Homotopy Theory of Higher Categories: From Segal Categories to n-Categories and BeyondCambridge University Press, 20 oct. 2011 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. |
Table des matières
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 |
618 | |
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Expressions et termes fréquents
adjoint applying cartesian model category cartesian product cell complexes cell(K Chapter cof(I cofibrant objects colimits commutative compatible considered construction Corollary corresponding defined definition delooping denote discussion element Enr(P enriched categories factorization fibrant objects fibrant replacement fully faithful FUNC gives global weak equivalence globular groupoid higher categories homotopy i-morphisms idempotent identity induces inj(I injective cofibrations invertible isomorphism iteration Lemma levelwise weak equivalence limit M-enriched minimal dimension model category monoidal morphism morphism f n-groupoid natural transformation notation notion Ob(A Ob(B obtain PC(M presheaf projective Proof Proposition pushout Reedy cofibration regular cardinal resp restricting retracts satisfies the Segal Section Seg(A Segal categories Segal condition Segal maps set of objects simplicial categories simplicial sets simplicial space strict n-category subcategory subset Suppose f surjective Tamsamani Theorem theory tractable left proper transfinite composition trivial cofibration trivial cofibrations trivial fibration truncation weak n-categories