The Homotopy Theory of (∞,1)-Categories

148 Pages·2014·1.07 MB·English

by  Magnus Hellstrøm-Finnsen

The Homotopy Theory of (∞,1)-Categories
Magnus Hellstrøm-Finnsen
Master of Science in Physics and Mathematics
Submission date: Januar 2014
Supervisor: Petter Andreas Bergh, MATH
Co-supervisor: Marius Thaule, IMF
Norwegian University of Science and Technology
Department of Mathematical Sciences

Norwegian University of Science and Technology
Department of Mathematical Sciences
TMA4900 Mathematics, Master Thesis
The homotopy theory
of (∞, 1)-categories
Supervisors:
Petter Andreas Bergh and Marius Thaule
Student:
Magnus Hellstrøm-Finnsen

Abstract
Thehomotopycategoryofastable(∞,1)-categorycanbeendowedwithatrian-
gulatedstructure. Themainobjectiveofthisthesisistogiveaproofofthisfact.
First it will be discussed some ideas of higher category theory, before (∞,1)-
categories and models of (∞,1)-categories will be studied. In particular, topo-
logicalcategoriesandsimplicialcategorieswillbementioned, butthemainfocus
willbeonquasi-categories,whichallaremodelsfor(∞,1)-categories. Thetheory
of (∞,1)-categories, which is required in order to define stable (∞,1)-categories,
isthendiscussed,inparticularfunctors,subcategories,joinconstructions,under-
categories,overcategories,initialobjects,terminalobjects,limitsandcolimitsare
formally discussed for quasi-categories. Finally, the definition of a stable (∞,1)-
category will be discussed. Then the main theorem will be proved, after the
required properties of stable (∞,1)-categories are discussed. Background theory
from ordinary categories and simplicial sets are collected in the appendices.
iii

Sammendrag
Homotopikategorientilenstabil(∞,1)-kategorikanbligittentriangulertstruk-
tur. Hovedm˚alet med denne oppgaven er˚a gi et bevis for dette faktumet. Først
vil det bli presentert noen ideer bak høyere kategoriteori, før (∞,1)-kategorier
og modeller for (∞,1)-kategorier vil bli studert. Spesielt er topologiske kate-
gorier og simplisielle kategorier nevnt, men hovedfokuset er p˚a teorien om kvasi-
kategorier. Teorien som kreves for˚a definere stabile (∞,1)-kategorier er deretter
diskutert, spesielt er funktorer, underkategorier, join konstruksjoner, kategorier
under,kategorierover,initielleobjekter,terminelleobjekter,grenserogkogrenser
formeltdiskutertforkvasikategorier. Tilsluttdiskuteresdefinisjonenavenstabil
(∞,1)-kategorisammenmednødvendigeegenskaperfor˚abevisehovedresultatet.
Bakgrunnsteori for ordinære kategorier og simplisielle mengder er samlet i ap-
pendiksene.
v

Contents
Abstract iii
Sammendrag v
Preface xi
Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapter 1 Higher categories 1
1.1 Introduction and plan . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Ideas of higher categories . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Ideas of higher morphisms . . . . . . . . . . . . . . . . . . . 5
1.2.2 Inductive interpretation of higher categories . . . . . . . . . 8
1.2.3 Higher invertible morphisms and (n,k)-categories . . . . . . 9
1.2.4 Higher categories as “weakly” enriched categories . . . . . . 12
Chapter 2 Models for (∞,1)-categories 17
2.1 Simplicial and topological categories . . . . . . . . . . . . . . . . . 18
2.1.1 Definition of topological categories . . . . . . . . . . . . . . 18
2.1.2 The homotopy category of a topological category . . . . . . 19
2.1.3 Simplicial categories . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Quasi-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Definition of quasi-categories . . . . . . . . . . . . . . . . . 24
2.2.2 The homotopy category of a quasi-category . . . . . . . . . 28
2.3 Model for (∞,1)-categories . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 Mapping spaces and higher morphisms . . . . . . . . . . . . 31
2.3.2 Functors and homotopy coherent diagrams . . . . . . . . . 37
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2.3.3 Contractable spaces of choices. . . . . . . . . . . . . . . . . 40
2.3.4 Comparison of quasi-categories with simplicial categories . 42
Chapter 3 Notions in the language of (∞,1)-categories 47
3.1 Functors and subcategories . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Functors of (∞,1)-categories . . . . . . . . . . . . . . . . . 48
3.1.2 Subcategories of quasi-categories . . . . . . . . . . . . . . . 49
3.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Join constructions . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 Overcategories and undercategories . . . . . . . . . . . . . . 54
3.2.3 Initial objects and terminal objects . . . . . . . . . . . . . . 57
3.2.4 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 4 Stable (∞,1)-categories 67
4.1 The notion of (∞,1)-categorical stability . . . . . . . . . . . . . . . 67
4.1.1 Pointed (∞,1)-categories . . . . . . . . . . . . . . . . . . . 68
4.1.2 Fibers and cofibers . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.3 Definition of stability . . . . . . . . . . . . . . . . . . . . . 74
4.2 Constructions of suspension functor and loop functor . . . . . . . . 75
4.2.1 Suspension functor . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.2 Loop functor . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.3 Suspension and loop functor on stable (∞,1)-categories . . 78
4.3 The homotopy category of stable (∞,1)-categories . . . . . . . . . 80
4.3.1 Additive structure . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.2 Triangulated structure . . . . . . . . . . . . . . . . . . . . . 90
4.3.3 Enclosing comments . . . . . . . . . . . . . . . . . . . . . . 100
Appendix A Notions in the ordinary categorical language 103
A.1 Monoidal and enriched categories . . . . . . . . . . . . . . . . . . . 103
A.1.1 Definition of monoidal categories . . . . . . . . . . . . . . . 103
A.1.2 Enriched categories . . . . . . . . . . . . . . . . . . . . . . . 105
A.2 Additive and triangulated categories . . . . . . . . . . . . . . . . . 108
A.2.1 Additive categories . . . . . . . . . . . . . . . . . . . . . . . 108
A.2.2 Definition of triangulated categories . . . . . . . . . . . . . 110
Appendix B Simplicial sets 115
B.1 Definition of simplicial sets . . . . . . . . . . . . . . . . . . . . . . 115
B.1.1 The delta-category . . . . . . . . . . . . . . . . . . . . . . . 115