Homotopy Theory of Higher Categories: From Segal Categories to n -Categories and Beyond

by Carlos Simpson

• ISBN13: 9780521516952
• ISBN10: 0521516951
• Format: Hardcover
• Copyright: 2011-12-30
• Publisher: Cambridge University Press

Summary

The study of higher categories is attracting growing interest for its many applications in topology, algebraic geometry, mathematical physics and category theory. In this highly readable book, Carlos Simpson develops a full set of homotopical algebra techniques and proposes a working theory of higher categories. Starting with a cohesive overview of the many different approaches currently used by researchers, the author proceeds with a detailed exposition of one of the most widely used techniques: the construction of a Cartesian Quillen model structure for higher categories. The fully iterative construction applies to enrichment over any Cartesian model category, and yields model categories for weakly associative n-categories and Segal n-categories. A corollary is the construction of higher functor categories which fit together to form the (n+1)-category of n-categories. The approach uses Tamsamani's definition based on Segal's ideas, iterated as in Pelissier's thesis using modern techniques due to Barwick, Bergner, Lurie and others.

Author Biography

Carlos Simpson is Directeur de Recherches in the CNRS in Nice, France.

Table of Contents

Preface	p. xi
Acknowledgements	p. xvii
Higher Categories	p. 1
History and motivation	p. 3
Strict n-categories	p. 21
Godement relations: the Eckmann-Hilton argument	p. 23
Strict n-groupoids	p. 25
The need for weak composition	p. 38
Realization functors	p. 39
n-groupoids with one object	p. 40
The case of the standard realization	p. 41
Nonexistence of strict 3-groupoids of 3-type S2	p. 43
Fundamental elements of n-categories	p. 51
A globular theory	p. 51
Identities	p. 54
Composition, equivalence and truncation	p. 54
Enriched categories	p. 57
The (n + l)-category of n-categories	p. 58
Poincaré n-groupoids	p. 60
Interiors	p. 61
The case n = ∞	p. 62
Operadic approaches	p. 65
May's delooping machine	p. 65
Baez-Dolan's definition	p. 66
Batanin's definition	p. 69
Trimble's definition and Cheng's comparison	p. 73
Weak units	p. 75
Other notions	p. 78
Simplicial approaches	p. 81
Strict simplicial categories	p. 81
Segal's delooping machine	p. 83
Segal categories	p. 86
Rezk categories	p. 91
Quasicategories	p. 93
Going between Segal categories and n-categories	p. 96
Weak enrichment over a cartesian model category: an introduction	p. 98
Simplicial objects in M	p. 98
Diagrams over ¿x	p. 99
Hypotheses on M	p. 100
Precategories	p. 101
Unitality	p. 102
Rectification of ¿x-diagrams	p. 104
Enforcing the Segal condition	p. 105
Products, intervals and the model structure	p. 107
Categorical Preliminaries	p. 109
Model categories	p. 111
Lifting properties	p. 112
Quillen's axioms	p. 113
Left properness	p. 116
The Kan-Quillen model category of simplicial sets	p. 119
Homotopy liftings and extensions	p. 121
Model structures on diagram categories	p. 124
Cartesian model categories	p. 129
Internal Hom	p. 132
Enriched categories	p. 135
Cell complexes in locally presentable categories	p. 144
Locally presentable categories	p. 146
The small object argument	p. 151
More on cell complexes	p. 154
Cofibrantly generated, combinatorial and tractable model categories	p. 168
Smith's recognition principle	p. 171
Injective cofibrations in diagram categories	p. 177
Pseudo-generating sets	p. 183
Direct left Bousfield localization	p. 192
Projection to a subcategory of local objects	p. 192
Weak monadic projection	p. 199
New weak equivalences	p. 208
Invariance properties	p. 211
New fibrations	p. 216
Pushouts by new trivial cofibrations	p. 218
The model category structure	p. 220
Transfer along a left Quillen functor	p. 222
Generators and Relations	p. 225
Precategories	p. 227
Enriched precategories with a fixed set of objects	p. 227
The Segal conditions	p. 229
Varying the set of objects	p. 230
The category of precategories	p. 232
Basic examples	p. 233
Limits, colimits and local presentability	p. 236
Interpretations as presheaf categories	p. 242
Algebraic theories in model categories	p. 251
Diagrams over the categories €(n)	p. 252
Imposing the product condition	p. 257
Algebraic diagram theories	p. 263
Unitality	p. 266
Unital algebraic diagram theories	p. 272
The generation operation	p. 273
Reedy structures	p. 274
Weak equivalences	p. 275
Local weak equivalences	p. 275
Unitalization adjunctions	p. 280
The Reedy structure	p. 282
Global weak equivalences	p. 289
Categories enriched over ho(M)	p. 292
Change of enrichment category	p. 294
Cofibrations	p. 297
Skeleta and coskeleta	p. 297
Some natural precategories	p. 302
Projective cofibrations	p. 304
Injective cofibrations	p. 307
A pushout expression for the skeleta	p. 308
Reedy cofibrations	p. 310
Relationship between the classes of cofibrations	p. 323
Calculus of generators and relations	p. 326
The T precategories	p. 326
Some trivial cofibrations	p. 329
Pushout by isotrivial cofibrations	p. 332
An elementary generation step Gen	p. 340
Fixing the fibrant condition locally	p. 343
Combining generation steps	p. 344
Functoriality of the generation process	p. 345
Example: generators and relations for 1-categories	p. 347
Generators and relations for Segal categories	p. 350
Segal categories	p. 350
The Poincaré-Segal groupoid	p. 352
Looping and delooping	p. 355
The calculus	p. 359
Computing the loop space	p. 370
Example: ¿3(S2)	p. 378
The Model Structure	p. 383
Sequentially free precategories	p. 385
Imposing the Segal condition on T	p. 385
Sequentially free precategories in general	p. 387
Products	p. 397
Products of sequentially free precategories	p. 397
Products of general precategories	p. 408
The role of unitality, degeneracies and higher coherences	p. 416
Why we can't truncate ¿	p. 419
Intervals	p. 421
Contractible objects and intervals in M	p. 422
Intervals for M-enriched precategories	p. 424
The versality property	p. 429
Contractibility of intervals for K-precategories	p. 432
Construction of a left Quillen functor K → M	p. 433
Contractibility in general	p. 435
Pushout of trivial cofibrations	p. 437
A versality property	p. 442
The model category of M-enriched precategories	p. 444
A standard factorization	p. 444
The model structures	p. 446
The cartesian property	p. 449
Properties of fibrant objects	p. 450
The model category of strict M-enriched categories	p. 450
Higher Category Theory	p. 453
Iterated higher categories	p. 455
Initialization	p. 456
Notation for n-categories	p. 457
Truncation and equivalences	p. 466
Homotopy types and higher groupoids	p. 469
The (n + l)-category nCAT	p. 477
Higher categorical techniques	p. 480
The opposite category	p. 481
Equivalent objects	p. 481
Homotopies and the homotopy 2-category	p. 484
Constructions with T	p. 489
Acyclicity of inversion	p. 495
Localization and interior	p. 499
Limits	p. 506
Colimits	p. 510
Invariance properties	p. 511
Limits of diagrams	p. 516
Limits of weak enriched categories	p. 527
Cartesian families	p. 528
The Yoneda embeddings	p. 531
Universe considerations	p. 536
Diagrams in quasifibrant precategories	p. 538
Extension properties	p. 543
Limits of weak enriched categories	p. 553
Cardinality	p. 563
Splitting idempotents	p. 567
Colimits of weak enriched categories	p. 576
Fiber products and amalgamated sums	p. 592
Stabilization	p. 596
Minimal dimension	p. 598
The stabilization hypothesis	p. 608
Suspension and free monoidal categories	p. 611
Epilogue	p. 616
References	p. 618
Index	p. 630
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