Model categories and their localizations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2003]
|
| Series: | Mathematical surveys and monographs
volume 99 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 457 Seiten Illustrationen |
| ISBN: | 0821832794 9780821849170 |
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MARC
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| 245 | 1 | 0 | |a Model categories and their localizations |c Philip S. Hirschhorn |
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| 264 | 4 | |c © 2003 | |
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Record in the Search Index
| _version_ | 1804129603520823296 |
|---|---|
| adam_text | Contents
Introduction ix
Model categories and their homotopy categories ix
Localizing model category structures xi
Acknowledgments xv
Part 1. Localization of Model Category Structures 1
Summary of Part 1 3
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 / localization functor 16
1.4. Concise description of the/ 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 A{/} 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. C local objects and C 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
4.1. Existence of left Bousfield localizations 71
4.2. Horns on S and 5 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
V
vi CONTENTS
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. X colocal objects and ./^ cellular objects 90
Chapter 6. Fiberwise Localization 93
6.1. 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
Summary of Part 2 103
Chapter 7. Model Categories 107
7.1. Model categories 108
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
9.1. 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 Transfmite 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
CONTENTS vii
10.6. Subcomplexes of relative / 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
11.1. Cofibrantly generated model categories 210
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
12.1. 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
14.1. The classifying space of a small category 254
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
viii CONTENTS
15.10. Cofibrant constants and fibrant constants 308
15.11. The realization of a bisimplicial set 312
Chapter 16. Cosimplicial and Simplicial Resolutions 317
16.1. Resolutions 318
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
Index 429
Bibliography 455
|
| any_adam_object | 1 |
| author | Hirschhorn, Philip S. 1952- |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.55 |
| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T19:07:46Z |
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| isbn | 0821832794 9780821849170 |
| language | English |
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| spelling | Hirschhorn, Philip S. 1952- Verfasser (DE-588)140253068 aut Model categories and their localizations Philip S. Hirschhorn Providence, Rhode Island American Mathematical Society [2003] © 2003 xv, 457 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs volume 99 Homotopia larpcal Topologia algébrica larpcal Homotopy theory Model categories (Mathematics) Homotopietheorie (DE-588)4128142-1 gnd rswk-swf Modelltheorie (DE-588)4114617-7 gnd rswk-swf Modelltheorie (DE-588)4114617-7 s Homotopietheorie (DE-588)4128142-1 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1326-2 Mathematical surveys and monographs volume 99 (DE-604)BV000018014 99 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010036273&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hirschhorn, Philip S. 1952- Model categories and their localizations Mathematical surveys and monographs Homotopia larpcal Topologia algébrica larpcal Homotopy theory Model categories (Mathematics) Homotopietheorie (DE-588)4128142-1 gnd Modelltheorie (DE-588)4114617-7 gnd |
| subject_GND | (DE-588)4128142-1 (DE-588)4114617-7 |
| title | Model categories and their localizations |
| title_auth | Model categories and their localizations |
| title_exact_search | Model categories and their localizations |
| title_full | Model categories and their localizations Philip S. Hirschhorn |
| title_fullStr | Model categories and their localizations Philip S. Hirschhorn |
| title_full_unstemmed | Model categories and their localizations Philip S. Hirschhorn |
| title_short | Model categories and their localizations |
| title_sort | model categories and their localizations |
| topic | Homotopia larpcal Topologia algébrica larpcal Homotopy theory Model categories (Mathematics) Homotopietheorie (DE-588)4128142-1 gnd Modelltheorie (DE-588)4114617-7 gnd |
| topic_facet | Homotopia Topologia algébrica Homotopy theory Model categories (Mathematics) Homotopietheorie Modelltheorie |
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