Preface ..................................................... xi
Acknowledgements .......................................... xvii
PART I HIGHER CATEGORIES ....................................... 1
1 History and motivation ....................................... 3
2 Strict n-categories ......................................... 21
2.1 Godement relations: the Eckmann-Hilton argument ........ 23
2.2 Strict n-groupoids ..................................... 25
2.3 The need for weak composition .......................... 38
2.4 Realization functors ................................... 39
2.5 n-groupoids with one object ............................ 40
2.6 The case of the standard realization ................... 41
2.7 Nonexistence of strict 3-groupoids of 3-type S2 ........ 43
3 Fundamental elements of n-categories ........................ 51
3.1 A globular theory ...................................... 51
3.2 Identities ............................................. 54
3.3 Composition, equivalence and truncation ................ 54
3.4 Enriched categories .................................... 57
3.5 The (n + l)-category of n-categories ................... 58
3.6 Poincare n-groupoids ................................... 60
3.7 Interiors .............................................. 61
3.8 The case n = ∞ ......................................... 62
4 Operadic approaches ......................................... 65
4.1 May's delooping machine ................................ 65
4.2 Baez-Dolan's definition ................................ 66
4.3 Batanin's definition ................................... 69
4.4 Trimble's definition and Cheng's comparison ............ 73
4.5 Weak units ............................................. 75
4.6 Other notions .......................................... 78
5 Simplicial approaches ....................................... 81
5.1 Strict simplicial categories ........................... 81
5.2 Segal's delooping machine .............................. 83
5.3 Segal categories ....................................... 86
5.4 Rezk categories ........................................ 91
5.5 Quasicategories ........................................ 93
5.6 Going between Segal categories and n-categories ........ 96
6 Weak enrichment over a cartesian model category:
an introduction ............................................. 98
6.1 Simplicial objects in  ............................... 98
6.2 Diagrams over Δχ ....................................... 99
6.3 Hypotheses on ...................................... 100
6.4 Precategories ......................................... 101
6.5 Unitality ............................................. 102
6.6 Rectification of Δχ-diagrams .......................... 104
6.7 Enforcing the Segal condition ......................... 105
6.8 Products, intervals and the model structure ........... 107
PART II CATEGORICAL PRELIMINARIES ............................. 109
7 Model categories ........................................... 111
7.1 Lifting properties .................................... 112
7.2 Quillen's axioms ...................................... 113
7.3 Left properness ....................................... 116
7.4 The Kan-Quillen model category of simplicial sets ..... 119
7.5 Homotopy liftings and extensions ...................... 121
7.6 Model structures on diagram categories ................ 124
7.7 Cartesian model categories ............................ 129
7.8 Internal Horn ......................................... 132
7.9 Enriched categories ................................... 135
8 Cell complexes in locally presentable categories ........... 144
8.1 Locally presentable categories ........................ 146
8.2 The small object argument ............................. 151
8.3 More on cell complexes ................................ 154
8.4 Cofibrantly generated, combinatorial and tractable
model categories ...................................... 168
8.5 Smith's recognition principle ......................... 170
8.6 Injecfive cofibrations in diagram categories .......... 177
8.7 Pseudo-generating sets ................................ 183
9 Direct left Bousfield localization ......................... 192
9.1 Projection to a subcategory of local objects .......... 192
9.2 Weak monadic projection ............................... 199
9.3 New weak equivalences ................................. 208
9.4 Invariance properties ................................. 211
9.5 New fibrations ........................................ 216
9.6 Pushouts by new trivial cofibrations .................. 218
9.7 The model category structure .......................... 220
9.8 Transfer along a left Quillen functor ................. 222
PART III GENERATORS AND RELATIONS ............................. 225
10 Precategories .............................................. 227
10.1 Enriched precategories with a fixed set of objects .... 227
10.2 The Segal conditions .................................. 229
10.3 Varying the set of objects ............................ 230
10.4 The category of precategories ......................... 232
10.5 Basic examples ........................................ 233
10.6 Limits, colimits and local presentability ............. 236
10.7 Interpretations as presheaf categories ................ 242
11 Algebraic theories in model categories ..................... 251
11.1 Diagrams over the categories  (n) .................... 252
11.2 Imposing the product condition ........................ 257
11.3 Algebraic diagram theories ............................ 263
11.4 Unitality ............................................. 266
11.5 Unital algebraic diagram theories ..................... 272
11.6 The generation operation .............................. 273
11.7 Reedy structures ...................................... 274
12 Weak equivalences .......................................... 275
12.1 Local weak equivalences ............................... 275
12.2 Unitalization adjunctions ............................. 280
12.3 The Reedy structure ................................... 282
12.4 Global weak equivalences .............................. 289
12.5 Categories enriched over ho() ....................... 292
12.6 Change of enrichment category ......................... 294
13 Cofibrations ............................................... 297
13.1 Skeleta and coskeleta ................................. 297
13.2 Some natural precategories ............................ 302
13.3 Projective cofibrations ............................... 304
13.4 Injective cofibrations ................................ 307
13.5 A pushout expression for the skeleta .................. 308
13.6 Reedy cofibrations .................................... 310
13.7 Relationship between the classes of cofibrations ...... 323
14 Calculus of generators and relations ....................... 326
14.1 The T precategories ................................... 326
14.2 Some trivial cofibrations ............................. 329
14.3 Pushout by isotrivial cofibrations .................... 332
14.4 An elementary generation step Gen ..................... 340
14.5 Fixing the fibrant condition locally .................. 343
14.6 Combining generation steps ............................ 344
14.7 Functoriality of the generation process ............... 345
14.8 Example: generators and relations for 1-categories .... 347
15 Generators and relations for Segal categories .............. 350
15.1 Segal categories ...................................... 350
15.2 The Poincaré-Segal groupoid ........................... 352
15.3 Looping and delooping ................................. 355
15.4 The calculus .......................................... 359
15.5 Computing the loop space .............................. 370
15.6 Example: π3(S2) ....................................... 378
PART IV THE MODEL STRUCTURE ................................... 383
16 Sequentially free precategories ............................ 385
16.1 Imposing the Segal condition on ϒ ..................... 385
16.2 Sequentially free precategories in general ............ 387
17 Products ................................................... 397
17.1 Products of sequentially free precategories ........... 397
17.2 Products of general precategories ..................... 408
17.3 The role of unitality, degeneracies and higher
coherences ............................................ 416
17.4 Why we can't truncate Δ ............................... 419
18 Intervals .................................................. 421
18.1 Contractible objects and intervals in .............. 422
18.2 Intervals for -enriched precategories ............... 424
18.3 The versality property ................................ 429
18.4 "Contractibility of intervals for X-precategories ..... 432
18.5 Construction of a left Quillen functor  → ......... 433
18.6 Contractibility in general ............................ 435
18.7 Pushout of trivial cofibrations ....................... 437
18.8 A versality property .................................. 442
19 The model category of -enriched precategories ........... 444
19.1 A standard factorization .............................. 444
19.2 The model structures .................................. 446
19.3 The cartesian property ................................ 449
19.4 Properties of fibrant objects ......................... 450
19.5 The model category of strict -enriched categories ... 450
PART V HIGHER CATEGORY THEORY ................................. 453
20 Iterated higher categories ................................. 455
20.1 Initialization ........................................ 456
20.2 Notation for n-categories ............................. 457
20.3 Truncation and equivalences ........................... 466
20.4 Homotopy types and higher groupoids ................... 469
20.5 The (n + l)-category nCAT ............................. 411
21 Higher categorical techniques .............................. 480
21.1 The opposite category ................................. 481
21.2 Equivalent objects .................................... 481
21.3 Homotopies and the homotopy 2-category ................ 484
21.4 Constructions with  .................................. 489
21.5 Acyclicity of inversion ............................... 495
21.6 Localization and interior ............................. 499
21.7 Limits ................................................ 506
21.8 Colimits .............................................. 510
21.9 Invariance properties ................................. 511
21.10 Limits of diagrams ................................... 516
22 Limits of weak enriched categories ......................... 527
22.1 Cartesian families .................................... 528
22.2 The Yoneda embeddings ................................. 531
22.3 Universe considerations ............................... 536
22.4 Diagrams in quasifibrant precategories ................ 538
22.5 Extension properties .................................. 543
22.6 Limits of weak enriched categories .................... 553
22.7 Cardinality ........................................... 563
22.8 Splitting idempotents ................................. 567
22.9 Colimits of weak enriched categories .................. 576
22.10 Fiber products and amalgamated sums .................. 592
Stabilization ......................................... 596
23.1 Minimal dimension ..................................... 598
23.2 The stabilization hypothesis .......................... 608
23.3 Suspension and free monoidal categories ............... 611
Epilogue ................................................... 616
References ................................................. 618
Index ...................................................... 630
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