Coherence in Three-Dimensional Category Theory

Cover......Page 1
Contents......Page 7
1  Tricategories......Page 11
2  Gray-monads......Page 13
3  An outline......Page 15
 Acknowledgements......Page 22
Part I Background......Page 23
1.1  Bicategorical conventions......Page 25
1.2  Mates in bicategories......Page 27
2.1  The Yoneda embedding......Page 31
2.2  Coherence for bicategories......Page 32
2.3  Coherence for functors......Page 39
3 Gray-categories......Page 45
3.1  The Gray tensor product......Page 46
3.2  Cubical functors......Page 48
3.3  The monoidal category Gray......Page 55
3.4  A factorization......Page 58
Part II Tricategories......Page 67
4.1  Basic definition......Page 69
4.2  Adjoint equivalences and tricategory axioms......Page 75
4.3  Trihomomorphisms and other higher cells......Page 76
4.4  Unpacked versions......Page 88
4.5  Calculations in tricategories......Page 93
4.6  Comparing definitions......Page 95
5.1  Primary example: Bicat......Page 96
5.2  Fundamental 3-groupoids......Page 99
6.1  Graphs......Page 107
6.2  The category of tricategories......Page 109
6.3  Free Gray-categories......Page 113
7 Basic structure......Page 116
7.1  Structure of functors......Page 117
7.2  Structure of transformations......Page 120
7.3  Pseudo-icons......Page 124
7.4  Change of structure......Page 133
7.5  Triequivalences......Page 137
8.1  Cubical tricategories......Page 139
8.2  Gray-categories......Page 143
9 Coherence via Yoneda......Page 148
9.1  Local structure......Page 149
9.2  Global results......Page 151
9.3  The cubical Yoneda lemma......Page 154
9.4  Coherence for tricategories......Page 164
10 Coherence via free constructions......Page 166
10.1  Coherence for tricategories......Page 167
10.2  Coherence and diagrams of constraints......Page 170
10.3  A non-commuting diagram......Page 171
10.4  Strictifying tricategories......Page 172
10.5  Coherence for functors......Page 181
10.6  Strictifying functors......Page 187
Part III Gray-monads......Page 191
11 Codescent in Gray-categories......Page 193
11.1  Lax codescent diagrams......Page 194
11.2  Codescent diagrams......Page 198
11.3  Codescent objects......Page 200
12 Codescent as a weighted colimit......Page 206
12.1  Weighted colimits in Gray-categories......Page 207
12.2  Examples: coinserters and coequifiers......Page 210
12.3  Codescent......Page 217
13 Gray-monads and their algebras......Page 219
13.1  Enriched monads and algebras......Page 220
13.2  Lax algebras and their higher cells......Page 223
13.3  Total structures......Page 229
14 The reflection of lax algebras into strict algebras......Page 237
14.1  The canonical codescent diagram of a lax algebra......Page 238
14.2  The left adjoint, lax case......Page 240
14.3  The left adjoint, pseudo case......Page 252
15 A general coherence result......Page 254
15.1  Weak codescent objects......Page 255
15.2  Coherence for pseudo-algebras......Page 276
Bibliography......Page 283
Index......Page 287