**Mathematical Surveys and Monographs**

Volume: 99;
2003;
457 pp;
Softcover

MSC: Primary 18; 55;

Print ISBN: 978-0-8218-4917-0

Product Code: SURV/99.S

List Price: $110.00

Individual Member Price: $88.00

**Electronic ISBN: 978-1-4704-1326-2
Product Code: SURV/99.S.E**

List Price: $110.00

Individual Member Price: $88.00

# Model Categories and Their Localizations

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*Philip S. Hirschhorn*

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.

#### Table of Contents

# Table of Contents

## Model Categories and Their Localizations

- Contents v6 free
- Introduction ix10 free
- Model categories and their homotopy categories ix10
- Localizing model category structures xi12 free
- Acknowledgments xv16 free
- Part 1. Localization of Model Category Structures 118 free
- Summary of Part 1 320
- Chapter 1. Local Spaces and Localization 522
- 1.1. Definitions of spaces and mapping spaces 522
- 1.2. Local spaces and localization 825
- 1.3. Constructing an f-localization functor 1633
- 1.4. Concise description of the f-localization 2037
- 1.5. Postnikov approximations 2239
- 1.6. Topological spaces and simplicial sets 2441
- 1.7. A continuous localization functor 2946
- 1.8. Pointed and unpointed localization 3148

- Chapter 2. The Localization Model Category for Spaces 3552
- Chapter 3. Localization of Model Categories 4764
- Chapter 4. Existence of Left Bousfield Localizations 7188
- Chapter 5. Existence of Right Bousfield Localizations 83100
- Chapter 6. Fiberwise Localization 93110

- Part 2. Homotopy Theory in Model Categories 101118
- Summary of Part 2 103120
- Chapter 7. Model Categories 107124
- 7.1. Model categories 108125
- 7.2. Lifting and the retract argument 110127
- 7.3. Homotopy 115132
- 7.4. Homotopy as an equivalence relation 119136
- 7.5. The classical homotopy category 122139
- 7.6. Relative homotopy and fiberwise homotopy 125142
- 7.7. Weak equivalences 129146
- 7.8. Homotopy equivalence 130147
- 7.9. The equivalence relation generated by "weak equivalence'' 133150
- 7.10. Topological spaces and simplicial sets 134151

- Chapter 8. Fibrant and Cofibrant Approximations 137154
- Chapter 9. Simplicial Model Categories 159176
- Chapter 10. Ordinals, Cardinals, and Transfinite Composition 185202
- 10.1. Ordinals and cardinals 186203
- 10.2. Transfinite composition 188205
- 10.3. Transfinite composition and lifting in model categories 193210
- 10.4. Small objects 194211
- 10.5. The small object argument 196213
- 10.6. Subcomplexes of relative I-cell complexes 201218
- 10.7. Cell complexes of topological spaces 204221
- 10.8. Compactness 206223
- 10.9. Effective monomorphisms 208225

- Chapter 11. Cofibrantly Generated Model Categories 209226
- 11.1. Cofibrantly generated model categories 210227
- 11.2. Cofibrations in a cofibrantly generated model category 211228
- 11.3. Recognizing cofibrantly generated model categories 213230
- 11.4. Compactness 215232
- 11.5. Free cell complexes 217234
- 11.6. Diagrams in a cofibrantly generated model category 224241
- 11.7. Diagrams in a simplicial model category 225242
- 11.8. Overcategories and undercategories 226243
- 11.9. Extending diagrams 228245

- Chapter 12. Cellular Model Categories 231248
- Chapter 13. Proper Model Categories 239256
- Chapter 14. The Classifying Space of a Small Category 253270
- 14.1. The classifying space of a small category 254271
- 14.2. Cofinal functors 256273
- 14.3. Contractible classifying spaces 258275
- 14.4. Uniqueness of weak equivalences 260277
- 14.5. Categories of functors 263280
- 14.6. Cofibrant approximations and fibrant approximations 266283
- 14.7. Diagrams of undercategories and overcategories 268285
- 14.8. Free cell complexes of simplicial sets 271288

- Chapter 15. The Reedy Model Category Structure 277294
- 15.1. Reedy categories 278295
- 15.2. Diagrams indexed by a Reedy category 281298
- 15.3. The Reedy model category structure 288305
- 15.4. Quillen functors 294311
- 15.5. Products of Reedy categories 294311
- 15.6. Reedy diagrams in a cofibrantly generated model category 296313
- 15.7. Reedy diagrams in a cellular model category 302319
- 15.8. Bisimplicial sets 303320
- 15.9. Cosimplicial simplicial sets 305322
- 15.10. Cofibrant constants and fibrant constants 308325
- 15.11. The realization of a bisimplicial set 312329

- Chapter 16. Cosimplicial and Simplicial Resolutions 317334
- Chapter 17. Homotopy Function Complexes 347364
- 17.1. Left homotopy function complexes 349366
- 17.2. Right homotopy function complexes 350367
- 17.3. Two-sided homotopy function complexes 352369
- 17.4. Homotopy function complexes 354371
- 17.5. Functorial homotopy function complexes 357374
- 17.6. Homotopic maps of homotopy function complexes 362379
- 17.7. Homotopy classes of maps 365382
- 17.8. Homotopy orthogonal maps 367384
- 17.9. Sequential colimits 376393

- Chapter 18. Homotopy Limits in Simplicial Model Categories 379396
- 18.1. Homotopy colimits and homotopy limits 380397
- 18.2. The homotopy limit of a diagram of spaces 383400
- 18.3. Coends and ends 385402
- 18.4. Consequences of adjoint ness 389406
- 18.5. Homotopy invariance 394411
- 18.6. Simplicial objects and cosimplicial objects 395412
- 18.7. The Bousfield-Kan map 396413
- 18.8. Diagrams of pointed or unpointed spaces 398415
- 18.9. Diagrams of simplicial sets 400417

- Chapter 19. Homotopy Limits in General Model Categories 405422
- 19.1. Homotopy colimits and homotopy limits 405422
- 19.2. Coends and ends 407424
- 19.3. Consequences of adjoint ness 411428
- 19.4. Homotopy invariance 414431
- 19.5. Homotopy pullbacks and homotopy pushouts 416433
- 19.6. Homotopy cofinal functors 418435
- 19.7. The Reedy diagram homotopy lifting extension theorem 423440
- 19.8. Realizations and total objects 426443
- 19.9. Reedy cofibrant diagrams and Reedy fibrant diagrams 427444

- Index 429446 free
- Bibliography 455472

#### 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) cross-referenced, and precise in all its details. In short, it is an important reference for the subject.

-- Zentralblatt MATH