Categorical homotopy theory

292 Pages·2017·1.401 MB·English

by  Emily Riehl

Categorical homotopy theory
Emily Riehl

Tomystudents,colleagues,friendswhoinspiredthiswork.

...whatwearedoingisfindingways
forpeopletounderstandandthink
aboutmathematics.
WilliamP.Thurston“Onproofand
progressinmathematics”[Thu94]

Contents
Preface xi
Prerequisites xiv
NotationalConventions xiv
Acknowledgments xv
PartI. Derivedfunctorsandhomotopy(co)limits 1
Chapter1. AllconceptsareKanextensions 3
1.1. Kanextensions 3
1.2. Aformula 5
1.3. PointwiseKanextensions 7
1.4. Allconcepts 9
1.5. Adjunctionsinvolvingsimplicialsets 10
Chapter2. Derivedfunctorsviadeformations 13
2.1. Homotopicalcategoriesandderivedfunctors 13
2.2. Derivedfunctorsviadeformations 18
2.3. Classicalderivedfunctorsbetweenabeliancategories 22
2.4. Previewofhomotopylimitsandcolimits 23
Chapter3. Basicconceptsofenrichedcategorytheory 25
3.1. Afirstexample 26
3.2. Thebaseforenrichment 26
3.3. Enrichedcategories 27
3.4. Underlyingcategoriesofenrichedcategories 30
3.5. Enrichedfunctorsandenrichednaturaltransformations 34
3.6. Simplicialcategories 36
3.7. Tensorsandcotensors 37
3.8. Simplicialhomotopyandsimplicialmodelcategories 42
Chapter4. Theunreasonablyeffective(co)barconstruction 45
4.1. Functortensorproducts 45
4.2. Thebarconstruction 47
4.3. Thecobarconstruction 48
4.4. Simplicialreplacementsandcolimits 49
4.5. Augmentedsimplicialobjectsandextradegeneracies 51
Chapter5. Homotopylimitsandcolimits: thetheory 55
5.1. Thehomotopylimitandcolimitfunctors 55
5.2. Homotopicalaspectsofthebarconstruction 57
vii
viii CONTENTS
Chapter6. Homotopylimitsandcolimits: thepractice 61
6.1. Convenientcategoriesofspaces 61
6.2. Simplicialmodelcategoriesofspaces 65
6.3. Warningsandsimplifications 66
6.4. Samplehomotopycolimits 67
6.5. Samplehomotopylimits 71
6.6. Homotopycolimitsasweightedcolimits 73
PartII. Enrichedhomotopytheory 77
Chapter7. Weightedlimitsandcolimits 79
7.1. Weightedlimitsinunenrichedcategorytheory 79
7.2. Weightedcolimitsinunenrichedcategorytheory 83
7.3. Enrichednaturaltransformationsandenrichedends 85
7.4. Weightedlimitsandcolimits 87
7.5. Conicallimitsandcolimits 88
7.6. Enrichedcompletenessandcocompleteness 90
7.7. Homotopy(co)limitsasweighted(co)limits 92
7.8. Balancingbarandcobarconstructions 93
Chapter8. Categoricaltoolsforhomotopy(co)limitcomputations 97
8.1. Preservationofweightedlimitsandcolimits 97
8.2. Changeofbaseforhomotopylimitsandcolimits 99
8.3. Finalfunctorsinunenrichedcategorytheory 101
8.4. Finalfunctorsinenrichedcategorytheory 103
8.5. Homotopyfinalfunctors 103
Chapter9. Weightedhomotopylimitsandcolimits 109
9.1. Theenrichedbarandcobarconstruction 109
9.2. Weightedhomotopylimitsandcolimits 111
Chapter10. Derivedenrichment 117
10.1. Enrichmentsencodedasmodulestructures 118
10.2. Derivedstructuresforenrichment 120
10.3. Weightedhomotopylimitsandcolimits,revisited 124
10.4. Homotopicalstructureviaenrichment 126
10.5. Homotopyequivalencesvs.weakequivalences 129
PartIII. Modelcategoriesandweakfactorizationsystems 131
Chapter11. Weakfactorizationsystemsinmodelcategories 133
11.1. Liftingproblemsandliftingproperties 133
11.2. Weakfactorizationsystems 137
11.3. ModelcategoriesandQuillenfunctors 138
11.4. Simplicialmodelcategories 143
11.5. WeightedcolimitsasleftQuillenbifunctors 144
Chapter12. Algebraicperspectivesonthesmallobjectargument 151
12.1. Functorialfactorizations 151
12.2. Quillen’ssmallobjectargument 152
CONTENTS ix
12.3. Benefitsofcofibrantgeneration 155
12.4. Algebraicperspectives 156
12.5. Garner’ssmallobjectargument 159
12.6. Algebraicweakfactorizationsystemsanduniversalproperties 164
12.7. Composingalgebrasandcoalgebras 168
12.8. Algebraiccellcomplexes 170
12.9. Epilogueonalgebraicmodelcategories 173
Chapter13. Enrichedfactorizationsandenrichedliftingproperties 175
13.1. Enrichedarrowcategories 176
13.2. Enrichedfunctorialfactorizations 177
13.3. Enrichedliftingproperties 179
13.4. Enrichedweakfactorizationsystems 183
13.5. Enrichedmodelcategories 185
13.6. Enrichmentascoherence 186
Chapter14. AbrieftourofReedycategorytheory 189
14.1. Latchingandmatchingobjects 190
14.2. ReedycategoriesandtheReedymodelstructures 191
14.3. Reedycofibrantobjectsandhomotopy(co)limits 194
14.4. Localizationsandcompletionsofspaces 198
14.5. Homotopycolimitsoftopologicalspaces 202
PartIV. Quasi-categories 205
Chapter15. Preliminariesonquasi-categories 207
15.1. Introducingquasi-categories 208
15.2. Closureproperties 209
15.3. Towardthemodelstructure 211
15.4. Mappingspaces 214
Chapter16. Simplicialcategoriesandhomotopycoherence 221
16.1. Topologicalandsimplicialcategories 221
16.2. Cofibrantsimplicialcategoriesaresimplicialcomputads 223
16.3. Homotopycoherence 224
16.4. UnderstandingthemappingspacesCX(x,y) 227
16.5. Agesturetowardthecomparison 231
Chapter17. Isomorphismsinquasi-categories 233
17.1. Joinandslice 234
17.2. IsomorphismsandKancomplexes 236
17.3. Invertingsimplices 239
17.4. Markedsimplicialsets 240
17.5. Invertingdiagramsofisomorphisms 243
17.6. Acontextforinvertibility 245
17.7. Homotopylimitsofquasi-categories 246
Chapter18. Asamplingof2-categoricalaspectsofquasi-categorytheory 249
18.1. The2-categoryofquasi-categories 249
18.2. Weaklimitsinthe2-categoryofquasi-categories 251
x CONTENTS
18.3. Arrowquasi-categoriesinpractice 253
18.4. Homotopypullbacks 254
18.5. Commaquasi-categories 255
18.6. Adjunctionsbetweenquasi-categories 256
18.7. Essentialgeometryofterminalobjects 260
Bibliography 263
GlossaryofNotation 267
Index 269