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201CoherenceinThree-DimensionalCategoryTheory
CAMBRIDGE TRACTS IN MATHEMATICS
GENERALEDITORS
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Coherence in Three-Dimensional
Category Theory
NICK GURSKI
UniversityofSheffield
CAMBRIDGE UNIVERSITY PRESS
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Informationonthistitle:www.cambridge.org/9781107034891
(cid:2)c NickGurski2013
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permissionofCambridgeUniversityPress.
Firstpublished2013
PrintedandboundintheUnitedKingdombytheMPGBooksGroup
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Gurski,Nick,1980–
Coherenceinthree-dimensionalcategorytheory/NickGurski,UniversityofSheffield.
pages cm.–(Cambridgetractsinmathematics;201)
ISBN978-1-107-03489-1(hardback)
1. Tricategories. I. Title.
QA169.G87 2013
512(cid:3).55–dc23
2012051079
ISBN978-1-107-03489-1Hardback
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Contents
Introduction page1
1 Tricategories 1
2 Gray-monads 3
3 Anoutline 5
Acknowledgements 12
Part I Background 13
1 Bicategoricalbackground 15
1.1 Bicategoricalconventions 15
1.2 Matesinbicategories 17
2 Coherenceforbicategories 21
2.1 TheYonedaembedding 21
2.2 Coherenceforbicategories 22
2.3 Coherenceforfunctors 29
3 Gray-categories 35
3.1 TheGraytensorproduct 36
3.2 Cubicalfunctors 38
3.3 ThemonoidalcategoryGray 45
3.4 Afactorization 48
Part II Tricategories 57
4 Thealgebraicdefinitionoftricategory 59
4.1 Basicdefinition 59
4.2 Adjointequivalencesandtricategoryaxioms 65
4.3 Trihomomorphismsandotherhighercells 66
4.4 Unpackedversions 78
v
vi Contents
4.5 Calculationsintricategories 83
4.6 Comparingdefinitions 85
5 Examples 86
5.1 Primaryexample:Bicat 86
5.2 Fundamental3-groupoids 89
6 Freeconstructions 97
6.1 Graphs 97
6.2 Thecategoryoftricategories 99
6.3 FreeGray-categories 103
7 Basicstructure 106
7.1 Structureoffunctors 107
7.2 Structureoftransformations 110
7.3 Pseudo-icons 114
7.4 Changeofstructure 123
7.5 Triequivalences 127
8 Gray-categoriesandtricategories 129
8.1 Cubicaltricategories 129
8.2 Gray-categories 133
9 CoherenceviaYoneda 138
9.1 Localstructure 139
9.2 Globalresults 141
9.3 ThecubicalYonedalemma 144
9.4 Coherencefortricategories 154
10 Coherenceviafreeconstructions 156
10.1 Coherencefortricategories 157
10.2 Coherenceanddiagramsofconstraints 160
10.3 Anon-commutingdiagram 161
10.4 Strictifyingtricategories 162
10.5 Coherenceforfunctors 171
10.6 Strictifyingfunctors 177
Part III Gray-monads 181
11 CodescentinGray-categories 183
11.1 Laxcodescentdiagrams 184
11.2 Codescentdiagrams 188
11.3 Codescentobjects 190
Contents vii
12 Codescentasaweightedcolimit 196
12.1 WeightedcolimitsinGray-categories 197
12.2 Examples:coinsertersandcoequifiers 200
12.3 Codescent 207
13 Gray-monadsandtheiralgebras 209
13.1 Enrichedmonadsandalgebras 210
13.2 Laxalgebrasandtheirhighercells 213
13.3 Totalstructures 219
14 Thereflectionoflaxalgebrasintostrictalgebras 227
14.1 Thecanonicalcodescentdiagramofalaxalgebra 228
14.2 Theleftadjoint,laxcase 230
14.3 Theleftadjoint,pseudocase 242
15 Ageneralcoherenceresult 244
15.1 Weakcodescentobjects 245
15.2 Coherenceforpseudo-algebras 266
Bibliography 273
Index 277
