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Model Categories and Their Localizations

Philip S. Hirschhorn Wellesley College, Wellesley, MA
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Softcover ISBN: 978-0-8218-4917-0
Product Code: SURV/99.S
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AMS Member Price: $93.60 Electronic ISBN: 978-1-4704-1326-2 Product Code: SURV/99.S.E List Price:$110.00
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AMS Member Price: $140.40 Click above image for expanded view Model Categories and Their Localizations Philip S. Hirschhorn Wellesley College, Wellesley, MA Available Formats:  Softcover ISBN: 978-0-8218-4917-0 Product Code: SURV/99.S  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$93.60
 Electronic ISBN: 978-1-4704-1326-2 Product Code: SURV/99.S.E
 List Price: $110.00 MAA Member Price:$99.00 AMS Member Price: $88.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$175.50 MAA Member Price: $157.95 AMS Member Price:$140.40
• Book Details

Mathematical Surveys and Monographs
Volume: 992003; 457 pp
MSC: Primary 18; 55;

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.

• Part 1 . Localization of model category structures
• 1. Local spaces and localization
• 2. The localization model category for spaces
• 3. Localization of model categories
• 4. Existence of left Bousfield localizations
• 5. Existence of right Bousfield localizations
• 6. Fiberwise localization
• Part 2. Homotopy theory in model categories
• 7. Model categories
• 8. Fibrant and cofibrant approximations
• 9. Simplicial model categories
• 10. Ordinals, cardinals, and transfinite composition
• 11. Cofibrantly generated model categories
• 12. Cellular model categories
• 13. Proper model categories
• 14. The classifying space of a small category
• 15. The reedy model category structure
• 16. Cosimplicial and simplicial resolutions
• 17. Homotopy function complexes
• 18. Homotopy limits in simplicial model categories
• 19. Homotopy limits in general model categories

• Reviews

• This book was many years in the writing, and it shows. It is very carefully written, exhaustively (even obsessively) cross-referenced, and precise in all its details. In short, it is an important reference for the subject.

Zentralblatt MATH
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 992003; 457 pp
MSC: Primary 18; 55;

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.

• Part 1 . Localization of model category structures
• 1. Local spaces and localization
• 2. The localization model category for spaces
• 3. Localization of model categories
• 4. Existence of left Bousfield localizations
• 5. Existence of right Bousfield localizations
• 6. Fiberwise localization
• Part 2. Homotopy theory in model categories
• 7. Model categories
• 8. Fibrant and cofibrant approximations
• 9. Simplicial model categories
• 10. Ordinals, cardinals, and transfinite composition
• 11. Cofibrantly generated model categories
• 12. Cellular model categories
• 13. Proper model categories
• 14. The classifying space of a small category
• 15. The reedy model category structure
• 16. Cosimplicial and simplicial resolutions
• 17. Homotopy function complexes
• 18. Homotopy limits in simplicial model categories
• 19. Homotopy limits in general model categories
• This book was many years in the writing, and it shows. It is very carefully written, exhaustively (even obsessively) cross-referenced, and precise in all its details. In short, it is an important reference for the subject.

Zentralblatt MATH
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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