The homotopy theory of -categories infty,1

288 Pages·2018·0.988 MB·English

by  Bergner, Julia Elizabeth

LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
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51 Stepsincommutativealgebra:Secondedition,R.Y.SHARP
52 FiniteMarkovchainsandalgorithmicapplications,OLLEHA¨GGSTRO¨M
53 Theprimenumbertheorem,G.J.O.JAMESON
54 Topicsingraphautomorphismsandreconstruction,JOSEFLAURI&RAFFAELESCAPELLATO
55 Elementarynumbertheory,grouptheoryandRamanujangraphs,GIULIANADAVIDOFF,
PETERSARNAK&ALAINVALETTE
56 Logic,inductionandsets,THOMASFORSTER
57 IntroductiontoBanachalgebras,operatorsandharmonicanalysis,GARTHDALESetal
58 Computationalalgebraicgeometry,HALSCHENCK
59 Frobeniusalgebrasand2Dtopologicalquantumfieldtheories,JOACHIMKOCK
60 Linearoperatorsandlinearsystems,JONATHANR.PARTINGTON
61 AnintroductiontononcommutativeNoetherianrings:Secondedition,K.R.GOODEARL&
R.B.WARFIELD,JR
62 Topicsfromone-dimensionaldynamics,KARENM.BRUCKS&HENKBRUIN
63 Singularpointsofplanecurves,C.T.C.WALL
64 AshortcourseonBanachspacetheory,N.L.CAROTHERS
65 ElementsoftherepresentationtheoryofassociativealgebrasI,IBRAHIMASSEM,
DANIELSIMSON&ANDRZEJSKOWRON´SKI
66 Anintroductiontosievemethodsandtheirapplications,ALINACARMENCOJOCARU&
M.RAMMURTY
67 Ellipticfunctions,J.V.ARMITAGE&W.F.EBERLEIN
68 Hyperbolicgeometryfromalocalviewpoint,LINDAKEEN&NIKOLALAKIC
69 LecturesonKa¨hlergeometry,ANDREIMOROIANU
70 Dependencelogic,JOUKUVA¨A¨NA¨NEN
71 ElementsoftherepresentationtheoryofassociativealgebrasII,DANIELSIMSON&
ANDRZEJSKOWRON´SKI
72 ElementsoftherepresentationtheoryofassociativealgebrasIII,DANIELSIMSON&
ANDRZEJSKOWRON´SKI
73 Groups,graphsandtrees,JOHNMEIER
74 RepresentationtheoremsinHardyspaces,JAVADMASHREGHI
75 Anintroductiontothetheoryofgraphspectra,DRAGOSˇCVETKOVIC´,PETERROWLINSON&
SLOBODANSIMIC´
76 NumbertheoryinthespiritofLiouville,KENNETHS.WILLIAMS
77 Lecturesonprofinitetopicsingrouptheory,BENJAMINKLOPSCH,NIKOLAYNIKOLOV&
CHRISTOPHERVOLL
78 Cliffordalgebras:Anintroduction,D.J.H.GARLING
79 IntroductiontocompactRiemannsurfacesanddessinsd’enfants,ERNESTOGIRONDO&
GABINOGONZA´LEZ-DIEZ
80 TheRiemannhypothesisforfunctionfields,MACHIELVANFRANKENHUIJSEN
81 Numbertheory,Fourieranalysisandgeometricdiscrepancy,GIANCARLOTRAVAGLINI
82 Finitegeometryandcombinatorialapplications,SIMEONBALL
83 Thegeometryofcelestialmechanics,HANSJO¨RGGEIGES
84 Randomgraphs,geometryandasymptoticstructure,MICHAELKRIVELEVICHetal
85 Fourieranalysis:PartI-Theory,ADRIANCONSTANTIN
86 Dispersivepartialdifferentialequations,M.BURAKERDOG˘AN&NIKOLAOSTZIRAKIS
87 Riemannsurfacesandalgebraiccurves,R.CAVALIERI&E.MILES
88 Groups,languagesandautomata,DEREKF.HOLT,SARAHREES&CLAASE.RO¨VER
89 AnalysisonPolishspacesandanintroductiontooptimaltransportation,D.J.H.GARLING
90 Thehomotopytheoryof(∞,1)-categories,JULIAE.BERGNER

LondonMathematicalSocietyStudentTexts90
The Homotopy Theory of
∞
( , 1)-Categories
JULIA E. BERGNER
UniversityofVirginia
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Contents
Preface pagexi
Acknowledgments xiii
Introduction 1
1 ModelsforHomotopyTheories 4
1.1 SomeBasicsinCategoryTheory 4
1.2 WeakEquivalencesandLocalization 13
1.3 ClassicalHomotopyTheory 16
1.4 ModelCategories 18
1.5 HomotopyCategories 22
1.6 EquivalencesBetweenModelCategories 25
1.7 AdditionalStructuresonModelCategories 28
2 SimplicialObjects 34
2.1 SimplicialSetsandSimplicialObjects 34
2.2 SimplicialSetsasModelsforSpaces 37
2.3 HomotopyLimitsandHomotopyColimits 39
2.4 SimplicialModelCategories 42
2.5 SimplicialSpaces 45
2.6 TheReedyModelStructureonSimplicialSpaces 47
2.7 CombinatorialModelCategories 54
2.8 LocalizedModelCategories 56
2.9 CartesianModelCategories 63
3 TopologicalandCategoricalMotivation 66
3.1 NervesofCategories 66
3.2 KanComplexesandGeneralizations 68
3.3 ClassifyingDiagrams 71
3.4 HigherCategories 74
3.5 HomotopyTheories 78
vii
viii Contents
4 SimplicialCategories 83
4.1 TheCategoryofSmallSimplicialCategories 83
4.2 Fixed-ObjectSimplicialCategories 84
4.3 TheModelStructure 86
4.4 ProofoftheExistenceoftheModelStructure 88
4.5 PropertiesoftheModelStructure 96
4.6 NervesofSimplicialCategories 99
5 CompleteSegalSpaces 101
5.1 SegalSpaces 102
5.2 SegalSpacesasCategoriesUptoHomotopy 107
5.3 CompleteSegalSpaces 110
5.4 CategoricalEquivalences 113
5.5 Dwyer–KanEquivalences 116
6 SegalCategories 124
6.1 BasicDefinitionsandConstructions 125
6.2 Fixed-ObjectSegalCategories 130
6.3 TheFirstModelStructure 138
6.4 TheEquivalenceWithCompleteSegalSpaces 145
6.5 TheSecondModelStructure 148
6.6 TheEquivalenceWithSimplicialCategories 151
7 Quasi-Categories 157
7.1 BasicDefinitions 157
7.2 PropertiesofAcyclicCofibrations 160
7.3 TheModelStructure 166
7.4 TheCoherentNerveandRigidificationFunctors 171
7.5 NecklacesandTheirRigidification 173
7.6 RigidificationofSimplicialSets 179
7.7 PropertiesoftheRigidificationFunctor 187
7.8 TheEquivalenceWithSimplicialCategories 194
7.9 TheEquivalenceWithCompleteSegalSpaces 207
8 RelativeCategories 213
8.1 BasicDefinitions 213
8.2 SubdivisionFunctors 218
8.3 TheModelStructureandEquivalenceWithComplete
SegalSpaces 221
9 ComparingFunctorstoCompleteSegalSpaces 233
9.1 ClassifyingandClassificationDiagrams 234
9.2 SomeResultsforSimplicialCategories 236
Contents ix
9.3 ComparisonofFunctors 239
9.4 CompleteSegalSpacesFromSimplicialCategories 242
10 Variantson(∞,1)-Categories 248
10.1 FiniteApproximations 248
10.2 Stable(∞,1)-Categories 251
10.3 DendroidalObjects 254
10.4 Higher(∞,n)-Categories 256
References 261
Index 267