The Homotopy Theory of (?,1)-Categories

An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them.

Julia E. Bergner (Author)

9781107499027, Cambridge University Press

Paperback / softback, published 15 March 2018

284 pages
22.8 x 15.1 x 1.6 cm, 0.41 kg

'The writing is accessible, even for students, and the ideas are clear. The author gives references for every claim and definition, with the added advantage that some technical [lengthy] points can be left out to avoid burying the ideas.' Najib Idrissi, zbMATH

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.

Preface
Acknowledgments
Introduction
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
References
Index.

Subject Areas: Algebraic topology [PBPD], Mathematical logic [PBCD]