Homotopy Theory of Higher Categories: From Segal Categories to n-Categories and Beyond (New Mathematical Monographs)
Cambridge University Press, 10/20/2011
EAN 9780521516952, ISBN10: 0521516951
Hardcover, 652 pages, 23.1 x 15.6 x 3.8 cm
Language: English
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.
Prologue
Acknowledgements
Part I. Higher Categories
1. History and motivation
2. Strict n-categories
3. Fundamental elements of n-categories
4. The need for weak composition
5. Simplicial approaches
6. Operadic approaches
7. Weak enrichment over a Cartesian model category
an introduction
Part II. Categorical Preliminaries
8. Some category theory
9. Model categories
10. Cartesian model categories
11. Direct left Bousfield localization
Part III. Generators and Relations
12. Precategories
13. Algebraic theories in model categories
14. Weak equivalences
15. Cofibrations
16. Calculus of generators and relations
17. Generators and relations for Segal categories
Part IV. The Model Structure
18. Sequentially free precategories
19. Products
20. Intervals
21. The model category of M-enriched precategories
22. Iterated higher categories
Part V. Higher Category Theory
23. Higher categorical techniques
24. Limits of weak enriched categories
25. Stabilization
Epilogue
References
Index.