You are here

Categorical Homotopy Theory

Emily Riehl
Cambridge University Press
Publication Date: 
Number of Pages: 
New Mathematical Monographs 24
[Reviewed by
Ittay Weiss
, on

My review of this excellently written book revolves around expanding on the author’s words from page xv: “Rather than present material that one could easily read elsewhere, we chart a less-familiar course that should complement the insights of the experienced and provide context for the naive student who might later read the classical accounts of this theory.”

Categorical homotopy theory started with Quillen’s formulation of a (closed) model category. The theory developed rapidly with numerous texts devoted to its various facets. The modern approach to the subject is somewhat different than how things started out. Consequently, the uninitiated student needs to navigate a considerable amount of information from sometimes conflicting, or unclear, directions.

What the book under review does very well is provide a clear perspective on the aims and techniques of the modern theory of categorical homotopy theory in a unified and self-contained fashion. The book only assumes some knowledge of category theory and devotes quite a few pages to developing the required concepts of (enriched) category theory, constantly motivated by homotopical considerations.

Model categories make their first appearance only in Part III, after the reader had seen the categorical machinery at work, and thus when the reader is primed to appreciate the usefulness of Quillen’s axioms. Part IV is then a treatment of quasi-categories, a subject at the forefront of current research.

Referring back to the author’s words, and thinking of students rather than experts as the intended readers of this review, the book is a welcome addition to the current literature on the subject of model categories and modern homotopy theory. It very successfully paves a welcoming road into the elaborate world of homotopy theory, in a way which is very suitable for students. There is much to gain from reading this book, either in tandem with the classical texts, or as a stand-alone text. It can serve as motivation for delving deeper, or, for those already in deeper water, for finding a clear perspective on the subject.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.

Part I. Derived Functors and Homotopy (Co)limits:
1. All concepts are Kan extensions
2. Derived functors via deformations
3. Basic concepts of enriched category theory
4. The unreasonably effective (co)bar construction
5. Homotopy limits and colimits: the theory
6. Homotopy limits and colimits: the practice

Part II. Enriched Homotopy Theory:
7. Weighted limits and colimits
8. Categorical tools for homotopy (co)limit computations
9. Weighted homotopy limits and colimits
10. Derived enrichment

Part III. Model Categories and Weak Factorization Systems:
11. Weak factorization systems in model categories
12. Algebraic perspectives on the small object argument
13. Enriched factorizations and enriched lifting properties
14. A brief tour of Reedy category theory

Part IV. Quasi-Categories:
15. Preliminaries on quasi-categories
16. Simplicial categories and homotopy coherence
17. Isomorphisms in quasi-categories
18. A sampling of 2-categorical aspects of quasi-category theory.