**Mathematical Surveys and Monographs**

Volume: 63;
1999;
209 pp;
Softcover

MSC: Primary 55;
Secondary 13; 16; 18; 20

Print ISBN: 978-0-8218-4361-1

Product Code: SURV/63.S

List Price: $67.00

Individual Member Price: $53.60

**Electronic ISBN: 978-1-4704-1290-6
Product Code: SURV/63.S.E**

List Price: $67.00

Individual Member Price: $53.60

# Model Categories

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*Mark Hovey*

*[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.*

—

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.

#### Table of Contents

# Table of Contents

## Model Categories

- Contents vii8 free
- Preface ix10 free
- Chapter 1. Model categories 114 free
- Chapter 2. Examples 2740
- Chapter 3. Simplicial sets 7386
- Chapter 4. Monoidal model categories 101114
- Chapter 5. Framings 119132
- Chapter 6. Pointed model categories 147160
- Chapter 7. Stable model categories and triangulated categories 177190
- Chapter 8. Vistas 195208
- Bibliography 201214
- Index 205218
- Errata 211224

#### Readership

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

#### Reviews

[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.

-- 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