From Categories to Homotopy Theory
From Categories to Homotopy Theory
£60.00
GBPCategory 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.
- Includes diagrammatical proofs, examples and exercises to encourage an active way of learning
- Provides enough background from category theory to make advanced topics such as K-theory, iterated loop spaces and functor homology accessible to readers
- Makes abstract concepts tangible, encouraging readers to learn what can be an intimidating subject
Reviews & endorsements
'It would be an excellent text for a graduate student just finishing introductory coursework and wanting to know about techniques in modern homotopy theory.' Julie Bergner
'… this book attempts to bridge the gap between the basic theory and the application of categorical methods to homotopy theory, which has been the subject of some recent exciting developments … the book would be very useful for beginner graduate students in homotopy theory.' Hollis Williams, IMA website
'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)
Product details
April 2020Hardback
9781108479622
400 pages
235 × 156 × 26 mm
0.68kg
115 exercises
Available
Table of Contents
- 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.
