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

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Cambridge University Press, 20 oct. 2011
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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.
 

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Table des matières

1 History and motivation
3
2 Strict ncategories
21
3 Fundamental elements of ncategories
51
4 Operadic approaches
65
5 Simplicial approaches
81
6 Weak enrichment over a cartesian model category an introduction
98
PART II Categorical preliminaries
109
7 Model categories
111
14 Calculus of generators and relations
326
15 Generators and relations for Segal categories
350
PART IV The model structure
383
16 Sequentially free precategories
385
17 Products
397
18 Intervals
421
19 The model category of mathscr Menriched precategories
444
PART V Higher category theory
453

8 Cell complexes in locally presentable categories
144
9 Direct left Bousfield localization
192
PART III Generators and relations
225
10 Precategories
227
11 Algebraic theories in model categories
251
12 Weak equivalences
275
13 Cofibrations
297
20 Iterated higher categories
455
21 Higher categorical techniques
480
22 Limits of weak enriched categories
527
23 Stabilization
596
Index
618
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À propos de l'auteur (2011)

Carlos Simpson is Directeur de Recherche in the CNRS in Toulouse and Nice, France.

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