rarr 119860119860 and composition map ∘ ∶ 119862119861times119861119860 rarr 119862119860 are defined to be the transposesof

1 times 119860 congminusrarr 119860 and 119862119861 times 119861119860 times 119860 idtimes120598minusminusminusminusrarr 119862119861 times 119861 120598minusrarr 119862the latter defined using the counit of the cartesian closure adjunction

Proof The task is to verify the commutative diagrams of (A21) in 119985 and then observe thatLemma A16 reveals that the underlying category of the 119985-category defined by the statement is the1-category119985 We leave the identity conditions to the reader and verify associativity

The definition of the composition map as an adjoint transpose implies that its adjoint transposethe top-right composite below is given by the left-bottom composite

119862119861 times 119861119860 times 119860 119862119860 times 119860

119862119861 times 119861 119862

∘timesid

idtimes120598 120598

120598

The associativity diagram below-left commutes if and only if the transposed diagram appearing as theouter boundary composite below-right commutes

119863119862 times 119862119860 119863119860

idtimes∘

∘timesid

119863119862 times 119862119861 times 119861119860 times 119860 119863119861 times 119861119860 times 119860

119863119862 times 119862119860 times 119860 119863119862 times 119862 119863

∘timesidtimes id

idtimes∘times id

idtimes idtimes120598 idtimes120598

∘timesid

idtimes120598 120598

which follows from bifunctoriality of times and two instances of the commutative square above

A24 Definition The free119985-category on a 1-category 119966 hasbull the same objects as 119966bull the hom-objects defined to be coproducts∐119966(119909119910) 1 of the terminal object 1 indexed by the hom-

set 119966(119909 119910)bull the identity map id119909 ∶ 1 rarr ∐

119966(119909119909) 1 given by the inclusion of the component indexed by theidentity arrow

bull the composition map given by acting by the composition function on the indexing sets

∐119966(119910119911) 1 times∐119966(119909119910) 1 cong ∐119966(119910119911)times119966(119909119910) 1 ∐

119966(119909119911) 1∐∘ 1

For example free119966119886119905-enriched categories are thosewith no non-identity 2-cells and free119982119982119890119905-enrichedcategories are those with no non-degenerate arrows in positive dimensions When context allows weuse the same name for the 1-category 119966 and the free 119985-category it generates using language to dis-ambiguate

A25 Definition A119985-enriched functor 119865∶ 119966 rarr 119967 is given bybull a mapping on objects that carries each 119909 isin 119966 to some 119865119909 isin 119967

bull for each pair of objects 119909 119910 isin 119966 an internal action on the 119985-objects of arrows given by a mor-phism 119865119909119910 ∶ 119966(119909 119910) rarr 119967(119865119909 119865119910) isin 119985 so that the119985-functoriality diagrams commute

119966(119910 119911) times 119966(119909 119910) 119966(119909 119911) 1 119966(119909 119909)

119967(119865119910 119865119911) times 119967(119865119909 119865119910) 119967(119865119909 119865119911) 119967(119865119909 119865119909)

119865119910119911times119865119909119910

119865119909119911

id119909

id119865119909119865119909119909

A prototypical example is given by the representable functors

A26 Example For any 119985-category 119966 and object 119888 isin 119966 the enriched representable 119985-functor119966(119888 minus) ∶ 119966 rarr 119985 is defined on objects by the assignment 119909 isin 119966 ↦ 119966(119888 119909) isin 119985 and whose internalaction on arrows

119966(119909 119910) 119966(119888 119910)119966(119888119909) 119966(119909 119910) times 119966(119888 119909) 119966(119888 119910)119966(119888minus)119909119910 ∘

is given by the adjoint transpose of the internal composition map for119966 The119985-functoriality diagramsare transposes of associativity and identity diagrams in 119966

The contravariant enriched representable functors are defined similarly see Exercise A2iiAs the next result explains an enriched representable functor can be thought of as a ldquotwo-steprdquo en-

richment of the corresponding unenriched representable functor the first step enriches the hom-setsto hom-objects in119985 and the second step enriches the composition function to an internal composi-tion map in119985

A27 Proposition To enrich a 1-category 119966 over119985 one must(i) First lift each of the unenriched representable functors through the underlying set functor to define an

unenriched functor that encodes the data of the objects of arrows in119985

119966 119985

119982119890119905119966(119888minus)

119966(119888minus)

(minus)0

119966 119985

119909 119966(119888 119909)

119966(119888minus)

(ii) Then enriched each of the unenriched119985-valued representable functors defined in (i) to a 119985-functor119966(119888 minus) ∶ 119966 rarr 119985 whose internal action on arrows encodes the data of the internal composition mapfor 119966

Proof The correspondence between the data in (i) and (ii) and the data of a119985-enriched categorywhose underlying category is 119966 is clear As discussed in Definition A22 there is no additional datarequired by the identity arrows in an enriched category Now the 119985-functoriality of the enrichedrepresentable functor 119966(119888 minus) ∶ 119966 rarr 119985 is expressed by commutative diagrams

119966(119910 119911) times 119966(119909 119910) 119966(119909 119911) 1 119966(119909 119909)

119966(119888 119911)119966(119888119910) times 119966(119888 119910)119966(119888119909) 119966(119888 119911)119966(119888119909) 119966(119888 119909)119966(119888119909)119966(119888minus)119910119911times119966(119888minus)119909119910

119966(119888minus)119909119911

id119909

id119966(119888119909)119966(119888minus)119909119909

mdash where the composition map in 119966 is the one that has just been defined mdash and these transpose to therequired associativity and identity axioms for the119985-category 119966

Both of the constructions of underlying unenriched categories and free categories extend functo-rially to functors see Exercise A2iii The relationship between these constructions is summarized bythe following proposition which we also decline to prove because it will be generalized by a resultthat we do provide a proof for in sectA6

A28 Proposition The free119985-category functor defines a fully faithful left adjoint to the underlying cate-gory functor Consequently a119985-category is free just when it is isomorphic to the free category on its underlyingcategory via the counit of this adjunction

Proof Exercise A2iv or see sectA6

Exercises

A2i Exercise Verify the unit condition left to the reader in the proof of Lemma A23

A2ii Exercise Define the opposite of a119985-category and dualize Example A26 to define contravari-ant enriched representable functors

A2iii Exercise(i) Define the underlying functor of an enriched functor(ii) Prove that the passage from enriched functors to underlying unenriched functors is functorial(iii) Define the free enriched functor on an unenriched functor(iv) Prove the passage from unenriched functors to free enriched functors is functorial

A2iv Exercise Prove A28

A3 Enriched natural transformations and the enriched Yoneda lemma

Recall an (unenriched) natural transformation 120572∶ 119865 rArr 119866 between parallel functors 119865119866∶ 119966 119967 is given bybull the data of an arrow 120572119909 isin 119967(119865119909 119866119909) for each 119909 isin 119966bull subject to the condition that for each morphism 119891 isin 119966(119909 119910) the diagram

119865119909 119866119909

119865119910 119866119910

119865119891

120572119909

119866119891

120572119910

commutes in119967This enriches to the notion of a119985-natural transformation whose data is exactly the same mdash a family ofarrows in the underlying category of119967 indexed by the objects of119966mdash but with a stronger119985-naturalitycondition expressed by internalizing the naturality condition (A31) given above

A32 Definition A119985-enriched natural transformation 120572∶ 119865 rArr 119866 between119985-enriched functors119865119866∶ 119966 119967 is given bybull an arrow 120572119909 ∶ 1 rarr 119967(119865119909 119866119909) for each 119909 isin 119966

bull so that for each pair of objects 119909 119910 isin 119966 the follow119985-naturality square commutes in119985

119966(119909 119910) 119967(119865119910 119866119910) times 119967(119865119909 119865119910)

119967(119866119909119866119910) times 119967(119865119909 119866119909) 119967(119865119909 119866119910)

120572119910times119865119909119910

119866119909119910times120572119909 ∘

A34 Example An arrow 119891∶ 1 rarr 119966(119909 119910) in the underlying category of a 119985-category 119966 defines a119985-natural transformation 119891lowast ∶ 119966(119910 minus) rArr 119966(119909 minus) between the enriched representable functors whosecomponent at 119911 isin 119966 is defined by evaluating the adjoint transpose of the composition map at 119891

1 119966(119909 119910) 119966(119910 119911)119966(119909119910)119891 119966(minus119911)119909119910

Evaluating one component of the associative diagram for 119966 at 119891 provides the required 119985-naturalitysquare

A35 Lemma(i) The vertical composite of119985-enriched natural transformations 120572∶ 119865 rArr 119866 and 120573∶ 119866 rArr 119867 both from119966 to119967 has component (120573 sdot 120572)119909 at 119909 isin 119966 defined by the composite

1 119967(119866119909119867119909) times 119967(119865119909 119866119909) 119967(119865119909119867119909)120573119909times120572119909 ∘

(ii) The horizontal composite of 120572∶ 119865 rArr 119866 from 119966 to119967 and 120574∶ 119867 rArr 119870 from119967 to ℰ has component(120574 lowast 120572)119909 at 119909 isin 119966 defined by the composite

1 119967(119865119909 119866119909) ℰ(119870119866119909119867119866119909) times ℰ(119867119865119909119867119866119909)

ℰ(119870119865119909 119870119866119909) times ℰ(119867119865119909 119870119865119909) 119967(119867119865119909 119870119866119909)

120572119909 120574119866119909times119867119865119909119866119909

119870119865119909119866119909times120574119865119909 ∘

which is well-defined by119985-naturality of 120574

Proof Exercise A3i

The data of the underlying natural transformation of a 119985-naturality transformation is given bythe same family of arrows The unenriched naturality condition (A31) is proven by evaluating theenriched naturality condition (A33) at an underlying arrow 119891∶ 1 rarr 119966(119909 119910) In particular the middlefour interchange rule for horizontal and vertical composition of 119985-natural transformations followsfrom the middle four interchange rule for horizontal and vertical composition of unenriched naturaltransformations for the data of the latter determines the data of the former Consequently LemmaA35 implies that

A36 Corollary For any cartesian closed category119985 there is a 2-category119985-119966119886119905 of119985-enriched cat-egories 119985-enriched functors and 119985-enriched natural transformations Moreoever the underlying categoryfunctor119985-119966119886119905 rarr 119966119886119905 and the free119985-category functor 119966119886119905 rarr 119985-119966119886119905 are both 2-functors

We now turn our attention to the119985-enriched Yoneda lemma which we present in several formsOne role of the Yoneda lemma is to give a representable characterization of isomorphic objects in 119966When 119966 is a119985-category this has several forms involving each of the three notions of ldquorepresentablerdquo

functor appearing in Proposition A27 The notion of 119985-natural isomorphism referred to in thefollowing result is defined to be a119985-natural transformation with an inverse for vertical composition

A37 Lemma For objects 119909 119910 in a119985-category 119966 the following are equivalent(i) 119909 and 119910 are isomorphic as objects of the underlying category of 119966(ii) The 119982119890119905-valued unenriched representable functors 119966(119909 minus)0 119966(119910 minus)0 ∶ 119966 119982119890119905 are naturally iso-

morphic(iii) The119985-valued unenriched representable functors119966(119909 minus) 119966(119910 minus) ∶ 119966 119985 are naturally isomorphic(iv) The119985-valued119985-functors 119966(119909 minus) 119966(119910 minus) ∶ 119966 119985 are naturally119985-isomorphic

Proof Applying the underlying category functor (minus)0 ∶ 119985-119966119886119905 rarr 119966119886119905 the fourth statementimplies the third The third statement implies the second by composing the underlying set functor(minus)0 ∶ 119985 rarr 119982119890119905 The second statement implies the first by the unenriched Yoneda lemma this isstill the main point Finally the first statement implies the last by a direct construction if 119891∶ 1 rarr119966(119909 119910) and 119892∶ 1 rarr 119966(119910 119909) define an isomorphism in the underlying category of119966 the correspondingrepresentable119985-natural transformations of Example A34 define a119985-natural isomorphism

Lemma A37 defines a common notion of isomorphism between two objects of an enriched cat-egory which turns out to be no different the usual unenriched notion of isomorphism This can bethought of as defining a ldquocheaprdquo form of the enriched Yoneda lemma The full form of the119985-Yonedalemma enriches the usual statement mdash a natural isomorphism between the set of natural transforma-tions whose domain is a representable functor to the set defined by evaluating the codomain at therepresenting object mdash to an isomorphism in119985 The first step to make this precise is to enrich the setof119985-enriched natural transformations between a parallel pair of119985-functors can to define an objectof119985

A38 Definition Let 119985 be a complete cartesian closed category and consider a parallel pair of119985-functors 119865119866∶ 119966 119967 with 119966 a small119985-category Then the119985-object of119985-natural transforma-tions is defined by the equalizer diagram

119985119966(119865 119866) prod119911isin119966

119967(119865119911 119866119911) prod119909119910isin119966

119967(119865119909119866119910)119966(119909119910)

where one map in the equalizer diagram is defined by projecting to119967(119865119909119866119909) applying the internalaction of 119866 on arrows and then composing while the other is defined by projection to 119967(119865119910119866119910)applying the internal action of 119865 on arrows and then composing

A39 Lemma The underlying set of the 119985-object of 119985-natural transformations 119985119966(119865 119866) is the set of119985-natural transformations from 119865 to 119866

Proof By its defining universal property elements of the underlying set of119985119966(119865 119866) correspondto maps120572∶ 1 rarr prod

119911isin119966119967(119865119911 119866119911) that equalize the parallel pair of maps described in Definition A38The map 120572 defines the components of a119985-natural transformation 120572∶ 119865 rArr 119866 and the commutativitycondition transposes to (A33)

TheYoneda lemma is usually expressed by the slogan ldquoevaluation at the identity is an isomorphismrdquobut since in the enriched context the enriched object of natural transformations is defined via a limitit is easier to define the map that induces a natural transformation instead Given an object 119886 isin 119964

in a small119985-category and a119985-functor 119865∶ 119964 rarr 119985 the internal action of 119865 on arrows transposes todefine a map that equalizes the parallel pair

119865119886 prod119911isin119964

119865119911119964(119886119911) prod119909119910isin119964

119865119910119865119909times119964(119909119910)119865119886minus (A310)

and thus induces a map 119865119886 rarr 119985119964(119964(119886 minus) 119865) in119985

A311 Theorem (enriched Yoneda lemma) For any small119985-category119964 object 119886 isin 119964 and119985-functor119865∶ 119964 rarr 119985 the canonical map defines an isomorphism in119985

119865119886 119985119964(119964(119886 minus) 119865)cong

that is119985-natural in both 119886 and 119865

Proof To prove the isomorphism it suffices to verify that (A310) is a limit cone To that endconsider another cone over the parallel pair

119881 prod119911isin119964

119865119911119964(119886119911) prod119909119910isin119964

119865119910119865119909times119964(119909119910)120582

and define a candidate factorization by evaluating the component 120582119886 at id119886

120582119886(id119886) ≔ 119881 119964(119886 119886) times 119881 119865119886id119886 times id 120582119886

To see that 120582119886(id119886) ∶ 119881 rarr 119865119886 indeed defines a factorization of 120582 through the limit cone it sufficesto show commutativity at each component 119865119911119964(119886 119911) of the product which we verify in transposedform

119964(119886 119911) times 119881

119964(119886 119911) times 119964(119886 119886) times 119881 119964(119886 119911) times 119881

119964(119886 119911) times 119865119886 119865119911

idtimes id119886 times id

idtimes120582119886

∘timesid

120582119911119865119886119911

The upper triangle commutes by the identity law for 119964 while the bottom square commutes because120582 defines a cone over the parallel pair Uniqueness of the factorization 120582119886(id119886) follows from the samediagram by taking 119911 = 119886 and evaluating at id119886

We leave the verification of119985-naturality to the reader or to [31 sect24]

Passing to underlying sets

A312 Corollary For any small119985-category119964 object 119886 isin 119964 and119985-functor 119865∶ 119964 rarr 119985 there is abijection between119985-natural transformations 120572∶ 119964(119886 minus) rArr 119865 and elements 1 rarr 119865119886 in the underlying setof 119865119886 implemented by evaluating the component at 119886 isin 119964 at the identity id119886

Since we have assumed our bases for enrichment to be cartesian closed the 2-category 119985-119966119886119905admits finite products allowing us to define multivariable 119985-functors The following result willimply that the structures characterized by 119985-natural isomorphisms in sectA4 and sectA5 assemble into119985-functors

A313 Proposition Let 119872∶ ℬop times 119964 rarr 119985 be a 119985-functor so that for each 119886 isin 119964 the 119985-functor119872(minus 119886) ∶ ℬop rarr119985 is represented by some 119865119886 isin ℬ meaning there exists a119985-natural isomorphism

ℬ(119887 119865119886) cong 119872(119887 119886)Then there is a unique way of extending the mapping 119886 isin 119964 ↦ 119865119886 isin ℬ to a119985-functor 119865∶ 119964 rarr ℬ so thatthe isomorphisms are119985-natural in 119886 isin 119964 as well as 119887 isin ℬ

Proof By the Yoneda lemma in the form of Corollary A312 to define a family of isomorphisms120572119887119886 ∶ ℬ(119887 119865119886) cong 119872(119887 119886) for each 119886 isin 119964 that are119985-natural in 119887 isin ℬ is to define a family of elements120578119886 ∶ 1 rarr 119872(119865119886 119886) for each 119886 isin 119964 By the 119985-naturality statement in the Yoneda lemma for theformer isomorphism to be 119985-natural in 119886 is equivalent to the family of elements 120578119886 ∶ 1 rarr 119872(119865119886 119886)being ldquoextraordinarilyrdquo119985-natural in 119886 What this means is that for any pair of objects 119886 119886prime isin 119964 theouter square commutes

119964(119886prime 119886) 119872(119865119886prime 119886)119872(119865119886119886)

ℬ(119865119886prime 119865119886)

119872(119865119886prime 119886)119872(119865119886prime119886prime) 119872(119865119886prime 119886)

119872(119865119886primeminus)119886prime119886

119865

119872(119865minus119886)119886119886prime

119872(119865119886prime119886)120578119886120572119865119886prime119886

119872(minus119886)

119872(119865119886prime119886)120578119886prime

The definition of the top-horizontal map119872(119865minus 119886)119886prime119886 depends on the internal action of 119865 on arrowswhich we seek to define but note that the composite of the other factor with the right vertical map isthe natural isomorphism 120572119865119886prime119886 Thus there is a unique way to define 119865119886prime119886 making the extraordinary119985-naturality square commute which is exactly the claim 119985-functoriality of these internal actionmaps for 119865 follows from119985-functoriality of119872 in the119964 variable

We close this section with some applications The correct notions of 119985-enriched equivalence or119985-enriched adjunction are given by interpreting the standard 2-categorical notions of equivalence andadjunction in the 2-category 119985 For sake of contrast we present both notions in an alternate formhere and leave it to the reader to apply Theorem A311 to relate these to the 2-categorical notions

A314 Definition A pair of 119985-categories 119966 and 119967 are 119985-equivalent if there exists a 119985-functor119865∶ 119966 rarr 119967 that isbull 119985-fully faithful ie each 119865119909119910 ∶ 119966(119909 119910) rarr 119967(119865119909 119865119910) is an isomorphism in119985 andbull essentially surjective on objects ie each 119889 isin 119967 is isomorphic to 119865119888 for some 119888 isin 119966

A315 Definition A 119985-enriched adjunction is given by a pair of 119985-functors 119865∶ ℬ rarr 119964 and119880∶ 119964 rarr ℬ together with isomorphisms

119964(119865119887 119886) cong ℬ(119887119880119886)that are119985-natural in both 119886 isin 119964 and 119887 isin ℬ

A316 Remark By Proposition A313 a119985-functor119880∶ 119964 rarr ℬ admits a119985-left adjoint if and onlyif each ℬ(119887119880minus) ∶ 119964 rarr 119985 is represented by some object 119865119887 isin 119964 in which case the data of the119985-natural isomorphism 119964(119865119887 minus) cong ℬ(119887119880minus) equips 119887 isin ℬ ↦ 119865119887 isin 119964 with the structure of a119985-functor Dual remarks construct enriched right adjoints

Exercises

A3i Exercise Prove Lemma A35

A3ii Exercise Use Corollary A312 to show that the notions of119985-equivalence and119985-adjunctiongiven in Definitions A314 and A315 are equivalent to the 2-categorical notions in119985-119966119886119905

A4 Tensors and cotensors

A 119985-category 119966 admits tensors just when for all 119862 isin 119966 the covariant representable functor119966(119862 minus) ∶ 119966 rarr 119985 admits a left119985-adjoint minus otimes 119862∶ 119985 rarr 119966 Dually a119985-category 119966 admits cotensorsjust when the contravariant representable functor 119966(minus 119862) ∶ 119966op rarr 119985 admits a mutual right adjoint119862minus ∶ 119985op rarr 119966 The aim in this section is to introduce both constructions formally In the nextsection we establish some useful formal properties of enriched categories that are either tensored orcotensored

A41 Definition A 119985-category 119966 is cotensored if for all 119881 isin 119985 and 119862 isin 119966 the 119985-functor119966(minus 119862)119881 ∶ 119966op rarr119985 is represented by an object 119862119881 isin 119966 ie there exists an isomorphism

119966(119883119862119881) cong 119966(119883 119862)119881

in119985 that is119985-natural in119883 By Proposition A313 the cotensor product defines a unique119985-functor

119966 times119985op (minus)minusminusminusminusrarr 119966

making the defining isomorphism119985-natural in all three variables

A42 Definition Dually a 119985-category 119966 is cotensored if for all 119881 isin 119985 and 119966 the 119985-functor119966(119862 minus)119881 ∶ 119966 rarr 119985 is represented by an object 119881 otimes 119862 isin 119966 ie there exists an isomorphism

119966(119881 otimes 119862119883) cong 119966(119862119883)119881

in119985 that is119985-natural in 119883 By Proposition A313 the tensor product defines a unique119985-functor

119985times119966 minusotimesminusminusminusminusrarr 119966making the defining isomorphism119985-natural in all three variables

Immediately from these definitions

A43 Lemma A119985-category119966 is tensored and cotensored if and only if the119985-functor119966(minus minus) ∶ 119966optimes119966 rarr119985 is part of a two-variable119985-adjunction

119966(119881 times 119860 119861)

119966(119860 119861119881) 119966(119860 119861)119881congcong

as expressed by the commutative triangle of119985-natural isomorphisms

A44 Lemma In any category 119966 that is enriched and cotensored over119985 there are119985-natural isomorphisms

1198621 cong 119862 and119862119880times119881

(119862119881)119880 (119862119880)119881congcong

for 119880119881 isin 119985 and 119862 isin 119966367

Proof By Lemma A37 to define the displayed 119985-natural isomorphisms it suffices to provethat these objects represent the same 119985-functors 119966op rarr 119985 By the defining universal property ofthe cotensor for any 119883 isin 119966

119966(1198831198621) cong 119966(119883 119862)1 cong 119966(119883119862)by Remark A19 Similarly by the defining universal property of the cotensor we have 119985-naturalisomorphisms in each of the three vertices of the triangle below

119966(119883119862119880times119881) cong 119966(119883 119862)119880times119881

119966(119883 (119862119881)119880) cong (119966(119883 119862)119881)119880 119966(119883 (119862119880)119881) cong (119966(119883 119862)119880)119881congcong

whence the connecting119985-natural isomorphisms are given by Lemma A17

Extending Lemma A23

A45 Lemma A cartesian closed category (119985times 1) is enriched tensored and cotensored over itself withtensors defined by the cartesian product and cotensors defined by the internal hom⁵

119881 otimes119882 ≔ 119881 times119882 and 119882119881 ≔ 119882119881

Proof Lemma A17 establishes the required isomorphisms (A18) in 119985 The proof of their119985-naturality is left to the reader or [31 sect18]

The presence of tensors and cotensors provides a convenient mechanism for testing whether an apriori unenriched adjunction may be enriched to a119985-adjunction

A46 Proposition Let 119880∶ 119964 rarr ℬ and 119865∶ ℬ rarr 119964 be unenriched adjoint functors between categories119964 and ℬ that are both enriched over119985

(i) If119964 andℬ are cotensored over119985 and119880∶ 119964 rarr ℬ is a cotensor-preserving119985-functor then 119865 ⊣ 119880may be enriched to a119985-adjunction

(ii) If119964 andℬ are tensored over119985 and 119865∶ ℬ rarr 119964 is a tensor-preserving119985-functor then 119865 ⊣ 119880 maybe enriched to a119985-adjunction

Proof Exercise A4i

There is a convenient device for turning an a priori unenriched category 119966 into a category that isboth enriched and cotensored over simplicial sets that is of utility in constructinginfin-cosmoi This isa specialization of the notion of closed119985-module of [54 143] to the case119985 = 119982119982119890119905 in which case lessdata is required to prove Proposition A48

A47 Definition An unenriched category 119966 is a right 119982119982119890119905-module if there exists a bifunctor

119966 times 119982119982119890119905op 119966(minus)minus

equipped with natural isomorphisms

119966 times 119982119982119890119905op times 119982119982119890119905op 119966 times 119982119982119890119905op

119966 times 119982119982119890119905op 119966

(minus)minustimesid

idtimes(minustimesminus) cong (minus)minus

(minus)minus

119966 119966 times 119982119982119890119905op

119966

(minus1)

id(minus)minuscong

⁵This justifies the abuse of exponential notation for both internal homs and cotensors

asserting that for 119880119881 isin 119982119982119890119905 and 119860 isin 119966119860119880times119881 cong (119860119880)119881 and 119860 cong 1198601

with these natural isomorphisms satisfying the manifest pseudo-algebra laws A right 119982119982119890119905-moduleis continuous if the bifunctor 119966 times 119982119982119890119905op rarr 119966 carries limits in its first variable to limits in 119966 andcarries colimits in its second variable to limits in 119966

A48 Proposition A continuous right 119982119982119890119905-module 119966 can be enriched to a 119982119982119890119905-category in a uniqueway so that there exist isomorphisms in 119982119982119890119905

119966(119860 119861119881) cong 119966(119860 119861)119881

for all 119860119861 isin 119966 and 119881 isin 119982119982119890119905 natural⁶ in all three variables

Proof We must construct an enrichment from the supplied right pseudo-action This is donestraightforwardly by defining the 119899-arrow in 119966(119860 119861) to be arrows 119860 rarr 119861120491[119899] in 119966 and using thepseudo-action laws and the canonical diagonal co-algebra structure on 120491[119899] to define an associativeand unital composition of such Note that the identity axiom of the right pseudo-action ensures thatarrows119860 rarr 119861120491[0] are in bijective correspondence with arrows119860 rarr 119861 and thus that this constructionreally does give an enrichment of 119966

Furthermore we must show that the pseudo-action supplies cotensors for this derived enrichmentthat is that

119966(119860 119861119881) cong 119966(119860 119861)119881

for all 119860119861 isin 119966 and 119881 isin 119982119982119890119905 natural in all three-variables which fact follows from one of theassumed continuity properties of (minus)minus ∶ 119966 times 119982119982119890119905op rarr 119966 using the fact that every simplicial set canbe expressed as a colimit of representables

We have not yet made use of continuity in of the right action bifunctor in its first variable Thereason for this hypothesis will explained in sectA5

Exercises

A4i Exercise Prove Proposition A46

A4ii Exercise Let119973 be a small unenriched category and let 119966 be a119985-category Prove that if 119966 istensored or cotensored then so is 119966119973

A5 Conical limits

Let 119985 be a cartesian closed category The conical limit of a 119985-enriched functor is the limitweighted by the terminal weight the 119985-functor that is constant at 1 isin 119985 In what follows wefocus on the slightly less general case of conical limits of unenriched diagrams mdash those indexed by1-categories instead of 119985-categories mdash as they will be of greatest interest here Conical limits nec-essarily define 1-categorical limits so we pay particular attention to what is required to enriched a1-categorical limit to a conical limit

⁶In fact these isomorphisms are 119982119982119890119905-natural

To motivate the universal property that characterizes conical limits consider a diagram 119865∶ 119973 rarr119985 indexed by a 1-category 119973 and valued in 119985 itself A cone 120582 over the diagram 119865 with summitsuggestively denoted lim 119865 isin 119985 is a limit cone if it satisfies the universal property

119985(119860 lim 119865) cong lim119895isin119973

119985(119860 119865119895)

an isomorphism of hom-sets The next lemma reveals that this isomorphism can always be enrichedto lie in119985 ie lifted along the underlying set functor of Definition A15

A51 Lemma If119985 is a cartesian closed category then any 1-categorical limit cone 120582∶ Δ lim 119865 rArr 119865 over adiagram 119865∶ 119973 rarr 119985 gives rise to an isomorphism in119985

(lim 119865)119860 cong lim119895isin119973(119865119895119860)

that is119985-natural in 119860 isin 119985

Proof For any 119860 isin 119985 the exponential (minus)119860 ∶ 119985 rarr 119985 is right adjoint to the product functorminus times 119860∶ 119985 rarr 119985 as such it preserves limits giving rise to the isomorphism of the statement

In terminology we now introduce Lemma A51 asserts that all 1-categorical limits in a cartesianclosed category119985 automatically enriched to define conical limits

A52 Definition Let 119966 be a 119985-category and let 119973 be a 1-category The conical limit of an unen-riched diagram 119865∶ 119973 rarr 119966 is given by an object 119871 isin 119966 and a cone 120582∶ Δ119871 rArr 119865 inducing a119985-naturalisomorphism of hom-objects in119985

119966(119883 119871) lim119895isin119973 119966(119883 119865119895) isin 119985120582lowastcong (A53)

for all 119883 isin 119966

The isomorphism (A53) looks identical to the usual isomorphism that characterizes 1-categoricallimits except for one very important difference it postulates an isomorphism in 119985 rather than anisomorphism in 119982119890119905 In the case where 119985 = 119982119982119890119905 the isomorphism of vertices that underlies thisisomorphism of simplicial sets describes the usual 1-categorical universal property To say that thelimit is conical and not merely 1-categorical is to assert that this universal property extends to all positivedimensions

Inspecting Definition A52 we see immediately that

A54 Proposition A 1-categorical limit cone is conical just when it is preserved by all 119985-valued repre-sentable functors 119966(119883 minus) ∶ 119966 rarr 119985

The proof of Lemma A51 generalizes to show

A55 Proposition If 119966 has tensors then all 1-categorical limits in 119966 are conical

Proof By Proposition A54 to show that a 1-categorical limit is conical it suffices to show thatit is preserved by the119985-valued representable functors 119966(119883 minus) ∶ 119966 rarr 119985 for all 119883 isin 119985 If 119966 admitstensors then each of these functors admits a left adjoint as right adjoints they necessarily preservethe 1-categorical limits of the statement

A56 Proposition Let119966 be enriched and cotensored over119985 A limit of an unenriched diagram119865∶ 119973 rarr 119966is a conical limit if and only if it is preserved by cotensors with all objects 119881 isin 119985

Proof Cotensors are 119985-enriched right adjoints which preserve conical limits The content isthat preservation by cotensors suffices to enriched a 1-categorical limit to a119985-categorical one

Recall that a 1-categorical limit cone (120582119895 ∶ 119871 rarr 119865119895)119895isin119973 is conical just when it is preserved by all119985-valued representable functors 119966(119883 minus) ∶ 119966 rarr 119985 To see that the natural map

119966(119883 119871) rarr lim119895isin119973

119966(119883 119865119895)

is an isomorphism in119985 we appeal to the (unenriched) Yoneda lemma and prove that for all 119881 isin 119985the map of hom-sets

119985(119881119966(119883 119871))0 rarr119985(119881 lim119895isin119973

119966(119883 119865119895))0 (A57)

is an isomorphism Since unenriched representables preserve 1-categorical limits

119985(119881 lim119895isin119973

119966(119883 119865119895))0 cong lim119895isin119973

119985(119881119966(119883 119865119895))0

By the unenriched universal property of cotensors maps 119881 rarr 119966(119883 119871) in 119985 correspond bijectivelyto maps 119883 rarr 119871119881 in 119966 So the map (A57) is isomorphic to the map of hom-sets

119966(119883 119871119881)0 rarr lim119895isin119973

119966(119883 119865119895119881)0 cong 119966(119883 lim119895isin119973(119865119895119881))0

where again the unenriched representable commutes with the 1-categorical limit To say that cotensorspreserve the limit 119871 cong lim119895isin119973 119865119895 means that lim119895isin119973(119865119895119881) cong (lim119895isin119973 119865119895)119881 cong 119871119881 so by the Yonedalemma the map of hom-sets

119966(119883 119871119881)0 rarr 119966(119883 lim119895isin119973(119865119895119881))0

is an isomorphism and thus (A57) and hence 119966(119883 119871) cong lim119895isin119973 119966(119883 119865119895) is an isomorphism as de-sired

A58 Proposition Suppose 119966 is a119985-category that admits conical limits indexed by a small category 119973Then the limit functor lim ∶ 119966119973 rarr 119966 is a119985-functor

Proof We first prove that the conical limit functor lim ∶ 119985119973 rarr119985 is a119985-functor The result fora general119985-category119966 then follows from the defining119985-natural isomorphism (A53) via PropositionA313

The advantage of this reduction is that we can make use of the fact that119985 and119985119973 are tensoredthe latter following from the former by Exercise A4ii Now by Proposition A46 to show that theconical limit is a 119985-functor it suffices to show that itrsquos unenriched left adjoint Δ∶ 119985 rarr 119985119973 is a119985-functor that preserves tensors and this is straightforward

Exercises

A5i Exercise Specialize the result of Proposition A55 to prove the following in any 2-category119966 that admits tensors with the walking-arrow category 120794 any 1-categorical limits that 119966 admits areautomatically conical⁷

⁷The statement asserts that the presence of tensors with 120794 implies that the universal property of 1-dimensional limitsautomatically has an additional 2-dimensional aspect See the discussion around Proposition 145 for an illustration ofthis and see [30 pp 306] for a proof

A6 Change of base

Change of base for enriched categories was first considered in [20] Their main result appearing asProposition A64 below is that a lax monoidal functor 119879∶ 119985 rarr119986 induces a change-of-base 2-functor119879lowast ∶ 119985-119966119886119905 rarr119986-119966119886119905

A61 Definition A (lax) monoidal functor between cartesian closed categories119985 and119986 is a func-tor 119879∶ 119985 rarr119986 equipped with natural transformations

119985times119985 119986times119986 120793 119985

119985 119986 119986

119879times119879

dArr120601times times

1 119879

119879

uArr1206010

so that the evident associativity and unit diagrams commute

A62 Definition A (lax) monoidal natural transformation between monoidal functors betweencartesian closed categories 119879119880∶ 119985 119986 is given by a natural transformation 120579∶ 119879 rArr 119880 so thatthe pasting diagrams commute

119985times119985 119986times119986 119985times119985 119986times119986 120793 119985 120793 119985

119985 119986 119985 119986 119986 119986

119879times119879

119880times119880

dArr120579times120579

dArr120601times times =

119879times119879

1119879 119880rArr120579 =

1 119880

119880

119879

119880

dArr120579

uArr1206010 uArr1206010

A63 Example A functor119879∶ 119985 rarr119986 between cartesian closed categories preserves finite productsjust when the natural maps defined for any 119860119861 isin 119985

119879(119860 times 119861) congminusrarr 119879119860 times 119879119861 and 1198791 congminusrarr 1are isomorphisms The inverse isomorphisms then make 119879 into a strong monoidal functor betweenthe cartesian closed categories 119985 and 119986 with the structure maps of Definition A61 given by nat-ural isomorphisms Moreover any natural transformation between product-preserving functors isautomatically a monoidal natural transformation

The following result was first stated as [20 II63] with the proof left to the reader We adoptthe same tactic and leave the diagram chases to the reader or to [14 424] and instead just give thedefinition

A64 Proposition A monoidal functor 119879∶ 119985 rarr119986 induces a change-of-base 2-functor

119879lowast ∶ 119985-119966119886119905 rarr119986-119966119886119905

Proof The construction of the change-of-base 2-functor is the important thing Let 119966 be a119985-category and define a 119986-category 119879lowast119966 to have the same objects and to have mapping objects119879lowast119966(119909 119910) ≔ 119879119966(119909 119910) The composition and identity maps are given by the composites

119879119966(119910 119911) times 119879119966(119909 119910) 119879(119966(119910 119911) times 119966(119909 119910)) 119879119966(119909 119911) 1 1198791 119879119966(119909 119909)120601 119879∘ 1206010 119879 id119909

which make use of the structure maps of the monoidal functor A straightforward diagram chasesverifies that 119879lowast119966 is a119986-category

If 119865∶ 119966 rarr 119967 is a 119985-functor then we define a 119986-functor 119879lowast119865∶ 119879lowast119966 rarr 119879lowast119967 to act on objectsby 119888 isin 119966 ↦ 119865119888 isin 119967 and with internal action on arrows defined by

119879119966(119909 119910) 119967(119879119909 119879119910)119879119865

Again a straightforward diagram chase verifies that 119879lowast119865 is119985-functorial It is evident from this defi-nition that 119879lowast(119866119865) = 119879lowast119866 sdot 119879lowast119865

Finally let 120572∶ 119865 rArr 119866 be a 119985-natural transformation between 119985-functors 119865119866∶ 119966 119967 anddefine a119986-natural transformation 119879lowast120572∶ 119879lowast119865 rArr 119879lowast119866 to have components

1 1198791 119879119967(119865119888 119866119888)1206010 119879120572119888

Another straightforward diagram chase verifies that 119879lowast120572 is119986-naturalIt remains to verify this assignment is functorial for both horizontal and vertical composition of

enriched natural transformations Consulting Lemma A35 we see that the component of 119879lowast(120573 sdot 120572) isdefined by the top-horizontal composite while the component of the vertical composite of 119879lowast120572 with119879lowast120573∶ 119879lowast119866 rArr 119879lowast119867 is defined by the bottom composite

1 1198791 119879(119967(119866119888119867119888) times 119967(119865119888 119866119888)) 119879119967(119865119888119867119888)

1198791 times 1198791 119879119967(119866119888119867119888) times 119879119967(119865119888 119866119888)

1206010

1206010times1206010

119879(120573119888times120572119888) 119879∘

120601

119879120573119888times119879120572119888

120601

By the unit axiom for the monoidal functor and naturality of120601 these composites agree The argumentfor functoriality of horizontal composites is similar

For any cartesian closed category 119985 the underlying set functor (minus)0 ∶ 119985 rarr 119982119890119905 is monoidal(Exercise A6i) Consequently

A65 Corollary For any cartesian closed category119985 the the underlying category construction defines a2-functor

(minus)0 ∶ 119985-119966119886119905 rarr 119966119886119905

A66 Remark In fact the change of base procedure is itself a 2-functor from the 2-category ofmonoidal categories monoidal functors and monoidal natural transformations to the 2-category of2-categories 2-functors and 2-natural transformations See [14 sect43] for a discussion and proof

If the lax monoidal functor 119879∶ 119985 rarr 119986 does not commute with the underlying set functorsfor 119985 and 119986 up to natural isomorphism the change-of-base 2-functor 119879lowast ∶ 119985-119966119886119905 rarr 119986-119966119886119905 willnot preserve underlying categories even up to natural isomorphism Consulting Definition A22 thefollowing condition on 119879 is required to ensure that underlying categories are preserved

A67 Lemma The change-of-base 2-functor induced by a monoidal functor (119879 120601 1206010) ∶ 119985 rarr119986 preservesunderlying categories if and only if for each 119881 isin 119985 the composite function on hom-sets

119985(1119881)0 119986(1198791 119879119881)0 119986(1 119879119881)0119879 minus∘1206010

is a bijection

Proof The displayed function defines the component at119881 isin 119985 of the unique monoidal naturaltransformation from the underlying set-functor for 119985 with the composite of 119879 with the underlyingset functor for119986 By Remark A66 if it defines a monoidal natural isomorphism then it induces a2-natural isomorphism between the underlying category 2-functor (minus)0 ∶ 119985-119966119886119905 rarr 119966119886119905 and the com-posite of the change of base 2-functor 119879lowast ∶ 119985-119966119886119905 rarr119986-119966119886119905 with the underlying category 2-functor(minus)0 ∶ 119986-119966119886119905 rarr 119966119886119905

Conversely this condition is necessary for the underlying category of the 119986-category 119879lowast119985 tocoincide with the underlying category of the cartesian closed category119985

One situation in which the condition of Lemma A67 is automatic is when the lax monoidalfunctor is the right adjoint of a monoidal adjunction which in this context might be thought of as achange-of-base adjunction The proof originally given in [29] is by a short diagram chase left to ExerciseA6ii

Another immediate consequence of Remark A66 is that the monoidal adjunctions of Exercise induce an adjunction between the corresponding change-of-base 2-functors Between cartesianclosed categories119985 and119986 the statement of this more general result simplifies by applying the ideasof Example A63 The right adjoint functor preserves finite products and so is automatically strongmonoidal and by Exercise (i) the left adjoint is as well Any natural transformation between product-preserving functors automatically defines a monoidal natural transformation Consequently

A68 Proposition Any adjunction comprised of finite-product preserving functors between cartesian closedcategories induces a change-of-base 2-adjunction

119985 119986 119985-119966119886119905 119986-119966119886119905119865

perp119880

119865lowast

perp119880lowast

A69 Example Both adjoints of the adjunction

119966119886119905 119982119982119890119905perph

of Proposition 1111 preserve finite products Hence by Proposition A68 induces a change-of-baseadjunction defined by the 2-functors

2-119966119886119905 119982119982119890119905-119966119886119905perp

hlowast

that act identically on objects and act by applying the homotopy category functor or nerve functorrespectively on homs Note h also satisfies the condition of Lemma A67 so both adjoints preserveunderlying categories as is evident from direct computation

Exercises

A6i Exercise For any cartesian closed category119985 prove that the underlying set functor (minus)0 ∶ 119985 rarr119982119890119905 is lax monoidal

A6ii Exercise Consider an adjunction in the 2-category ofmonoidal categories monoidal functorsand monoidal natural transformations

(i) Prove that the left adjoint is strong monoidal⁸(ii) Prove that the right adjoint is ldquopro-normalrdquo satisfying the condition of Lemma A67

⁸Hint take the mates of the structure maps of the monoidal right adjoint

APPENDIX B

Introduction to 2-category theory

2-categories and double categories were first introduced by Charles Ehresmann A notable earlyexpository account appeared in [32]

B1 2-categories and the calculus of pasting diagrams

The category 119966119886119905 of small categories is cartesian closed with exponentials 119861119860 defined to be thecategory of functors and natural transformations from 119860 to 119861 and terminal object 120793 Exploiting thework in Appendix A we can concisely define a 2-category to be a category enriched over this cartesianclosed category Unpacking this definition we see it contains a considerable amount of structure

B11 Definition (2-category) A 2-category 119966 is a category enriched in 119966119886119905 Explicitly it hasbull a collection of objects 119860119861bull a collection of arrows 119891∶ 119860 rarr 119861 frequently called 1-cells these being the objects of the hom-

categories 119966(119860 119861) and

bull a collection of arrows between arrows 119860 119861119891

119892dArr120572 called 2-cellssup1 these being the morphisms

of the hom-categories 119966(119860 119861) from 119891 to 119892so that

(i) For each fixed pair of objects 119860119861 isin 119966 the 1-cells and 2-cells form a category In particularbull A pair of 2-cells as below-left admits a vertical composite as below-right

119860 119861 ≕ 119860 119861

119891dArr120572119892

ℎdArr120573

119891

dArr120573sdot120572

bull Each 1-cell 119891∶ 119860 rarr 119861 has an identity 2-cell 119860 119861119891

119891

dArrid119891

(ii) The objects and 1-cells define a category in the ordinary sense in particular each object hasan identity arrow id119860 ∶ 119860 rarr 119860

(iii) The objects and 2-cells form a category In particular

sup1Implicitly in this graphical representation is the requirement that a 2-cell 120572 has a domain 1-cell 119891 and a codomain1-cell 119892 and these 1-cells have a common domain object 119860 and codomain object 119861 The objects 119860 and 119861 may be referredto as the 0-cell source and 0-cell target of 120572

bull A pair of 2-cells as below-left admits a horizontal composite as below-right

119860 119861 119862 ≕ 119860 119862119891

119892dArr120572

119895

119896

dArr120574

119895119891

119896119892

dArr120574lowast120572

bull The identity 2-cells on identity 1-cells

119860 119860id119860

id119860

dArridid119860

define identities for horizontal composition(iv) Finally the horizontal composition is functorial with respect to the vertical composition

bull The horizontal composite of the identity 2-cells is an identity 2-cell

119860 119861 119862 = 119860 119862119892

119892dArrid119892

119896

dArrid119896

119896119892

dArrid119896119892

bull In the situation below the horizontal composite of the vertical composites coincides withthe vertical composite of the horizontal composites a property referred to as middle-fourinterchange

119860 119861 119862

119891dArr120572119892

ℎdArr120573

119895dArr120574119896

ℓdArr120575

Adegenerate special case of horizontal composition inwhich all but one of the 2-cells is an identityid119891 on its boundary 1-cell 119891 is called ldquowhiskeringrdquo

B12 Definition (whiskering) The whiskered composite of a 2-cell 119860 119861119891

119892dArr120572 with a pair of

1-cells 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119861 rarr 119884 is defined by the horizontal composite

119883 119884119887119891119886

119887119892119886

dArr119887sdot120572sdot119886 ≔ 119883 119860 119861 119884119886119891

119892dArr120572 119887 ≔ 119883 119860 119861 119884

119886

119886dArrid119886

119891

119892dArr120572

119887

dArrid119887

As the following lemma reveals horizontal composition can be recovered from vertical composi-tion and whiskering Our primary interest in this result however has to with a rather prosaic conse-quence appearing as the final part of the statement which will be applied shockingly often

B13 Lemma (naturality of whiskering) Any horizontally-composable pair of 2-cells in a 2-category givesrise to a commutative square in the hom-category from the domain object to the codomain object whose diagonal

defines the horizontal composite

119966 ni 119860 119861 119862119891

119892dArr120572

119896

dArr120573 ℎ119891 119896119891

ℎ119892 119896119892ℎ120572

120573119891

120573lowast120572 119896120572

120573119892

isin 119966(119860119862)

In particular if any three of the four whiskered 2-cells ℎ120572 119896120572 120573119891 and 120573119892 are invertible so is the fourth

Proof By middle-four interchange

120573119892sdotℎ120572 = (120573lowastid119892)sdot(idℎ lowast120572) = (120573sdotidℎ)lowast(id119892 sdot120572) = 120573lowast120572 = (id119896 sdot120573)lowast(120572sdotid119891) = (id119896 lowast120572)sdot(120573lowastid119891) = 119896120572sdot120573119891

The operations of horizontal and vertical composition are special cases of composition by pastingan operation first introduced by Beacutenabou [2] The main result proven in the 2-categorical context byPower [44] is that a well-formed pasting diagram such as

119861 119865

119860 119862 119864 119867 119885

119863 119866

119889

119887

dArr120574 dArr120572 119892

119888

dArr120573119886

119894

119895dArr120575

119890

119891

dArr120598 ℎ

119896

119898

119899

dArr120577

has a unique 2-cell compositesup2 We leave the formal statement and proof of this result to the literatureand instead describe informally how such pasting composites should be interpreted

B15 Digression (how to read a pasting diagram) A pasting diagram in a 2-category 119966 represents aunique composite 2-cell defining a morphism in one of the hom-categories between a pair of objectsTo identify these objects look at the underlying directed graph of objects and 1-cells in the pastingdiagram If the pasting diagram is well-formed that graph should have a unique source object 119860 anda unique target object 119885 This indicates that the pasting diagram defines a 2-cell in the hom-category119966(119860 119885) The object 119860 is its source 0-cell and the object 119885 is its target 0-cell

The next step is to identify the source 1-cell and the target 1-cell of the pasting diagram Theseshould both be objects of 119966(119860 119885) ie 1-cells in the 2-category from 119860 to 119885 Again if the pastingdiagram is well-formed the source 1-cell should be the unique composable path of 1-cells none ofwhich occur as codomains of any 2-cells in the pasting diagram In the diagram (B14) these are the1-cells whose labels appear above the arrow and their composite is 119888119887119886 Dually the target 1-cell shouldbe the unique composable path of 1-cells none of which occur as domains of any 2-cells in the pastingdiagram In (B14) these are the 1-cells whose labels appear below the diagram and their compositeis 119899ℓ119895

The final step is to represent the pasting diagram as a vertical composite of 2-cells each of whichis a map between a pair of composite 1-cells from 119860 to 119885 that trace a composable path through thedirected graph of the pasting diagram Each 2-cell in the pasting diagram will label precisely one of

sup2This result was generalized to the bicategorical context by Verity [62] in which case the composite 2-cell is well-defined once its source and target 1-cells are specified

the 2-cells of this composite The expressions of these vertical 2-cell composites are not necessarilyunique and may not necessarily pass through every possible composable path of 1-cells (though therewill be some vertical composite of 2-cells that does pass through each path of 1-cells)

To start pick any 2-cell in the pasting diagram whose 1-cell source can be found as a subsequenceof the source 1-cell in the (B14) either 120572 or 120573 can be chosen first Whisker it so that it defines a 2-cellfrom the source 1-cell 119888119887119886 to another path of composable 1-cells from 119860 to 119885 through the pastingdiagram Then this whiskered composite forms the first step in the sequence of composable 2-cellsRemove this part of the pasting diagram and repeat until you arrive at the target 1-cell In the exampleabove there are three possible ways to express the composite pasted cell (B14) as vertical compositesof whiskered 2-cell represented by the three paths through the following commutative diagram in thecategory 119966(119860 119885)

119888119887119886 119888119891119890119889119886 119888119891119890119894

ℎ119892119887119886 ℎ119892119891119890119889119886 ℎ119892119891119890119894 ℎ119892119891119896119895 ℎ119898ℓ119895 ℎℓ119895

119888120572119886

120573119887119886

119888119891119890120574

120573119891119890119889119886 120573119891119890119894

ℎ119892120572119886 ℎ119892119891119890120574ℎ119892119891120575 ℎ120598119895 120577ℓ119895

isin 119966(119860 119885)

B16 Digression (notions of sameness inside a 2-category) From the point of view of 2-categorytheory the most natural notion of ldquosamenessrdquo for two objects of a 2-category is equivalence 119860 and 119861are to be regarded as the same if there exist 1-cells 119891∶ 119860 rarr 119861 and 119892∶ 119861 rarr 119860 together with invertible2-cells 120572∶ id119860 cong 119892119891 and 120573∶ 119891119892 cong id119861

From the point of view of 2-category theory the most natural notion of ldquosamenessrdquo for a parallelpair of morphisms in a 2-category is isomorphism ℎ 119896 ∶ 119860 119861 are to be regarded as the same if thereexists an invertible 2-cell 120574∶ ℎ cong 119896

From the point of view of 2-category theory the most natural notion of ldquosamenessrdquo for a pair of2-cells with common boundary is equality Because a 2-category lacks any higher dimensional mor-phisms to mediate between 2-cells there is no weaker notion of sameness available

A 2-category has four duals including itself each of which have the same objects 1-cells and2-cells but with some domains and codomains swapped

B17 Definition (op and co duals) Let 119966 be a 2-categorybull Its op-dual 119966op is the 2-category with 119966op(119860 119861) ≔ 119966(119861119860) This reverses the direction of the

1-cells but not the 2-cellsbull Its co-dual 119966co is the 2-category with 119966co(119860 119861) ≔ 119966(119860 119861)op This reverses the direction of the

2-cells but not the 1-cellsbull Its coop-dual 119966coop is the 2-category with 119966coop(119860 119861) ≔ 119966(119861119860)op This reverses the direction

of both the 2-cells and the 1-cells

Further details about the specification of 119966op 119966co and 119966coop as 119966119886119905-enriched categories are left toExercise B1iii

The data of an equivalence in 119966 also defines an equivalence in each of its dualsIn Digression 122 we saw that the data of a simplicial category could be expressed as a diagram of

a particular type valued in 119966119886119905 A small 2-category can be similarly encoded in fact in two differentways as a category defined internally to the category of categories

B18 Definition (internal category) Let ℰ be any category with pullbacks An internal category inℰ is given by the data

1198621 times11986201198621

1198621 1198621

1198620

120587ℓ 120587119903

119889 119888

1198621 times11986201198621 1198621 1198620∘

119889

119888119894

subject to commutative diagrams that define the domains and codomains of composites and identitiesand encode the fact that composition is associative and unital The details are left to Exercise B1i orthe literature

For example a double category is a category internal to 119966119886119905 A 2-category can be realized as aspecial case of this construction in the following two ways

B19 Digression (2-categories as category objects) A 2-category may be defined to be an internalcategory in 119966119886119905

11986212 times119862011986212 11986212 1198620∘

119889

119888119894

in which the category 1198620 is a set namely the set of objects of the 2-category The 1- and 2-cells occuras the objects and arrows of the category 11986212 The functors 119889 119888 ∶ 11986212 rarr 1198620 send 1- and 2-cells totheir domain and codomain 0-cells The functor 119894 ∶ 1198620 rarr 11986212 sends each object to its identity 1-cellThe action of the functor ∘ ∶ 11986212 times1198620 11986212 rarr 11986212 on objects defines composition of 1-cells and theaction on morphisms defines the horizontal composition on 2-cells Vertical composition on 2-cells isthe composition inside the category 11986212 Functoriality of this map encodes middle-four interchangeThis definition motivates the Segal category model of (infin 1)-categories described in Appendix E

Dually a 2-category may be defined to be an internal category in 119966119886119905

11986202 times1198620111986202 11986202 11986201∘

119889

119888119894

in which the categories 11986201 and 11986202 have the same set of objects and all four functors are identity-on-objects Here the common set of objects defines the objects of the 2-category and the arrowsof 11986201 and 11986202 define the 1- and 2-cells respectively The functors 119889 119888 ∶ 11986212 11986202 define thedomain and codomain 1-cells for a 2-cell which the functor ∘ ∶ 11986202times1198620111986202 rarr 11986202 encodes verticalcomposition of 2-cells The composition inside the category 11986202 defines horizontal composition of2-cells Functoriality of this map encodes middle-four interchange

Exercises

B1i Exercise Complete the definition of an internal category sketched in Definition B18

B1ii Exercise Define a duality involution on double categories that exchanges the two expressionsof a 2-category as an internal category appearing in Digression B19

B1iii Exercise For any 2-category 119966 define 119966119886119905-enriched categories 119966op 119966co and 119966coop along thelines specified by Definition B17

B2 The 3-category of 2-categories

Ordinary 1-categories form the objects of a 2-category of categories functors and natural transfor-mations Similarly 2-categories form the objects of a 3-category of 2-categories 2-functors 2-naturaltransformations and modifications In this section we briefly introduce all of these notions

Recall from Definition B11 that a 2-category is a category enriched in 119966119886119905 Similarly 2-functorsand 2-natural transformations are precisely the119966119886119905-enriched functors and119966119886119905-enriched natural trans-formations of Appendix A By Corollary A36 2-categories 2-functors and 2-natural transformationsassemble into a 2-category The 3-dimensional cells between 2-categories mdash the modifications mdash aredefined using the 2-cells of a 2-category like the 2-dimensional cells between 1-categories mdash the nat-ural transformations mdash are defined using the 1-cells in a 1-category

B21 Definition A 2-functor 119865∶ 119966 rarr 119967 between 2-categories is given bybull a mapping on objects 119966 ni 119909 ↦ 119865119909 isin 119967bull a functorial mapping on 1-cells 119966 ni 119891∶ 119909 rarr 119910 ↦ 119865119891∶ 119865119909 rarr 119865119910 isin 119967 respecting domains and

codomains andbull a mapping on 2-cells

119966 ni 119909 119910 ↦ 119865119909 119865119910119891

119892dArr120572

119865119891

119865119892

dArr119865120572 isin 119967

that respects 0- and 1-cell sources and targets that is functorial for both horizontal and verticalcomposition and horizontal and vertical identities

B22 Definition A 2-natural transformation 119966 119967119865

119866

dArr120601 between a parallel pair of 2-functors

119865 and 119866 is given by a family of 1-cells (120601119888 ∶ 119865119888 rarr 119866119888)119888isin119966 in 119967 indexed by the objects of 119966 that arenatural with respect to the 1-cells 119891∶ 119909 rarr 119910 in 119966 in the sense that the square

119865119909 119865119910

119866119909 119866119910

120601119909

119865119891

120601119910

119866119891

commutes in 119967 and also natural with respect to the 2-cells 119909 119910119891

119892dArr120572 in 119966 in the sense that the

top-right and bottom-left whiskered composites

119865119909 119865119910

119866119909 119866119910

120601119909

119865119891

119865119892

dArr119865120572

120601119910119866119891

119866119892

dArr119866120572

are equal ie 120601119910 sdot 119865120572 = 119866120572 sdot 120601119909B23 Definition A modification Ξ∶ 120601 120595

119966 119967Ξ

119865

119866

120601 120595

between parallel 2-natural transformations is given by a family of 2-cells in119967

119865119888 119866119888120601119888

120595119888

dArrΞ119888

indexed by the objects 119888 isin 119966 with the property that for any 1-cell 119891∶ 119909 rarr 119910 in 119966 the whiskeredcomposites Ξ119910 sdot 119865119891 = 119866119891 sdot Ξ119909 are equal in 119967 and for any 2-cell 120572∶ 119891 rArr 119892 in 119967 the horizontalcomposites in119967

119865119909 119865119910 119866119910119865119891

119865119892

dArr119865120572

120601119910

120595119910

dArrΞ119910 = 119865119909 119866119909 119866119910120601119909

120595119909

dArrΞ119909

119866119891

119866119892

dArr119866120572

are equal

Finally the category of 2-categories is cartesian closed with internal hom ℬ119964 defined to bethe 2-category of 2-functors 2-natural transformations and modifications So now we can definea 3-category to be a category enriched in 2-categories

Exercises

B2i Exercise For the reader who has a lot of blank paper unpack the definition of a 3-category justgiven

B3 Adjunctions and mates

B31 Proposition (equivalence invariance of adjointness) Suppose given an essentially commutativesquare

119861 119861prime 119861 119861prime

119860 119860prime 119860 119860prime

119887

119891 cong120572 119891prime

119887

cong119906prime119886120598sdot119906prime120572119906sdot120578prime119906⊢

119886

119906prime⊣

119906

119886

119906prime

Then 119891 admits a right adjoint 119906 if and only 119891prime admits a right adjoint 119906prime in which case the mate of theisomorphism 120572 is an isomorphism

Proof An exercise for now

B32 Lemma Suppose given a triple of adjoint functors ℓ ⊣ 119894 ⊣ 119903 Then the counit of ℓ ⊣ 119894 is invertible ifand only if the unit of 119894 ⊣ 119903 is invertible

B4 Right adjoint right inverse adjunctions

B41 Lemma Suppose we are given a pair of 1-cells 119906∶ 119860 rarr 119861 and 119891∶ 119861 rarr 119860 and a 2-isomorphism119891119906 cong id119860 in a 2-category If there exists a 2-cell 120578prime ∶ id119861 rArr 119906119891 with the property that 119891120578prime and 120578prime119906 are2-isomorphisms then 119891 is left adjoint to 119906 Furthermore in the special case where 119906 is a section of 119891 then 119891 isleft adjoint to 119906 with the counit of the adjunction an identity

Proof For now see [49 412]

B5 A bestiary of 2-categorical lemmas

B51 Lemma Suppose 119891 ⊣ 119906 is an adjunction under 119861 in the sense thatbull the triangles involving both adjoints commute

119861

119862 119860

119886119888

119906⊤119891

bull and 120578119886 = id119886 and 120598119888 = id119888Then if 119888 admits a right adjoint 119903 with counit 120584∶ 119888119903 rArr id119862 then 119906120584 exhibits 119903 as an absolute right lifting of119906 through 119886

119861

119862 119860dArr119906120584

119886

119906

119903

B52 Example For example consider the diagram of adjoint functors

120491 120491+ 120491perp

120793

[minus1] perp

For now see the proof of [49 531]

APPENDIX C

Abstract homotopy theory

C1 Lifting properties weak factorization systems and Leibniz closure

C11 Lemma Any class of maps characterized by a right lifting property is closed under composition productpullback retract and limits of towers see Lemma C11

Proof For now see [47 1114] and dualize

On account of the dual of Lemma C11 any set of maps in a cocomplete category ldquocellularly gen-eratesrdquo a larger class of maps with the same left lifting property

C12 Definition The class of maps cellularly generated by a set of maps is comprised of those mapsobtained as sequential composites of pushouts of coproducts of those maps

Appendix of Semantic Considerations

APPENDIX D

The combinatorics of simplicial sets

D1 Simplicial sets and markings

D11 Definition Write 120491le119899 sub 120491 for the full subcategory of the simplex category of 111 spannedby the ordinals [0] hellip [119899] Restriction and left and right Kan extension define adjunctions

119982119890119905120491op

119982119890119905120491ople119899res119899

ran119899perp

lan119899perp

inducing an idempotent comonad sk119899 ≔ lan119899 ∘ res119899 and an idempotent monad cosk119899 ≔ ran119899 ∘ res119899 on119982119982119890119905 that are adjoint sk119899 ⊣ cosk119899 The counit and unit of this comonad and monad define canonicalmaps

sk119899119883 119883 cosk119899119883120598 120578

relating a simplicial set119883 with its 119899-skeleton and 119899-coskeleton We say119883 is 119899-skeletal or 119899-coskeletalif the former or latter of these maps respectively is an isomorphism

The next lemma proves that the monomorphisms are cellularly generated by the simplex boundaryinclusions 120597Δ[119899] Δ[119899] for 119899 ge 0

D12 Lemma Any monomorphism of simplicial sets decomposes canonically as a sequential composite ofpushouts of coproducts of the maps 120597Δ[119899] Δ[119899] for 119899 ge 0

D13 Definition ((left-right-inner-)anodyne extensions)bull The set of horn inclusions Λ119896[119899] Δ[119899] for 119899 ge 1 and 0 le 119896 le 119899 cellularly generates the

anodyne extensionsbull The set of left horn inclusions Λ119896[119899] Δ[119899] for 119899 ge 1 and 0 le 119896 lt 119899 cellularly generates the

left anodyne extensionsbull The set of right horn inclusions Λ119896[119899] Δ[119899] for 119899 ge 1 and 0 lt 119896 le 119899 cellularly generates the

right anodyne extensionsbull The set of inner horn inclusions Λ119896[119899] Δ[119899] for 119899 ge 2 and 0 lt 119896 lt 119899 cellularly generates the

inner anodyne extensions

APPENDIX E

Examples ofinfin-cosmoi

For now see sectIV2 of [51]

APPENDIX F

Compatibility with the analytic theory of quasi-categories

The aim in this section is to prove that the synthetic theory of quasi-categories is compatible withthe analytic theory pioneered by Andreacute Joyal Jacob Lurie and many others We also discuss somespecial features of theinfin-cosmos of quasi-categories proving in particular that universal properties inthisinfin-cosmos are determined pointwise

For now seebull for limits Proposition 432 I526-8 of [49]bull for adjunctions Proposition 526 IV4120-22 of [51] VIII336 of [52]bull for cartesian fibrations IV4124 of [51]

For one instantiation of the pointwise-detection of universal properties in 119980119966119886119905 see sect61 of [49]

Bibliography

[1] J M Beck Triples Algebras and Cohomology PhD thesis Columbia University 1967[2] J Beacutenabou Introduction to bicategories In Reports of the Midwest Category Seminar number 47 in Lecture Notes in

Math pages 1ndash77 Springer-Verlag Berlin 1967[3] J E Bergner A characterization of fibrant Segal categories Proceedings of the AMS 2007[4] J E Bergner A model category structure on the category of simplicial categories Transactions of the American Mathe-

matical Society 3592043ndash2058 2007[5] J E Bergner Three models for the homotopy theory of homotopy theories Topology 46(4)397ndash436 September 2007[6] G J Bird G M Kelly A J Power and R H Street Flexible limits for 2ndashcategories J Pure Appl Algebra 61 November

1989[7] A Blumberg and M Mandell Algebraic K-theory and abstract homotopy theory Advances in Mathematics 2263760ndash

3812 2011[8] J Boardman and R Vogt Homotopy Invariant Algebraic Structures on Topological Spaces volume 347 of Lecture Notes in

Mathematics Springer-Verlag 1973[9] F BorceuxHandbook of Categorical Algebra 1 Basic Category Theory Encyclopedia of Mathematics and its Applications

Cambridge University Press 1994[10] F Borceux Handbook of Categorical Algebra 2 Categories and Structures Encyclopedia of Mathematics and its Applica-

tions Cambridge University Press 1994[11] A K Bousfield and D M Kan Homotopy limits completions and localizations Lecture Notes in Mathematics Vol 304

Springer-Verlag Berlin 1972[12] J Climent Vidal and J Soliveres Tur Kleisli and eilenberg-moore constructions as parts of biadjoint situations Ex-

tracta mathematicae 25(1)1ndash61 2010[13] J-M Cordier and T Porter Vogtrsquos theorem on categories of homotopy coherent diagrams Mathematical Proceedings

of the Cambridge Philosophical Society 10065ndash90 1986[14] G S H Cruttwell Normed Spaces and the Change of Base for Enriched Categories PhD thesis Dalhousie University

December 2008[15] G S H Cruttwell and M A Shulman A unified framework for generalized multicategories Theory Appl Categ

24(21)580ndash655 2010[16] D Dugger and D I Spivak Rigidification of quasi-categories Algebr Geom Topol 11(1)225ndash261 2011[17] W Dwyer and D Kan Calculating simplicial localizations J Pure Appl Algebra 1817ndash35 1980[18] W Dwyer D Kan and J Smith Homotopy commutative diagrams and their realizations J Pure Appl Algebra 575ndash24

1989[19] W G Dwyer and D M Kan Simplicial localizations of categories J Pure Appl Algebra 17(267ndash284) 1980[20] S Eilenberg and G Kelly Closed categories In Proceedings of the Conference on Categorical Algebra (La Jolla 1965) pages

421ndash562 Springer-Verlag 1966[21] P Gabriel and M Zisman Calculus of Fractions and Homotopy Theory Number 35 in Ergebnisse der Math Springer-

Verlag 1967[22] K Hess A general framework for homotopic descent and codescent arXiv10011556 2010[23] A Hirschowitz and C Simpson Descente pour les n-champs arXivmath9807049 2011[24] A Joyal Quasi-categories and Kan complexes Journal of Pure and Applied Algebra 175207ndash222 2002[25] A Joyal The theory of quasi-categories and its applications Quadern 45 vol II Centre de Recerca Matemagravetica Barcelona

httpmatuabcat~kockcrmhocatadvanced-courseQuadern45-2pdf 2008

[26] A Joyal and M Tierney Quasi-categories vs Segal spaces In A D et al editor Categories in Algebra Geometry andMathematical Physics (StreetFest) volume 431 of Contemporary Mathematics pages 277ndash325 American MathematicalSociety 2007

[27] G Kelly On Mac Lanersquos conditions for coherence of natural associativities commutativities etc J Algebra 1397ndash4021964

[28] G Kelly and S Mac Lane Coherence in Closed Categories J Pure Appl Algebra 1(1)97ndash140 1971[29] G M Kelly Doctrinal adjunction In Category Seminar (Proc Sem Sydney 19721973) volume Vol 420 of Lecture Notes

in Mathematics pages 257ndash280 Springer Berlin 1974[30] G M Kelly Elementary observations on 2-categorical limits Bulletin of the Australian Mathematical Society 49301ndash317

1989[31] G M Kelly Basic concepts of enriched category theory Reprints in Theory and Applications of Categories 10 2005[32] G M Kelly and R H Street Review of the Elements of 2-Categories volume 420 of Lecture Notes in Mathematics pages

75ndash103 Springer-Verlag 1974[33] S Lack Homotopy-theoretic aspects of 2-monads Journal of Homotopy and Related Structures 2(2)229ndash260 2007[34] S Lack Icons Applied Categorical Structures 18(3)289ndash307 2010[35] F W Lawvere Ordinal sums and equational doctrines In Sem on Triples and Categorical Homology Theory (ETH Zuumlrich

196667) pages 141ndash155 Springer Berlin 1969[36] J Lurie Higher Topos Theory volume 170 of Annals of Mathematical Studies Princeton University Press Princeton New

Jersey 2009[37] J Lurie Higher Algebra httpwwwmathharvardedu~luriepapersHApdf September 2014[38] S Mac Lane Categories for the Working Mathematician volume 5 ofGraduate Texts in Mathematics Springer-Verlag 1971[39] S Mac Lane Categories for the Working Mathematician Graduate Texts in Mathematics Springer-Verlag second edition

edition 1998[40] M Makkai and R Pareacute Accessible Categories The Foundations of Categorical Model Theory volume 104 of Contemporary

Mathematics American Mathematical Society 1989[41] J P May and K Ponto More concise algebraic topology Localization completion and model categories Chicago Lectures

in Mathematics University of Chicago Press Chicago IL 2012[42] J-P Meyer Bar and cobar constructions I J Pure Appl Algebra 33(2)163ndash207 1984[43] R Pellissier Cateacutegories enrichies faibles PhD thesis Universiteacute de Nice-Sophia Antipolis 2002[44] A J Power A 2-categorical pasting theorem Journal of Algebra 129439ndash445 1990[45] C Rezk A model for the homotopy theory of homotopy theory Transactions of the American Mathematical Society

353(3)973ndash1007 2001[46] E Riehl On the structure of simplicial categories associated to quasi-categories Math Proc Cambridge Philos Soc

150(3)489ndash504 2011[47] E Riehl Categorical Homotopy Theory volume 24 of New Mathematical Monographs Cambridge University Press 2014[48] E Riehl Category Theory in Context Aurora Modern Math Originals Dover Publications 2016[49] E Riehl and D Verity The 2-category theory of quasi-categories Adv Math 280549ndash642 2015[50] E Riehl and D Verity Completeness results for quasi-categories of algebras homotopy limits and related general

constructions Homol Homotopy Appl 17(1)1ndash33 2015[51] E Riehl and D Verity Fibrations and Yonedarsquos lemma in aninfin-cosmos J Pure Appl Algebra 221(3)499ndash564 2017[52] E Riehl and D Verity Cartesian exponentiation and monadicity In preparation 2018[53] S Schanuel and R H Street The free adjunction Cahiers de Topologie et Geacuteomeacutetrie Diffeacuterentielle Cateacutegoriques 2781ndash83

1986[54] M Shulman Homotopy limits and colimits and enriched category theory Arxiv preprint arXivmath0610194

[mathAT] 2009[55] R Street Fibrations and Yonedarsquos lemma in a 2-category In Category Seminar (Proc Sem Sydney 19721973) volume

Vol 420 of Lecture Notes in Math pages 104ndash133 Springer Berlin 1974[56] R H Street The formal theory of monads Journal of Pure and Applied Algebra 2149ndash168 1972[57] R H Street Elementary cosmoi I In Category Seminar (Proc Sem Sydney 19721973) volume Vol 420 of Lecture Notes

in Math pages 134ndash180 Springer Berlin 1974[58] R H Street Fibrations in bicategories Cahiers de Topologie et Geacuteomeacutetrie Diffeacuterentielle Cateacutegoriques XXI-2111ndash159 1980

[59] R H Street Correction to ldquofibrations in bicategoriesrdquo Cahiers topologie et geacuteomeacutetrie diffeacuterentielle cateacutegoriques 28(1)53ndash56 1987

[60] Y J Sulymainfin-categorical monadicity and descent New York Journal of Mathematics 23749ndash777 2017[61] D Verity Weak complicial sets I basic homotopy theory Advances in Mathematics 2191081ndash1149 September 2008[62] D Verity Enriched categories internal categories and change of base Reprints in Theory and Application of Categories

201ndash266 June 2011[63] M Weber Yoneda Structures from 2-toposes Applied Categorical Structures 15(3)259ndash323 2007[64] R J Wood Abstract proarrows I Cahiers topologie et geacuteomeacutetrie diffeacuterentielle cateacutegoriques XXIII-3279ndash290 1982[65] D Zaganidis Towards an (infin 2)-category of homotopy coherent monads in aninfin-cosmos PhD thesis Eacutecole Polytechnique

Feacutedeacuterale de Lausanne 2017

Preface

Part I Basic infinity-category theory

Chapter 1 infinity-Cosmoi and their homotopy 2-categories

11 Quasi-categories

12 infinity-Cosmoi



Chapter 2 Adjunctions limits and colimits I





Chapter 3 Weak 2-limits in the homotopy 2-category


32 infinity-categories of arrows


34 Representable comma infinity-categories


Chapter 4 Adjunctions limits and colimits II


42 infinity-categories of cones


44 Loops and suspension in pointed infinity-categories

Chapter 5 Fibrations and Yonedas lemma






Part II Homotopy coherent category theory

Chapter 6 Simplicial computads and homotopy coherence




Chapter 7 Weighted limits in infinity-cosmoi




74 More infinity-cosmoi


Chapter 8 Homotopy coherent adjunctions and monads





Chapter 9 The formal theory of homotopy coherent monads



93 Limits and colimits in the infinity-category of algebras


95 Monadic descent


Part III The calculus of modules

Chapter 10 Two-sided fibrations


102 The infty -cosmos of two-sided fibrations



Chapter 11 The calculus of modules





Chapter 12 Formal category theory in a virtual equipment

121 Exact squares



124 Limits and colimits in cartesian closed infinity-cosmoi

Part IV Change of model and model independence

Chapter 13 Cosmological biequivalences


132 Biequivalences of infinity-cosmoi as change-of-model functors



Chapter 14 Proof of model independence



Appendix A Basic concepts of enriched category theory





A5 Conical limits

A6 Change of base

Appendix B Introduction to 2-category theory






Appendix C Abstract homotopy theory



Appendix D The combinatorics of simplicial sets


Appendix E Examples of infty -cosmoi

Appendix F Compatibility with the analytic theory of quasi-categories

Bibliography