Mathematical Surveys and Monographs 2003; 457 pp; softcover Volume: 99 ISBN10: 0821849174 ISBN13: 9780821849170 List Price: US$110 Member Price: US$88 Order Code: SURV/99.S
 The aim of this book is to explain modern homotopy theory in a manner accessible to graduate students yet structured so that experts can skip over numerous linear developments to quickly reach the topics of their interest. Homotopy theory arises from choosing a class of maps, called weak equivalences, and then passing to the homotopy category by localizing with respect to the weak equivalences, i.e., by creating a new category in which the weak equivalences are isomorphisms. Quillen defined a model category to be a category together with a class of weak equivalences and additional structure useful for describing the homotopy category in terms of the original category. This allows you to make constructions analogous to those used to study the homotopy theory of topological spaces. A model category has a class of maps called weak equivalences plus two other classes of maps, called cofibrations and fibrations. Quillen's axioms ensure that the homotopy category exists and that the cofibrations and fibrations have extension and lifting properties similar to those of cofibration and fibration maps of topological spaces. During the past several decades the language of model categories has become standard in many areas of algebraic topology, and it is increasingly being used in other fields where homotopy theoretic ideas are becoming important, including modern algebraic \(K\)theory and algebraic geometry. All these subjects and more are discussed in the book, beginning with the basic definitions and giving complete arguments in order to make the motivations and proofs accessible to the novice. The book is intended for graduate students and research mathematicians working in homotopy theory and related areas. Readership Graduate students and research mathematicians. Reviews "This book was many years in the writing, and it shows. It is very carefully written, exhaustively (even obsessively) crossreferenced, and precise in all its details. In short, it is an important reference for the subject."  Zentralblatt MATH Table of Contents Part 1. Localization of model category structures  Summary of part 1
 Local spaces and localization
 The localization model category for spaces
 Localization of model categories
 Existence of left Bousfield localizations
 Existence of right Bousfield localizations
 Fiberwise localization
Part 2. Homotopy theory in model categories  Summary of part 2
 Model categories
 Fibrant and cofibrant approximations
 Simplicial model categories
 Ordinals, cardinals, and transfinite composition
 Cofibrantly generated model categories
 Cellular model categories
 Proper model categories
 The classifying space of a small category
 The Reedy model category structure
 Cosimplicial and simplicial resolutions
 Homotopy function complexes
 Homotopy limits in simplicial model categories
 Homotopy limits in general model categories
 Index
 Bibliography
