From Categories to Homotopy Theory

Bridge the gap between category theory and its applications in homotopy theory with this guide for graduate students and researchers.

Birgit Richter (Author)

9781108479622, Cambridge University Press

Hardback, published 16 April 2020

400 pages, 115 exercises
23.5 x 15.6 x 2.6 cm, 0.68 kg

'The book has been thoughtfully written with students in mind, and contains plenty of pointers to the literature for those who want to pursue a subject further. Readers will find themselves taken on an engaging journey by a true expert in the field, who brings to the material both insight and style.' Daniel Dugger, MathSciNet (https://mathscinet.ams.org)

Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.

Introduction
Part I. Category Theory: 1. Basic notions in category theory
2. Natural transformations and the Yoneda lemma
3. Colimits and limits
4. Kan extensions
5. Comma categories and the Grothendieck construction
6. Monads and comonads
7. Abelian categories
8. Symmetric monoidal categories
9. Enriched categories
Part II. From Categories to Homotopy Theory: 10. Simplicial objects
11. The nerve and the classifying space of a small category
12. A brief introduction to operads
13. Classifying spaces of symmetric monoidal categories
14. Approaches to iterated loop spaces via diagram categories
15. Functor homology
16. Homology and cohomology of small categories
References
Index.

Subject Areas: Algebraic topology [PBPD]