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

Develops a full set of homotopical algebra techniques dedicated to the study of higher categories.

Carlos Simpson (Author)

9780521516952, Cambridge University Press

Hardback, published 20 October 2011

652 pages, 35 b/w illus.
23.5 x 15.8 x 3.8 cm, 1.05 kg

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.

Subject Areas: Topology [PBP], Discrete mathematics [PBD]