Higher Categories and Homotopical Algebra

At last, a friendly introduction to modern homotopy theory after Joyal and Lurie, reaching advanced tools and starting from scratch.

Denis-Charles Cisinski (Author)

9781108473200, Cambridge University Press

Hardback, published 2 May 2019

448 pages
23.4 x 15.7 x 2.9 cm, 0.75 kg

'As someone doing research and advising graduate students in a closely-related area, I am happy to see a book like this in the literature. It will help readers to learn this subject, and to gain a deep understanding of the foundational ideas.' Julie Bergner, MAA Reviews

This book provides an introduction to modern homotopy theory through the lens of higher categories after Joyal and Lurie, giving access to methods used at the forefront of research in algebraic topology and algebraic geometry in the twenty-first century. The text starts from scratch - revisiting results from classical homotopy theory such as Serre's long exact sequence, Quillen's theorems A and B, Grothendieck's smooth/proper base change formulas, and the construction of the Kan–Quillen model structure on simplicial sets - and develops an alternative to a significant part of Lurie's definitive reference Higher Topos Theory, with new constructions and proofs, in particular, the Yoneda Lemma and Kan extensions. The strong emphasis on homotopical algebra provides clear insights into classical constructions such as calculus of fractions, homotopy limits and derived functors. For graduate students and researchers from neighbouring fields, this book is a user-friendly guide to advanced tools that the theory provides for application.

Preface
1. Prelude
2. Basic homotopical algebra
3. The homotopy theory of ?-categories
4. Presheaves: externally
5. Presheaves: internally
6. Adjoints, limits and Kan extensions
7. Homotopical algebra
References
Notation
Index.

Subject Areas: Algebraic topology [PBPD], Algebraic geometry [PBMW]