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Full Description
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.
Contents
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.
