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Model categories and their localizations

Author: Philip S Hirschhorn
Publisher: Providence, RI : American Mathematical Society, ©2003.
Series: Mathematical surveys and monographs, no. 99.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:

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. This book explains  Read more...

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Genre/Form: Electronic books
Additional Physical Format: Print version:
Hirschhorn, Philip S. (Philip Steven), 1952-
Model categories and their localizations.
Providence, RI : American Mathematical Society, ©2003
(DLC) 2002027794
(OCoLC)50802638
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Philip S Hirschhorn
ISBN: 9781470413262 1470413264 9780821849170 0821849174
OCLC Number: 870332554
Description: 1 online resource (xv, 457 pages) : illustrations.
Contents: Model categories and their homotopy categories ix --
Localizing model category structures xi --
Part 1. Localization of Model Category Structures 1 --
Chapter 1. Local Spaces and Localization 5 --
1.1. Definitions of spaces and mapping spaces 5 --
1.2. Local spaces and localization 8 --
1.3. Constructing an f-localization functor 16 --
1.4. Concise description of the f-localization 20 --
1.5. Postnikov approximations 22 --
1.6. Topological spaces and simplicial sets 24 --
1.7. A continuous localization functor 29 --
1.8. Pointed and unpointed localization 31 --
Chapter 2. The Localization Model Category for Spaces 35 --
2.1. The Bousfield localization model category structure 35 --
2.2. Subcomplexes of relative [Lambda]{f}-cell complexes 37 --
2.3. The Bousfield-Smith cardinality argument 42 --
Chapter 3. Localization of Model Categories 47 --
3.1. Left localization and right localization 47 --
3.2. [characters not reproducible]-local objects and [characters not reproducible]-local equivalences 51 --
3.3. Bousfield localization 57 --
3.4. Bousfield localization and properness 65 --
3.5. Detecting equivalences 68 --
Chapter 4. Existence of Left Bousfield Localizations 71 --
4.2. Horns on S and S-local equivalences 73 --
4.3. A functorial localization 74 --
4.4. Localization of subcomplexes 76 --
4.5. The Bousfield-Smith cardinality argument 78 --
4.6. Proof of the main theorem 81 --
Chapter 5. Existence of Right Bousfield Localizations 83 --
5.1. Right Bousfield localization: Cellularization 83 --
5.2. Horns on K and K-colocal equivalences 85 --
5.3. K-colocal cofibrations 87 --
5.4. Proof of the main theorem 89 --
5.5. K-colocal objects and K-cellular objects 90 --
Chapter 6. Fiberwise Localization 93 --
6.2. The fiberwise local model category structure 95 --
6.3. Localizing the fiber 95 --
6.4. Uniqueness of the fiberwise localization 98 --
Part 2. Homotopy Theory in Model Categories 101 --
Chapter 7. Model Categories 107 --
7.2. Lifting and the retract argument 110 --
7.3. Homotopy 115 --
7.4. Homotopy as an equivalence relation 119 --
7.5. The classical homotopy category 122 --
7.6. Relative homotopy and fiberwise homotopy 125 --
7.7. Weak equivalences 129 --
7.8. Homotopy equivalences 130 --
7.9. The equivalence relation generated by "weak equivalence" 133 --
7.10. Topological spaces and simplicial sets 134 --
Chapter 8. Fibrant and Cofibrant Approximations 137 --
8.1. Fibrant and cofibrant approximations 138 --
8.2. Approximations and homotopic maps 144 --
8.3. The homotopy category of a model category 147 --
8.4. Derived functors 151 --
8.5. Quillen functors and total derived functors 153 --
Chapter 9. Simplicial Model Categories 159 --
9.2. Colimits and limits 163 --
9.3. Weak equivalences of function complexes 164 --
9.4. Homotopy lifting 167 --
9.5. Simplicial homotopy 170 --
9.6. Uniqueness of lifts 175 --
9.7. Detecting weak equivalences 177 --
9.8. Simplicial functors 179 --
Chapter 10. Ordinals, Cardinals, and Transfinite Composition 185 --
10.1. Ordinals and cardinals 186 --
10.2. Transfinite composition 188 --
10.3. Transfinite composition and lifting in model categories 193 --
10.4. Small objects 194 --
10.5. The small object argument 196 --
10.6. Subcomplexes of relative I-cell complexes 201 --
10.7. Cell complexes of topological spaces 204 --
10.8. Compactness 206 --
10.9. Effective monomorphisms 208 --
Chapter 11. Cofibrantly Generated Model Categories 209 --
11.2. Cofibrations in a cofibrantly generated model category 211 --
11.3. Recognizing cofibrantly generated model categories 213 --
11.4. Compactness 215 --
11.5. Free cell complexes 217 --
11.6. Diagrams in a cofibrantly generated model category 224 --
11.7. Diagrams in a simplicial model category 225 --
11.8. Overcategories and undercategories 226 --
11.9. Extending diagrams 228 --
Chapter 12. Cellular Model Categories 231 --
12.2. Subcomplexes in cellular model categories 232 --
12.3. Compactness in cellular model categories 234 --
12.4. Smallness in cellular model categories 235 --
12.5. Bounding the size of cell complexes 236 --
Chapter 13. Proper Model Categories 239 --
13.1. Properness 239 --
13.2. Properness and lifting 243 --
13.3. Homotopy pullbacks and homotopy fiber squares 244 --
13.4. Homotopy fibers 249 --
13.5. Homotopy pushouts and homotopy cofiber squares 250 --
Chapter 14. The Classifying Space of a Small Category 253 --
14.2. Cofinal functors 256 --
14.3. Contractible classifying spaces 258 --
14.4. Uniqueness of weak equivalences 260 --
14.5. Categories of functors 263 --
14.6. Cofibrant approximations and fibrant approximations 266 --
14.7. Diagrams of undercategories and overcategories 268 --
14.8. Free cell complexes of simplicial sets 271 --
Chapter 15. The Reedy Model Category Structure 277 --
15.1. Reedy categories 278 --
15.2. Diagrams indexed by a Reedy category 281 --
15.3. The Reedy model category structure 288 --
15.4. Quillen functors 294 --
15.5. Products of Reedy categories 294 --
15.6. Reedy diagrams in a cofibrantly generated model category 296 --
15.7. Reedy diagrams in a cellular model category 302 --
15.8. Bisimplicial sets 303 --
15.9. Cosimplicial simplicial sets 305 --
15.10. Cofibrant constants and fibrant constants 308 --
15.11. The realization of a bisimplicial set 312 --
Chapter 16. Cosimplicial and Simplicial Resolutions 317 --
16.2. Quillen functors and resolutions 323 --
16.3. Realizations 324 --
16.4. Adjointness 326 --
16.5. Homotopy lifting extension theorems 331 --
16.6. Frames 337 --
16.7. Reedy frames 342 --
Chapter 17. Homotopy Function Complexes 347 --
17.1. Left homotopy function complexes 349 --
17.2. Right homotopy function complexes 350 --
17.3. Two-sided homotopy function complexes 352 --
17.4. Homotopy function complexes 354 --
17.5. Functorial homotopy function complexes 357 --
17.6. Homotopic maps of homotopy function complexes 362 --
17.7. Homotopy classes of maps 365 --
17.8. Homotopy orthogonal maps 367 --
17.9. Sequential colimits 376 --
Chapter 18. Homotopy Limits in Simplicial Model Categories 379 --
18.1. Homotopy colimits and homotopy limits 380 --
18.2. The homotopy limit of a diagram of spaces 383 --
18.3. Coends and ends 385 --
18.4. Consequences of adjointness 389 --
18.5. Homotopy invariance 394 --
18.6. Simplicial objects and cosimplicial objects 395 --
18.7. The Bousfield-Kan map 396 --
18.8. Diagrams of pointed or unpointed spaces 398 --
18.9. Diagrams of simplicial sets 400 --
Chapter 19. Homotopy Limits in General Model Categories 405 --
19.1. Homotopy colimits and homotopy limits 405 --
19.2. Coends and ends 407 --
19.3. Consequences of adjointness 411 --
19.4. Homotopy invariance 414 --
19.5. Homotopy pullbacks and homotopy pushouts 416 --
19.6. Homotopy cofinal functors 418 --
19.7. The Reedy diagram homotopy lifting extension theorem 423 --
19.8. Realizations and total objects 426 --
19.9. Reedy cofibrant diagrams and Reedy fibrant diagrams 427.
Series Title: Mathematical surveys and monographs, no. 99.
Responsibility: Philip S. Hirschhorn.

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