The Crossed Menagerie (14 chapter version):

an introduction to crossed gadgetry and cohomology in algebra and

topology.

(Notes, in part, prepared for the XVI Encuentro Rioplatense de

Álgebra y Geometŕıa Algebraica, in Buenos Aires, 12 - 15

December 2006, and for course MATH5312, Spring - Summer term

2007, University of Ottawa)1

Timothy Porter

September 23, 2019

1Working document: Edited September 23, 2019

2

Introduction

These notes were originally intended to supplement lectures given at the Buenos Aires meetingin December 2006, were extended to give a lot more background for a course in cohomology atOttawa (Summer term 2007) and then grew to include a lot more, including material on non-Abelian cohomology, stacks and gerbes, and related areas of TQFT and HQFT theory.

They introduce some of the family of crossed algebraic gadgetry that have their origins in com-binatorial group theory in the 1930s and ‘40s, then were pushed much further by Henry Whiteheadin the papers on Combinatorial Homotopy, and, in particular, in [277]. Since about 1970, moreinformation and more examples have come to light, initially in the work of Ronnie Brown and PhilHiggins, (for which a useful central reference will be the forthcoming, [64]), in which crossed com-plexes were studied in depth. Explorations of crossed squares by Loday and Guin-Valery, [143, 186]and from about 1980 onwards indicated their relevance to many problems in algebra and algebraicgeometry, as well as to algebraic topology have become clear. More recently in the guise of 2-groups,they have been appearing in parts of differential geometry, [16, 52] and have, via work of Breen andothers, [48–51], been of central importance for non-Abelian cohomology. This connection betweenthe crossed menagerie and non-Abelian cohomology is almost as old as the crossed gadgetry itself,dating back to Dedecker’s work in the 1960s, [104]. Yet the basic message of what they are, whythey work, how they relate to other structures, and how the crossed menagerie works, still needrepeating, especially in that setting of non-Abelian cohomology in all its bewildering beauty.

The original notes have been augmented by lots of additional material, since the link withnon-Abelian cohomology was worth pursuing in much more detail. These notes thus contain anintroduction to the way ‘crossed gadgetry’ interacts with non-Abelian cohomology and areas suchas topological and homotopical quantum field theory. This entails the inclusion of a fairly detailedintroduction to torsors, gerbes, etc. This is based in part on Larry Breen’s beautiful Minneapolisnotes, [51].

If this is the first time you have met this sort of material, then some words of warning andwelcome are in order.

There is much too much in these notes to digest in one go!

There is probably a lot more than you will need in your continuing research. For instance, thematerial on torsors, etc., is probably best taken at a later sitting and the chapter ‘Beyond 2-types’is not directly used until a lot later, so can be glanced at. The notes are also designed with a‘dipper’ in mind. I know that I do not always read a textbook or set of notes in a sequentialmanner, so after looking at crossed modules, you might need to look at sheaves of crossed modules,only to find you did need some ideas from the sections on crossed complexes as well. There arethe, now usual, hyperlinks to help navigation, but also some comments are repeated, especially ifthey are ‘warnings’, say about terminology or notation, so do not think, ‘he should have editedthat out’ if you find some such repeated point. That repetition is possibly, even probably, there fora purpose.

There are bits of bold text ‘for the reader’. Both the original audience in Buenos Aires andlater the students of MATH 5312 in Ottawa had varied backgrounds, so some quite routine ideashad not been met by a subset of them. Different subsets for different ideas. The use of boldindicates where it might be a good idea to write down some more details for yourself, if you havenot seen the idea before, or perhaps have not used it recently. It may indicate an exercise, althoughthere are few formal exercises. Typically going through an easy example of some idea or throughthe ‘routine’ verification of some properties can help build the intuition as to the interpretation and

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meaning of a concept. ‘Meaning’ and ‘interpretation’ are highly dependent on the use the reader,i.e., you, will make of an idea, so it is not for me to impose my meaning on you. I can howeverprovide indications of intuition that has been helpful to me or to others, especially when at anentry point for a sequence of new ideas.

To start with I have concentrated on the group theoretic and geometric aspects of cohomology,since the non-Abelian theory is better developed there, but it is easy to attack other topics such asLie algebra cohomology, once the basic ideas of the group case have been mastered and applicationsin differential geometry do need the torsors, etc. I have emphasised approaches using crossedmodules (of groups). Analogues of these gadgets do exist in the other settings (Lie algebras, etc.),and most of the ideas go across without too much pain. If handling a non-group based problem(e.g. with monoids or categories), then the internal categorical aspect - crossed module as internalcategory in groups - would replace the direct method used here. Moreover the group based theoryhas the advantage of being central to both algebraic and geometric applications.

The aim of the notes is not to give an exhaustive treatment of cohomology and related concepts.That would be impossible. If, at the end of reading the relevant sections, the reader feels that theyhave some intuition on the meaning and interpretation of cohomology classes in their own area,and that they can more easily attack other aspects of cohomological and homotopical algebra bythemselves, then the notes will have succeeded for them.

Although not ‘self contained’, I have tried to introduce topics such as sheaf theory as and whennecessary, so as to give a natural development of the ideas. Some readers will already have beenintroduced to these ideas and they need not read those sections in detail. Such sections are, Ithink, clearly indicated. They do not give all the details of those areas, of course. For a start, thosedetails are not needed for the purposes of the notes, but the summaries do try to sketch in enough‘intuition’ to make it reasonable clear, I hope, what the notes are talking about!

I have not assumed much formal knowledge of category theory, at least to start with. Theidea of category, a functor and a natural transformation will be used and not defined. Mention ofpullbacks, pushouts and similar notions, if the reader has not met them before, should send them‘hot foot’ to a category theory textbook or to web-sources such as Wikipedia. Only a passingacquaintance is needed to start with, as hopefully, as such ideas are used they will become clearer.Later on many more areas related to category theory will be introduced, but it will be assumedthat by that point in the notes, the reader is more independently able to search out and read othersources for details when needed.

I have not tried to cover everything possible and would point the noses of researchers, who wantto find out more, in the direction of the nLab. This is a ‘wiki’ devoted to, I quote:collaborative work on Mathematics, Physics and Philosophy especially insofar as these subjectstouch on n-categories and related ‘higher algebraic structures’.The home page can be found at: http://ncatlab.org/nlab/show/HomePage, [221].

Acknowledgements

These notes were started as backup for the lectures at the XVI Encuentro Rioplatense deÁlgebra y Geometŕıa Algebraica, in Buenos Aires, 12-15 December 2006. That meeting, and thusmy visit to Argentina, was supported by several organisations there, CONICET, ANCPT, andthe University of Buenos Aires, and in Uruguay, CSIC and PDT, and by a travel grant from theLondon Mathematical Society. The visit would not have been possible without the assistance ofGabriel Minian and his colleagues and students, who provided an excellent environment for researchdiscussions and, of course, the meeting itself.

http://ncatlab.org/nlab/show/HomePage

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The notes were continued and expanded for course MATH 5312 in the Spring of 2007 duringa visit as a visiting professor to the Department of Mathematics and Statistics of the Universityof Ottawa. Thanks are due to Rick Blute, Pieter Hofstra, Phil Scott, Paul-Eugene Parent, BarryJessup and Jonathan Scott for the warm welcome and the mathematical discussions on some of thematerial and the students of MATH 5312 for their interest and constructive comments. Financialassistance from the University of Ottawa and the NSERC is also gratefully acknowledged.

Some parts of the notes formed part of my session “Classifying spaces of categorical groups,and relations with non-Abelian cohomology” at the Workshop on Categorical Groups, Barcelona,(June 16th – 20th, 2008), and this resulted in some reworking of material, and additional materialbeing added, as I learnt or relearnt various aspects of the theory of categorical groups that I hadnot seen recently, forgotten, never seen or whatever. Thanks are due to the organisers and theparticipants who together contributed to a delightful and rewarding meeting and thus, incidently,to the revision of the notes.

Again the first few chapters of the notes were used for a ‘TQFT Club’ minicourse in theCAMGSD, in the IST, Lisbon, (3 and 4 December, 2008), entitled The ‘Crossed’ technology andits applications. The feedback and questions that resulted again motivated certain additions andchanges. Thanks are due to Roger Picken and João Faria Martins for the invitation, and, inaddition, the participants, in particular Pedro Resende and Tim Van der Linden, for useful questionsand comments, and the warm welcome. Since then I have benefitted from feedback, discussionsand further ideas in visits to Lisbon, Louvain-la-Neuve, and more recently Leeds.

At Lyon, I used parts of the description of Schreier theory as a basis for a series of talks to theworking group, Invariants algébriques en informatique, in the Spring of 2010, during a visit to theInstitut Camille Jordan, Lyon 1, (UMR 5208 du CNRS). Acknowledgement is due for the supportfrom the CNRS and, of course, thanks for the warm welcome, and for the stimulating discussionson possible applications and analogues of these ideas in various areas, with Philippe Malbos andYves Guiraud. More recently, they and others invited me to give a minicourse on the connectionsbetween ‘Rewriting and Homotopy ’ at a meeting (week 5 of LI2012) held at the C.I.R.M. at Luminy.(Thanks to the organisers and the staff at the C.I.R.M. for a very enjoyable visit.) The notes forthis needed some additional material on syzygies and were a good occasion / excuse to preparemore material relating to the Abels and Holz methods of calculating syzygies, the links with workon algebraic K-theory and Haefliger’s complexes of groups. This fits in well and hopefully will beuseful for some reader. (I will put some of this on the nLab.)

Tim Porter, ex-Bangor, and other places including Ottawa, Paris, Granada, Lisbon, Lyon,Savoie, Leeds, Louvain-la-Neuve, ..., 2007 - 2018.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Preliminaries 15

1.1 Groups and Groupoids, some basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 A very brief introduction to cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.3 Homology and Cohomology of spaces. . . . . . . . . . . . . . . . . . . . . . . 21

1.2.4 Betti numbers and Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2.6 The bar resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3 Simplicial things in a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.1 Simplicial Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.2 Simplicial Objects in Categories other than Sets . . . . . . . . . . . . . . . . 30

1.3.3 The Moore complex and the homotopy groups of a simplicial group . . . . . 34

1.3.4 Kan complexes and Kan fibrations . . . . . . . . . . . . . . . . . . . . . . . . 35

1.3.5 Simplicial groups are Kan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3.6 T -complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.3.7 Group T-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Crossed modules - definitions, examples and applications 41

2.1 Crossed modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.1 Algebraic examples of crossed modules . . . . . . . . . . . . . . . . . . . . . . 41

2.1.2 Topological Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.1.3 Restriction along a homomorphism ϕ/ ‘Change of base’ . . . . . . . . . . . . 46

2.2 Group presentations, identities and 2-syzygies . . . . . . . . . . . . . . . . . . . . . . 46

2.2.1 Presentations and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.2 Directed graphs, quivers and their reflexive ‘cousins’ . . . . . . . . . . . . . . 48

2.2.3 Cayley quiver of a ‘generated group’ . . . . . . . . . . . . . . . . . . . . . . . 51

2.2.4 Free crossed modules and identities . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3 Cohomology, crossed extensions and algebraic 2-types . . . . . . . . . . . . . . . . . 55

2.3.1 Cohomology and extensions, continued . . . . . . . . . . . . . . . . . . . . . . 55

2.3.2 Not really an aside! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3.3 Perhaps a bit more of an aside ... for the moment! . . . . . . . . . . . . . . . 59

2.3.4 Automorphisms of a group yield a 2-group . . . . . . . . . . . . . . . . . . . . 60

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6 CONTENTS

2.3.5 Back to 2-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 Crossed complexes 67

3.1 Crossed complexes: the Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.1.1 Examples: crossed resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.1.2 The standard crossed resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2 Crossed complexes and chain complexes: I . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.1 Semi-direct products and derivations. . . . . . . . . . . . . . . . . . . . . . . 71

3.2.2 Derivations and derived modules. . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.4 Derivation modules and augmentation ideals . . . . . . . . . . . . . . . . . . 73

3.2.5 Generation of I(G). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2.6 (Dϕ, dϕ), the general case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.2.7 Dϕ for ϕ : F (X)→ G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Associated module sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Homological background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.2 The exact sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.3 Reidemeister-Fox derivatives and Jacobian matrices . . . . . . . . . . . . . . 79

3.4 Crossed complexes and chain complexes: II . . . . . . . . . . . . . . . . . . . . . . . 83

3.4.1 The reflection from Crs to chain complexes . . . . . . . . . . . . . . . . . . . 83

3.4.2 Crossed resolutions and chain resolutions . . . . . . . . . . . . . . . . . . . . 85

3.4.3 Standard crossed resolutions and bar resolutions . . . . . . . . . . . . . . . . 86

3.4.4 The intersection A ∩ [C,C]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.5 Simplicial groups and crossed complexes . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5.1 From simplicial groups to crossed complexes . . . . . . . . . . . . . . . . . . . 87

3.5.2 Simplicial resolutions, a bit of background . . . . . . . . . . . . . . . . . . . . 89

3.5.3 Free simplicial resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.5.4 Step-by-Step Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.5.5 Killing Elements in Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . 92

3.5.6 Constructing Simplicial Resolutions . . . . . . . . . . . . . . . . . . . . . . . 93

3.6 Cohomology and crossed extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6.1 Cochains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6.2 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6.3 Huebschmann’s description of cohomology classes . . . . . . . . . . . . . . . . 94

3.6.4 Abstract Kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.7 2-types and cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.7.1 2-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.7.2 Example: 1-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.7.3 Algebraic models for n-types? . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.7.4 Algebraic models for 2-types. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.8 Re-examining group cohomology with Abelian coefficients . . . . . . . . . . . . . . . 99

3.8.1 Interpreting group cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.8.2 The Ext long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.8.3 From Ext to group cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.8.4 Exact sequences in cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 105

CONTENTS 7

4 Syzygies, and higher generation by subgroups 1094.1 Back to syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1.1 Homotopical syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.1.2 Syzygies for the Steinberg group . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2 A brief sideways glance: simple homotopy and algebraic K-theory . . . . . . . . . . 1124.2.1 Grothendieck’s K0(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.2.2 Simple homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.2.3 The Whitehead group and K1(R) . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2.4 Milnor’s K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.2.5 Higher algebraic K-theory: some first remarks . . . . . . . . . . . . . . . . . 123

4.3 Higher generation by subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.3.1 The nerve of a family of subgroups . . . . . . . . . . . . . . . . . . . . . . . . 1244.3.2 n-generating families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.3.3 A more complex family of examples . . . . . . . . . . . . . . . . . . . . . . . 1274.3.4 Volodin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.5 The two nerves of a relation: Dowker’s construction . . . . . . . . . . . . . . 1314.3.6 Barycentric subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.3.7 Dowker’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.3.8 Flag complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.3.9 The homotopy type of Vietoris - Volodin complexes . . . . . . . . . . . . . . 1364.3.10 Back to the Volodin model ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.3.11 The case of van Kampen’s theorem and presentations of pushouts . . . . . . 147

4.4 Group actions and the nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.4.1 The G-action on N(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.4.2 The G-action on V (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.4.3 Group actions on simplicial complexes . . . . . . . . . . . . . . . . . . . . . . 155

4.5 Complexes of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.5.1 Simplicial complexes, barycentres and scwols . . . . . . . . . . . . . . . . . . 1614.5.2 Complexes of groups: introduction . . . . . . . . . . . . . . . . . . . . . . . . 1634.5.3 Graphs of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.5.4 The fundamental group(oid) of a graph of groups . . . . . . . . . . . . . . . . 1674.5.5 A graph of 2-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.5.6 A brief glance at the Bass-Serre theory . . . . . . . . . . . . . . . . . . . . . . 1714.5.7 Fundamental group(oid) of a complex of groups . . . . . . . . . . . . . . . . . 1734.5.8 Haefliger’s theorem on developable complexes of groups . . . . . . . . . . . . 1754.5.9 Over to scwols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.5.10 Quotient of a scwol by an action . . . . . . . . . . . . . . . . . . . . . . . . . 1774.5.11 Coverings of small categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.5.12 Fundamental group(oid) of a scwol . . . . . . . . . . . . . . . . . . . . . . . . 1794.5.13 Constructing a simply connected covering of a connected category . . . . . . 1814.5.14 The groupoid case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

4.6 Complexes of groups on a scwol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.6.1 General complexes of groups revisited . . . . . . . . . . . . . . . . . . . . . . 1864.6.2 Morphisms of complexes of groups . . . . . . . . . . . . . . . . . . . . . . . . 1874.6.3 Homotopy of morphisms of complexes of groups . . . . . . . . . . . . . . . . 1884.6.4 The category associated to a complex of groups . . . . . . . . . . . . . . . . . 188

8 CONTENTS

4.7 Orbifolds, orbihedra and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.7.1 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5 Beyond 2-types 191

5.1 n-types and decompositions of homotopy types . . . . . . . . . . . . . . . . . . . . . 191

5.1.1 n-types of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

5.1.2 n-types of simplicial sets and the coskeleton functors . . . . . . . . . . . . . . 198

5.1.3 Postnikov towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.1.4 Whitehead towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

5.2 Crossed squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

5.2.1 An introduction to crossed squares . . . . . . . . . . . . . . . . . . . . . . . . 212

5.2.2 Crossed squares, definition and examples . . . . . . . . . . . . . . . . . . . . 212

5.3 2-crossed modules and related ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.3.1 Truncations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.3.2 Truncated simplicial groups and the Brown-Loday lemma . . . . . . . . . . . 215

5.3.3 1- and 2-truncated simplicial groups . . . . . . . . . . . . . . . . . . . . . . . 216

5.3.4 2-crossed modules, the definition . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.3.5 Examples of 2-crossed modules . . . . . . . . . . . . . . . . . . . . . . . . . . 219

5.3.6 Exploration of trivial Peiffer lifting . . . . . . . . . . . . . . . . . . . . . . . . 220

5.3.7 2-crossed modules and crossed squares . . . . . . . . . . . . . . . . . . . . . . 221

5.3.8 2-crossed complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.4 Catn-groups and crossed n-cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.4.1 Cat2-groups and crossed squares . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.4.2 Interpretation of crossed squares and cat2-groups . . . . . . . . . . . . . . . . 226

5.4.3 Catn-groups and crossed n-cubes, the general case . . . . . . . . . . . . . . . 230

5.5 Loday’s Theorem and its extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.5.1 Simplicial groups and crossed n-cubes, the main ideas . . . . . . . . . . . . . 233

5.5.2 Squared complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

5.6 Crossed N-cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2385.6.1 Just replace n by N? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2385.6.2 From simplicial groups to crossed n-cube complexes . . . . . . . . . . . . . . 240

5.6.3 From n to n− 1: collecting up ideas and evidence . . . . . . . . . . . . . . . 241

6 Classifying spaces, and extensions 245

6.1 Non-Abelian extensions revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

6.2 Classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.2.1 Simplicially enriched groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6.2.2 Conduché’s decomposition and the Dold-Kan Theorem . . . . . . . . . . . . . 251

6.2.3 W and the nerve of a crossed complex . . . . . . . . . . . . . . . . . . . . . . 254

6.3 Simplicial Automorphisms and Regular Representations . . . . . . . . . . . . . . . . 256

6.4 Simplicial actions and principal fibrations . . . . . . . . . . . . . . . . . . . . . . . . 258

6.4.1 More on ‘actions’ and Cartesian closed categories . . . . . . . . . . . . . . . . 258

6.4.2 G-principal fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6.4.3 Homotopy and induced fibrations . . . . . . . . . . . . . . . . . . . . . . . . . 263

6.5 W , W and twisted Cartesian products . . . . . . . . . . . . . . . . . . . . . . . . . . 265

6.6 More examples of Simplicial Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

CONTENTS 9

7 Non-Abelian Cohomology: Torsors, and Bitorsors 271

7.1 Descent: Bundles, and Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 271

7.1.1 Case study 1: Topological Interpretations of Descent. . . . . . . . . . . . . . 272

7.1.2 Case Study 2: Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 275

7.1.3 Case Study 3: Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

7.1.4 Change of Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

7.2 Descent: simplicial fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.2.1 Fibre bundles, the simplicial viewpoint . . . . . . . . . . . . . . . . . . . . . . 282

7.2.2 Atlases of a simplicial fibre bundle . . . . . . . . . . . . . . . . . . . . . . . . 283

7.2.3 Fibre bundles are T.C.P.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

7.2.4 . . . and descent in all that? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

7.3 Descent: Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.3.1 Introduction and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.3.2 Presheaves and sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

7.3.3 Sheaves and étale spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

7.3.4 Covering spaces and locally constant sheaves . . . . . . . . . . . . . . . . . . 296

7.3.5 A siting of Grothendieck toposes . . . . . . . . . . . . . . . . . . . . . . . . . 297

7.3.6 Hypercovers and covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

7.3.7 Simplicial approaches to descent data . . . . . . . . . . . . . . . . . . . . . . 301

7.3.8 Base change at the sheaf level . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

7.3.9 Pullback and pushforward in the spatial case . . . . . . . . . . . . . . . . . . 305

7.3.10 Change of base for general presheaves . . . . . . . . . . . . . . . . . . . . . . 309

7.3.11 Change of index and geometric morphisms . . . . . . . . . . . . . . . . . . . 310

7.3.12 Change of base and descent data . . . . . . . . . . . . . . . . . . . . . . . . . 312

7.4 Descent: Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7.4.1 Torsors: definition and elementary properties . . . . . . . . . . . . . . . . . . 313

7.4.2 Torsors and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

7.4.3 Change of base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

7.4.4 Contracted Product and ‘Change of Groups’ . . . . . . . . . . . . . . . . . . 320

7.4.5 Simplicial Description of Torsors . . . . . . . . . . . . . . . . . . . . . . . . . 325

7.4.6 Torsors and exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

7.5 Bitorsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

7.5.1 Bitorsors: definition and elementary properties . . . . . . . . . . . . . . . . . 328

7.5.2 Bitorsor form of Morita theory (First version): . . . . . . . . . . . . . . . . . 330

7.5.3 Twisted objects: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

7.5.4 Cohomology and Bitorsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

7.5.5 Bitorsors, a simplicial view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

7.5.6 Cleaning up ‘Change of Base’ . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7.6 Relative M-torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

7.6.1 Relative M-torsors: what are they? . . . . . . . . . . . . . . . . . . . . . . . . 346

7.6.2 An alternative look at Change of Groups and relative M-torsors . . . . . . . . 351

7.6.3 Examples and special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7.6.4 Change of crossed module bundle for ‘bitorsors’. . . . . . . . . . . . . . . . . 355

7.6.5 Representations of crossed modules. . . . . . . . . . . . . . . . . . . . . . . . 356

10 CONTENTS

8 Hypercohomology and exact sequences 359

8.1 Hyper-cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

8.1.1 Classical Hyper-cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

8.1.2 Čech hyper-cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

8.1.3 Non-Abelian Čech hyper-cohomology. . . . . . . . . . . . . . . . . . . . . . . 363

8.2 Mapping cocones and Puppe sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 365

8.2.1 Mapping Cylinders, Mapping Cones, Homotopy Pushouts, Homotopy Cok-ernels, and their cousins! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

8.2.2 Puppe exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

8.3 Puppe sequences and classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 376

8.3.1 Fibrations and classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . 376

8.3.2 WG is a Kan complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

8.3.3 Loop spaces and loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

8.3.4 Applications: Extensions of groups . . . . . . . . . . . . . . . . . . . . . . . . 383

8.3.5 Applications: Crossed modules and bitorsors . . . . . . . . . . . . . . . . . . 384

8.3.6 Examples and special cases revisited . . . . . . . . . . . . . . . . . . . . . . . 387

8.3.7 Devissage: analysing M−Tors . . . . . . . . . . . . . . . . . . . . . . . . . . 387

9 Non-Abelian Cohomology: Stacks 389

9.1 Fibred Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

9.1.1 The structure of Sh(B) and Tors(G) . . . . . . . . . . . . . . . . . . . . . . . 3909.1.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

9.1.3 Fibred Categories and pseudo-functors . . . . . . . . . . . . . . . . . . . . . . 392

9.2 The Grothendieck construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

9.2.1 The basic Grothendieck construction and its variants . . . . . . . . . . . . . . 394

9.2.2 Fibred categories as Grothendieck fibrations . . . . . . . . . . . . . . . . . . . 397

9.2.3 From pseudo-functors to fibrations . . . . . . . . . . . . . . . . . . . . . . . . 402

9.2.4 . . . and back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

9.2.5 Two special cases and a generalisation . . . . . . . . . . . . . . . . . . . . . . 406

9.3 Fibred category theory: some elementary results . . . . . . . . . . . . . . . . . . . . 408

9.3.1 Fibred subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

9.3.2 Fibred categories: a categorification of presheaves and a simplicial view . . . 409

9.3.3 More structure: 2-cells, equivalences, etc. . . . . . . . . . . . . . . . . . . . . 412

9.3.4 The fibred Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

9.3.5 An interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

9.3.6 Change of base for fibrations / fibred categories? First thoughts. . . . . . . . 421

9.4 Back to the Grothendieck construction ... and lax, op-lax and pseudo things . . . . . 423

9.4.1 The Grothendieck construction as a (op-)lax colimit . . . . . . . . . . . . . . 423

9.4.2 Presenting the Grothendieck construction / op-lax colimit . . . . . . . . . . . 429

9.5 Prestacks: sheaves of local morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 432

9.5.1 Sh(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4339.5.2 Tor(B;G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4369.5.3 Prestackification! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

9.5.4 Change of base for prestacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

9.6 From prestacks to stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

9.6.1 The descent category, Des(U, F ) . . . . . . . . . . . . . . . . . . . . . . . . . 437

CONTENTS 11

9.6.2 Simplicial interpretations of Des(U, F ): first steps . . . . . . . . . . . . . . . 4389.6.3 Stacks - at last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

9.6.4 Back to Sh(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4429.6.5 Stacks of Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

9.6.6 Strong and weak equivalences: stacks and prestacks . . . . . . . . . . . . . . 443

9.6.7 ‘Stack completion’ aka ‘stackification’ . . . . . . . . . . . . . . . . . . . . . . 444

9.6.8 Stackification and Pseudo-Colimits . . . . . . . . . . . . . . . . . . . . . . . . 447

9.6.9 Stacks and sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

9.6.10 What about stacks of bitorsors? . . . . . . . . . . . . . . . . . . . . . . . . . 450

9.6.11 Stacks of equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

9.6.12 Morita theory revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

9.7 Base change for stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

9.8 Locally constant stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

10 Non-Abelian Cohomology: Gerbes 457

10.1 Gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

10.1.1 Definition and elementary properties of Gerbes . . . . . . . . . . . . . . . . . 458

10.1.2 G-gerbes and the semi-local description of a gerbe . . . . . . . . . . . . . . . 460

10.1.3 Some examples and non-examples of gerbes . . . . . . . . . . . . . . . . . . . 461

10.2 Geometric examples of gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

10.2.1 Line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

10.2.2 Line bundle gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

10.2.3 From bundles gerbes to gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . 477

10.2.4 Bundle gerbes and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

10.3 Cocycle description of gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

10.3.1 The local description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

10.3.2 From local to semi-local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

11 Homotopy Coherence and Enriched Categories. 491

11.1 Case study: examples of homotopy coherent diagrams . . . . . . . . . . . . . . . . . 491

11.2 Simplicially enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

11.2.1 Categories with simplicial ‘hom-sets’ . . . . . . . . . . . . . . . . . . . . . . . 495

11.2.2 Examples of S-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

11.2.3 From simplicial resolutions to S-categories . . . . . . . . . . . . . . . . . . . . 499

11.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

11.3.1 The ‘homotopy’ category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

11.3.2 Tensoring and Cotensoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

11.4 Nerves and Homotopy Coherent Nerves . . . . . . . . . . . . . . . . . . . . . . . . . 506

11.4.1 Kan and weak Kan complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

11.4.2 Categorical nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

11.4.3 Quasi-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

11.4.4 Homotopy coherent diagrams and homotopy coherent nerves . . . . . . . . . 511

11.4.5 Simplicial coherence and models for homotopy types . . . . . . . . . . . . . . 517

11.5 Useful examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

11.5.1 G-spaces: discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

11.5.2 Lax, Op-lax and pseudo- functors . . . . . . . . . . . . . . . . . . . . . . . . . 520

12 CONTENTS

11.5.3 ...and nerves for 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

11.5.4 Lax, oplax and pseudo-natural transformations . . . . . . . . . . . . . . . . . 531

11.5.5 Modifications between pseudo-natural transformations . . . . . . . . . . . . 533

11.5.6 Somewhat of an aside on lax, and pseudo-limits and colimits . . . . . . . . . 535

11.5.7 Back to the nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

11.5.8 Weak actions of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

11.5.9 Čech and Vietoris complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

11.6 Two nerves for 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

11.6.1 The 2-category, X(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54811.6.2 The geometric nerve, Ner(X(C)) . . . . . . . . . . . . . . . . . . . . . . . . . 54911.6.3 W (H) in functional composition notation . . . . . . . . . . . . . . . . . . . . 551

11.6.4 Visualising W (K(C)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55311.7 Pseudo-functors between 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

11.7.1 Weak maps between crossed modules . . . . . . . . . . . . . . . . . . . . . . . 556

11.7.2 The simplicial description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

11.7.3 The conjugate loop groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

11.7.4 Identifying M(G, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

11.7.5 Cofibrant replacements for crossed modules . . . . . . . . . . . . . . . . . . . 572

11.7.6 Weak maps: from cofibrant replacements to the algebraic form . . . . . . . . 576

11.7.7 Butterflies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

11.7.8 . . . and the strict morphisms in all that? . . . . . . . . . . . . . . . . . . . . . 584

12 Other enrichments, other versions of (homotopy) coherence? 589

12.1 Other enrichments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

12.1.1 Enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

12.1.2 A little bit of enriched category theory . . . . . . . . . . . . . . . . . . . . . . 590

12.1.3 Changing the enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

12.1.4 Enrichment over Simp.K−Mod . . . . . . . . . . . . . . . . . . . . . . . . . 59512.1.5 Enrichment over chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . 596

12.2 From simplicially enriched to chain complex enriched . . . . . . . . . . . . . . . . . . 598

12.2.1 Shuffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

12.2.2 Shuffles in more detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

12.2.3 The Eilenberg - Mac Lane map: proof of concept . . . . . . . . . . . . . . . . 601

13 More simplicial and categorical constructions! 607

13.1 Total complex constructions: part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

13.2 Ordinal sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

13.2.1 Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

13.2.2 Dec and diag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

13.3 First Interlude: Augmentations, Décalage and Resolutions . . . . . . . . . . . . . . . 613

13.3.1 Augmented Simplicial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 613

13.3.2 Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

13.3.3 Back towards the décalage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

13.3.4 Monads and Comonads, Algebras and Coalgebras . . . . . . . . . . . . . . . . 627

13.3.5 More on comonadic resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 631

13.4 Second Interlude: Ends and coends . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

CONTENTS 13

13.4.1 From sets of natural transformations towards ‘ends’ . . . . . . . . . . . . . . 638

13.4.2 Dinatural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

13.4.3 More examples of ends and coends and their uses: . . . . . . . . . . . . . . . 643

13.5 Ordinal sum, revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

13.5.1 The right adjoint of Dec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

13.5.2 The Artin-Mazur codiagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

13.6 Total complex constructions: part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

13.6.1 Total complexes of cosimplicial simplicial sets . . . . . . . . . . . . . . . . . . 657

13.6.2 ∇ and Tot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65813.6.3 The Alexander-Whitney map and its applications . . . . . . . . . . . . . . . . 659

13.6.4 The classical Eilenberg-Zilber theorem . . . . . . . . . . . . . . . . . . . . . . 662

13.7 Crs as an S-category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664

13.7.1 Crossed complexes recalled and revisited . . . . . . . . . . . . . . . . . . . . . 665

13.7.2 Tensor products of crossed complexes . . . . . . . . . . . . . . . . . . . . . . 670

13.8 Complicial and weak complicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

13.8.1 Stratified simplicial sets in the sense of Verity . . . . . . . . . . . . . . . . . . 674

14 Indexed / weighted limits and colimits 677

14.1 Enriched Limits and Colimits? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

14.1.1 Bousfield Kan Homotopy Limits . . . . . . . . . . . . . . . . . . . . . . . . . 678

14.1.2 Homotopy pullback, a simple case study . . . . . . . . . . . . . . . . . . . . . 679

14.2 Profunctors / Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

14.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

14.2.2 Composition? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

14.2.3 The bicategory of V-profunctors . . . . . . . . . . . . . . . . . . . . . . . . . 684

14.2.4 Right Kan extensions and profunctors . . . . . . . . . . . . . . . . . . . . . . 684

14.3 Indexed / weighted limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . 685

14.3.1 Indexed cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

14.3.2 Indexed / weighted limits: the definition . . . . . . . . . . . . . . . . . . . . . 687

14.3.3 Some first examples: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

14.3.4 Total category of a cosimplicial category . . . . . . . . . . . . . . . . . . . . . 689

14.3.5 Lax and pseudo-(co)ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

14.4 Simplicial Replacement Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699

14.4.1 The simple case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699

14.4.2 The enriched case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

14.5 Homotopy limits as indexed limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704

14.6 Homotopy colimits and simplicial replacement . . . . . . . . . . . . . . . . . . . . . . 706

14.6.1 Homotopy colimit, the enriched case . . . . . . . . . . . . . . . . . . . . . . . 706

14.6.2 Simplicial replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706

14.6.3 The Grothendieck construction and homotopy colimits . . . . . . . . . . . . . 707

14.6.4 The homotopy orbit space / Borel construction . . . . . . . . . . . . . . . . . 707

14.6.5 G-spaces and fibrations over BG . . . . . . . . . . . . . . . . . . . . . . . . . 707

14.7 Homotopy Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

14.7.1 Simplicially coherent ends and coends. . . . . . . . . . . . . . . . . . . . . . . 708

14.7.2 Coherent ends and Kan complexes. . . . . . . . . . . . . . . . . . . . . . . . . 714

14.7.3 Coherent transformations and the coherent Yoneda lemma. . . . . . . . . . . 716

14 CONTENTS

Bibliography 725

Index 743

Chapter 1

Preliminaries

1.1 Groups and Groupoids, some basic ideas

Before launching into crossed modules, we need a word on groupoids.

1.1.1 Groupoids

By a groupoid, we mean a small category in which all morphisms are isomorphisms. (If you havenot formally met categories then do not worry, the idea will come through without that specificformal knowledge, although a quick glance at Wikipedia for the definition of a category mightbe a good idea at some time soon. You do not need category theory as such at this stage.)These groupoids typically arise in three situations (i) symmetry objects of a fibered structure,(ii) equivalence relations, and (iii) group actions. It is worth noting that several of the initialapplications of groups were thought of, by their discoverers, as being more naturally this type ofgroupoid structure.

For the first, assume we have a family of sets {Xa : a ∈ A}. Typically we have a functionf : X → A and Xa = f−1a for a ∈ A. We form the symmetry groupoid of the family by takingthe index set, A, as the set of objects of the groupoid, G, and, if a, a′ ∈ A, then G(a, a′), the set ofarrows in our symmetry groupoid from a to a′, is the set Bijections(Xa, Xa′). This G will containall the individual symmetry groups / permutation groups of the various Xa, but will also recordcomparison information between different Xas.

Of course, any group is a groupoid with one object and, conversely, if G is any groupoid, wehave, for each object a of G, a group G(a, a), of arrows that start and end at a. This is the‘automorphism group’, autG(a), of a within G. It is also referred to as the vertex group of G at a,and denoted G(a). This later viewpoint and notation emphasise more the combinatorial, graph-likeside of G’s structure. Sometimes the notation G[1] may be used for G as the process of regardinga group as a groupoid is a sort of ‘suspension’ or ‘shift’. It is one aspect of ‘categorification’, cf.Baez and Dolan, [15].

That combinatorial side is strongly represented in the second situation, equivalence relations.Suppose that R is an equivalence relation on a set X. Going back to basics, R is a subset of X×Xsatisfying:

(a) if a, b, c ∈ X and (a, b) and (b, c) ∈ R, then (a, c) ∈ R, i.e., R is transitive;

(b) for all a ∈ X, (a, a) ∈ R, alternatively the diagonal ∆ ⊆ R, i.e., R is reflexive;

15

16 CHAPTER 1. PRELIMINARIES

(c) if a, b ∈ X and (a, b) ∈ R, then (b, a) ∈ R, i.e., R is symmetric.

Two comments might be made here. The first is ‘everyone knows that!’, the second ‘that is not theusual order to put them in! Why the change?’

It is a well known, but often forgotten, fact that from R, you get a groupoid (which we willdenote by R). The objects of R are the elements of X and R(a, b) is a singleton if (a, b) ∈ R andis empty otherwise. (There is really no need to label the single element of R(a, b), when this isnon empty, but it is sometimes convenient to call it (a, b) at the risk of over using the ordered pairnotation.) Now transitivity of R gives us a composition function: for a, b, c ∈ X,

◦ : R(a, b)×R(b, c)→ R(a, c).

(Remember that a product of a set with the empty set is itself always empty, and that for any set,there is a unique function with domain ∅ and codomain the set, so checking that this compositionworks nicely is slightly more subtle than you might at first think. This is important when handlingthe analogues of equivalence relations in other categories., then you cannot just write (a, b)◦(b, c) =(a, c), or similar, as ‘elements’ may not be obvious things to handle.) Of course this compositionis associative, but if you have not seen the verification, it is important to think about it, lookingfor subtle points, especially concerning the empty set and empty function and how to do the proofwithout ‘elements’.

This composition makes R into a category, since (a) gives the existence of identities for eachobject. (Ida = (a, a) in ‘elementary’ notation.) Finally (c) shows that each (a, b) is invertible, soR is a groupoid. (You now see why that order was the natural one for the axioms. You cannotprove that (a, a) is an identity until you have a composition, and similarly until you have identities,inverses do not make sense.) We may call R, the groupoid of the equivalence relation R.

This shows how to think of R as a groupoid, R. The automorphism groups, R(a), are allsingletons as sets, so are trivial groups1. Conversely any groupoid, G, gives a diagram

Arr(G)s //t// Ob(G)

ioo

with s = ‘source’, t = ‘target’ and i= ‘identity’ as it picks out the identity arrow on each object.It thus gives a function,

Arr(G)(s,t) // Ob(G)×Ob(G) .

The image of this function is an equivalence relation as is easily checked. We will call this equivalencerelation, R, for the moment. If G is a groupoid such that each G(a) is a trivial group, then eachG(a, b) has at most one element (check it), so (s, t) is a one-one function and it is then trivial tonote that G is isomorphic to the groupoid of the equivalence relation, R.

Examples: (i) We denote by I, the interval groupoid which has two objects, 0 and 1, andmorphisms ι : 0 → 1 and its inverse, together with, of course, the identity arrows at each object.This is the groupoid that corresponds to the equivalence relation on {0, 1} which makes 0 and 1equivalent. It is used within the theory of groupoids as the analogue of the topological unit interval,[0, 1].

1The groupoid is simply connected.

1.1. GROUPS AND GROUPOIDS, SOME BASIC IDEAS 17

(ii) If X is any set, we get two ‘extreme’ forms of equivalence relation on X, and thus two‘extreme’ examples of groupoids.

Definition: a) The discrete groupoid on the set, X, denoted Disc(X), has X as its set ofobjects and, for each ordered pair, (x1, x2), of elements of X,

Disc(S)(x1, x2) =

{{idx1} if x1 = x2∅ if x1 6= x2

b) The codiscrete groupoid on a set, X, denoted Codisc(X), has X as its set of objects and,for each ordered pair, (x1, x2), of elements of X, Codisc(X)(x1, x2) is a singleton set, the uniqueelement of which is conveniently denoted (x1, x2).

The two extreme examples of equivalence relations on X are, of course, (i) the relation ofequality, or identity, in which R is just ∆, the diagonal of X × X, and (a, b) ∈ R if, and only if,x = y, and (ii) the relation in which everything is related, so R = X ×X. The groupoid, Disc(X),corresponds to the first, whilst Codisc(X) corresponds to the second. The example, I, given in (i),above, is the case Codisc({0, 1}).

We have looked at these simple cases in some detail as in applications of the basic ideas,especially in algebraic geometry, arguments using elements are quite tricky to give and the initialintuition coming from these set-based examples can easily be forgotten.

The third situation, that of group actions, is also a common one in algebra and algebraicgeometry. Equivalence relations often come from group actions. If G is a group and X is a G-setwith (left) G-action,

G×X // X

(g, x) g · x

,

(i.e., a function act(g, x) = g · x, which must satisfy the rules 1 · x = x and for all g1, g2 ∈ G,g1 · (g2 · x) = (g1g2) · x, a sort of associativity law), then we get a groupoid ActG(X), that will becalled the action groupoid of the G-set, as follows:

• the objects of ActG(X) are the elements of X;

• if a, b,∈ X,ActG(X)(a, b) ∼= {g | g · a = b}.

An important word of caution is in order here. Logical complications can occur here if ActG(X)(a, b)is set equal to {g | g · a = b}, since then a g can occur in several different ‘hom-sets’. A good wayto avoid this is to take

ActG(X)(a, b) = {(g, a) | g · a = b}.

This is a non-trivial change. It basically uses a disjoint union, but although very simple, it isfundamental in its implications. We could also do it by taking ArrG(X) = G×X with source andtarget maps s(g, x) = x, t(g, x) = g ·x. (It is useful, if you have not seen this before, to see how thevarious parts of the definition of an action match with parts of the structural rules of a groupoid.This is important as it indicates how, much later on, we will relax those rules in various ways.)

18 CHAPTER 1. PRELIMINARIES

We will sometimes use the notation, Gy X, when discussing a left action of a group G on X.

In a groupoid, G, we say two objects, x and y are in the same connected component of G, ifG(x, y) is not empty. This gives an equivalence relation on the set of objects of G, as you caneasily check. The equivalence classes re called the connected components of G and the set ofconnected components is usually denoted π0(G), by analogy with the usual notion for the set ofconnected components of a topological space.

We have not discussed morphisms of groupoids. These are straightforward to define and towork with. Together groupoids and the morphisms between them form a category, the category ofgroupoids, which will be denoted Grpd.

(As we introduced structures of various types, we will usually introduce a corresponding formof morphism and it will be rare that the resulting ‘context’ of objects and morphisms does not forma category. It is important to look up the definition of categories and functors, but for the momentyou will not need to know very much ‘category theory’ to read the notes. It will suffice to get togrips with that as we go further and have good motivating examples for what is needed.)

Most of the concepts that we will be handling in what follows exist in many-object, groupoidversions as well as single-object, group based ones. For simplicity we will often, but not always,give concepts in the group based form, and will leave the other many-object form ‘to the reader’.The conversion is usually not that difficult.

For more details on the theory of groupoids, two good sources are Ronnie Brown’s book, [59]and Phil Higgins’ monograph, now reprinted as [151].

1.2 A very brief introduction to cohomology

Partially as a case study, at least initially, we will be looking at various constructions that relateto group cohomology. Later we will explore a more general type of (non-Abelian) cohomology,including ideas about the non-Abelian cohomology of spaces, but that is for later. To start withwe will look at a simple group theoretic problem that will be used for motivation at several placesin what follows. Much of what is in books on group cohomology is the Abelian theory, whilst wewill be looking more at the non-Abelian one. If you have not met cohomology at all, take a look atthe Wikipedia entries for group cohomology. You may not understanding everything, but there areideas there that will recur in what follows, and some terms that are described there or on linkedentries, that will be needed later.

1.2.1 Extensions.

Given a group, G, an extension of G by a group K is a group E with an epimorphism p : E → Gwhose kernel is isomorphic to K (i.e., a short exact sequence of groups

E : 1→ K → E p→ G→ 1.

As we asked that K be isomorphic to Ker p, we could have different groups E perhaps fitting intothis, yet they would still be essentially the same extension. We say two extensions, E and E′, are

1.2. A VERY BRIEF INTRODUCTION TO COHOMOLOGY 19

equivalent if there is an isomorphism between E and E′ compatible with the other data. We candraw a diagram

E

��

1 // K //

=��

E //

∼=��

G //

=��

1

E′ 1 // K // E′ // G // 1

A typical situation might be that you have an unknown group E′ that you suspect is really E (i.e.,is isomorphic to E). You find a known normal subgroup K of E is isomorphic to one in E′ andthat the two quotient groups are isomorphic,

1 // K //

∼=��

E //

?��

G //

∼=��

1

1 // K ′ // E′ // G′ // 1

(But always remember, isomorphisms compare snap shots of the two structures and once chosencan make things more ‘rigid’ than perhaps they really ‘naturally’ are. For instance, we might haveG a cyclic group of order 5 generated by an element a, and G′ one generated by b. ‘Naturally’we choose an isomorphism ϕ : G → G′ to send a to b, but why? We could have sent a to anynon-identity element of G′ and need to be sure that this makes no difference. This is not just‘attention to detail’. It can be very important. It stresses the importance of Aut(G), the group ofautomorphisms of G in this sort of situation.)

A simple case to illustrate that the extension problem is a valid one, is to consider K = C3 =〈a | a3〉, G = C2 = 〈b | b2〉.

We could take E = S3, the symmetric group on three symbols, or alternatively D3 (also calledD6 to really confuse things, but being the symmetry group of the triangle). This has a presentation〈a, b | a3, b2, (ab)2〉. But what about C6 = 〈c | c6〉? This has a subgroup {1, c2, c4} isomorphic to Kand the quotient is isomorphic to G. Of course, S3 is non-Abelian, whilst C6 is. The presentation ofC6 needs adjusting to see just how similar the two situations are. This group also has a presentation〈a, b | a3, b2, aba−1b〉, since we can deduce aba−1b = 1 from [a, b] = 1 and b2 = 1 where in termsof the old generator c, a = c2 and b = c3. So there is a presentation of C3 which just differs by asmall ‘twist’ from that of S3.

How could one be sure if S3 and C6 are the ‘only’ groups (up to isomorphism) that we couldput in that central position? Can we classify all the extensions of G by K?

These extension problems were one of the impetuses for the development of a ‘cohomological’approach to algebra, but they were not the only ones.

1.2.2 Invariants

Another group theoretic input is via group representation theory and the theory of invariants. IfG is a group of n × n invertible matrices then one can use the simple but powerful tools of linearalgebra to get good information on the elements of G and often one can tie this information in tosome geometric context, say, by identifying elements of G as leaving invariant some polytope orpattern, so G acts as a subgroup of the group of the symmetries of that pattern or object.

If, therefore, we use the group Gl(n,K) of such invertible matrices over some field K, then wecould map an arbitrary G into it and attempt to glean information on elements of G from the

20 CHAPTER 1. PRELIMINARIES

corresponding matrices. We thus consider a group homomorphism

ρ : G→ Gl(n,K),

then look for nice properties of the ρ(g). of course, ρ need not be a monomorphism and then wewill loose information in the process, but in any case such a morphism will make G act (linearly)on the vector space Kn. We could, more generally, replace K by a general commutative ring R, inparticular we could use the ring of integers, Z, and then replace Kn by a general module, M , overR. If R = Z, then this is just an Abelian group. (If you have not formally met modules look up adefinition. The theory feels very like that of vector spaces to start with at least, but as elementsin R need not have inverses, care needs to be taken - you cannot cancel or divide in general, sorx = ry does not imply x = y! Having looked up a definition, for most of the time you can think ofmodules as being vector spaces or Abelian groups and you will not be far wrong. We will shortlybut briefly mention modules over a group algebra, R[G], and that ring is not commutative, butagain the complications that this does cause will not worry us at all.)

We can thus ‘represent’ G by mapping it into the automorphism group of M . This gives M thestructure of a G-module. We look for invariants of the action of G on M - what are they? Supposethat G is some group of symmetries of some geometric figure or pattern, that we will call X, inRn, then for each g ∈ G, gX = X, since g acts by pushing the pattern around back onto itself. Aninvariant of G, considered as acting on M , or, to put it more neatly, of the G-module, M , is anelement m in M such that g.m = m for all g ∈ G. These form a submodule,

MG = {m | gm = m for all g ∈ G}.

Clearly, it will help in our understanding of the structure of G if we can calculate and analysethese modules of invariants. Now suppose we are looking at a submodule N of M , then NG

is a submodule of MG and we can hope to start finding invariants, perhaps by looking at suchsubmodules and the corresponding quotient modules, M/N . We have a short exact sequence

0→ N →M →M/N → 0,

but, although applying the (functorial) operation (−)G does yield

0→ NG →MG → (M/N)G,

the last map need not be onto so we may not get a short exact sequence and hence a nice simpleway of finding invariants!

Example: Try G = C2 = {1, a}, M = Z, the Abelian group of integers, with G action,a.n = −n, and N = 2Z, the subgroup of even integers, with the same G action. Now calculate theinvariant modules MG and NG; they are both trivial, but M/N ∼= Z2, and ..., what is (M/N)G forthis example?

The way of studying this in general is to try to to continue the exact sequence further to the rightin some universal and natural way (via the theory of derived functors). This is what cohomologydoes. We can get a long exact sequence,

0→ NG →MG → (M/N)G → H1(G,N)→ H1(G,M)→ H1(G,M/N)→ H2(G,N)→ . . . .

1.2. A VERY BRIEF INTRODUCTION TO COHOMOLOGY 21

But what are these Hk(G,M) and how does one get at them for calculation and interpretation?In fact, what is cohomology in general?

Its origins lie within Algebraic Topology as well as in Group Theory and that area providessome useful intuitions to get us started, before asking how to form group cohomology.

1.2.3 Homology and Cohomology of spaces.

Naively homology and cohomology give methods for measuring the holes in a space, holes of differentdimensions yield generators in different (co)homology groups. The idea is easily seen for graphsand low dimensional simplicial complexes.

First we recall the definition of simplicial complex as we will need to be fairly precise aboutsuch objects and their role in relation to triangulations and related concepts.

Definition: A simplicial complex, K, is a set of objects, V (K), called vertices and a set, S(K),of finite non-empty subsets of V (K), called simplices. All singletons are simplices and simplicessatisfy the condition that if σ ⊂ V (K) is a simplex and τ ⊂ σ, τ 6= ∅, then τ is also a simplex.

We say τ is a face of σ. If σ ∈ S(K) has p+ 1 elements, it is said to be a p-simplex. The set ofp-simplices of K is denoted by A. The dimension of K is the largest p such that Kp is non-empty.

We will sometimes use the notation, P(X), for the power set of a set X, i.e., the set of subsetsof X. Later when there are more demands on the notation, we will often use a substitute notation,writing VK for V (K), and SK for S(K), thus freeing up the notation used here for other uses. Thisshould cause very little confusion, but does need noting.

Suppose, now, that X = {0, . . . , p}, then there is a simple example of a simplicial complex,known as the standard abstract p-simplex, ∆[n], (or sometimes ∆n),with vertex set, V (∆[n]) = Xand with S(∆[n]) = P(X) \ {∅}, in other words all non-empty subsets of X are to be simplices. (Ifyou have not met simplicial complexes before this is a good example to work with workingout what it looks like and ‘feels like’ for n = 0, 1, 2 and 3. It is too regular to be general, so wewill, below, see another example which is perhaps a bit more typical.

Comment on notation: We have alternative notations, ∆[n] and ∆n, for a good reason. Thetopic of simplicial complexes naturally leads on to that of simplicial sets both in these notes andhistorically. The notation ∆[n] can stand for both the simplicial complex version of the standard n-simplex or the analogous simplicial set. Most of the time this reuse is harmless, but once in a while,especially when comparing what happens in the two settings, it will be necessary to distinguishthem in which case we will use ∆[n] for the simplicial set, with ∆n for the simplicial complex. Thislatter notation will also serve for the geometric n-simplex. In any case, we will try to make it clearwhich context is the relevant one if this is not immediately clear from the surrounding discussion.

When thinking about simplicial complexes, it is important to have a picture in our minds ofa triangulated space (probably a surface or similar, a wireframe as in computer graphics). Thesimplices are the triangles, tetrahedra, etc., and are determined by their sets of vertices. Not everyset of vertices need be a simplex, but if a set of vertices does correspond to a simplex then all its

22 CHAPTER 1. PRELIMINARIES

non-empty subsets do as well, as they give the faces of that simplex. Here is an example:

4

ss

2

1

3

.................................................................................................................................................................................................................................................................

..............................................

............................................

............................................

...............................................................................................................................................................................

0 sss

Here V (K) = {0, 1, 2, 3, 4} and S(K) consists of {0, 1, 2}, {2, 3}, {3, 4} and all the non-emptysubsets of these. Note the triangle {0, 1, 2} is intended to be solid, (but I did not work out how todo it on the Latex system I was using!)

Simplicial complexes are a natural combinatorial generalisation of (undirected) graphs. Theynot only have vertices and edges joining them, but also possible higher dimensional simplicesrelating paths in that low dimensional graph. It is often convenient to put a (total) order on theset V (K) of vertices of a simplicial complex as this allows each simplex to be specified as a listσ = 〈v0, v1, . . . , vn〉 with v0 < v1 < . . . < vn, instead of as merely a set {v0, v1, . . . , vn} of vertices.This, in turn, allows us to talk, unambiguously, of the kth face of such a simplex, being the listwith vk omitted, so the zeroth face is 〈v1, . . . , vn〉, the first is 〈v0, v2, . . . , vn〉 and so on.

Although strictly speaking different types of object, we tend to use the terms ‘vertex’ and ‘0-simplex’ interchangeably and also use ‘edge’ as a synonym for ‘1-simplex’. We will usually write K0for V (K) and may write K1 for the set of edges of a graph, thought of as a 1-dimensional simplicialcomplex.

An abstract simplicial complex is a combinatorial gadget that models certain aspects of a spatialconfiguration. Sometimes it is useful, perhaps even necessary, to produce a topological space fromthat data in a simplicial complex.

Definition: To each simplicial complex K, one can associate a topological space called thepolyhedron of K often also called or geometric realisation of K and denoted |K|.

This can be constructed by taking a copy K(σ) of a standard topological p-simplex for eachp-simplex of K and then ‘gluing’ them together according to the face relations encoded in K.

Definition: The standard (topological) p-simplex is usually taken to be the convex hull of thebasis vectors e1, e2, . . . , ep+1 in Rp+1, to represent each abstract p-simplex, σ ∈ S(K), and then‘gluing’ faces together, so whenever τ is a face of σ we identify K(τ) with the corresponding faceof K(σ). This space is usually denoted ∆p.

There is a canonical way of constructing |K| as follows: |K| is the set of all functions fromV (K) to the closed interval [0, 1] such that

• if α ∈ |K|, the set{v ∈ V (K) | α(v) 6= 0}

is a simplex of K;

1.2. A VERY BRIEF INTRODUCTION TO COHOMOLOGY 23

• for each v ∈ V (K),∑α∈|K|

α(v) = 1.

We can put a metric d on |K| by

d(α, β) =( ∑v∈V (K)

(pv(α)− pv(β))2) 1

2.

This however gives |K| as a subspace of R#(V (K)), and so is usually of much higher dimension thenmight seem geometrically significant in a given context. For instance, the above example would berepresented as a subspace of R5, rather than R2, although that is the dimension of the picture wegave of it.

Given two simplicial complexes, K, L, then a function on the vertex sets, f : V (K) → V (L)is a simplicial map if it preserves simplices. (But that needs a bit of care to check out its exactmeaning! ... for you to do. Look it up, or better try to see what the problem might be, try toresolve it yourself and then look it up! )

1.2.4 Betti numbers and Homology

One of the first sorts of invariant considered in what was to become Algebraic Topology was thefamily of Betti numbers. Given a simple shape, the most obvious piece of information to note wouldbe the number of ‘pieces’ it is made up of, or more precisely, the number of components. The ideais very well known, at least for graphs, and as simplicial complexes are closely related to graphs,we will briefly look at this case first.

For convenience we will assume the vertices V = V (Γ) of a given finite graph, Γ, are ordered, sofor each edge e of Γ, we can assign a source s(e) and a target t(e) amongst the vertices. (This givesus a ‘directed graph’ or ‘quiver’2). Two vertices v and w are said to be in the same component ofΓ if there is a sequence of edges, e1, . . . , ek, of Γ joining them

3. There are, of course, several waysof thinking about this, for instance, define a relation ∼ on V by : for each e, s(e) ∼ t(e). Extend∼ to an equivalence relation on V in the standard way, then v ∼ w if and only if they are in thesame component. The zeroth Betti number, β0(Γ), is the number of components of Γ.

The first Betti number, β1(Γ), somewhat similarly, counts the number of cycles of Γ. We haveordered the vertices of Γ, so have effectively also directed its edges. If e is an edge, going from uto v, (so u < v in the order on Γ0), we write e also for the path going just along e and −e forthat going backwards along it, then extend our notation so s(−e) = t(e) = v, etc. Adding in these‘negative edges’ corresponds to the formation of the symmetric closure of ∼. For the transitiveclosure we need to concatenate these simple one-edge paths: if e′ is an edge or a ‘negative edge’from v to w, we write e+ e′ for the path going along e then e′. Playing algebraically with s and tand making them respect addition, we get a ‘pseudo-calculation’ for their difference ∂ = t− s:

∂(e+ e′) = t(e+ e′)− s(e+ e′) = t(e) + t(e′)− s(e)− s(e′) = t(e′)− s(e) = u− w,

since t(e) = v = s(e′). In other words, defined in a suitable way, we would get that ∂, equal to‘target minus source’, applies nicely to paths as well as edges, so that, for instance, two vertices

2We will return to discuss directed graphs aka quivers, in section 2.2.3.3In fact here, the ordering we have assumed on the vertices complicates the exposition a little, but it is useful

later on so will stick with it here.

24 CHAPTER 1. PRELIMINARIES

would be related in the transitive closure of ∼ if there was a ‘formal sum’ of edges that mappeddown to their ‘difference’. We say ‘formal sum’ as this is just what it is. We will need ‘negativevertices’ as well as ‘negative edges’.

We set this up more formally as follows: LetC0(Γ) = the set of formal sums,

∑v∈Γ0 avv with av ∈ Z, the additive group of integers, (an

alternative form is to take av ∈ R.;C1(Γ) = the set of formal sums,

∑e∈Γ1 bee with be ∈ Z,

where Γ1 denotes the set of edges of Γ, and ∂ : C1(Γ)→ C0(Γ) defined by extending additively themapping given on the edges by ∂ = t− s.

The task of determining components is thus reduced to calculating when integer vectors differ bythe image of one in C1(Γ). The Betti number β0(Γ) is just the rank of the quotient C0(Γ)/Im(∂),that is, the number of free generators of this commutative group. This would be exactly thedimension of this ‘vector space’ if we had allowed real coefficients in our formal sums not justinteger ones.

Having reformulated components and ∼ in an algebraic way, we immediately get a pay-off inour determination of cycles. A cycle is a path which starts and ends at the same vertex; a path isbeing modelled by an element in C1(Γ), so a cycle is an element x in C1(γ) satisfying ∂(x) = 0.With this we have β1(Γ) = rank(Ker(∂)), a similar formulation to that for β0. The similarity iseven more striking if we replace the graph Γ by a simplicial complex K. We can then define ingeneral and in any dimension p, Cp(K) to be the commutative group of all formal sums

∑σ∈Kp aσσ.

We next need to get an analogue of the ∂ = t − s formula. We want this to correspond tothe boundary of the objects to which it is applied. For instance, if σ was the triangle / 2-simplex,〈v0, v1, v2〉, we would want ∂σ to be 〈v1, v2〉+ 〈v0, v1〉 − 〈v0, v2〉, since going (clockwise) around thetriangle, that cycle will be traced out:

〈v1〉

〈v0〉 〈v2〉....................................................................................................................................................................................................................................................................................................................................................

〈v0, v1〉 〈v1, v2〉

〈v0, v2〉

If we write, in general, diσ for the ith face of a p-simplex σ = 〈v0, . . . , vp〉, then in this 2-

dimensional example ∂σ = d0σ − d1σ + d2σ, changing the order for later convenience. This is thesum of the faces with weighting (−1)i given to diσ. This is consistent with ∂ = t− s in the lowerdimension as t = d0 and s = d1. We can thus suggest that

∂ = ∂p : Cp(K)→ Cp−1(K)

be defined on p-simplices by

∂pσ =

p∑i=0

(−1)idiσ,

and then extended additively to all of Cp(K).

As an example of what this does, look at a square K, with vertices v0, v1, v2, v3, edges 〈vi, vi+1〉for i = 0, 1, 2 and 〈v0, v2〉, and 2-simplices σ1 = 〈v0, v1, v2〉 and σ2 = 〈v0, v2, v3〉. As the square

1.2. A VERY BRIEF INTRODUCTION TO COHOMOLOGY 25

has these two 2-simplices, we can think of it as being represented by σ1 + σ2 in C2(K), then∂(σ1 + σ2) = 〈v0, v1〉 + 〈v1, v2〉 + 〈v2, v3〉 − 〈v0, v3〉, as the two occurrences of the diagonal 〈v0, v2〉cancel out as they have opposite sign, and this is the path around the actual boundary of thesquare.

It is important to note that the boundary of a boundary is always trivial, that is, the compositemapping

Cp(K)∂p−→ Cp−1(K)

∂p−1−−−→ Cp−2(K)

is the mapping sending everything to 0 ∈ Cp−1(K).The idea of the higher Betti numbers, βp(K), is that they measure the number of p-dimensional

‘holes’ in K. Imagine we has a tunnel-shaped hole through a space K, then we would have a cyclearound the hole at one end of the tunnel and another around the hole at the other end. If wemerely count cycles then we will get at least two such coming from this hole, but these cycles arelinked as there is the cylindrical hole itself and that gives a 2 dimensional element with boundarythe difference of the two cycles. In general, a p-cycle will be an element x of Cp(K) with trivialboundary, i.e., such that ∂x = 0, and we say that two p-cycles x and x′ are homologous if there isan element y in Cp+1(K) such that ∂y = x − x′. The ‘holes’ correspond to classes of homologouscycles as in our tunnel.

The number of ‘independent’ cycle classes in the various dimensions give the correspondingBetti number. Using some algebra, this is easier to define rigorously, but, at the same time, thegeometric insights from the vaguer description are important to try to retain. (They are not alwaysput in a central enough position in textbooks!) This algebraic approach identifies βp(K) as the(torsion free) rank of a certain commutative group formed as follows: the pth homology group ofK is defined to be the quotient:

Hp(K) =Ker(∂p : Cp(K)→ Cp−1(K))Im(∂p : Cp+1(K)→ Cp(K))

,

and then βp(K) = rank(Hp(K)).Thus far we have from K built a sequence of modules, C(K)n, generated by the n-simplices

of K and with homomorphisms ∂p : Cp(K) → Cp−1(K) satisfying ∂p−1∂p = 0.. (We abstract thisstructure calling it a chain complex. We will look at in more detail at several places later in thesenotes.)

Exercises: Try to investigate this homology in some very simple situations perhaps includingsome of the following:(a) V (K) = {0, 1, 2, 3}, S(K) = P(V (K)) \ {∅, {0, 1, 2, 3}}. This is an empty tetrahedron so oneexpects one 3-dimensional hole., i.e., β3(K) = 1 but the others are zero.(b) ∆[2] is the (full) triangle and ∂∆[2] its boundary, so is an empty triangle. Find the homologyof ∂∆[2]× ∂∆[2], which is a triangulated torus.(c) Find the homology of ∆[1]× ∂∆[2], which is a cylinder.

Note, it is up to you to find the meaning of product in this context. Remember the discussionof the square, above, which is, of course ∆[1]×∆[1].

Often cohomology is more use than homology. Starting with K and a module M work outCn(K,M) = Hom(C(K)n,M). Now the boundary maps increase (upper) degree by one. Thecohomology is Hn(K,M) = Ker ∂n/Im∂n−1. Again this measures ‘holes’ detectable by M ! What

26 CHAPTER 1. PRELIMINARIES

does that mean? The cohomology groups are better structured than the homology ones, but howare these invariants be interpreted?

A simplicial map, f : K → L, will induce a map on cohomology groups. Try it! We canequally well do this for chain or ‘cochain complexes’. There is a notion of chain map between chaincomplexes, say, ϕ : C → D and such a map will induce maps on both homology ad cohomology.Of special interest is when the induced maps are isomorphisms. The chain map is then called aquasi-isomorphism.

1.2.5 Interpretation

The question of interpretation is a very crucial one but, rather than answering it now, we will returnto the cohomology of groups. The terminology may seem a bit strange. Here we have been talkingabout measuring holes in a space, so how does that relate to groups. The idea is that one builds aspace from a group in such a way as the properties of the space reflect those of the group in somesense. The simplest case of this is an Eilenberg - Mac Lane space, K(G, 1). The defining propertyof such a space is that its fundamental group is G whilst all other homotopy groups are trivial.Eilenberg and Mac Lane showed that however such a space was constructed its cohomology could begot just from G itself and that cohomology was related with the extension problem and the invariantmodule problem. Their method was to build a chain complex that would copy the structure of thechain complex on the K(G, 1). This chain complex, the bar resolution, was very important becausealthough in the group case there was an alternative route via the topological space K(G, 1), formany other types of algebraic system (Lie algebras, associative algebras, commutative algebras,etc.), the analogous basic construction could be used, and in those contexts no space was available.Thus from G, we want to construct a nice chain complex directly. The construction is reasonablysimple. It gives a natural way of getting a chain complex, but it does not exploit any particularfeatures of the group so if the group is infinite, the modules will be infinitely generated, which willoccupy us later, as we use insights from combinatorial group theory to construct smaller modelsfor equivalent resolutions, and better still look at ‘crossed’ versions.

For the moment we just need the definition (adapted from the account given in Wikipedia):

1.2.6 The bar resolution

The input data is a group G and a module M with a left G-action (i.e., a left G-module).For n ≥ 0, we let Cn(G,M) be the group of all functions from the n-fold product Gn to M :

Cn(G,M) = {ϕ : Gn →M}

This is an Abelian group; its elements are called the n-cochains. We further define group homo-morphisms

∂n : Cn(G,M)→ Cn+1(G,M)

by

∂n(ϕ)(g0, . . . , gn) = g0 · ϕ(g1, . . . , gn)

+

n−1∑i=0

(−1)i+1ϕ(g0, . . . , gi−1, gigi+1, gi+2, . . . , gn)

+(−1)n+1ϕ(g0, . . . , gn−1)

1.3. SIMPLICIAL THINGS IN A CATEGORY 27

These are known as the coboundary homomorphisms. The crucial thing to check here is ∂n+1 ◦∂n =0, thus we have a chain complex and we can ‘compute’ its cohomology. For n ≥ 0, define the groupof n-cocycles as:

Zn(G,M) = Ker ∂n

and the group of n-coboundaries as{B0(G,M) = 0

Bn(G,M) = Im(∂n−1) n ≥ 1

and

Hn(G,M) = Zn(G,M)/Bn(G,M).

Thinking about this topologically, it is as if we had constructed a sort of space / simplicial complex,K, out of G by taking Kn = G

n. We will see this idea many times later on. This cochain complexis often called the bar resolution. It exists in a normalised and a unnormalised form. This is theunnormalised one. It can also be constructed via a chain complex, sometimes denoted βG, so thatthis C(G,M) is formed by taking Hom(βG,M), in a suitable sense.

There are lots of properties that are easy to check here. Some will be suggested as exercises foryou to do. For others, you can refer to some of the standard textbooks that deal with introductionsto group cohomology, for instance, K. Brown’s [56].

One further point is that this cohomology used a module, and so encodes ‘commutative’ orAbelian information. We will be also looking at the non-Abelian case.

Before we leave this introduction to cohomology, it should be mentioned that in the topologicalcase, if we do not have a simplicial complex to start with, we either use the singular complex (seenext section) which is a simplicial set and not a simplicial complex, but the theory extends easilyenough, or we use open covers of the space to build a system of simplicial complexes approximatingto the space. We will see this later as Čech cohomology. This is most powerful when the moduleM of coefficients is allowed to vary over the various points of the space. For this we will need thenotion of sheaf, which will be discussed in some detail later.

1.3 Simplicial things in a category

1.3.1 Simplicial Sets

Simplicial objects