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Mathematical Surveys and Monographs
1999; 209 pp; softcover
List Price: US$63
Member Price: US$50.40
Order Code: SURV/63.S
[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.
Graduate students and research mathematicians working in algebraic topology, algebraic geometry, \(K\)-theory, and commutative algebra.
"[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
"This book provides a thorough and well-written guide to Quillen's model categories. To read this book one requires only a basic knowledge of category theory and some familiarity with chain complexes and topological spaces. This makes the text not only a volume for experts, but also usable in a classroom setting. Model Categories is written in a very clear style. The text fills a gap in the mathematical literature."
-- Mathematical Reviews
"The book under review gives a modern and accessible account of the basic facts; and even if it is not intended to be a textbook, it should be a good starting point for students, as well as a reference for active researchers. The book fills a vacant niche in the literature and ... may well become a standard reference."
-- Zentralblatt MATH
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