Model categories and their localizations / Philip S. Hirschhorn.

Format:

Book

Author/Creator:

Hirschhorn, Philip S. (Philip Steven), 1952- author.

Series:

Mathematical surveys and monographs ; no. 99.

Mathematical surveys and monographs, 0076-5376 ; volume 99

Language:

English

Subjects (All):

Model categories (Mathematics).

Homotopy theory.

Physical Description:

1 online resource (478 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2003]

Summary:

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.

Contents:

""Contents""; ""Introduction""; ""Model categories and their homotopy categories""; ""Localizing model category structures""; ""Acknowledgments""; ""Part 1. Localization of Model Category Structures""; ""Summary of Part 1""; ""Chapter 1. Local Spaces and Localization""; ""1.1. Definitions of spaces and mapping spaces""; ""1.2. Local spaces and localization""; ""1.3. Constructing an f-localization functor""; ""1.4. Concise description of the f-localization""; ""1.5. Postnikov approximations""; ""1.6. Topological spaces and simplicial sets""; ""1.7. A continuous localization functor""

""4.1. Existence of left Bousfield localizations""""4.2. Horns on S and S-local equivalences""; ""4.3. A functorial localization""; ""4.4. Localization of subcomplexes""; ""4.5. The Bousfield-Smith cardinality argument""; ""4.6. Proof of the main theorem""; ""Chapter 5. Existence of Right Bousfield Localizations""; ""5.1. Right Bousfield localization: Cellularization""; ""5.2. Horns on K and K-colocal equivalences""; ""5.3. K-colocal cofibrations""; ""5.4. Proof of the main theorem""; ""5.5. K-colocal objects and K-cellular objects""; ""Chapter 6. Fiberwise Localization""

""6.1. Fiberwise localization""""6.2. The fiberwise local model category structure""; ""6.3. Localizing the fiber""; ""6.4. Uniqueness of the fiberwise localization""; ""Part 2. Homotopy Theory in Model Categories""; ""Summary of Part 2""; ""Chapter 7. Model Categories""; ""7.1. Model categories""; ""7.2. Lifting and the retract argument""; ""7.3. Homotopy""; ""7.4. Homotopy as an equivalence relation""; ""7.5. The classical homotopy category""; ""7.6. Relative homotopy and fiberwise homotopy""; ""7.7. Weak equivalences""; ""7.8. Homotopy equivalence""

""7.9. The equivalence relation generated by ""weak equivalence''""""7.10. Topological spaces and simplicial sets""; ""Chapter 8. Fibrant and Cofibrant Approximations""; ""8.1. Fibrant and cofibrant approximations""; ""8.2. Approximations and homotopic maps""; ""8.3. The homotopy category of a model category""; ""8.4. Derived functors""; ""8.5. Quillen functors and total derived functors""; ""Chapter 9. Simplicial Model Categories""; ""9.1. Simplicial model categories""; ""9.2. Colimits and limits""; ""9.3. Weak equivalences of function complexes""; ""9.4. Homotopy lifting""

""9.5. Simplicial homotopy""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 455-457) and index.

Description based on print version record.

ISBN:

1-4704-1326-4