*[The book] starts with an account of the
definitions, and a development of the homotopy theory of model
categories. This is probably the first time in which the important
notion of cofibrant generation has appeared in a book, and the
consideration of the 2-category of model categories and Quillen
adjunctions is another interesting feature.*

—**Bulletin of the London Mathematical
Society**

Model categories are used as a tool for inverting certain
maps in a category in a controllable manner. As such, they are useful
in diverse areas of mathematics. The list of such areas is continually
growing.

This book is a comprehensive study of the relationship between a
model category and its homotopy category. The author develops the
theory of model categories, giving a careful development of the main
examples. One highlight of the theory is a proof that the homotopy
category of any model category is naturally a closed module over the
homotopy category of simplicial sets.

Little is required of the reader beyond some category theory and
set theory, which makes the book accessible to advanced graduate
students. The book begins with the basic theory of model categories
and proceeds to a careful exposition of the main examples, using the
theory of cofibrantly generated model categories. It then develops the
general theory more fully, showing in particular that the homotopy
category of any model category is a module over the homotopy category
of simplicial sets, in an appropriate sense. This leads to a
simplification and generalization of the loop and suspension functors
in the homotopy category of a pointed model category. The book
concludes with a discussion of the stable case, where the homotopy
category is triangulated in a strong sense and has a set of small weak
generators.

Readership

Graduate students and research mathematicians working
in algebraic topology, algebraic geometry, $K$-theory, and commutative
algebra.