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From Categories to Homotopy Theory

Birgit Richter
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics
[Reviewed by
Julie Bergner
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The purpose of this book is to give an introduction to category theory with an eye toward its usage in modern homotopy theory. It is divided into two parts: the first an introduction to category theory, and the second consisting of several different applications to algebraic topology. 
There are a number of good introductions to category theory available, but very often they are written by researchers in category theory itself, so they are interested in various constructions for their own sake. In contrast, the author of this book has written the first part of the book in answer to the question of which aspects of category theory are most important for modern homotopy theory. While still not exactly short at around 200 pages, the book takes an efficient route through many key concepts such the Yoneda Lemma, limits and colimits, and Kan extensions. Several sections are then devoted to various kinds of categories with more structure: abelian categories, symmetric monoidal categories, and enriched categories. These sections are by no means comprehensive treatments of these topics (which can be found elsewhere), but rather intended to provide a good working introduction to them.
The second part of the book provides a tour through many topics in homotopy theory in which the category theory developed in the first part plays a key role. The first two, and longest, chapters in this section develop simplicial objects and nerves of categories, two notions that have become central in modern homotopy theory. In particular, the author introduces some classical results about simplicial sets, for example that they provide a combinatorial model for topological spaces, but also includes topics that are of much current interest, such as quasi-categories, Segal sets, and symmetric spectra. The author then introduces a number of other significant categorical constructions in homotopy theory, such as operads, iterated loop spaces, and functor homology. 
The book does require some background, or at least mathematical maturity, on the part of the reader. A good introductory course in algebra would probably suffice for the first section, but some topology background would be helpful for the second part, at least for motivation if not for explicit mathematical techniques. 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 is a professor of mathematics at the University of Virginia and her research is in homotopy theory.