Dold-Kan equivalence andhomotopy

Paula Verdugo

Supervisor: Lawrence Breen

Memoire M2 Mathematiques Fondamentales

Universite Paris 13

Abstract

In this dissertation we study the well known Dold-Kan correspondence betweensimplicial abelian groups and non-negative chain complexes and its behaviour withrespect to the classical model structures in the involved categories. To this end, we gothrough basics concepts on abstract homotopy theory and we apply this concepts inthe particular case of the Dold-Kan correspondence. Throughout the text we use theconcept of simplicial objects; we briefly review related concepts, with certain details forthe category of sets.

We claim no original content, the bibliography used will be described when pertinent.

Contents

Introduction 5

1 Abstract homotopy theory 7

1.1 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Consequences of the axioms . . . . . . . . . . . . . . . . . . . . . . 9

1.1.2 Small object argument . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Cofibrantly generated model categories . . . . . . . . . . . . . . . . . . . . 13

1.2.1 Basics on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Galois connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.3 Small object argument . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.4 Another definition of model categories . . . . . . . . . . . . . . . . 22

1.2.5 Cofibrantly generated model categories . . . . . . . . . . . . . . . . 27

1.3 Homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Quillen adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.5 Quillen equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Simplicial generalities 36

2.1 The simplex category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.1 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.2 Singular functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.3 Adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.4 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4 Simplicial homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 The Dold-Kan correspondence 49

3.1 From sR-Mod to chain complexes . . . . . . . . . . . . . . . . . . . . . . 49

3.1.1 Building up complexes from a simplicial module . . . . . . . . . . 49

3.1.2 Relations between the different chain complexes . . . . . . . . . . 52

3.2 From Ch+(R) to simplicial modules . . . . . . . . . . . . . . . . . . . . . 55

3.3 The equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Homotopies and the Dold-Kan correspondence . . . . . . . . . . . . . . . 58

4 Model category structure in Ch+(R) 61

4.1 MC1 is satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 MC2 is satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 MC3 is satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 MC4 is satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 MC5 is satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Model structure on sSet 71

5.1 Cofibrantly generated model structure . . . . . . . . . . . . . . . . . . . . 71

5.2 Towards the completion of the structure. . . . . . . . . . . . . . . . . . . . 82

5.2.1 Anodyne extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.2 Minimal fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.3 Fibrations and geometric realization . . . . . . . . . . . . . . . . . 86

6 Model category structure in sAb 88

6.1 Adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2 Transferring the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 The Dold-Kan correspondence revisited 91

Bibliography 93

Introduction

Model categories were first introduced by Quillen in [Qui67] and they have become afundamental concept in homotopy theory. This concept allows us to invert certain classesof arrows in a category, as we often wish to do. Once solved the problem of invertingarrows one wish to study functors preserving the structure, and there is when theconcepts of Quillen adjunctions and Quillen equivalences arise. For our brief treatmentof these themes we have mostly relied on [DS95], [Hir03], [Hov99] and [GJ99]. Whilethe first and third have a similar approaches -although [Hov99] develops the theory ofcofibrantly generated model categories- the second one concentrates more on localizationof categories and [GJ99] works out simplicial examples.

Historically the motivations and development of homotopy theory in these termshave been within the framework of algebraic topology; however, they have been used inseveral areas of Mathematics such as homological algebra and abstract algebra (see forexample [Hov02]), algebraic K-theory and derived algebraic geometry.

The Dold-Kan correspondence, proven independently in 1958 by A. Dold in [Dol58]and D. Kan in [Kan58], is an equivalence of categories between simplicial objeccts in anabelian category and non-negative chain complex in such category; we will work in theparticular case of the category of abelian groups. This correspondence preserves certainmodel structures on the involved categories, as we will also show.

We will now provide an overview of this dissertation. In the first chapter we go overthe original definition of (closed) model categories we present the small object argumentas worked in [DS95]. After that we give an alternative equivalent definition to Modelcategories via weak factorization systems and we present a more general version of thesmall object argument following [Hov99]. Once we have that covered, we present thebasics on cofibrantly generation of model categories and we show how it facilitates therecognition of model categories itself and of Quillen adjunction and Quillen equivalencestogether with a way to transfer such structure via adjunctions.

The second chapter has as aim the introduction of simplicial objects, with emphasison simplicial sets and its homotopy groups; here are introduced the Kan complexes andits relation with simplicial abelian groups.

Chapter three is devoted to the construction of the Dold-Kan correspondence andthe proof that it effectively is an equivalence of categories.

5

The following three chapters, the fourth, fifth and sixth, deal with the endowmentof model structure on non-negative chain complexes of modules over commutative rings,simplicial sets and simplicial abelian groups respectively. In the last two cases, these areassumed to be cofibrantly generated.

In the seventh chapter we go back to the Dold-Kan correspondence in order to proveit is actually a Quillen equivalence.

6

Chapter 1

Abstract homotopy theory

1.1 Model categories

Definition 1.1.1. Given a commutative square diagram

A X

B Y

f

i p

g

(1.1)

a lift or lifting in such diagram is an arrow h : B → X such that hi = f and ph = g, i.e.the following diagram commutes

A X

B Y

f

i ph

g

When a diagram like 1.1 has a lifting, we say that i has the left lifting property(LLP) with respect to p and p has the right lifting property (RLP) with respect to i. Orequivalently we call (i, p) a lifting extension pair.

Proposition 1.1.2. Let C and D be two categories and F : C → D and U : D → Ctwo functors such that (F,U) is an adjoint pair. Given two maps i : A → B in C andp : X → Y in D, the pair (Fi, p) is a lifting extension pair if and only if so is (i, Up).

Proof. The adjointness of F and U implies the existence of a one-to-one correspondencebetween the commutative solid diagrams below:

FA X

FB Y

Fi ph and

A UX

B UY

i Uph

7

Again by the adjointness of F and U , under this correspondence, the existence of thearrow h in the diagram on the left is equivalent to the existence of the arrow h on theright one.

Definition 1.1.3. A model category is a category C with three distinguished classesof arrows, namely weak equivalences (WE), fibrations (Fib) and cofibrations (Cofib)satisfying certain axioms. Fibrations that are at the same time weak equivalences arecalled acyclic fibrations and cofibrations which are weak equivalences are called acycliccofibrations. The axioms which we require are:

MC1 C is complete and cocomplete,

MC2 (two out of three) given f and g two composable maps, if two of the three mapsf , g and gf are weak equivalences, then so is the third,

MC3 (retracts) if f and g are maps in C such that g belongs to one of the distinguishedclasses and f is a retract of g (as objects in the arrow category), then f belongs tothe same class. Explicitly, the condition that f is a retract of g means that thereexist ι, ι′, r, r′ such that the diagram below commutes:

X Y X

X ′ Y ′ X ′

ι

idX

f

r

g f

ι′

idX′

r′

MC4 (lifting) given a commutative diagram as 1.1 there exists a lift in the diagram ineither of the following two cases:

1. i is a cofibration and p an acyclic fibration,

2. i is an acyclic cofibration and p a fibration,

MC5 (factorisation) any arrow f in C admits two factorisations, namely:

1. f = pi, where i is a cofibration and p an acyclic fibration,

2. f = pi, where i is an acyclic cofibration and p a fibration.

We will sometimes denote a model category as in the definition by (C,WE,Fib,Cofib).

If we denote the class of all morphisms that have the left lifting property with respectto acyclic fibrations by LLP (Fib∩WE), and all the morphisms that have the right liftingproperty with respect to acyclic cofibrations by RLP (Cofib∩WE), then axiom MC4can be reformulated as

Cofib ⊂ LLP (Fib∩WE) and Fib ⊂ RLP (Cofib∩WE)

8

1.1.1 Consequences of the axioms

Remark 1.1.4. The axioms MC1-MC5 are symmetric with respect to fibrations andcofibrations and thus if (C,WE,Fib,Cofib) is a model category then (Cop,WE,Cofib,Fib)is again a model category.

Remark 1.1.5. A model category has initial and final objects (as consequence of MC1).

Proposition 1.1.6. If (C,WE,Fib,Cofib) is a model category, then:

1. Cofib = LLP (Fib∩WE),

2. Cofib∩WE = LLP (Fib),

3. Fib = RLP (Cofib∩WE),

4. Fib∩WE = RLP (Cofib).

Proof. We will prove points 1 and 2, for 3 and 4 are very similar.

1. Since by hypothesis the inclusion Cofib ⊂ LLP (Fib∩WE) holds (axiom MC4), itremains to prove that LLP (Fib∩WE) ⊂ Cofib.

Let f : X → Y be an arrow in LLP (Fib∩WE). By axiom MC5 , f admits adecomposition f = pi where i is a cofibration and p is an acyclic fibration.

Being that by hypothesis f has got the right left lifting property with respect toacyclic fibrations we have a lifting as showed in the following diagram:

X A

Y Y

i

f pg (1.2)

By means of the diagram 1.2 we can construct the commutative diagram thatfollows:

X X X

Y A Y

f i f

idY

g p

from which conclude that f is a retract of i and thus by axiom MC5, f is acofibration.

2. Let us prove in a first instance the inclusion Cofib∩WE ⊂ LLP (Fib). Letf : X → Y be an arrow in Cofib∩WE and f ′ : X ′ → Y ′ a fibration. Then, byMC4 a lifting as follows exists

X X ′

Y Y ′

f f ′g

9

This is, f has the left lifting property with respect to fibrations that is what welooked for.

For the inverse inclusion, let f be an arrow with the left lifting property withrespect to fibrations. By axiom MC4 f admits a decomposition f = pi where i isan acyclic cofibration and p a fibration.

Then, by hypothesis we have the following lifting:

X A

Y Y

i

f pg

from which we can construct the commutative diagram

X X X

Y A Y

f i f

idY

g p

We deduce then that f is a retract of i, implying that f is an acyclic cofibration.

1.1.2 Small object argument

Quillen’s small object argument is a construction that allows us to factorize a map ina category starting from a set of maps in such category. It was introduced by Quillenin [Qui67] while proving the model structure on the category of topological spaces, andit is especially useful for endowing categories with a model structure in the cofibrantlygenerated context that we will describe in what follows.

We present here a basic version following [DS95], and we will present a bit later amore general version following [Hov99].

For a further discussion, see B. Chorny in [Cho06] and R. Garner in [Gar07]; thelatter focusses on improving aspects such as the verification of a universal property andthe former generalizes it to be able to apply the construction in the context of properclasses and not only in sets.

During this section C will denote a category with all small colimits.

Definition 1.1.7. An object A in C is said to be sequencially small if for any functorF : Z+ → C, being Z+ the category associated to the ordered set of non negative integers,the natural map

colimn HomC(A,F (n)) HomC(A, colimn F (n))

is a bijection.

10

We recall the construction of the morphism in the definition 1.1.7.

Applying the functor HomC(A,−) to the diagrams:

F (n) F (m)

colimF

we obtain commutative diagrams:

HomC(A,F (n)) HomC(A,F (m))

HomC(A, colimF )

And thus, by the universal property of colimits, we obtain the dashed arrow ϕ:

HomC(A,F (n)) HomC(A,F (m))

colim HomC(A,F (−))

HomC(A, colimF )

ϕ

Theorem 1.1.8 (Small object argument). Let F = {fi : Ai → Bi}i∈I be set of maps inC such that each Ai is sequentially small, and let f : X → Y be another distinguishedarrow in C. Then there exists a factorization f = p ◦ i such that the map p has the rightlifting property with respect to each of the maps in F .

Proof. The first thing we shall do is to construct a factorization. In order to do that,consider for each i the set Si defined to be the set of all pairs of maps (g, h) such thatthe following diagram commutes:

Ai X

Bi Y

fi

g

f

h

(1.3)

From here we define G1(F , f) to be the push-out object of the solid diagram written

11

below: ∐(f,g)∈Sii∈I

Ai X

∐(f,g)∈Sii∈I

Bi G1(F , f)

∐g

∐fi

i1(1.4)

With the diagrams in 1.3 and the universal property of push outs, we obtain a mapp1 : G1(F , f)→ Y as follows:

∐(fi,g)∈Sii∈I

Ai X

∐(fi,g)∈Sii∈I

Bi G1(F , f)

Y

∐g

∐fi

i1

f

∐h

p1

(1.5)

The map p1 : G(F , f)→ Y we have just built verifies p1i1 = f .

Now we inductively define objects Gk(F , f) and maps pk : Gk(F , f)→ Y by settingGk(F , f) = G1(F , pk−1) and pk = (pk−1)1.

The previous considerations lead to a commutative diagram as the following:

X G1(F , f) · · · Gk(F , f) Gk+1(F , f) · · ·

Y

f

i1

p1

i2

pk

ik

pk+1

ik+1

(1.6)

Let G∞(F , f) be the colimit of the direct limit associated to the upper row in thediagram 1.6. Then we have a natural arrow i : X → G∞(F , f).

Note that we have actually defined a functor

Z+ C

k Gk(F , f)

G(F ,f)

and G∞(F , f) is its colimit.

12

Finally, using the commutativity of 1.6 and the universal property of colimits, wehave an arrow p : G∞(F , f)→ Y such that f = p ◦ i.

It remains to prove that p has the right lifting property with respect to all the mapsin the family F . Let us consider then a commutative square representing the liftingproblem we want to solve for fi : Ai → Bi in F :

Ai G∞(F , f)

Bi Y

g

fi p

h

(1.7)

Since by hypothesis Ai is sequentially small, the isomorphism of the definition 1.1.7with the functor G give us:

colimn HomC(Ai, Gn(F , f))

∼−→ HomC(Ai, G∞(F , f))

Then, there exists an integer k such that the map g : Ai → G∞(F , f) can be identifiedwith the composite of a map g′ : Ai with the canonical map Gk(F , f) → G∞(F , f).Using this we can modify diagram 1.7 to abtain the following, also commutative, one:

Ai Gk(F , f) Gk+1(F , f) G∞(F , f)

Bi Y Y Y

g

g′

fi pk

ik

pk+1 p

h id id

By construction, the pair (g′, h) is in the Si used for the definition of Gk+1(F , f)from Gk(F , f) and therefore there exists a map Bi → Gk+1(F , f) such that the diagram

Ai Gk(F , f)

Bi Gk+1(F , f)

g′

fi ik+1

is commutative.

Composing such map with the canonical arrow Gk+1(F , f) → G∞(F , f) we obtainthe lifting for the diagram 1.7.

1.2 Cofibrantly generated model categories

We start this section by giving concepts that will be useful for the development of thetheme of the section: we start with basic concepts on set theory and we continue withGalois connections.

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1.2.1 Basics on set theory

We first recall the definition of an ordinal, since we will be working with them.

Definition 1.2.1. A set α is called transitive if every element of α is also a subset of α.

Definition 1.2.2 (Ordinal). A set α is called an ordinal if in addition to being transitive,it is well ordered by the membership relation.

Definition 1.2.3. Given a set A, we define the cardinality of A, noted |A|, to be thesmallest ordinal for which there is a bijection |A| → A.

Definition 1.2.4 (Cardinal). A cardinal is an ordinal κ such that κ = |κ|.

Example 1.2.5. Ordinals that are easily shown to be such are ∅, {∅}, {∅, {∅}}.

It is known that every element of an ordinal is an ordinal and that given ordinalsα and β we have α = β, α ∈ β or β ∈ α (moreover, the cases are mutually exclusive);a proof of these facts can be found in [Kri98]. We will often think of an ordinal as acategory where there is a unique map from α to β if and only if α ∈ β.

Definition 1.2.6. Let C be a category admitting all small colimits and λ an ordinal. Aλ-sequence in C is a colimit-preserving functor X : λ→ C. We represent such functor as:

X0 X1 · · · Xβ · · ·

Definition 1.2.7. Let X be a λ-sequence,

• the map X0 → Colimβ<γ Xβ is called the composition of the λ sequence,

• if D is a collection of morphisms of C such that for every β + 1 < λ the mapXβ → Xβ+1 is in D, then the morphism X0 → Colimβ<γ Xβ is called a transfinitecomposition of maps in D.

Remark 1.2.8. If λ = ℵ0, then a λ-sequence is just an ordinary sequence.

Definition 1.2.9. Let γ be a cardinal and α a limit ordinal, we say that α is γ-filteredif A ⊂ γ and |A| < α implies A < α.

We now generalize definition 1.1.7.

Definition 1.2.10. Let C be a category with all small colimits, D a collection of mor-phisms of C, A an object of C and κ a cardinal.

• We say that A is κ-small relative to D if for every κ-filled ordinal λ and everyλ-sequence X : λ→ C such that the arrow Xβ → Xβ+1 is in D every time β < λ,the canonical map

colimβ<λ HomC(A,Xβ) −→ HomC(A, colimβ<λXβ)

is an isomorphism.

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• A is said to be small relative to D if it is κ-small relative to D for some κ.

• A is said to be small if it is small relative to C itself.

Theorem 1.2.11. Every set is small.

Proof. See [Hov99, Example 2.1.5].

1.2.2 Galois connections

Definition 1.2.12. We define a Galois connection as a tetrad (A,B,F ,G) where (A,6),(B,4) are partially ordered sets and F : A→ B, G : B → A are order-reversing functionsthat satisfy:

• a 6 GFa for all a ∈ A

• b 4 FGb for all b ∈ B

What we have defined as a Galois connection is usually called in the literature acoGalois connection or antitone Galois connection, and the term Galois connection isreserved for a similar definition where the functions F and G are monotone instead oforder-reversing.

Notation. A Galois connection (A,B,F ,G) as in the previous definition will also berepresented as in the following diagram:

(A,6) (B,4)

F

G

Remark 1.2.13. It is interesting to note that a Galois connection (A,B,F ,G) can beinterpreted as a dual adjunction.

Indeed, each partially ordered set (X,≤) can naturally be seen as a category whoseobjects are the elements of the set X and, given x, x′ ∈ X, there exists an (unique) arrowfrom x to x′ if and only if x ≤ x′. With this identification, the functions F and G fromthe Galois connection are contravariant functors between the associated categories tothe involved partially ordered sets. We shall make an abuse of notation representing byX both (X,≤) and its associated category.

In this context, F and G are dual adjoints. This is, for each a ∈ A and b ∈ B we havean isomorphism HomB(b,Fa) ' HomA(a,Gb) that is natural in both variables. Thebijection is immediate:

• b 4 Fa⇒ GFa 6 Gb⇒ a 6 Gb, for we always have a 6 GFa because it is a Galoisconnection,

15

• a 6 Gb⇒ FGb 4 Fa⇒ b 4 Fa, for we always have b 4 FGb because it is a Galoisconnection.

The naturality of the bijection is not illuminating nor difficult and it is left as exercise.

Definition 1.2.14. Given a partially ordered set (X,≤), a closure operator over X is afunction cl : (X,≤) −→ (X,≤) verifying the following conditions:

• X ′ ≤ cl(X ′) for all X ′ ∈ X

• X1 ≤ X2 ⇒ cl(X1) ≤ cl(X2) for all X1, X1 ∈ X

• cl(cl(X ′)) = cl(X ′) for all X ′ ∈ X

We will see that every Galois connections induces in a natural way two closureoperators, for this we need the following proposition.

Proposition 1.2.15. Given a Galois connection

(A,6) (B,4)

F

G

the equalities below hold:

1. F = FGF

2. G = GFG

Proof. We start showing assertion 1. For any a ∈ A we have a 6 GFa and henceFGFa 4 Fa. Furthermore, we also know that for any b ∈ B, b 4 FGb holds. Inparticular, considering b = Fa, we obtain Fa 4 FGFa.

We conclude then that Fa = FGFa for all a ∈ A, as we wanted.

The proof for 2 is analogous.

We are now able to prove what we had claimed.

Theorem 1.2.16. Given a Galois connection as in the previous proposition, operatorsFG : (A,6) −→ (A,6) and GF : (B,4) −→ (B,4) are closure operators.

Proof. The first two conditions in the definition of closure operator are trivially verifiedbecause we are dealing with a Galois connection, whereas the third follows from theprevious proposition.

16

Notation. For the sake of clarity we introduce the following notation.

• FG b = b for all b ∈ B

• GF a = a for all a ∈ A

• C(X) = {x ∈ X such that x = x}

Proposition 1.2.17. In the previous context, both functions F : A −→ C(B) andG : B −→ C(A) are surjective.

Proof. We will show it for F : A→ C(B); the proof for G is completely analogous.

Firstly, we note that for every a ∈ A, Fa is effectively an element of C(B). Onone hand we know FG(Fa) = Fa, but for proposition 1.2.15 is FGFa = Fa. Then,Fa = Fa.

Show the surjectivity of F is easy: for any b ∈ C(B) we have b = b= FGb.

1.2.3 Small object argument

In this subsection we work towards a generalization of the small object argument statedbefore, for this we follow [Hov99], where one can find for example lemmas 1.2.24 and1.2.25.

Definition 1.2.18. Let I be a class of maps in a category C.

1. A map is I-injective if it has the right lifting property with respect to every mapin I. We will denote the class of I-injective maps by I-inj.

2. A map is I-projective if it has the left lifting property with respect to every mapin I. We will denote the class of I-projective maps by I-proj.

3. A map is said to be an I-fibration if it has the right lifting property with respectto every I-injective map. The class of I-fibrations is the class (I-proj)-inj and isdenoted I-fib.

4. A map is said to be an I-cofibration if it has the left lifting property with respectto every I-projective map. The class of I-cofibrations is the class (I-inj)-proj andis denoted I-cof.

Although we have used previously a different notation for the maps having the right(left) lifting property, we will use this new one from this point.

Remark 1.2.19. It is easy to observe that with the above definitions we have a Galoisconnection as follows:

17

(P(Arr(C)),⊆) (P(Arr(C)),⊆)

−inj

−proj

Explicitly, if I ⊂ J , then I− inj ⊃ J− inj and I−proj ⊃ J−proj (and in consequenceI− cof ⊂ J− cof and I−fib ⊂ J−fib). Furthermore, we have I ⊂ I− cof and I ⊂ I−fib(the converses are true if C is a model category).

We also have (I − cof)− inj = I − inj and (I − fib)− proj = I − proj.

The following lemma is some times useful.

Lemma 1.2.20. Let C and D be two categories, I a class of maps in C, J a class ofmaps in D and L : C → D and R : D → C functors such that (L,R) is an adjoint pair.Then:

1. R(LI − inj) ⊂ I − inj,

2. L(I − cof) ⊂ LI − cof,

3. L(RJ − proj) ⊂ J − proj,

4. R(J − fib) ⊂ RJ − fib.

Proof. We will prove only the first two assertions, for the last ones are dual to those.

For the first, let g ∈ L(I)− inj and f ∈ I; then g has the right lifting property relativeto L(f). By adjointness, this implies that R(g) has the right lifting property relative tof and thus R(g) ∈ I − inj.

In order to prove the second one, let f be an I-cofibration and g ∈ L(I)− inj. By 1.we deduce that R(g) ∈ I − inj and in consequence f has the left lifting property withrespect to R(g). By adjointness, L(f) has the left lifting property with respecto to g andthus L(f) ∈ L(I)− cof.

Definition 1.2.21. Let C be a category containing all the small colimits, and let I be aset of arrows in C. A relative I-cell complex is a transfinite composition of pushouts ofelements of I. The collection of all relative I-cell complexes is denoted by I-cell.

Explicitly, a map f : A → B in C is a relative I-cell complex if there is an ordinalλ and a λ-sequence X : λ → C such that for every β verifying β + 1 < λ, there is apush-out square as follows:

Cβ Xβ

Dβ Xβ+1

gβ

where the maps Xβ → Xβ+1 are in I and f is the composition of X.

18

Definition 1.2.22. Let C be a category containing all the small colimits, I a set ofarrows in C. We say that an object A is an I-cell complex if the map 0→ A is a relativeI-cell complex.

Remark 1.2.23. The identity map at A is the composition of the trivial 1-sequence A,thus identity maps are relative I-cell complexes. In fact, if f : A→ B is an isomorphism,then f is also the composition of the 1-sequence A, so f is a relative I-cell complex.

Lemma 1.2.24. Let C be a category admitting all small colimits and I a class of mapsin it, then we have the inclusion I − cell ⊂ I − cof.

Proof. It suffices to prove that the class I − cof is closed under pushouts and transfinitecompositions.

We shall firstly consider the case of pushouts. We begin by considering a pushoutdiagram:

A X

B Y

i x

where i is a map in I-cof. We wil prove that x is also an I-cofibration by showing it hasthe left lifting property with respect to every map in I-inj. Let then j be a map in I-inj,we want to show that any commutative diagram as the following has a lifting:

X X

Y D

a

x (1.8)

The left lifting property of i with respect to j give us the dashed arrow in the followingdiagram:

A X C

B Y D

X

i

a

xjh

Now, by the universal property of the pushout we obtain the dashed arrow in thediagram below, that give us a the lifting of diagram 1.8:

A X

B Y

C

i xa

h

This conclude the case of pushouts. We will now prove the closeness under transfinitecompostions.

19

Let X : λ→ C be a λ-sequence such that the correspondent maps in the category Care I-cofibrations and let c be its transfinite composition. We wish to show that thereexist the dashed arrow in the solid commutative diagram below:

X0 A

Xλ B

c f (1.9)

where f is I-injective.

If λ is a successor ordinal the result is direct, for a composition of two I-cofibrationsis again an I-cofibration.

Let α ≤ λ be a limit ordinal and assume that the composition up to α (excluded) isalways an I-cofibration.

Thus, given a commutative diagram as follows:

X0 X1 X2 · · · Xα = colimβ<αXβ

A B

c0

c

c1 c2

f

where the dashed arrows are the given by the induction hypothesis, we can use theuniversal property of colimits to obtain the lift of the diagram 1.9. We conclude fromthis that c is an I-cofibration.

Next we prove an useful property of relative I − cell complexes.

Lemma 1.2.25. Suppose C is a cocomplete category and I is a set of maps in C. Thenany push-out of coproducts of maps in I is in I − cell.

Proof. Suppose we have a push-out diagram:∐k∈K

Ck X

∐k∈K

Dk Y

∐gk

f

where K is a set and every gk : Ck → Dk is a map of I. We must show that f is arelative I − cell complex.

20

Since every set is isomorphic to an ordinal, we assume that K is an ordinal λ. Weconstruct a λ-sequence setting X0 = X and Xβ+1 as the push-out Xβ

∐CβDβ for succesor

ordinals and Xβ = colimα<β Xα for limit ordinals.

One easily checks that the transfinite composition X → Xλ is isomorphic to the mapf , from we conclude that f is a relative I-cell complex.

Definition 1.2.26. Let C be a category. A pair of classes of arrows (L,R) is said to bea weak factorization system if it satisfies:

1. any arrow f in C can be factorized as f = r ◦ l where r ∈ R and l ∈ L,

2. L = R− proj and R = L− inj.

Definition 1.2.27. A weak factorization system (R,L) is said to be functorial if thereexist two endofunctors on the category of arrows, γ and δ, such that the factorization in1.2.26 of any arrow f is given by f = δ(f) ◦ γ(f).

We present now a result that allows us to construct a functorial factorization ofall arrows in a category. Such a factorization will later be used to construct a weakfactorization system.

This is a generalization of theorem 1.1.8 and we will not give a proof this time, butwe remit to the interested reader to a precise reference.

Theorem 1.2.28. Let C be a cocomplete category, I a set of arrows of C with domainssmall relative to I− cell. Then there exist two functors γ and δ on the category of arrowsof C such that γ(f) ∈ I − cell, δ(f) ∈ I − inj and f = δ(f) ◦ γ(f) for every arrow f in C.

Proof. See [Hov99, Theorem 2.1.14].

Using this theorem we construct the announced weak factorization system.

Theorem 1.2.29 (The small object argument). Let C be a cocomplete category, I a setof arrows of C with domains small relative to I-cell. Then the pair (I − cof, I − inj) isa (functorial) weak factorization system.

Proof. By theorem 1.2.28, we have a functorial factorization of every arrow in C withone of the elements in I − cell ⊂ I − cof (lemma 1.2.24). Thus (I − cof, I − inj) is aweak factorization system, for it satisfies the required lifting properties.

Definition 1.2.30. Let C be a category and I a set of maps in C. We say that I permitsthe small object argument if all the domains of the elements of I are small relative toI-cell.

21

Corollary 1.2.31. Let C be a cocomplete category and I a set of maps in C that permitsthe small object argument, let also f : X → Y be an I-cofibration. Then f is a retractof a relative I-cell complex.

Proof. The small object argument (1.2.29) gives us a factorization f = p ◦ i, wherei ∈ I − cell and p ∈ I − inj. But we know by hypothesis that f is an I-cofibration, thusf has the left lifting property with respect to p.

We conclude by lemma 1.2.32 that f is a retract of i, which concludes the proof.

1.2.4 Another definition of model categories

We will now give a new definition of model category, and we will prove that it is equivalentto our original definition (1.1.3).

Definition (Model category (def. 2)). A model category is a bicomplete category C withthree distinguished classes of arrows, namely weak equivalences (WE), fibrations (Fib)and cofibrations (Cofib) such that

• C is complete and cocomplete,

• (two out of three) given f and g two composable maps, if two of the three maps f ,g and gf are weak equivalences, then so is the third,

• (Cofib ∩WE,Fib) and (Cofib, F ib ∩WE) are weak factorization systems.

In what follows we will prove different lemmas in order to proof what we had antici-pated before, that is the following theorem:

Theorem. The two given definitions of model category (namely, 1.1.3 and 1.2.4) areequivalent.

Lemma 1.2.32 (The retract argument). Let f be an arrow in C admitting a factorizationf = pi, then:

1. if f has the left lifting property with respect to p, then f is a retract of i,

2. dually, if f has the right lifting property with respect to i, then f is a retract of p.

Proof. We shall prove statement 1, the proof of 2 is similar.

When we consider the diagram

X Z

Y Y

f

i

p

idY

22

we have a lift r : Y → Z and then f is a retract of i as the diagram displayed belowshows:

X X X

Y Z Y

id

f

id

i f

r p

Lemma 1.2.33. Let I be a set of maps in a cocomplete category C such that the domainof every map in I is small relative to I-cell. Then every map in I-cof is the retract ofsome map in I-cell with the same domain.

Proof. Then, by the small object argument (1.2.29), we have a factorization f = p ◦ iwhere i ∈ I − cell and p ∈ I − inj. Since we know that f is an I-cofibration, this isf ∈ (I − inj)− proj, the retract argument (1.2.32) tells us that f is a retract of i, whichconcludes the proof.

Lemma 1.2.34. Let C be a model category (in the sense of 1.1.3), then the pairs(Cofib, F ib ∩WE) and (Cofib ∩WE,Fib) are weak factorization systems.

Proof. By axiom MC4 we know that every cofibration does have the left lifting propertywith respect to acyclic fibrations, i.e. Cofib ⊂ (Fib ∩WE)− proj.

For the other inclusion, suppose f is an arrow of C in (Fib ∩WE)− proj, considernow the decomposition f = pi where p is an acyclic fibration and i a cofibration givenby axiom MC5.

Since f has the left lifting property with respect to p, the retract argument (1.2.32)implies that f is a retract of i and therefore, by axiom MC3, f is a cofibration.

In order to finish the proof that (Cofib, F ib ∩WE) is a weak factorization systemwe need to prove that Fib ∩WE = Cofib− inj.

The inclusion Fib ∩WE ⊂ Cofib− inj follows directly from axiom MC4.

Suppose now that f is an arrow in Cofib − inj, considering the decomposition wetook before and the retract argument we obtain that f is a retract of i and therefore anacyclic fibration.

The proof for (Cofib ∩WE,Fib) to be a weak factorization system is dual.

This proves that if C is a model category in the sense of 1.1.3 then it is so in thesense of definition 1.2.4.

The converse follows from the lemmas we shall prove hereafter.

Lemma 1.2.35. Let C be a category and A and B two classes of morphisms such thatA = B − proj and B = A− inj, then A and B are closed under retracts.

23

Proof. Let g be a map in A and f a retract of g, thus we have a commuative diagramas follows:

X Y X

X ′ Y ′ X ′

ι

idX

f

r

g f

ι′

idX′

r′

(1.10)

We shall prove that f is in A by showing that it has the left lifting property for everyb ∈ B.

Given b ∈ B and a commutative diagram:

X •

X ′ •

f b(1.11)

We can splice both diagrams, 1.10 and 1.11, and obtain the following:

X Y X •

X ′ Y ′ X ′ •

ι

idX

f

r

g f

ι′

idX′

r′h

(1.12)

where the dashed arrow h is given by the fact that g is in A.

Finally, considering the composition h ◦ ι′ we obtain the desired lift for diagram 1.11.

The proof that B is closed under retracts is similar.

Corollary 1.2.36. If C is a model category in the sense of 1.2.4, the classes of fibrations,cofibrations, acyclic fibrations and acyclic cofibrations are closed under retracts.

Lemma 1.2.37. Let C be a model category in the sense of 1.2.4, then the class of weakequivalence is closed under retracts.

Proof. Let w be a map in WE and f a retract of it, then we have the commutative

24

diagram:

X Y X

X ′ Y ′ X ′

ι

idX

f

r

w f

ι′

idX′

r′

(1.13)

By definition of model category, we can factorize the arrow f as f = p◦ i with p ∈ Fiband i ∈ Cofib∩WE, and thus we can rewrite the diagram 1.13 as follows:

X Y X

Z Z

X ′ Y ′ X ′

ι

idX

i

r

w

i

p p

ι′

idX′

r′

(1.14)

We can now consider the push-out diagram that appears below:

X Y

Z Z ∪X Y

i i1

i2

By the universal property of the push-out we obtain a decomposition of the weakequivalence, w = p1 ◦ i1, as illustrated below:

X Y

Z Z ∪X Y

X ′ Y ′

i

w

i1

i2

p p1

Using again the universal property of the push-out we obtain the dashed arrow in

25

the following diagram:

X Y X

Z Z ∪X Y Z

idX

i i1 i

p1

idZ

From this construction, we can build a new commutative diagram, which shows thati1 is a retract of i:

Y X Y

Z ∪X Y Z Z ∪X Y

i1

id

i i1

id

(1.15)

Since i is an acyclic cofibration, corollary 1.2.36 implies that i1 is also an acycliccofibration.

By the 2-out-of-3 property we deduce that the map p1 is a weak equivalence. It isleft to prove that p is a weak equivalence, in order to conclude from a new applicationof the 2-out-of-3 property that so is f .

We build, from the lower row of diagram 1.15, this new one:

Z Z ∪X Y Z

X ′ Y ′ X ′

i2

id

p p1 p

ι′

id

r′

We can consider a factorization of p1 given by p1 = q ◦ j being q a fibration and jan acyclic cofibration; this factorization allows us to build the following commutativediagram:

Z ∪X Y Z

Z ′ Y ′ X ′

j ph′

q r′

26

where the dashed arrow follows from the fact that the pair (Cofib∩WE,Fib) is a weakfactorization system. We use this lift to show that p is a retract of q, as illustrated inthe following commutative diagram:

Z Z ∪X Y Z ′ Z

X ′ Y ′ X ′

id

p

j h′

q p

id

r′

By lemma 1.2.36, p is an acyclic cofibration, in particular, it is a weak equivalence.

1.2.5 Cofibrantly generated model categories

Definition 1.2.38. We say that a model category C is cofibrantly generated if there aresets I and J of maps such that:

1. the domains of the maps of I are small relative to I-cell,

2. the domains of the maps of J are small relative to J-cell,

3. the class of fibrations is J-inj,

4. the class of acyclic fibrations is I-inj.

We refer to I as the set of generating cofibrations, and to J as the set of generatingacyclic cofibrations.

Note that given the two generating sets, we can reconstruct the sets of cofibrationsand acyclic cofibrations. Those are given by I-cof and J-cof respectively. Indeed,

Cofib = (Fib ∩WE)− proj = (I − inj)− proj = I − cof

and we proceed similarly for acyclic cofibrations.

Remark 1.2.39. If a category is cofibrantly generated, then it follows directly from thesmall object argument (1.2.29) that the weak factorization systems (Cofib, F ib ∩WE)and (Cofib ∩WE,Fib) are functorial.

Remark 1.2.40. One could define a dual notion of fibrantly generated model categories;indeed, a model category is fibrantly generated if and only if its opposite model categoryis cofibrantly generated.

27

However, this new notion it is not as useful as the original since the examples ofmodel categories that natural arise have no “cosmall” objects. A clear example of thisis the category of sets. While any set is small (1.2.11), the only cosmall objects in thecategory of sets are the empty set and the singletons; for details on this see [Hov99, page35].

Theorem 1.2.41. Let C be a complete and cocomplete category. Consider W a class ofmorphisms and I and J sets of maps in C. The following statements are equivalent:

1. There is a cofibrantly generated model category structure on C with I as the setof generating cofibrations, J as generating acyclic cofibrations and W as weakequivalences.

2. The following conditions are satisfied:

(a) W satisfies the 2-out-of-3 property and is closed under retracts,

(b) the domains of the maps in I and J are small relative to I-cell and J-cellrespectively,

(c) J − cell ⊂ (I − cof ∩W ),

(d) I − inj ⊂ (J − inj∩W ),

(e) either (I − cof ∩W ) ⊂ J − cof or (J − inj∩W ) ⊂ I − inj.

Proof. Assume assertion 1 holds. Then by definition of model structure, W satisfies the2-out-of-3 property and by lemma 1.2.37 it is closed under retracts. The domains of themaps in both I and J are small by defnition of cofibrantly generated model category. Bylemma 1.2.24, J − cell ⊂ J − cof = I − cof ∩W . By definition of cofibrantly generatedmodel category, we also know I − inj = J − inj∩W .

Conversely, suppose now that 2 holds. We define the weak equivalences to be thefamily W , the fibrations the set J − inj and cofibrations the set I − cof.

A similar proof of which we gave for lemma1.2.35 proves the closeness under retractsof fibrations and cofibrations. Since the class of weak equivalences is also closed underfibrations, we can deduce that so are the classes Fib ∩WE and Cofib ∩WE.

By hypothesis, every map in I-inj is an acyclic fibration; and every map in J-cell isan acyclic cofibration.

Applying the small object argument (1.2.28) to I and J (and conditions c and d fromthe hypothesis) we obtain the factorizations we need; there are still lifting properties toprove for what we will use the last assertion of the hypothesis.

Suppose first that W ∩ I − cof ⊆ J − cof. On one hand, every acyclic cofibration is inJ − cof and thus it has the left lifting property with respect to the class J − inj, whichis exactly the class of fibrations. Now let p : X → Y be an acyclic fibration, we need toprove that it has the right lifting property with respect to the class of cofibrations (or,what is the same, with respect to I).

We can factor p as p = j ◦ i where i is a cofibration and j is I-injective. By the2-out-of-3 property of W we know that i has to be an acyclic cofibration, this together

28

with what we have just proved implies that p has the right lifting property with respectto i. Thus, p is a retract of j (by the retract argument), and so p is I-injective as wewanted to prove.

The case assuming (J − inj ∩W ) ⊆ I − inj is similar and we will not write the proof.

We will now give a result that allow us to transfer the model structure in a categoryto another in presence of an adjunction.

Theorem 1.2.42. Let C and D be two categories such that C is a cofibrantly generatedmodel category with I the set of generating cofibrations and J the generating acycliccofibrations and D admits all small limits and colimits. Let (F,U) be an adjoint pair offunctors. If the following conditions hold:

1. both FI and FJ permit the small object argument,

2. the functor U takes relative FJ-cell complexes to weak equivalences,

then there exists a cofibrantly generated model structure on D such that FI is the set ofgenerating cofibrations, FJ the set of generating acyclic cofibrations and the weak equiv-alences are the maps such that its image via U are weak equivalences in C. Furthermore,the adjoint pair (F,U) is a Quillen pair with respect to those model structures.

Proof. Let W be the maximal class of arrows in D such that U(W ) are weak equivalencesin C. We will show that W , FI and FJ satisfy the conditions of the second statementin theorem 1.2.41.

Since any functor preserves retracts and compositions, W is closed under retractsand satisfy the two out of three property; this is the condition a of the the statement 2.Part b is just one of our hypothesis.

For condition c, our hypothesis imply that relative FJ-cell complexes are in W .Since every I-injective map is a J-injective map, we know that every FJ-cofibration is aFI-cofibration; this, together with lemma 1.2.24, leads to the condition b.

To show condition d we first note that every map that is FI-injective is also a FJ-injective. On the other hand, proposition 1.1.2 implies that the functor U takes everyFI-injective to an acyclic fibration in C, so FI − inj ⊂W which conclude the proof.

We shall now prove the last condition, e. Suppose g : X → Y is in FJ − inj∩W , oneapplication of proposition 1.1.2 implies that Ug is J-injective and a weak equivalence inthe category C and thus Ug an acyclic fibration and in consequence it is an I-injective map.A new application of proposition 1.1.2 leads to the conclusion that g is an FI-injectivemap.

Finally, since left adjoints preserve colimits we have that F takes all relative I-cellcomplexes to relative FI-cell complexes and all relative J-cell complexes to relativeFJ-cell complexes. And since any functor preserves retracts, we deduce from 1.2.31 thatF is a left Quillen functor.

29

1.3 Homotopy category

Definition 1.3.1. Let C be a model category and W ⊆ Arr(C) the class of weakequivalences. We define the homotopy category of C, denoted Ho(C), as the localizationof C with respect to W , i.e. Ho(C) = C[W−1].

Since it is a localization, it is determinated by the following universal property: letC → Ho(C) be the localization functor, then for any functor F : C → D such that mapsevery arrow in W to an isomorphism in D there exists a functor F : Ho(C) → D suchthat the following diagram commutes:

C D

Ho(C)

F

F

A priori the localization with respect to a class of arrows is not a category, but it isthe case in model categories.

Definition 1.3.2. Let C be a model category and X an object in C; we write 0 and 1its initial and final object respectively. Then

• X is said to be fibrant if the unique map X → 1 is a fibration,

• X is said to be cofibrant if the unique map 0→ X is a cofibration.

Remark 1.3.3. Given a model category C, an arrow f in it is a weak equivalence if andonly if it is an isomorphism in the category Ho(C).

Definition 1.3.4. Let C be a model category. Then

• a fibrant replacement is a functor P : C → C such that assigns to any X a fibrantobject in C, together with a natural transformation idC ⇒ P which is a weakequivalence for every object,

• a cofibrant replacement is a functor Q : C → C such that assigns to any X acofibrant object in C, together with a natural transformation Q⇒ idC which is aweak equivalence for every object.

Lemma 1.3.5. If C is a cofibrantly generated model category then there exist fibrant andcofibrant replacement.

Proof. Let X be an object of C, we can factor it in the following way:

X 1

P (X)

iX pX

30

where i is a trivial cofibration and p a fibration.

Since we are in a model category, this decomposition is functorial and thus theassociation X 7→ P (X) is a functor. The family {iX}X form a natural transformationverifying that it is a weak equivalence for every object.

The cofibrant replacement is defined in a similar way.

It can be shown using these replacements that when C is a model category, Ho(C) iseffectively a category (see [GJ99, Chapter II, theorem 1.11 and remark 1.12]).

1.4 Quillen adjunctions

Definition 1.4.1. Let C and D two model categories.

• a functor L : C → D is a left Quillen functor if it is a left adjoint and it preservescofibrations and acyclic cofibrations,

• a functor R : D → C is a right Quillen functor if it is a right adjoint and it preservesfibrations and acylclic fibrations.

Definition 1.4.2. Let C and D be two model categories and let L : C → D andR : D → C two functors such that (L,R) is an adjoint pair. The adjunction is calledQuillen adjunction if L is a left Quillen functor and R is a right Quillen functor.

Lemma 1.4.3. Let C and D be two model categories and let L : C → D and R : D → Ctwo functors such that (L,R) is an adjoint pair. Then the following statements areequivalent:

1. R is a right Quillen functor,

2. L is a left Quillen functor,

3. (L,R) is a Quillen adjunction.

Proof. We will begin by proving that 1. implies 2. Since we already know that L is a leftadjoint, it remains to prove that it preserves cofibrations and acyclic cofibrations.

Recall that we have Cofib = (Fib∩WE)−proj and Cofib∩WE = Fib−proj. Andlet ϕ be the natural isomorphism ϕ : HomD(L−,−) =⇒ HomC(−, R−) given with theadjoint pair (L,R).

Consider now f to be a cofibration in C and p an acyclic fibration in D -and since Ris a right Quillen functor R(p) is an acyclic fibration in C. Thus, for every commutingsolid diagram as follows:

X R(Z)

Y R(W )

k

f R(p)a

l

31

there exists the dashed lifting.

By naturality of ϕ we can deduce from the previous the following commutativediagram:

L(X) Z

L(Y ) W

ϕ−1(k)

L(f) p

ϕ−1(l)

ϕ−1(a)

This implies that L(f) has the left lifting property with respect to acyclic fibrations,meaning that L(f) is a cofibration in D.

Analogously, we prove that 2. implies 1. Once we have the equivalence between thefirst two conditions the proof is finished.

In case we are dealing with cofibrantly generated model categories, it is easier toshow that an adjunction is Quillen. We prove a result in that direction.

Lemma 1.4.4. Let C and D be cofibrantly generated model categories with a set ofgenerating cofibrations I and a set of generating acyclic cofibrations J . The followingare equivalent:

1. (L,R) is a Quillen adjunction,

2. for every i ∈ IC and j ∈ JC, we have that L(i) is a cofibration and L(j) is anacyclic cofibration.

Proof. It follows directly from the definiton of Quillen adjunction that 1. implies 2. Nowassume that 2. holds, we will show that L is a left Quillen functor, which is enough bylemma 1.4.3.

Now, by lemma 1.2.20 we knoe that L(IC − cof) ⊆ L(IC)− cof. By hypothesis we alsoknow that L(IC) is included in the cofibrations of D, this is in ID − cof. Thus we have

L(IC − cof) ⊆ L(IC)− cof ⊆ ID − cof

We have showed that L preserves cofibrations. Similarly, we could show that it preservesacyclic cofibrations.

Given a Quillen adjunction (L,R) between model categories

C DL

R

32

we can construct an adjunction (called the derived adjunction) in the respectivehomotopy categories as indicated below

Ho(C) Ho(D)

LQ

RP

For details on this we refer the reader to [DS95, theorem 9.7].

The functors involved in the new adjunction are called the derived functors of R andL.

Remark 1.4.5. We will observe the unit and counit of the new adjunction.

Let X be a cofibrant object in Ho(C), then the unit at X is given by the composition

X RL(X) RPL(X)η R(iLX)

where η is the unit of the original adjunction (L,R), and iLX : L(X) → PL(X) is themap given by the fibrant replacement of L(X).

For a X not necessarily cofibrant, the unit is given by

X Q(X) RPLQ(X)∼

where the left arrow is the map from the cofibrant replacement of X. Note that it is aweak equivalence and therefore it is invertible in the homotopy category.

The counit is constructed in a similar way.

1.5 Quillen equivalences

Definition 1.5.1. Let C and D be model categories. A Quillen equivalence betweenC and D is a Quillen adjunction (L,R) with the natural isomorphism associatedϕ : HomD(L−,−) =⇒ HomC(−, R−) satisfying that for every cofibrant objecct X ∈ Cand every fibrant object Y ∈ D, a map f : L(X)→ Y is a weak equivalence in C if andonly if ϕ(f) : X → R(Y ) is a weak equivalence in D.

Theorem 1.5.2. Let (L,R) be a Quillen adjunction. Then the following are equivalent:

1. (L,R) is a Quillen equivalence,

2. the categories Ho(C) and Ho(D) are equivalent categories. Moreover, the equivalenceis given by the derived adjunction (LQ,RP ).

Proof. The first thing we should do is to recall that if (F,G) is an adjoint pair suchthat both F and G are full and faithful functors, they are equivalences of categories.And that F is full and faithful is and only if the unit of the adjunction is a natural

33

isomorphism, and G is full and faithful if and only if the counit of de adjunction is anatural isomorphism.

Assume 1. holds and let X be a cofibrant object. We know that the map

LX PLXiLX (1.16)

is a weak equivalence.

Since LX is cofibrant (because L is a left Quillen functor) and PLX is fibrant and(L,R) is a Quillen adjunction, we have that the associated map to 1.16:

X RPLX

given by the adjunction is also a weak equivalence.

This implies that for every object X, the unit of the adjunction (LQ,RP ) at X (givenin ??) is a weak equivalence in C, and thus an isomorphism in Ho(C).

Dually, we show that the counit of the adjunction (LQ,RP ) is also an isomorphism.By the recall we did at the beginning, this implies that LQ and RP are equivalences ofcategories.

We proceed now to prove that 2. implies 1. Let X ∈ C be a cofibrant object, Y ∈ Dfibrant and f : LX → Y a weak equivalence in D. We shall prove that the map ϕ(f)

X RLX RYη R(f)

given by the adjunction is also a weak equivalence.

Thus we have the following commutative diagram:

X RLX RY

X RPLX RPY

η R(f)

R(iLX) R(iY )

∼ RP (f)

(1.17)

where iLX and iY are obtained from the fibrant replacement (and thus they are weakequivalences), and the map X → RPLX is a weak equivalence by hypothesis (since itis an equivalence of categories, the unit and counit of the adjunction of 2. are actuallynatural isomorphism).

Now, since f is a weak equivalence, so is P (f), as we illustrate in the upper squareof the following diagram:

LX Y

PLX PY

1 1

f

Cofib∩WE3 ∈Cofib∩WE

P (f)

Fib3 ∈Fib

(1.18)

34

Since R is a right Quillen functor it preserves weak equivalences between fibrantobjects and thus RP (f) (see diagram 1.18) and R(iY ) are weak equivalences. Finally,we conclude applying the 2-out-of-3 property of WE in diagram 1.17.

Remark 1.5.3. If (L,R) is a Quillen adjunction which is also an equivalence of categoriesthen it is a Quillen equivalence.

This follows directly from the proof of theorem 1.5.2 (in which we worked with theadjunctions) and the fact that R and L preserves weak equivalences between fibrantobjects and cofibrant objects respectively.

D. Dugger and B. Shipley gave in [DS09] an example showing that not every equiva-lence between homotopy categories of model categories lifts to a Quillen equivalence.

35

Chapter 2

Simplicial generalities

For this chapter, besides the bibliography already mentioned, we have also made use of[May67], which is a standard text in the subject, and [Fri12].

2.1 The simplex category

In order to define the notion of simplicial objects we have to construct a category whichencodes, in an elementary way, the structure we wish to obtain.

We define the simplex category, ∆, as the category whose objects are the finitetotally ordered sets [n] = 0, 1, . . . , n (with the usual order) for every n ≥ 0, and whosemorphisms are the order preserving maps α : [m]→ [n].

Remark 2.1.1. We have two interpretations of the category ∆ that might be insightful:

1. we can interpret it as the category of finite ordinal numbers, therefore a fullsubcategory of the category of all linearly orderer sets, Ord,

2. and we also can interpret it as a full subcategory of Cat, taking each object [n] asthe category induced by the preorder in [n].

Remark 2.1.2. An arrow α in ∆ is a monomorphism (resp. epimorphism) if and only ifthe function α is injective (resp. surjective).

In what follows we will regard some particular maps or characteristics in the category∆ that will leads to relevant characteristics in our definition of simplicial objects.

The simplex category is combinatoric in essence and therefore it can be studiedin that way. For example, one can easily observe that the cardinality of the hom-setHom∆([i], [n]) is the combinatoric number

(n+i+1i+1

).

A priori, one could think that hom-sets are a complete chaos in this category; withthe aim to dissolve this impression we will introduce special maps that will allow usto make sense out of this apparent chaos. Fixed n, for each i we define the face map

36

εi : [n− 1]→ [n] as the unique injective map in ∆ whose image skips i; analogously, foreach i we define the degeneracy map ηi : [n+ 1]→ [n] as the unique surjective map in ∆such that maps two elements to i.

Explicitly, we have defined the face and degeneracy maps as follows:

εi(j) =

{j if j < i

j + 1 if j ≥ i, ηi(j) =

{j if j ≤ ij − 1 if j > i

(2.1)

Lemma 2.1.3. Every arrow α : [m]→ [n] in ∆ admits a unique epi-mono decomposition

m n

p

α

η ε

where the monomorphism ε is a composition of faces ε = εi1 · · · εil and the epimorphismη is a composition of degeneracies η = ηj1 · · · ηjt, verifying 0 ≤ il < · · · < i1 < n and0 ≤ j1 < · · · < jt < m.

Proof. It is easy to check that any monotone function α is determined by its image (asubset of {0, . . . , n}) and by the set of those elements j ∈ {1, . . . ,m} at which α doesnot increase.

Let {i1, . . . , il} the elements of [n] not in the image of α ordered in the inverse senseand {j1, . . . , jt} the element of [m] where α does not increase. Then is straightforwardthe equality α = εi1 · · · εil ◦ ηj1 · · · ηjt .

Applying the previous lemma to the composition of any two degeneracy or face mapswe obtain the following result, which will be useful later on:

Corollary 2.1.4 (Cosimplicial identities). In the category ∆ the following equalities aresatisfied:

1. εjεi = εiεj−1 if i < j,

2. ηjηi = ηiηj+1 if i ≤ j

3. ηjεi =

εiηj−1 if i < j

id if i = j or i = j + 1

εi−1ηj if i > j + 1

Proof. The proof of these identities is a direct application of lemma 2.1.3.

Remark 2.1.5. The identities of corollary 2.1.4 can be proven directly from the explicitdescription of faces and degeneracies given in 2.1.

37

Proposition 2.1.6. The category ∆ is generated in arrows by the face maps εni : [n]→[n− 1] and degeneracy maps ηnj : [n+ 1]→ [n] verifying the relations stated in corollary2.1.4.

Proof. It is clear, being that such relations allow us to put any arrow in ∆ in the formof the composition in lemma 2.1.3.

2.2 Simplicial objects

Definition 2.2.1. Let C be a category. A simplicial object A in C is a contravariantfunctor from ∆ to C, this is, a functor A : ∆op → C.

Remark 2.2.2. A simplicial object in C is simply a presheaf on the simplex category withvalues over C.

Remark 2.2.3. For any category C, we can consider the category sC whose objects areall the simplicial objects in C and its morphisms natural transformations between them(i.e. the category of presheaves on ∆ with values over C).

Analogously, we have the definition of a cosimplicial object (and the associatedcategory):

Definition 2.2.4. Let C be a category. A cosimplicial object C in C is a functor fromC : ∆→ C.

Notation. Given a simplicial object A we will write An = A([n]), and for a cosimplicialobject C we will write Cn = C([n]).

Example 2.2.5 (Constant simplicial object). Let C be an object in a category C . Thesimplest example of a simplicial object in C is the constant contravariant functor withvalue C.

Example 2.2.6 (Topological simplices). In this example we will show the widely usedn-simplices as a cosimplicial object in the category Top. We have a functor:

∆ Top

n r([n])

where r([n]) represents the topological standard n-simplex, this is,

r([n]) = {(t0, . . . , tn) ∈ Rn+1 :∑

ti = 1, ti ≥ 0}

with the relative topology.

38

In arrows, for any arrow α : [m] → [n] in ∆ we define α∗ : r([m]) → r([n]) byα∗(t0, . . . , tm) = (s0, . . . , sn), where:

si =

0 if α−1(i) = ∅∑j∈α−1(i)

tj if α−1(i) 6= ∅

To check that the above effectively defines a functor is an easy exercise.

The next theorem show which is the relevant information in the category ∆ that wewant to keep in the definition of simplicial object, giving us at the same time the classicaland more explicit definition.

Theorem 2.2.7. The set of simplicial objects A in a category set C is in bijection withthe triples ({An}, {∂i : An → An−1}i∈{0,1,...,n}, {σi : An → An+1}i∈{0,1,...,n})n∈N satisfyingthe following conditions:

1. ∂i∂j = ∂j−1∂i if i < j,

2. σiσj = σj+1σi if i ≤ j

3. ∂iσj =

σj−1∂i if i < j

id if i = j or i = j + 1

σj∂i−1 if i > j + 1

Proof. If A is a simplicial object in C, we can obtain the data in the triples above bysetting An = A([n]), ∂ni = A(εni ) and σni = A(ηni ). It follows directly from corollary 2.1.4that the equations in the statement are satisfied.

Conversely, we define a contravariant functor A : ∆→ C by setting A([n]) = An andfor each map in ∆ written in the form of the proof of the lemma 2.1.3, α = εii · · · ηjt ,A(α) = σjt · · · ∂i1 .

Terminology. We will call faces the maps ∂i and degeneracies the maps σi; the elementsof An will be called n-simplices.

Notation. Given a simplicial set A : ∆op → C, we will use A to refer both the functorand the triple in the previous proposition.

Remark 2.2.8. In the light of this characterization (or new and equivalent definition) ofsimplicial objects, one could wish to interpret in the same way what morphisms betweensimplicial objects are. As we have said earlier, given two simplicial objects over a categoryC, let say A : ∆op → C and A′ : ∆op → C, morphisms between A and A′ in the categorysC are just the natural transformations τ : A→ A′.

This translates in the following: a morphism between two simplicial sets in C,f : A → A′, is a family {fn : An → A′n} of arrows in C such that commutes with

39

faces and degeneracies, this is: {fn∂i = ∂ifn+1

fnσi = σifn−1

If we dualize the above theorem, we get cosimplicial objects:

Theorem 2.2.9. The set of cosimplicial objects C in a category set C is in bijection withthe triples ({Cn}, {∂i : Cn−1 → Cn}i∈{0,1,...,n}, {σi : An+1 → An}i∈{0,1,...,n})n∈N satisfyingthe following conditions:

1. ∂j∂i = ∂i∂j−1 if i < j,

2. σjσi = σiσj+1 if i ≤ j

3. σj∂i =

∂iσj−1 if i < j

id if i = j or i = j + 1

∂i−1σj if i > j + 1

2.3 Simplicial sets

In this section we will study a particular case of simplicial objects, the simplicial objectswith values in the category Set.

Definition 2.3.1. A simplicial set is a simplicial object in the category Set, i.e. afunctor ∆op → Set (or, equivalently, a triple as in theorem 2.2.7).

Example 2.3.2. [Standard n-simplex] For each [n] ∈ ∆ we can define the simplicial set∆n = Hom∆(−, [n]), defined obviously on arrows.

This simplicial set, called standard n-simplex is nothing but the representable con-travariant functor on ∆ with representative [n].

In the following proposition we give a description for the n-simplices of a simplicialset X.

Proposition 2.3.3. Let X be a simplicial set, then for each n we have a natural iso-morphism Xn = HomsSet(∆

n, X).

Proof. This is an immediate corollary of Yoneda’s lemma, since it states a naturalisomorphism

HomFct(Hom∆(−, [n]), X) ' X([n])

So far we have seen a particular example of simplicial sets with a natural and simpledefinition that ends up giving a way to classify the n-simplexes of any simplcial set.However, the particularity and relevance of these objects is deeper than that, fact whichis apparent from the following proposition:

40

Proposition 2.3.4. Any simplicial set is canonically a colimit of standard n-simplices.

Proof. We know that any presheaf on a category C is canonically a colimit of representablepresheaves, the result follows if we consider C = ∆ from the comments on the example2.3.2.

In what follows we define intuitive subsumplicial sets and certain relevant functors.

Definition 2.3.5 (Boundary). The boundary of the simplicial set ∆[n], denoted ∂∆[n],is defined to be the coequalizer gluing all (n−1)-simplices together along their boundariesaccording to the face maps:∐

[n−2]→[n]inj.

∆ [n− 2]∐

[n−1]→[n]inj.

∆ [n− 1] ∂∆ [n]

Definition 2.3.6 (k-th horn). Given the standard n-simplex ∆n,with n ≥ 1, for each0 ≤ k ≤ n we define its k-th horn, denoted by Λk[n], to be the coequalizer gluing togetherall (n− 1)-simplices as follows:

∐[n−2]→[n]inj.k in the image

∆ [n− 2]∐

[n−1]→[n]inj.k in the image

∆ [n− 1] Λk [n]

This is, the k-th horn of the simplicial set ∆[n] is the sub-simplicial set obtained byconsidering the image of face maps {∂j(idn)}j 6=k.

2.3.1 Geometric realization

Definition 2.3.7 (End). Let C and D be categories, being D complete. For a functorF : Cop × C → D we define its end as the following equalizer:

∫c∈C

F (c, c)∏c∈C

F (c, c)∏

ϕ:c→c′F (c, c′)

F ∗

F∗

Where the product over ϕ : c → c′ should be viewed as a double product (over theobjects c, c′ ∈ C and over the arrows ϕ between these two fixed objects); and the arrowsF ∗, F∗ are the ones easily obtained considering the arrows whose (ϕ, c, c′)-component isF (id, ϕ) and F (ϕ, id) respectively.

Remark 2.3.8. Given two functors F,G : C → D we can consider a new functor as follows:

HomD(F−, G−) : Cop × C Set

Since Set is a complete category, we can consider the end of this functor. And it followsdirectly from the definition that:∫

c∈CHomD(Fc,Gc) = Nat(F,G)

41

where Nat(F,G) is the set of natural transformations from F to G.

We have the analogous notion of coends, defined as follows.

Definition 2.3.9 (Coend). Let C and D be categories, being D cocomplete. For afunctor F : Cop × C → D we define its coend as the following coequalizer:

∐ϕ:c→c′

F (c′, c)∐c∈C

F (c, c)

∫ c∈CF (c, c)

F ∗

F∗

where the arrows and indexation are analogous to the ones in end’s definition.

We are able now to define the geometric realization functor, that associates to eachsimplicial set a topological space.

First, given a simplicial set K, we define a functor as follows:

∆op ×∆ Top

([n] , [m]) Kn × r([m])

FK

where we are endowing Kn with the discrete topology, r([m]) with the induced topologyby Rn+1 and considering the product topology in Kn × r([m]).

Definition 2.3.10. Given a simplicial object K we define its geometrical realization asthe coend of the functor FK , this is

|K| =∫ [n]∈∆

FK([n] , [n])

Note that we actually have a functor, | − | : sSet→ Top.

Remark 2.3.11. Since the geometric realization is actually the coequalizer:

∐ϕ:[n]→[m]

Km × r([n])∐

[n]∈∆

Kn × r([n])

∫ [n]∈∆

Kn × r([n])F ∗

F∗

we know the explicit form of it. Indeed, we have:

|K| =∐n

Kn × r([n])/∼

where the relation ∼ is given by: (x, s) ∈ Kn×r([n]) and (y, t) ∈ Km×r([m]) are relatedif and olny if there exists ϕ : [n]→ [m] such that K(ϕ)(y) = x and r(ϕ)(s) = t; with thequotient topology.

42

Another important remark is that the following diagram is commutative:

∆ sSet

Top

∆[−]

r |−|

2.3.2 Singular functor

Now we will go in the opposite direction, this is we will build a simplicial set out of atopological space.

We define the functor Sing : Top → sSet as follows. Let X be a topologicalspace, firstly, we consider as n-simplices the set HomTop(r([n]), X), or what is the same,Hom Top(|∆[n]|, X). Then we define face and degeneracy operators (at level n) to be:

∂i(f)(t0, . . . , tn−1) = f(t0, . . . , ti−1, 0, ti, . . . , tn−1)

σi(f)(t0, . . . , tn+1) = f(t0, . . . , ti−1, ti + ti+1, ti+2, . . . , tn+1)

The functor is defined obviously on arrows.

The fact that we are taking as simplices the continuous functions (this is, the arrowsin the category Top) is of great significance in order to distinguish between topologicalspaces with the same underlying set. We illustrate this in the following example.

Example 2.3.12. Let us consider the set X = {0, 1}.

• If we consider X with the indiscrete topology then every map r([n]) → X iscontinuous, thus Sing(X)n = {0, 1}r[n].

• However, if we consider X with the topology whose open sets are ∅, X and {0} weobtain that Sing(X)n is in bijection with the open subsets of r([n]), correspondingto the possible preimages of 0.

2.3.3 Adjunction

These two functors form in fact an adjoint pair, namely (| − |, Sing)1.

Let K be a simplicial set and X a topological space. The following chain of equalities

1Moreover, this is a Quillen adjunction and it could be used to prove the Quillen equivalence betweenthe categories sSet and Top -using the same tools we will use later with simplicial modules and chaincomplexes.

43

and isomorphisms prove the adjunction:

HomsSet(K,Sing(X)) '∫

[n]∈∆

HomSet(Kn, Sing(X)n) (remark 2.3.8)

=

∫[n]∈∆

HomSet(Kn,HomTop(r([n]), X))

'∫

[n]∈∆

HomTop(Kn × r[n], X) (2.2)

' Hom>

(∫ [n]∈∆

Kn × r[n], X

)| − | is a colimit

HomSet(|K|, X)

Where the isomorphism of 2.2is given by associating to each arrow f in the setHomSet(Kn,HomTop(r([n]), X)), the element g ∈ HomTop(Kn × r[n], X) defined byg(k, x) = f(k)(x).

2.3.4 Homotopy groups

Definition 2.3.13 (Kan condition). The simplicial setX satisfies the Kan condition if forany collection of (n−1)-simplices x0, . . . , xk−1, xk+1, . . . , xn in X such that ∂iXj = ∂j−1xifor any i < j with i 6= k and j 6= k, there is an n-simplex x ∈ X such that ∂ix = xi forall i 6= k.

Although the Kan condition as defined in 2.3.13 has advantages in terms of concisenessfor encoding combinatorial information, it lacks of conceptual conciseness. We invert theroles with the following equivalent formulation:

Definition (Kan condition). The simplicial set X satisfies the Kan condition if anymorphism of simplicial sets Λnk → X can be extended to a simplicial morphism ∆n → X.

Definition 2.3.14. A simplicial set satisfying the Kan condition is called fibrant.

We will now give two important cases of simplicial sets satisfying the Kan condition.

Proposition 2.3.15. Given a topological space X, the simplicial set Sing(X) does satisfythe Kan condition.

Proof. Consider a morphism of simplicial sets f : Λnk → Sing(X). This is equivalentto associate to each n − 1 face, ∂i∆

n with i 6= k, of ∆n a n − 1-simplex of Sing(Y ),ηn : |∆n−1| → Y .

Every other simplex of Λnk is a face or a degeneracy of a face of one of these (n− 1)-simplices, and therefore the rest of the map f is determined by this data.

By the adjunction 2.3.3 we have a map g : |Λnk | → X. Now, since the pair (|∆n|, |Λnk |)is homeomorphic to (In−1 × I, In−1 × 0), we can consider a continuous retractionπ : |∆n| → |Λnk |. Defining η = gπ : |Λn| → X we obtain a n-simplex of Sing(X) such

44

that ∂iη are precisely the simplices ηi for all i 6= k. Applying the adjunction 2.3.3 onemore time we obtain the desired extension.

Proposition 2.3.16. If G is a simplicial object in the category of groups, then theunderlying simplicial set is fibrant. In consequence, the same holds for simplicial abeliangroups and simplicial R-modules.

Proof. Consider elements x0, . . . , xk−1, xk+1, . . . , xn ∈ Gn−1 such that ∂ixj = ∂j−1xi fori < j with i 6= k.

We will reason by induction: we use induction on r to find gr ∈ Gn such that∂i(gr) = xi for all i ≤ r with i 6= k.

For the base case we set g−1 = e ∈ Gn and suppose that we have defined g = gr−1.If r = k, we simply set gr = g; if r 6= k we consider gr = g(σku)−1, where u = x−1

r (∂rg).

Note that in the late case, if i < r and i 6= k we have ∂i(u) = e and in consequencegr satisfies the inductive hypothesis.

The element gn−1 thus satisfies ∂i(y) = xi for all i 6= k.

Simplicial homotopy groups

Definition 2.3.17. Let X be a simplicial set. We say that two n-simplices x, x′ arehomotopic if:

• ∂ix = ∂ix′ for 0 ≤ i ≤ n, and

• there exists a simplex y ∈ Xn−1 such that

– ∂ny = x,

– ∂n+1y = x′, and

– ∂iy = σn−1∂ix = σn−1∂ix′ for all 0 ≤ i ≤ n− 1.

The idea here is that we ask for a simplex y that we can imagine as having x onone face and x′ on another face, both of those faces sharing an edge and their othercorresponding edges are collapsed together as degeneracies.

We now prove a lemma that will be fundamental in the definition of simplicialhomotopy groups.

Lemma 2.3.18. If X is a fibrant simplicial set, the relation ∼ of homotopy of simplicesis an equivalence relation.

Proof. The reflexivity is immediate; indeed, given x ∈ Xn we take y = σn(x) and thuswe have

∂nσn(x) = x = ∂n+1σn(x)

45

and ∂iσn(x) = σn−1∂i(x) for all 0 ≤ i < n.

We will show the transitivity and reflexivity at once. Suppose x ∼ x′ and x ∼ x′′, wemust show x′ ∼ x′′.

Let y′ an homotopy between x and x′, and let y′′ an homotopy between x and x′′.Then the n+ 2 (n+ 1)-simplices

∂0σnσnx′, · · · , ∂n−1σnσnx

′, y′, y′′

verify the compatibility condition on definition 2.3.4, thus being that X is fibrant, thereexists an (n + 2)-simplex z such that ∂i(z) = ∂iσnσn(x′) for 0 ≤ i < n, ∂n(z) = y′ and∂n+1(z) = y∂∂ .

It follows from it that

• ∂i∂n+2(z) = σn−1∂i(x′) for all 0 ≤ i < n,

• ∂n∂n+2(z) = x′, and

• ∂n+1∂n+2(z) = x′′

This means that z is an homotopy between x′ and x′′.

We are now able to define the homotopy groups for fibrant simplicial sets.

Definition 2.3.19. Suppose X is a fibrant simplicial set and ∗ ∈ X0 a basepoint. Byabuse of notation, we write ∗ for the element σn0 (∗) ∈ Xn; and we also set

Zn = {x ∈ Xn : ∂i(x) = ∗ for all i− 0, . . . , n}

We define, for each n > 0, the n-th homotopy group of X with basepoint ∗ as

πn(X, ∗) = Zn� ∼

where ∼ is the relation of homotopy in simplices.

Example 2.3.20. Given a simplicial set X, looking at the case n = 0 we have

π0(X) = X0� ∼

where for each y ∈ X1 is verified ∂0(y) ∼ ∂1(y).

This is not the unique definition of simplicial homotopy groups, but we have chosenthis one for being purely combinatorial and thus suits well with the presentation we havebeen doing. For a further discussion on different ways to define the simplicial homotopygroups we refer the reader to [Fri12] and [May67].

We could define a group structure in the homotopy groups for n ≥ 1, but we will notdo it. Instead, we will state later a result from which we can deduce it.

46

Remark 2.3.21. In the category of simplicial R-modules, we note that πn(A) = Hn(N(A)).

We refer the interested reader to [May67, thm. 17.4] and [GJ99, p. 153] for proofsof this fact when considering different definitions of the homotopy groups.

Example 2.3.22 (Classifying space). Let G be a group, then we can define a simplicialset BG defined by BG0 = {e}, BG1 = G, · · · , BGn = Gn, · · · . With face maps definedby

∂i(g1, · · · , gn) =

(g2, · · · , gn) if i = 0

(g1, · · · , gigi+1, · · · , gn) if 0 < i < n

(g1, · · · , gn−1) if i = n

and degeneracy maps

σi(g1, · · · , gn) = (g1, · · · , gi, 1, gi+1, · · · , gn)

We call to BG the classifying space of the group G.

Example 2.3.23. Consider a group G and its classifying space BG. Then when itsnormalized Moore complex is built we obtain:

N(BG)i =

Ker(d0 : G→ {e}) = G if i = 1i−1⋂k=0

Ker(dk : BGk → BGk−1) = {e} otherwise

In consequence, the homotopy groups are

πn(BG) = Hn(N(BG)) =

{G if n = 1

{e} if n 6= 1

Definition 2.3.24 (Eilenberg-MacLance space). If G is a group, an Eilenberg-MacLanespace of type K(G,n) is a fibrant simplicial set K such that πn(K) = G and πi(K) = 0for i 6= n.

Remark 2.3.25. The classifying space BG is an Eilenberg-MacLane space of type K(G, 1).

2.4 Simplicial homotopies

Definition 2.4.1. Let A and B two simplicial objects in a category C. Two simplicialmaps f, g : A→ B are said to be homotopic if for each n there is a family of morphismin C, hi : An → Bn+1 for i = 0, . . . , n such that:

· ∂0h0 = f ,

· ∂n+1hn = g,

47

· ∂ihj =

hj−1∂i if i < j

∂ihi−1 if i = j 6= 0

hj∂i−1 if i > j + 1

· σihj =

{hj+1σi if i ≤ jhjσi−1 if i > j

When C is an abelian category or we are working with simplicial sets we have a clearerway to see simplicial homotopies, we state that in the following proposition.

Proposition 2.4.2. Suppose that C is either an abelian category or Set. Let A,B besimplicial objects and f, g : a→ B two simplicial maps.

There is a bijection with simplicial homotopies between f and g and simplicial mapsA×∆1 → B such that the following diagram commutes:

A A×∆1 A

B

ϕ0

fh

ϕ1

g

where ϕ : A→ A×∆1 is the composite A A×∆0 A×∆1' , with thesecond map the one induced by the map ε0 : [0] → [1] in the simplex category; ϕ1 isdefined analogously from ε1 : [0]→ [1].

Proof. This follows directly from the definitions.

48

Chapter 3

The Dold-Kan correspondence

The aim of this chapter is to prove the well known correspondence quoted in the title,which is an equivalence between the category of simplicial abelian groups and the categoryof non negative chain complexes over Ab. In fact, instead of Ab we can consider thecategory of (right or left) modules over a ring with unity R without increasing either theconceptual nor technical difficulties of the proof, and that is what we will do.

The relevance of this equivalence is apparent since it relates the simplicial world(although we will prove it for the category of modules over a ring, it holds for simplicialobjects in any abelian category) with the widely developed world of homological alge-bra. Besides that, we can interpret the Dold-Kan equivalence as a suggestion to studysimplicial objects in non-abelian context.

After the proof, we will see how the Dold-Kan correspondence enable us to constructthe known Eilenberg-MacLane spaces K(G,n) for any group G and non negative integern.

The strategy for the proof will be to construct functors sR-Mod → Ch+(R) andCh+(R)→ sR-Mod such that the composites are the respective identities.

3.1 From sR-Mod to chain complexes

Let R be a unital ring and R-Mod the category of (left or right) modules over R.

There are several ways of make a chain complex out of a simplicial abelian R-module.In this section we will show four of them which are closely related in a sense we willmake more precise later. One of the four constructions will be the chosen one to provethe equivalence we wish to establish.

3.1.1 Building up complexes from a simplicial module

In this subsection, A will be a simplicial R-module.

Definition 3.1.1 (Alternating face maps complex). We define the complex C(A) by

49

putting the n-simplices at level n, this is C(A)n = An, and setting the differentials

dn : An → An−1 to be dn =n∑i=0

(−1)i∂i.

Remark 3.1.2. The verification that C(A) is effectively a complex is easy from thesimplicial identities (conditions 1, 2 and 3 in 2.2.7).

Thus we have defined a functor (with the obvious behavior on arrows) given by:

sR-Mod Ch+(R-Mod)

A C(A)

C

Example 3.1.3 (Hochschild homology). Let k be a commutative ring and R a k-algebrawith unity. Let M be an R-bimodule. We can thus define a simplicial k-module A bysetting An = M ⊗R⊗n (with A0 = M) and degeneracies and faces as:

∂i(m⊗ r1 ⊗ · · · ⊗ rn) =

r1m⊗ r2 ⊗ · · · ⊗ rn if i = 0

m⊗ r1 ⊗ · · · ⊗ riri+1 ⊗ · · · ⊗ rn if 0 < i < n

r1 ⊗ · · · ⊗ rn−1 ⊗mrn if i = n

for all ri ∈ R and m ∈M , and

σi(m⊗ r1 ⊗ · · · ⊗ rn) = m⊗ r1 ⊗ · · · ⊗ ri ⊗ 1⊗ ri+1 ⊗ · · · ⊗ rn

for all ri ∈ R and m ∈M .

Standard computations show that these maps verify the simplicial identities. And onecan observe that the homology of the unnormalised complex associated to this simplicialk-module is the Hochschild homology.

Remark 3.1.4. Analogously, we can see Hochschild cohomology as the cohomology of theassociated cochain for a special cosimplicial objecct.

Definition 3.1.5 (Moore or normalised complex). We call normalised (or Moore) chaincomplex associated to A, noted N(A), to the chain complex defined by

Nn(A) =

n−1⋂i=0

ker(∂i : An → An−1)

with differential dn = (−1)n∂n.

This is, N(A)n is the submodule of An of the elements such that are annihilated forevery face map ∂i for all i < n.

This time we have defined a functor (again defined obviously on arrows)

sR-Mod Ch+(R-Mod)

A N(A)

N

50

Remark 3.1.6. By construction, N(A) is a chain subcomplex of the alternating face mapscomplex associated to A; in fact, what we have with this inclusion as a subcomplex is a

natural transformation R-Mod Ch+(R)

N

C

ι

This fact is easy to observe since for each arrow f : A→ B in the category S R-Mod,the following diagram trivially commutes:

A N(A) C(A)

B N(B) C(B)

f

ιA

N(f) C(f)

ιB

We used the face maps to construct the chain subcomplex N(A) ⊂ C(A), we willnow rely on the degeneracies to construct a new subcomplex of C(A).

In order to do that we shall prove the following lemma:

Lemma 3.1.7. Given a simplicial module A, the differential in the unnormalised complexC(A) can be restricted to the submodules Dn ⊂ An generated by the degenerate n-simplexes of A.

Proof. We have taken the submodules Dn =n−1∑j=0

Im(σi : An−1 → An); what we want to

show is that for any n and b ∈ Dn we have dn(b) ∈ Dn−1, where dn is the differential inthe unnormalised complex associated to A.

It suffices to prove it for elements of the form y = σi(a) for some i and a ∈ An−1,and that is what we will show.

Let consider then a ∈ An−1 and σi : An−1 → An, then we have:

dn(σi(a)) =

n∑j=0

(−1)j∂j

σi(a)

=i−1∑j=0

(−1)jσi−1∂j(a) + (−1)ia+ (−1)i+1a+n∑

j=i+2

(−1)jσi∂j−1(a) (th. 2.2.7)

=i−1∑j=0

(−1)jσi−1∂j(a) +n∑

j=i+2

(−1)jσi∂j−1(a)

This proves what we were looking for, since it is obvious that dn(σi(a)) is a sum ofdegenerate n− 1-simplexes and thus it is in Dn−1.

51

Definition 3.1.8 (Degenerate complex). We call degenerate chain complex associatedto A, denoted D(A), the chain complex defined by

D(A)n =n−1∑i=0

Im(σi : An−1 → An)

with differential the restriction of the unnormalised complex C(A).

In the light of lemma 3.1.7 we know that the previous definition makes sense.

As we did before, we can construct from here a functor (with the obvious behavioron arrows):

sR-Mod Ch+(R-Mod)

A D(A)

D

And it is direct from the construction the fact that we have, just as in the case of thenormalised complex, an inclusion ι : D(A)→ C(A) resulting in a natural transformationι : D ⇒ C.

Summarizing, so far we have the following constructions:

R-Mod Ch+(R)

N

C

ι R-Mod Ch+(R)

D

C

ι

Note that lemma 3.1.7 implies that for each n, the differential dn of the unnormalisedcomplex C(A) induces a differential dn : C(A)n/D(A)n → dn : C(A)n−1/D(A)n−1.

This allows us to construct a new complex (C/D) given by (C/D)(A)n = C(A)n/D(A)nand with differential the one induced by dn.

Again (using the fact that a morphism between simplicial objects commutes withfaces and degeneracies) this mapping give us a functor (C/D) : sR-Mod → Ch+(R);and the projection maps C(A)n → C(A)n−1/D(A)n−1 conform a natural transformationπ : C ⇒ (C/D), this time with source C.

The relevant functor for our equivalence will be N .

3.1.2 Relations between the different chain complexes

In this section we shall establish insightful relations between the previous constructions,we shall start with the following proposition.

Proposition 3.1.9. The composite π ◦ ι : N =⇒ (C/D) is a natural isomorphism.

Proof. We already know this is a natural transformation, so we need only check thatgiven a simplicial module A, the composite:

N(A) C(A) (C/D)(A)ιA πA

52

is an isomorphism.

We shall show this by induction, for each k < n we define the following submodulesof An:

NkAn :=

k⋂i=0

ker(∂i), DkAn :=

k∑i=0

Im(σi)

Note that N(A)n = Nn−1An and D(A)n = Dn−1An, thus when finished the inductionover k we will have proven the statement.

Let’s call φk to the composite NkAn An An/(DkAn) which we want

to show to be an isomorphism.

In the case k = 0, what we want is φ0 to be the isomorphism ker(∂0) ' An/ Im(σ0).

But this is easy, indeed, we have maps An−1 Anσ0

∂0, where σ0 is a split monomor-

phism with section ∂0, thus ker(∂0)⊕ Im(σ0) = An.

Now let us suppose we have established the isomorphism Nk−1An ' An/(Dk−1An),we shall construct from this the sought one for k.

First, we note that we have a split exact sequence:

0 An−1/Dk−1An−1 An/Dk−1An An/DkAn 0σk (3.1)

with σk the map induced by σk (σk(Dk−1An−1) ⊂ Dk−1An on account to the equalityσkσj = σjσk−1 for all j < k). The exactness is direct and the section is given by the facemap ∂k.

Similarly, we have another short exact sequence, although this time not so evident:

0 Nk−1An−1 Nk−1An NkAn 0σk (3.2)

By assembling the short exact sequences 3.1 and 3.2 we obtain the following commu-tative diagram:

0 Nk−1An−1 Nk−1An NkAn 0

0 An−1/Dk−1An−1 An/Dk−1An An/DkAn 0

σk

' 'σk

Since the first two vertical arrows are isomorphisms by the inductive hypothesis, so isthe third.

Corollary 3.1.10. Given a simplicial module A, for each n we have an isomorphism

N(A)n ⊕D(A)n = C(A)n

53

Summarizing again, in order to illustrate this new result, we have the followingcommutative diagram where the arrows are natural transformations between the functorswe have built:

C/D

N C D

'

ι

π

ι

πι=0

Proposition 3.1.11. Let A be a simplicial module. The inclusion ι : N(A)→ C(A) isa chain homotopy equivalence.

Proof. Given a simplicial module A, we can define for each j ≥ 0 a subcomplex NjA ofits unnormalized Moore complex NA as follows:

NjAn =

j⋂i=0

ker ∂i if n ≥ j + 2

NAn if n ≤ j + 1

The simplicial identities ∂k∂i = ∂i−1∂k that hold for i > k allow us to verify thatthose groups forms a chain subcomplex; this means we can verify that for any x ∈ NjAnwith n ≥ j + 2 and any j ≤ j it is

∂k

n∑i=j+1

(−1)i∂i(x)

= 0

The first step is to construct a chain map f : C(A)→ N(A) such that f ◦ ι : N(A)→N(A) is the identity map.

We set N−1 = C(A). We have inclusions ιj : Nj+1A ↪→ NjA, and observe that it alsoholds N(A) = ∩j≥0NjA.

Now for each j and n we define group homomorphisms fj : NjAn → Nj+1An by:

fj(x) =

{x− σj+1∂j+1(x) if n ≥ j + 2

x if n ≤ j + 1

The simplicial identities imply that fj effectively goes to Nj+1 and that the collection offj define a chain map fj : NjA→ Nj+1A. One can also check that the composite fj ◦ ιjis the identity.

We define now new homomorphisms tj : NjAn → NjAn+1 by:

tj(x) =

{(−1)jσj+1(x) if n ≥ j + 1

0 otherwise

We could also check that for all n we have the equality:

id−ιjfj = dtj + tjd

54

Finally, we have that the compositions fn−1 ◦ · · · ◦ f0 : C(A)n → N(A)n define achain map f : C(A)→ N(A) that we were looking for.

Now, defining a new collection of maps τn : C(A)n → C(A)n+1 by

τn = ι0 · · · ιn−2tn−1fn−2 · · · f0 + ι0 · · · ιn−3tn−2fn−3 · · · f0 + · · ·+ ι0t1f0 + t0

we obtain a chain homotopy between ι ◦ f and the identity map ni C(A).

3.2 From Ch+(R) to simplicial modules

As the title of the section suggests, in what follow we will endeavour to construct theinverse of the functor N .

Let C∗ be a non-negative chain complex of R-modules, we want to construct asimplicial R-module out of C, i.e. a functor K(C) : ∆op → R-Mod.

In order to do that, for each n we define:

K(C)n =⊕k≤n

⊕ν:[n]�[k]

Cν

where the inner sum is taken over all surjective maps ν : [n]� [k] and Cν is just a copyof Ck.

Now, given α : [n] → [m] an arrow in ∆ we wish to define a new arrow K(C)(α) :K(C)m → K(C)n, we will do it in each summand Cν . First, we take the epi-monodecomposition of ν ◦ α given in 2.1.3, that gives us a commutative square as follows:

[n] [m]

[q] [k]

α

η ν

ε

(3.3)

From this, we define K(C)(α) in each summand as:

K(C)(α)ν =

idCk if k = q

dk if k = q + 1 and ε = εk

0 otherwise

where dk : Ck → Ck−1 is the differential of the complex C, εk is the one defined in 2.1and ε the map in 3.3.

We still have to verify that this definition leads to a simplicial object. Let α : [l]→ [m]and β : [m] → [n] two composable morphisms in ∆, the fact that K(C)(β ◦ α) =K(C)(α) ◦K(C)(β) follows from the following commutative diagram (in fact, the familyof them), which is obtained assembling diagrams built like in 3.3:

55

[l] [m] [n]

[q] [q′] [p]

α β

ν

This ends the proof that K(C) is a simplicial object.

We shall now define our functor K on arrows. For any chain map f = (fn) : C∗ → D∗,we define K(f) : K(C)→ K(D) at level n by K(f)n = (ινfm)ν:[n]�[m] where the mapιν : Dm → K(D)n is the inclusion map into the coordinate corresponding to ν.

Thus, we have defined a functor:

Ch+(R-Mod) sR-Mod

C K(C)

K

3.3 The equivalence

Theorem 3.3.1 (Dold-Kan correspondence). The functors

N : sR-Mod→ Ch+(R) and K : Ch+(R)→ sR-Mod

form an equivalence of categories.

Proof. We shall prove there exist natural isomorphisms Ψ : idCh+(R) =⇒ NK andΦ : KN =⇒ idsR-Mod.

Let us look first Ψ : idCh+(R) =⇒ NK, for which we will use the isomorphismN =⇒ C/D (proposition 3.1.9).

Let C∗ be a chain compex. For each surjection ν : [n] → [k] we can decomposeν = ηi1 · · · ηit (2.1.3). Thus, if k 6= n we can write Cν = (σit · · ·σi1)Cidk where σij is thedegeneracy map of the simplicial object K(C) corresponding to ηij . Last thing impliesthat such Cν lies in the degenerate complex D(K(C)).

Then, by the isomorphism 3.1.9 we have N(K(C))n = Cidn = Cn. A naive attemptis to define ΨC∗ to be idCn in each level, the first thing to note is that it effectively is amorphism of chain complexes, i.e. the following diagram commutes:

C∗ · · · Cn+1 Cn Cn−1 · · ·

NK(C∗) · · · Cn+1 Cn Cn−1 · · ·

ΨC∗

dn+1

id

dn

id id

fact that is obvious from the definitions of the functor at maps level.

56

To end the proof we shall show that Ψ defined as we did is natural. This is, given amap of chain complexes f : C∗ → D∗ we have a commutative diagram:

C∗ C∗ NK(C∗)

D∗ D∗ NK(D∗)

f

ΨC∗

f NK(f)

ΨD∗

Standard computations show that this effectively commutes.

The construction of the natural isomorphism Φ : KN =⇒ id is subtler.

The first step is to define for each n a morphism KN(A)n → An, we know that

KN(A)n =⊕k≤n

⊕ν:[n]�[k]

N(A)ν

We shall describe the map ΦA : KN(A)n → An in each summand; given ν : [n]� [k]we know N(A)ν = N(A)k, thus we define (ΦA)ν as the composite

N(A)k Ak An

ν∗

A(ν)(3.4)

It is clear from the definition that ΦA is a morphism of simplicial objects.

Since we have explicit definitions, a routine check shows that the family {ΦA} leadsto a natural transformation, this is, for any simplicial morphism f : A→ B the followingdiagram commutes:

A KN(A) A

B KN(B) B

f

ΦA

KN(f) f

ΦB

Now that we have the natural transformation we wish to conclude the proving thateach ΦA is an isomorphism of simplicial modules by showing that it is so at each level.

Let us start with the surjectivity, which we shall prove by induction in n.

The base case, namely n = 0, is easy. Indeed, we have N(A)0 = A0 and being thatthe only surjective map [0]→ [0] is id, we get (ΦA)0 = idA0 . We will note (ΦA)k = Φk

from now on.

Suppose now that Φm is an isomorphism for all m < n. By corollary 3.1.10 weknow that An = N(A)n ⊕ D(A)n. On one hand, N(A)n ⊂ Im(Φn) (because of thesummand N(A)n). On the other hand, by the inductive hypothesis we know thatAn−1 ⊂ Im(Φn−1), since the map ΦA commutes with degeneracies the former assertionimplies that D(A)n ⊂ Im(Φn). Thus we have proven that the map Φn is surjective.

57

We shall now show the injectivity of Φn. Let (aν) be an element in KN(A)n =⊕k≤n

⊕ν:[n]�[k]

N(A)ν such that Φn((aν)) = 0; we should show that all aν are zero.

We are then assuming ∑ν:[n]�[k]

ν∗(aν) = 0 (3.5)

The first thing we note is that (id[n])∗(aid) = aid = 0, as it is the only term that

might not be in D(A)n when considering the decomposition An ' N(A)n ⊕D(A)n incorollary 3.1.10.

We will use the argument just above together with the inductive hypothesis in orderto prove aν = 0 for all ν.

Suppose there exists aν 6= 0 for certain ν0 : [n]� [k0], let µ : [k0]→ [n] be a sectionof ν0.

Applying A(µ) to equation 3.5 we obtain:

0 = A(µ) ◦∑

ν:[n]�[k]

ν∗(aν)

=∑

ν:[n]�[k]

(ν ◦ µ)∗(aν)

= Φk0((µ∗aν))

By the inductive hypothesis and the argument we made before, it suffices to provethat we can take µ to be a section only for ν0 (because in that case, the only componentrelated to id : [k0]→ [k0] will be aν0).

The first thing to observe is that we can assume ν0 to be minimal with respect theusual order of functions (ν ≤ ν ′ iff ν(a) ≤ ν ′(a) for each a ∈ [n]). Take a maximalelement µ which is not a section of any ν ′ > ν0 (such a maximal element exists), by theway we chose µ, it is not a section for any ν ′ < ν either.

3.4 Homotopies and the Dold-Kan correspondence

The Dold-Kan correspondence is well behaved regarding homotopies, we write this inthe theorem below.

Theorem 3.4.1. Under the Dold-Kan equivalence, simplicial homotopies correspond tohomology and simplicial homotopies between maps corresponds to homotopies of maps ofchain complexes.

Proof. The first part follows from 2.3.21.

We shall prove that both, N and K, preserve homotopies of maps.

58

Let A and B simplicial R-modules and f, g : A→ B two homotopical simplicial mapsand let h an homotopy between them, we wish to prove that N(f) and N(g) are chainhomotopic. Since our category is additive, it is equivalent to prove that if f is simplicialhomotopic to zero, then N(f) is chain homotopic to zero.

For each n we define θn : N(A)n → N(B)n+1 by θn =n∑i=0

(−1)ihi (the restrictions

actually); in what follows we will show that this family of morphisms form a chainhomotopy between N(f) and zero.

The first thing to observe is that for each n, θn is well defined, i.e. θn(N(A)n) ⊂N(B)n+1 (this is a straightforward verification). Thus, we have:

N(A) · · · N(A)n+1 N(A)n N(A)n−1 · · ·

N(B) · · · N(B)n+1 N(B)n N(B)n−1 · · ·

N(f) 0

dAn+1

N(f)n+1 0

θ

dAn

N(f)n 0

θ

N(f)n−1 0

dBn+1 dBn

The following computation shows that {θn} effectively form a chain homotopy betweenN(f) and the zero map:

θn−1dAn + dBn+1θn = (−1)nθn−1∂

An + (−1)n+1∂Bn+1θn

= (−1)n

[n−1∑i=0

(−1)ihi ◦ ∂An +

n∑i=0

(−1)i+1∂Bn+1 ◦ hi

]

= (−1)n

[n−1∑i=0

(−1)ihi ◦ ∂An +

n−1∑i=0

(−1)i+1hi ◦ ∂An+1 + (−1)n+1∂Bn+1 ◦ hn

]= −fn= −N(f)n

Now we wish to prove that the functor K sends chain homotopies to simplicialhomotopies. Let C and D be two non-negative complexes of R-modules and f, g : C → Dtwo chain homotopic maps, and let the family {θn} a homotopy between them.

We will build a simplicial homotopy between the maps K(f) and K(g), i.e. for eachn we shall define a family {hi} with hi : K(C)n → K(D)n+1 verifying the conditions in2.4.1. Let fix n, for each i ≤ n we define hi,⊕

ν:[n]�[p]

Cν⊕

ν′:[n+1]�[p]

Dν′

on the summands of K(C) by induction on n− p.

59

Firstly, on Cid[n]= Cn we define

hi|Cn =

σifn if i < n− 1

σn−1fn−1 − σnθn−1d if i = n− 1

σn(fn − θn−1d)− θn if i = n

where d is the differential of the complex C.

Now, let ν : [n] � [p] be such that p 6= n and let j be the largest element of [n]such that ν(j) = ν(j + 1). Then we have ν = ν ′ηj for some ν ′ and σj : Cν ' Cν′ is anisomorphism.

By the inductive hypothesis we have defined hi for Cν′ , we set hi = hi ◦ σj and thenwe define hi on Cν as follows:

hi|Cν =

{σj ˆhi−1 if j < i

σj+1hi if j ≥ i

Standard computations show that this define a simplicial homotopy between K(f)and K(g).

Example 3.4.2 (Eilenberg-MacLane spaces). As we annunciated earlier, we will buildEilenberg-MacLane spaces via the Dold-Kan correspondence.

Let G be an abelian group and n ≥ 0, we can consider G[n], the chain complex beingG concentrated in degree n. The homology of this chain complex is:

Hi(G [n]) =

{0 if i 6= n

G if i = n

From the Dold-Kan correspondence, we can associate a simplicial abelian group K(G[n]).Since homology corresponds to homotopy under the map K (see thm. 3.4.1), K(G[n])is an Eilenberg-MacLane space of type K(G,n).

60

Chapter 4

Model category structure inCh+(R)

The aim of this chapter is to give a structure of model category to the category of non-negative chain complexes of modules over the ring R. There are two classical (and dual)model structures, called injective and projective, we will work on the later, following[DS95].

A proof for the existence of this structure, moreover its cofibrantly generated nature,can also be found in [Hov99, chapter 2]; where it is also discussed with references toprevious works the case of unbounded chain complexes.

We shall establish the three distinguished classes of morphisms. A map f : M → Nin Ch+(R) is:

• a weak equivalence if it induces isomorphisms in the homology, Hk(M)→ Hk(N)for all k ≥ 0,

• a cofibration if fk : Mk → Nk is a monomorphism and whose cokernel is a projectiveR-module for all k ≥ 0,

• a fibration if fk : Mk → Nk is an epimorphism for all k > 0.

The first thing to note is that the three classes are closed under composition. It isclear for WE and Fib; for cofibrations, although the injectivity of the composition isclear, we need to give an intermediate step in order to show the additional property ofthe cokernel.

Let f, g be two composable cofibrations we have exact sequences

0 Coker(fk) Coker(gkfk) Coker(gk) 0 (4.1)

where the injectivity of the first map is given by the injectivity of fk.

Since Coker(gk) is a projective R-module (by hypothesis), the sequence 4.1 splits andthus Coker(gkfk) = Coker(fk)⊕Coker(gk). The R-module Coker(gkfk) is then projectivefor it is direct sum of projective modules.

61

We will start now to verify the required axioms in definition 1.1.3.

4.1 MC1 is satisfied

Since limits and colimits in the category of chain complexes are computed degreewise itsuffices to prove that the category R-Mod admits finite limits and colimits.

But it is direct, in one hand R-Mod admits limits for it as a terminal object, it admitsproducts of two objects and for any pair of parallel arrows its kernel exists. On the otherhand R-Mod also admits colimits being that it has initial object, admits coproducts oftwo objects and the cokernel of any pair of parallel arrows exists.

4.2 MC2 is satisfied

It is direct from the fact that Hk(gf) = Hk(g) Hk(f) and composition of isomorphims isan isomorphism again.

4.3 MC3 is satisfied

We will resume part of what we want to prove in the following lemma:

Lemma 4.3.1. Let f : X → X ′ a retract of g : Y → Y ′. Then,

1. if g is an epimorphism, so is f ,

2. if g is a monomorphism, so is f ,

3. if g is an isomorphism, so is f .

Proof. Since f is a retract of g, we know there exist arrows i, i′, r, r′ in R-Mod such thatthe following is a commutative diagram

X Y X

X ′ Y ′ X ′

ι

idX

f

r

g f

ι′

idX′

r′

Let us prove 1. For that, let ϕ1, ϕ2 be two arrows in R-Mod such that ϕ1 ◦f = ϕ2 ◦f .We have then:

62

ϕ1f = ϕ2f ⇒ ϕ1fr = ϕ2fr

⇒ ϕ1r′g = ϕ2r

′g

⇒ ϕ1r′ = ϕ2r

′ (g is an epimorphism)

⇒ ϕ1r′ι′ = ϕ2r

′ι′

⇒ ϕ1 idX′ = ϕ2 idX′

⇒ ϕ1 = ϕ2

So we conclude that f is an epimorphism.

The proof of 2 is similar. Let ϕ1, ϕ2 be two arrows in R-Mod such that f ◦ϕ1 = f ◦ϕ2.We have then:

fϕ1 = fϕ2 ⇒ ι′fϕ1 = ι′fϕ2

⇒ gιϕ1 = gιϕ2

⇒ ιϕ1 = ιϕ2 (g is a monomorphism)

⇒ rιϕ1 = rιϕ2

⇒ ϕ1 = ϕ2

Thus, f is a monomorphism.

Assertion 3 is a direct consequence of the two precedent ones.

We are able now to properly start with the verification of axiom MC3. For thatconsider g : M → N and f : M ′ → N ′ two arrows in Ch+(R), being f a retract of g.

It is clear that since f is a retract of g, H(f) is a retract of H(g). Hence, if g isa weak equivalence, H(g) is an isomorphism and thus by lemma 4.3.1 H(f) is so. Weconclude from that that f is also a weak equivalence.

That belonging to Fib is closed under retracts is straightforward from lemma 4.3.1.

Let now g be a cofibration. From lemma 4.3.1 we know that f is also a monomorphism,so it remains to prove the condition of projectivity over the cokernel of each fk.

Since we know that f is a retract of g, by staring at the ”slice” at level k we deducethat fk is a retract of gk; taking now cokernels in the corresponding diagram at level k

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we obtain the commutative diagram below:

M ′k Mk M ′k

N ′k Nk N ′k

Coker(fk) Coker(gk) Coker(fk)

ιk

idM′k

fk

r

gk fk

ι′k

idN′k

r′k

idCoker(fk)

(4.2)

We can deduce from the diagram 4.2 that Coker(fk) is a direct sum of Coker(gk),since the latter is a projective R-module so is the former.

With this we have finished to prove that axiom MC3 is verified with our election ofdistinguished classes.

4.4 MC4 is satisfied

Let us prove the condition 1 of axiom MC4. So, we consider a commutative square

A X

B Y

f

i p

g

(4.3)

where i is a cofibration and p is an acyclic fibration. We wish to prove that the squareconsidered admits a lifting h : B → X.

Lemma 4.4.1. The chain complex Ker(p) is acyclic.

Proof. The first thing we will show is that the acyclicity of p implies the surjectivity ofthe map p0, converting p into an epimorphism.

We can construct the following commutative diagram with exact rows:

Ker(d1) X1 X0 H0(X) 0

Ker(d′1) Y1 Y0 H0(Y ) 0

d1

p1

π

p0 H0(p)

d′1 π

(4.4)

Since p1 is an epimorphism (p is a fibration), H0(p) is also an epimorphism (in fact, itis an isomorphism because p is a weak equivalence) and 0→ 0 is a monormophism, thefive lemma give us that p0 is an epimorphism.

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Since p is then an epimorphism in the category Ch+(R) we have a short exactsequence of chain complexes:

0 Ker(p) X Y 0p

(4.5)

We will note Ker(p) = K, constructing the long exact sequence of homology for theshort exact sequence 4.5 we obtain te exact sequence:

· · · Hn(K) Hn(X) Hn(Y ) Hn−1(K) · · ·'

From which we can deduce that K is an acyclic chain complex.

We are now able to begin to define the morphism h : B → X we are looking for. Wewill do such definition by induction.

· Definition of h0.

Since i is a fibration, we have that P0 = Coker(i0) is a projective module and thusthe short exact sequence below splits:

0 A0 B0 P0 0i0 coker

We know then that B0 ' A0⊕P0 (from now on we will identify those modules withoutcarrying the isomorphism), and let s : P0 → B0be a section such that coker ◦s = idP0 .

We will use again the projectivity of P0 to construct an arrow ϕ0 : P0 → X0 asillustrated in the following diagram:

P0

X0 Y0 0

ϕ0 g0s

p0

Finally, we define h0 : B0 = A0 ⊕ P0 → X0 by

h0(a, t) := f0(a) + ϕ0(t)

With this definition of h0,

A0 X0

B0 Y0

f0

i0 p0

g0

h0 (4.6)

is by construction a commutative diagram.

· The inductive step.

Assume k > 0 and that we have constructed hj : Bj → Xj for all j < k, verifyingtwo properties:

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• for each j the analogous diagram of 4.6 commutes

• that they can be part of a morphism between chain complexes, this is:

djhj = hj−1∂′j (4.7)

where d and ∂′ are the differentials of X and B respectively.

In the same way we proceeded to construct h0, we write Bk = Ak ⊕Pk and we definehk : Bk → Xk such that the following diagram is commutative:

Ak Xk

Bk Yk

fk

ik pk

gk

hk

This map hk is almost the one we are looking for but a priori it does not verify thecondition for being part of a chain map (4.4). We have to slightly modify it in order toarrive to our required h.

We define a morphism measuring the failure of hk to verify 4.4, this is we defineε : Bk → Xk−1 to be the difference ε = dkhk − hk−1∂

′k.

Claim. The newly built map ε : Bk → Xk−1 verifies:

1. dk−1ε = 0,

2. pk−1ε = 0,

3. εik = 0.

Proof. Assertion 1 is direct, indeed

dk−1ε = dk−1(dkhk − hk−1∂′k) = 0− dk−1hk−1∂

′k = hk−2∂

′k−1∂

′k = 0

The following equalities, where d′ is the differential of Y , prove the second assertion:

pk−1ε = pk−1(dkhk − hk−1∂′k)

= d′kpkhk − gk−1∂′k

= dkfk − fk−1∂k

= 0

Assertion 3 holds because:

εik = (dkhk − hk−1∂′k)ik

= dkfk − hk−1ik−1∂k

= dkfk − fk−1∂k

= 0

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From the rest of the section, given a chain complex (D, δ) we will denote Ker(δk) byZk(D).

The claim 4.4 implies that ε induces a morphism

ε′ :Bk

Im(ik)→ Zk−1(K)

Since we have an isomorphism BkIm(ik) ' Pk, we will consider ε′ : Pk → Zk−1(K).

Now, K is acylic (by lemma 4.4.1) and then the map δk : Kk → Zk−1(K) is anepimorphism, we are able then to use the projectivity of Pk as follows:

Pk

Kk Zk−1(K) 0

ε′′ε′

The last step before defining hk consists in considering the composition:

Bk Yk Pk Kk Xkgk

ϕk

coker(pk) ε′′

Finally, we define hk = hk − ϕk. Simple verifications show that hk is a morphismverifying what we want.

We will now proceed to prove the part 2 of the axiom MC4.

Given an integer n we can define two functors given by

Ch+(R) R-Mod R-Mod Ch+(R)

X Xn M Dn(M)

Fn Dn

where the chain complex Dn(M) is given by Dn(M)k =

{M if k = n, n− 1

0 otherwisewith

differential equal to the identity at level n and zero in the rest.

Lemma 4.4.2. For each n, the pair (Dn, Fn) is an adjoint pair. This is, we have anatural isomorphism

HomCh+(Dn(M), N) ' HomR(M,Fn(N))

Proof. The proof is immediate from the definition taking the isomorphism to send a mapof chain complex f : Dn(M)→ N to its coordinate fn.

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Corollary 4.4.3. If M is a projective R-module, then Dn(M) is a projective chaincomplex.

We start with a diagram

A X

B Y

f

i p

g

(4.8)

where i is an acyclic cofibration and p is a fibration.

We can construct the following short exact sequence

0 A B Coker(i) 0i (4.9)

Since i is a weak equivalence, from the long exact sequence of homology associatedto 4.9 we deduce that Coker(i) is an acyclic chain complex. This fact in addition withthe fact that for each k the R-module Coker(i)k = Coker(ik) is projective, implies thatCoker(i) is a projective object in the category Ch+(R).

Given that Coker(i) is projective, the short exact sequence 4.9 splits and in conse-quence we have B ' P ⊕A.

Finally, proceding as we did to construct h0 while verifying the first part of thisaxiom, we obtain a lifting h : B → X in the diagram 4.8.

4.5 MC5 is satisfied

In order to verify this axiom we will use the small object argument described in 1.1.2.

Given n ≥ 1, Dn will denote the chain complex Dn(R) and we will use D0 for thecomplex with R at level 0 and zero elsewhere. Also, for each n ≥ 0 we will write Sn

for the complex with R at level n and 0 elsewhere, while we set S−1 for the zero chaincomplex.

We have a unique map j0 : S−1 → D0 and evident inclusions jn : Sn−1 → Dn, whichare the identity at level n− 1.

Lemma 4.5.1. For each n, the chain complex Sn is sequentially small.

Proof. This is direct from the well known natural isomorphism HomR(R,M) 'M .

Lemma 4.5.2. Let f : X → Y be a map of chain complexes. Then:

1. f is a fibration if and only if it has the right lifting property with respect to themaps 0→ Dn for all n ≥ 1,

2. f is an acyclic fibration if and only if it has the right lifting property with respectto the maps jn : Sn−1 → Dn for all n ≥ 0.

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Proof. Regarding the statement 1, if f is a fibration we know that it is almost anepimorphism (we do not have that f0 is an epimorphism) but since we are consideringn to be greater or equal to 1, we can replace momentously f0 : X0 → Y0 by 0 → 0 (letf · : X · → Y · be the modified morphism) and use then the projectivity of the chaincomplex Dn as illustrated in the following diagram:

0 X ·

Dn Y ·

0

f ·

to prove that f has the right lifting property with respect to maps 0→ Dn.

The converse follows from a standard argument of diagram chasing.

Consider a map f : X → Y in Ch+(R), and concentrate first in the factorization ofMC5 1.

As we anticipated, we shall use the small object argument. Fixed the family of mapsF = {jn}n≥0, since lemma 4.5.1 assure the sequentially smallness of the disks Dn, wecan apply theorem 1.1.8 to obtain a factorization of f as follows:

X Y

G∞(F , f)

i

f

p

We know that p has the right lifting property with respect to any arrow on F andhence, by lemma 4.5.2, p is an acyclic fibration.

It remains to prove that i is a cofibration. For that we will return to the proposition1.1.6, and we will prove then that i have the left lifting property with respect to acyclicfibrations. Let g : A→ B be an acyclic fibration and a commutative diagram as follows:

X A

G∞(F , f) B

i g (4.10)

By construction, for each k, we know that at level n the chain complex Gk+1(F , f)is a direct sum of a copy of Gk(F , f) with copies of R. Thus, G∞(F , f) at level n is adirect sum of Xn with copies of R. This allows us to write G∞(F , f) = X ⊕

⊕n≥0⊕Sn.

Then we can extend diagram 4.10, to the following one:

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X A

X ⊕⊕n≥0⊕Sn A

X ⊕⊕n≥0⊕Dn+1 B

i id

id⊕jng

(4.11)

Now, lemma 4.5.2 guaranties that g has the right lifting property with respect to theinclusions jn and in consequence we have a lifting h : X ⊕

⊕n≥0⊕Dn+1 → A for the lower

square diagram in 4.11.

The composite h ◦ (id⊕jn) is a lift for 4.10. This ends the proof of that i is acofibration.

We shall now proceed to prove the second part of the axiom MC5. We consider onceagain the arrow f : X → Y in Ch+(R), but this time we look for de decomposition of1.1.8 with the family F ′ = {0→ Dn}n≥1. We obtain then a new factorization

X Y

G∞(F , f)

i

f

p

As we did earlier, theorem 1.1.8 and lemma 4.5.2 ensure that p is a fibration. This timewe want to prove in addition that i is an acyclic cofibration.

The reasoning for this is very similar to the one we did before, relying this time inthe fact that Cofib∩WE = LLP (Fib) (proposition 1.1.6).

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Chapter 5

Model structure on sSet

The aim of this chapter is to endow the category sSet with (cofibrantly generated)model structure; we will do this via the theorem 1.2.41. Although we will not proveall the results we need (in particular for the proof of the last required condition in thementioned theorem), we give for those we left without proof a precise reference.

For this chapter we have followed mostly [Hov99] and [GJ99], which use results inthe category of topological spaces. An intrinsic proof of this model category structure onsimplicial sets can be found in the original paper by Quillen [Qui67] as well as in [Cis07]and [JT99].

5.1 Cofibrantly generated model structure

We consider the following sets of arrows in the category sSet:

- WsSet is the set of arrows f : X → Y such that |f | : |X| → |Y | verifies that

πn(|f |, x) : πn(|X|, x)→ πn(|Y |, |f |(x))

is a group isomorphism for every n ≥ 0 and every x ∈ |X|,

- IsSet = {∂∆n ↪→ ∆n : n ≥ 0},

- JsSet = {Λr[n] ↪→ ∆n : n ≥ 0, 0 ≤ r ≤ n},

- FsSet = JsSet-inj,

- CsSet = IsSet-cof.

In the literature the elements JsSet − inj are called Kan extensions. We will writethe explicit definition for historical reasons.

Definition (Kan fibration). A map of simplicial sets f : X → Y is a Kan fibration if

71

for every commutative diagram in sSet like the solid diagram that follows:

Λnk X

∆n Y

ι ph

where ι : Λnk → ∆n is the obvious inclusion, there exists a map h : ∆n → X such that thetotal diagram commute.

Definition 5.1.1. A map of simplicial sets is an anodyne extension if it is in JsSet-cof.

Theorem 5.1.2. (sSet, CsSet, FsSet,WsSet) is a model category where FsSet is the set offibrations, CsSet cofibrations and WsSet weak equivalences. It is cofibrantly generated withIsSet as set of generating cofibrations and JsSet as set of generating acyclic cofibrations.

Proof. As said earlier, we will prove this statement via theorem 1.2.41. Since this proofis rather long, for the sake of clarity we will separate it in smaller results.

First of all, it is easy to observe that sSet is a complete and cocomplete categoryfor Set is so and limits and colimits are computed levelwise.

An overview of the structure for the rest of the proof is:

1. WsSet satisfies the 2-out-of-3 property and it is closed under retracts (5.1.3),

2. IsSet and JsSet admits the small object argument (5.1.4),

3. JsSet − cell ⊂ (IsSet − cof ∩WsSet) (proposition 5.1.8),

4. IsSet − inj ⊂ (JsSet − inj ∩WsSet) (proposition 5.1.13),

5. (JsSet − inj ∩WsSet) ⊂ IsSet − inj.

Throughout the rest of the chapter, when we refer to “condition 1” (or 2, 3, 4, 5) wewill refer to the respective condition listed above.

Lemma 5.1.3. The set WsSet satisfies the 2-out-of-3 property and it is closed underretracts.

Proof. Let us start with the 2-out-of-3 property. Let f, g be two composable arrows insSet such that two out of h = g ◦ f, f, g are in WsSet.

For clarity we set |f | : X → Y , |g| : Y → Z, and obviously |h| : X → Z.

If either f and g are the ones in WsSet or g and h then it is obvious that the third oneis also in WsSet. There could arise a problem when the case is that f and h are elementsin WsSet; indeed, there could exists an element y ∈ Y such that it is not in the image off and thus we would not a priori be able to prove that πn(g, y) is bijective for n ≥ 1. But

72

it is not a real problem, since if y1, y2 ∈ Y are in the same path connected component,then πn(Y, y1) ' πn(Y, y2) and that we do know that π0(g) is an isomorphism.

Let now suppose that f is a retract of some w ∈WsSet. This is, we have the followingcommutative diagram:

X Y X

X ′ Y ′ X ′

ι

idX

|f |

r

|w| |f |

ι′

idX′

r′

We can apply the functor πn to the diagram above -where we ignore basepoints inorder to relieve the notation and do other abuses of notations with the same purpose-,obtaining:

πn(X) πn(Y ) πn(X)

πn(X ′) πn(Y ′) πn(X ′)

ι

id

πn|f |

r

πn|w| πn|f |

ι′

id

r′

We want to build an inverse for πn(|f |), which we will call ϕ. Let y ∈ Y , since πn(|w|)is known to be an isomorphism, we can take a ∈ πn(Y ) such that w(a) = ι′(y). Wedefine ϕ(y) = r(a), thus:

(πn|f |) ◦ ϕ(y) = (πn|f |)r(a) = r′w(a) = r′ι′(a) = y

Proposition 5.1.4. Every simplicial set is small relative to the inclusions.

Proof. See [Hir03, Example 10.4.4]

In order to prove the condition (3) we shall use the following lemma, which give us acharacterization of the set IsSet − cof.

Lemma 5.1.5. A map f in sSet is in IsSet − cof if and only if it is injective.

Proof. The first thing to note to show that every map in IsSet − cof is injective is thatevery map in IsSet is injective. Now, since the set of injective maps is closed underpush-outs and transfinite compositions, we have that any IsSet − cell is injective.

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Now, corollary 1.2.33 together with the fact that injective maps are closed underretracts (fact that follows easily looking at the relevant diagram) leads to our firstinclusion: every map in IsSet − cof is injective.

Conversely, let f : K → L be an injective map of simplicial sets; we shall show f asa transfinite composition of maps in push-outs of arrows in IsSet, that is, we will showthat f ∈ IsSet − cell and thus in IsSet − cof (lemma 1.2.24).

We will build by induction a N-sequence such that f is its transfinite composition.

For the base case we set X0 = K and f0 = f and S0 = L0\ Im(f0). By proposition2.3.3 we know that each element s in S0 has naturally associated a map ϕs : ∆0 → L,notice that each ϕs factors uniquely through X0, obtaining the following commutativediagram:

∆0 L

X0

ϕs

ϕs

Now we define X1 and f1 through the following (universal property of) push-outdiagram:

∅ =∐s∈S0

∂∆0 X0

∐s∈S0

∆0 X1

L

α0

f0

f1

(5.1)

where the upper horizontal map is the coproduct of the restriction to the boundary ofϕs.

Notice that by construction the map f1 is injective (since both α0 and f0 are) andsurjective on the 0-simplices.

Assume now we have constructed simplicial sets Xk and maps fk : Xk → L for k ≤ nsuch that every fk is injective it is also surjective on the simplices of dimension less thank. Then, using a diagram similar to 5.1, namely:

∅ =∐s∈Sn

∂∆n Xn

∐s∈Sn

∆n Xn+1

L

αn

fn

fn+1

where Sn = Ln\ Im(fn), and the upper horizontal map is the given by the restriction to

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the boundaries of the unique factorization through Xn of the maps ∆n → L associatedto each s ∈ Sn. Again, fn+1 is injective and it is surjective on k-simplices for k ≤ n.

It is easy to show that L is the colimit searched and f the transfinite composition ofthe N-sequence built.

In order to continue we will use some properties of special maps in the category Top.We define the following sets of arrows in this category:

- ITop = {Sn−1 ↪→ Dn : n ≥ 0},

- JTop = {Dn ↪→ Dn × I, x 7→ (x, 0) : n ≥ 0},

- WTop is the set of maps f : X → Y in Top such that X 6= ∅ and the induced mapπn(f, x) : πn(X,x)→ πn(Y, f(x)) is a group isomorphism for every n ≥ 0 and everyx ∈ X, and the identity map in the empty set.1

Remark 5.1.6. It can be shown that JTop − cof ⊂ ITop − cof. For details see after thedefinition 2.4.3 of [Hir03].

Lemma 5.1.7. The following inclusion holds ITop − inj ⊂WTop

Proof. Let p : X → Y a map in ITop−inj and x0 ∈ X. We show that p ∈WTop by showingthat π0(p) : π0(X) → π0(Y ) is bijective and so is πn(p, x0) : πn(X,x0) → πn(Y, p(x0))for any n > 0.

Let us start showing that π0 is surjective by showing that actually p is surjective.Since p is ITop-injective and the map ∅ → ∗ is in ITop, for every solid diagram as below,we have the dashed filling:

∅ X

∗ Y

p

So, p is necessarily surjective and thus so is π0(p).

Now let us go for the injectivity of π0(p). Let x, y ∈ X be two points such that p(x)and p(y) are in the same path connected component, and left γ be a path in Y betweenthem. Since p is ITop-injective, we have a filling for the following solid diagram:

{0, 1} S0 X

I D1 Y

p

γ

where the upper horizontal arrow maps 0 ↪→ x and 1 ↪→ y. This shows that x and y areactually in the same path connected component of X.

1The notation used is not whimsical as it might seems but it relates to a cofibrantly generated modelstructure that can be introduced in Top similar to the one we are working on on sSet.

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We will now pass to the case n > 0, starting with the surjectivity of the inducedmap πn(p, x0) : πn(X,x0)→ πn(Y, p(x0)).

Let f : (Sn, ∗)→ (Y, p(x0)) and element in πn(Y, p(x0)), we shall construct an arrowg : (Sn, x0) → (X,x0) in πn(X,x0) such that f = p ◦ g by making use of the liftingproperties.

The first thing to note is that we have the following push-out diagram:

Sn−1 ∗

Dn Sn

(5.2)

For this, recall that the push-out in Top is obtained by taking the quotient of Dn∐∗

by the image of Sn−1, i.e. identifying the boundary of Dn with the point ∗, which leadsto Sn−1.

The next step is to prove that the map ∗ → Sn belongs to ITop − cof. Indeed, leti ∈ ITop − inj and a commutative diagram as follows:

∗ A

Sn B

i (5.3)

Now gluing together diagrams 5.2 and 5.3 we can construct:

Sn−1 ∗ A

Dn Sn B

id

where d is the filler given by the fact that Sn−1 ↪→ Dn is in ITop and the dashed arrow(which is the filling we are looking for) is the map induced by the universal property ofthe push-out. This conclude the proof that ∗ → Sn is an ITop − cofibration.

Finally, we consider the diagram:

∗ X

Sn Y

pg

f

where the g is the arrow given by the lifting property leading by the fact that p ∈ ITop−injand ∗ → Sn is in ITop − cof = (ITop − inj)− proj.

It remains to prove the last step: the injectivity of πn(p, x0) for each n > 0.

Let us consider two arrows f, g : (Sn, ∗)→ (X,x0) such that its images are the same,this is [p ◦ f ] = [p ◦ g] in πn(Y, p(x0)). We have then a basepoint preserving homotopy

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H : Sn × I → Y between the two maps p ◦ f and p ◦ g; we can see this homotopy as amap H : Sn ∧ I+ → Y -where Sn ∧ I+ = (Sn × I)�(∗ × I).

Notice that the maps f and g induce a map (f, g) : Sn ∨ Sn → X -where Sn ∨ Sn =(Sn × {0, 1})�(∗ × {0, 1}).

Since the inclusion Sn ∨ Sn → Sn ∧ I+ is in ITop − cof, we have a lifting as showedbelow:

Sn ∨ Sn X

Sn ∧ I+ Y

(f,g)

p

H

This filling give us the homotopy we looked for and then f and g represent the sameelement of πn(X,x0).

Proposition 5.1.8. Every anodyne extension is in IsSet − cof ∩WsSet.

Proof. Firstly, we recall that the anodyne extensions are the set JsSet− cof. Since everymap in JsSet − cof is injective, lemma 5.1.5 implies that they are in IsSet − cof. Theremarks in 1.2.19 leads to the inclusion JsSet − cof ⊂ IsSet − cof.

We shall prove now that JsSet ⊂WsSet. Let f be a map in JsSet, i.e. an inclusionΛr[n]→ ∆[n]. Notice that we have homeomorphisms |Λr[n]| and |∆n| ' Dn ' Dn− 1×Iand that we can chose those maps in a way that the following diagram commutes:

|Λr [n] | Dn−1

|∆n| Dn−1 × I

∼

|f |

∼

Now let g be an arrow in JTop − inj, then the diagram below illustrate that |f | is anJTop-cofibration:

|Λr [n] | Dn−1 •

|∆n| Dn−1 × I •

∼

|f | g

∼

This means that |JsSet| ⊂ JTop − cof, and hence:

|JsSet − cof| ⊂ |JsSet| − cof ⊂ JTop − cof ⊂ ITop − cof ⊂WTop

where we have used proposition 1.2.20, remark 5.1.6 and lemma 5.1.7 for the first, thirdand fourth inclusions respectively.

Finally, we are able to conclude JsSet ⊂WsSet.

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We proceed now to prove the condition 4, we will do this by showing that the inclusionIsSet − inj ⊂ JsSet ∩WsSet holds. For doing it we will need some previous results.

Proposition 5.1.9. The functor geometric realization preserves finite products.

Proof. See [Hov99, lemma 3.1.8]

Proposition 5.1.10. The functor geometric realization preserves finite limits.

Proof. See [Hov99, lemma 3.2.4]

Lemma 5.1.11. The class of sets JTop − inj is closed under retracts.

Proof. Let g be an arrow in JTop − inj and f a retract of it. Suppose now that we havea commutative diagram as follows:

Dn X

Dn × I Y

a

f

b

(5.4)

So we can construct the commutative diagram displayed below:

Dn X Y X

Dn × I X ′ Y ′ X ′

a ι

idX

f

r

g f

b ι′

idX′

r′

where the right part of it is obtained from the fact of f being a retract of g and thedashed arrow is given by the right lifting property of g with respect to the element inJTop.

Finally, the composition of the dashed arrow with r give us a filling for 5.4

Proposition 5.1.12. If f is an IsSet − injection, then its geometric realization |f | is aJTop − injection.

Proof. Suppose f : K → L is as in the hypothesis. In lemma 5.1.5 we have seen that theset of IsSet-cofibrations is the set of injective arrows, so f has the right lifting propertywith respect to any injective map of simplicial sets. Using this, we have the dashed fillingin the following commutative diagram:

K K

K × L L

(id,f) f

pL

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where pL is the projecction onto L.

From here we can easily note that f is a retract of πL, fact illustrated in the commu-tative diagram below:

K K × L K

L L L

(id,f)

id

f

h

pL f

id

And thus, since the geometric realization preserves finite products (proposition5.1.9), |f | is a retract of |pL|, which is again the projection on the second factor,|pL| = p|L| : |K| × |L| → |L|.

Observe now that p|L| is in JTop − inj. Indeed, given the solid commutative diagramthat follows:

Dn |K| × |L|

Dn × I |L|

a

p|L|

b

we can define the dashed filling, h : Dn × I → |K| × |L|, by h(x, t) = (p|L| ◦ a(x), b(x, t)).

We have proven then that |f | is a retract of a map in JsTop − inj, thus by lemma5.1.11, |f | also belongs to JsTop − inj.

Proposition 5.1.13. The inclusion IsSet − inj ⊂ JsSet − inj ∩WsSet holds.

Proof. By lemma 5.1.5 we know that JsSet ⊂ IsSet−cof, and thus, using the observationson 1.2.19, we know the inclusion IsSet − inj ⊂ JsSet − inj also holds.

Let f : K → L be an arrow in IsSet − inj, we shall prove now that f is in WsSet, i.e.that its geometric realization |f | induces isomorphisms in the homotopy groups.

Let v be an element in L0 (we will call also v to the associated map ∆[0]→ L), andset F = f−1(v). Then we have the following push out diagram:

F K

∆[0] L

f

v

(5.5)

Note that the map F → ∆[0] is in IsSet − inj. Indeed, given the commutativesquare involved to prove that (the square at the left in 5.6) we can write the following

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commutative diagram:

∂∆[n] F = f−1(v) K

∆[n] ∆[0] L

f

v

(5.6)

where the dashed arrow is induced by the lifting property of f , which shows that theimage of ∆[n] in K is contained in F . Thus, we are able to define a filling ∆[n]→ F .

This implies, by lemma 5.1.5, that f has the right lifting property with respect toany injective map. We can deduce also that F has a 0-simplex w.

We now consider the composite F → ∆[0]w−→ F , collapsing all of F to w; we call it

cw. Since F → ∆[0] is IsSet − injective, there exists a lifting as showed:

F × ∂∆[1] F

F ×∆[1] ∆[0]

(id,cw)

H (5.7)

To continue the proof we will study the information obtained from applying thegeometric realization functor to what we have until now.

In the first place, since this functor preserves finite limits (proposition 5.1.10), thepull-back diagram 5.5 leads to a new pull-back diagram:

|F | |K|

∗ |L|

|f ||v|

which implies that |F | is the fiber of |f | over the point |v|(∗).

By proposition 5.1.12, |f | is in JTop − inj. Thus the maps |F | ↪→ |K| |f |−→ |L| inducea long exact sequence in homotopy (for further details see [Hat01, theorem 4.4.1]):

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· · · πn(|F |, |w|) πn(|K|, |w|) πn(|L|, |v|)

πn−1(|F |, |w|) πn−1(|K|, |w|) πn−1(|L|, |v|)

πn−2(|F |, |w|) · · · π1(|L|, |v|)

π0(|F |) π0(|K|) π0(|L|)

|f |∗

|f |∗

π0(|f |)

(5.8)

where exactness at the level π0 means that d(π0(|F |)) = π0(|f |)−1(|v|).

We shall show that |F | is contractible, in order to deduce the isomorphisms in thehomotopy groups for n ≥ 1.

Applying the geometric realization functor to the diagram 5.7, we obtain -ignoringthe singleton in the lower right corner:

|F | × {0, 1} |F |

|F | × I

(id,|w|)

|H|

Thus |H| : |F | × I → |F | is a homotopy between the identity map id|F | and theconstant map |w|; hence |F | is contractible.

In consequence πn(|F |, |w|) is trivial for every n > 0, and hence we deduce from thelong exact sequence 5.8 the isomorphisms for n ≥ 1.

It remains to show that π0(|f |) : π0(|K|) → π0(|L|) is also an isomorphism. It isclearly injective, since we have the exactness at level 0 of the long exact sequence 5.8 forevery |v|.

For the surjectivity, note that since f has the right lifting property relative to allinclusions the following solid commutative diagram has a filling:

∅ K

∆[0] L

f

v

for every element v ∈ L0, thus there is an element w ∈ K0 mapped to v.

By observing that every element in |L| is in the same path connected component ofa 0-simplex of L we conclude the proof.

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Now we wish to prove the condition 5, which we will do by proving the converseof 5.1.13. This will use some auxiliary concepts and it will require to promenade on arelative tortuous path. For the sake of clarity, we will start here a new subsection.

5.2 Towards the completion of the structure.

5.2.1 Anodyne extensions

The theory of anodyne extensions was introduced by Gabriel-Zisman in [GZ67], it encodesand partially suppresses most of the combinatorial manipulations utilized before instatements about fibrations.

Definition 5.2.1. A class M of monomorphisms of sSet is said to be saturated if thefollowing conditions are satisfied:

S1 All isomorphisms are in M ,

S2 M is closed under push-outs, this is, for any pusho-out diagram:

A C

B B ∪A C

i i∗

if i ∈M then so is i∗,

S3 each retract of an element of M is in M ,

S4 M is closed under countable compositions and arbitrary direct sums, meaning re-spectively:

S4a) given

X1 X2 X3 · · ·i1 i2 i3

with ij ∈M , the canonical map X1 → lim←−Xi is in M ,

S4b) given ij : Xj → Yj in M , with j ∈ I, the arrow∐ij :

∐j∈I

Xj∐j∈I

Yj

is also in M .

Definition 5.2.2. Given a class of monomorphisms B, we call MB the saturated classgenerated by B. It is the intersection of all saturated classes containing B. One alsosays that MB is the saturation of B.

Hereinafter we consider three classes of monomorphisms:

- B1 := {Λnr ↪→ ∆n : 0 ≤ r ≤ n, n > 0},

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- B2 := {Λ1 × ∂∆n ∪∂∆n×Λε[1] ∆n × Λε[1] ↪→ ∆1 ×∆n : n > 0, ε = 0, 1},

- B3 := {(∆1 × Y ) ∪ (Λε[1]×X) ↪→ ∆×X : ε = 0, 1, Y ⊂ X}

Remark 5.2.3. We now observe something that will be important later. The set MB1

is included in the set of anodyne extensions. Indeed, the class JsSet − cof of anodyneextensions is a saturated class and it includes B1, thus MB1 ⊂ JsSet − cof.

This actually can be shown to be an equality.

Proposition 5.2.4. The classes B1, B2 and B2 have the same saturation.

Proof. See [GJ99, proposition 4.2].

Let i : K ↪→ L be an anodyne extension and f : Y ↪→ X an arbitrary inclusion. Thuswe can consider the following push-out diagram and the induced dashed arrow by theuniversal property:

K × Y K ×X

L× Y (L× Y )×K×Y (K ×X)

L×X

id×f

i×id i×id

id×f

(5.9)

Corollary 5.2.5. Let i : K ↪→ L be an anodyne extension and Y ↪→ X an arbitraryinclusion. Then the induced map

(K ×X) ∪ (L× Y ) L×X

(dashed in 5.9) is an anodyne extension.

Proof. See [GJ99, corollary 4.6].

We shall introduce now a new object that will play a fundamental role in the maintheorem of this subsection.

We define a functor:

sSet×sSet sSet

(X,Y ) Hom(X,Y )

Hom

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where the simplicial set Hom(X,Y ) is the functor ∆op → Set defined in objects by

Hom(X,Y )n = HomsSet(X ×∆n, Y )

and in arrows as we describe: for a map θ : [m]→ [n] in the category ∆op we define

θ∗ : Hom(X,Y )n → Hom(X,Y )m

as the one sending a map f : X ×∆n → Y to the composite

X ×∆m X ×∆n Yid×θ f

We can define the natural evaluation map as follows:

X ×Hom(X,Y ) Y

(x, f) f(x, idn)

ev

Routine verifications shows that ev is effectively a map of simplicial sets.

Proposition 5.2.6 (Exponential law). The pair (−×−,Hom(−,−)) is an adjunctionpair.

Proof. We will prove this adjunction by showing that the function

ev∗ : HomsSet(K,Hom(X,Y ))→ HomsSet(X ×K,Y )

which is defined by sending the simplicial map g : K → Hom(X,Y ) to the composite

X ×K X ×Hom(X,Y ) Yid×g ev

is a bijection natural in K, X and Y .

We explicitly define its inverse, as the map

HomsSet(X ×K,Y ) HomsSet(K,Hom(X,Y ))

by sending a map g : X ×K → Y to the map g∗ : K → Hom(X,Y ); where g∗ mapsevery x ∈ Kn to the composite

X ×∆n X ×K Yid×ιx g

being ιx the map ιx : ∆n → K such that ιx(idn) = x (see 2.3.3).

The fact that are mutually inverses functions is direct from the definitions, whereasthe fact that are maps of simplicial sets is proven with standard calculations.

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Proposition 5.2.7. Suppose that i : K ↪→ L is an arbitrary inclusion and p : X → Yan arrow in JsSet − inj. Then the map

Hom(L,X) Hom(K,X)×Hom(K,Y ) Hom(L, Y )(i∗,p∗)

which is induced by the diagram below is in JsSet − inj:

Hom(L,X)

Hom(K,X)×Hom(K,Y ) Hom(L, Y ) Hom(L, Y )

Hom(K,X) Hom(K,Y )

p∗

i∗i∗

p∗

Proof. Let us consider the following commutative diagram:

Λnk Hom(L,X)

∆n Hom(K,X)×Hom(K,Y ) Hom(L, Y )

(i∗,p∗) (5.10)

We wish to show that there exists a lifting for the considered diagram.

The exponential law (see 5.2.6) allows us to identify each diagram like 5.10 with oneas follows:

(Λnk × L) ∪Λnk×K (∆n ×K) X

∆n × L Y

j p (5.11)

By corolary 5.2.5 the map j is an anodyne extension, thus we have a filling for thediagram 5.11 which reflects in one for the diagram 5.10.

5.2.2 Minimal fibrations

Definition 5.2.8. Suppose p : X → Y is a map in JsSet-inj. We say that p is locally

trivial if, for every n-simplex ∆[n]y−→ Y of Y , the pullback fibration ∆[n]×Y X

y∗p−−→ ∆[n]

is isomorphic over ∆[n] to a product fibration ∆[n]× F p1−→ ∆[n].

Definition 5.2.9 (Minimal fibration). A Kan fibration p : X → Y of simplicial sets iscalled minimal fibration in case that for each n ≥ 0, every path connected component ofevery fiber of the fibration

Hom(∆n, X) Hom(∂∆n, X)×Hom(∂∆n,Y ) Hom(∆n, Y ) (5.12)

85

has only one 0-simplex.

Theorem 5.2.10. Let p : X → Y be an element in JsSet-inj. Then we can factor p as

X Y

X ′r

p

p′

where p′ is a minimal fibration and r is a retraction onto a subsimplicial set X ′ of Xsuch that r ∈ IsSet-inj.

Proof. See [Hov99, theorem 3.5.9].

Corollary 5.2.11. Suppose p is an element in JsSet-inj such that every fiber of p isnon-empty and has no non-trivial homotopy groups. Then p is in IsSet-inj.

Proof. See [Hov99, corollary 3.5.10].

Note that one could be tempted to use corollary 5.2.11 to prove that every map inJsSet− inj∩WsSet is also in IsSet but we need the hypothesis over the homotopy groups.In the next section we develop a relation between being in WsSet and homotopy groupsthat will allows us to use it.

5.2.3 Fibrations and geometric realization

In this section we complete the proof that sSet is a model category.

Proposition 5.2.12. Let p be a map in JsSet-inj that is locally trivial. Then |p| is inJTop-inj.

Proof. See [Hov99, proposition 3.6.1].

Corollary 5.2.13. Let p be an element in JsSet-inj, thus |p| is in JTop-inj with domainand codomain compactly generated topological spaces.

Proof. See [Hov99, proposition 3.6.2].

Proposition 5.2.14. Suppose X is a fibrant simplicial set and v an element in X0.Then there is a natural isomorphism πn(X, v) ' πn(|X|, |v|)

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Proof. See [Hov99, proposition 3.6.3]

Theorem 5.2.15. Suppose p : X → Y is an element of both WsSet and JsSet − inj.Then p has the right lifting property with respect to IsSet − inj.

Proof. By corollary 5.2.11 it suffices to show that the fibers of p are non empty and havenon trivial homotopy groups. Let F be a fiber of p over an element v ∈ Y0, then bycorollary 5.2.13, |F | is the fiber of |p| over |v|.

Since p is in WsSet, we have that |F | is non empty and it has no non trivial homotopygroups.

Finally, proposition 5.2.14 implies that F is non empty and that it has no non trivialhomotopy groups, as we wanted to have.

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Chapter 6

Model category structure in sAb

We will use theorem 1.2.42 and the (cofibrantly generated) model structure in the categoryof simplicial sets described in chapter 5 to endow with (cofibrantly generated) modelstructure to the category of simplicial abelian groups.

6.1 Adjunction

To accomplish that task we shall show the existence of an adjunction as in the statementof theorem 1.2.42. For that, we will recall a classic example of adjunction betweenfunctors F : Set→ Ab and U : Ab→ Set.

Definition 6.1.1. A concrete category is a pair (C, U) where U : C → Set is a faithfulfunctor, called forgetful functor.

When a left adjoint functor exists for U we call it free functor; suppose it exists andit is called F : Set → C. If X is a set, the object FX is called free object of basis X.We have this situation:

Set CF

U

In this case, is the characterization of adjoint pair via a unity, namely a naturaltransformation η : idSet ⇒ UF , verifying that for every objects X ∈ Set, C ∈ C andevery arrow f : X → UC there exists a unique map f : FX → C such that Uf makesthe following diagram commutative:

X UC C

UFX FX

f

ηXUf f

(6.1)

the one that will result us more familiar.

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Indeed, this is the well known “universal property of the free object” we are usedto from the categories Grp,Ab,R-Mod, k− alg, where U is the usual forgetful functorgiven by consider the underlying sets without the additional structure.

The free object (free group, free abelian group, free R-module, free k-algebra) withbasis X, FX, verifies that for every object G (group, abelian group, R-module, k-algebra) and any map f : X → G there exists a unique morphism f : FX → G makingthe following diagram commutative:

X G

FX

f

ιXf

(6.2)

The diagram 6.1 is indeed equal to this diagram, where we have ignored the factthat the arrows f and ιX have got as codomains UG and UFX respectively. Notethat diagram 6.2 has actually no sense, for there are arrows between objects in diferentcategories and it must be understood as 6.1.

This said, the usual “canonical inclusion” ιX : X → FX is nothing but the componentfor the object X of the unity of the adjoint pair.

As we have quickly said, an instance of the previous construction is the well knownadjunction when we take the free abelian group functor, that results in an adjoint pair

Set Ab

F

U

Once can easily construct this adjunction a similar one with its respective simplicialcategories, obtaining an adjoint pair:

sSet sAb

F

U

(6.3)

6.2 Transferring the structure

Taking the adjoint pair of 6.3 we will proceed as we have announced: we will verify thatthe hypothesis of theorem 1.2.42 holds for the functors F and U .

The first thing we wish to show is that FI and FJ admit the small object argument,being I and J the sets chosen in section 5.1. This is a straightforward consequence ofthe fact that all simplicial sets are small (5.1.4) and that F is a left adjoint functor andso it preserves colimits.

For the second hypothesis of theorem 1.2.42 we note the following inclusion chain:

U(FJsSet − cell) ⊆ U(FJsSet − cof) ⊂ J − cof

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But since J − cof is the set of acyclic cofibrations, we have just proved the inclusionU(FJsSet − cell) ⊆WsSet.

Explicitly, the model structure on sAb is given by the distinguished classes ofmorphisms we describe next; given f : A→ B a morphism in sAb, it is:

• a weak equivalence if the induced map between the normalized Moore complexesf∗ : N(A)→ N(B) is a quasi-isomoprhism,

• a fibration if f : A→ B is a Kan fibration between the underlying simplicial sets,

• a cofibration if it has the left lifting property with respect to all maps which areweak equivalences and fibrations.

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Chapter 7

The Dold-Kan correspondencerevisited

As we announced before, we will show that the Dold-Kan correspondence is a Quillenequivalence.

Since the functors K and N are mutually inverse equivalences of categories, by remark1.5.3 it suffices to prove that (K,N) or (N,K) is a Quillen adjunction.

It is trivial that N maps weak equivalences onto weak equivalences. In order to seethat it maps fibrations in fibrations we want to show that Kan fibrations are mapped tochain maps that are surjective in all positive degree.

Let f : A→ B be a fibration. The existence of liftings for diagrams of the form:

Λnn A

∆n B

0

f

implies that N(f)n : N(A)n → N(B)n is surjective for all n ≥ 1.

Thus, the adjunction (K,N) is a Quillen adjunction. It actually can be shown that(N,K) also is a Quillen adjunction (and thus, in this case, a Quillen equivalence); see[SS03, section 4.1].

In the previous paper, [SS03], the authors study also the behaviour of the Dold-Kancorrespondence regarding the monoidal structures involved.

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