The Homotopy Theory of (?,1)-Categories: 90 (London Mathematical Society Student Texts, Series Number 90)

The Homotopy Theory of (?,1)-Categories: 90 (London Mathematical Society Student Texts, Series Number 90)

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Julia E. Bergner
Cambridge University Press, 3/15/2018
EAN 9781107101364, ISBN10: 1107101360

Hardcover, 284 pages, 23.6 x 15.5 x 2 cm
Language: English

The notion of an (∞,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. This book provides a relatively self-contained source of the definitions of the different models, the model structure (homotopy theory) of each, and the equivalences between the models. While most of the current literature focusses on how to extend category theory in this context, and centers in particular on the quasi-category model, this book offers a balanced treatment of the appropriate model structures for simplicial categories, Segal categories, complete Segal spaces, quasi-categories, and relative categories, all from a homotopy-theoretic perspective. Introductory chapters provide background in both homotopy and category theory and contain many references to the literature, thus making the book accessible to graduates and to researchers in related areas.

1. Models for homotopy theories
2. Simplicial objects
3. Topological and categorical motivation
4. Simplicial categories
5. Complete Segal spaces
6. Segal categories
7. Quasi-categories
8. Relative categories
9. Comparing functors to complete Segal spaces
10. Variants on (∞, 1)-categories