Parametrized homotopy theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2006]
|
| Series: | Mathematical surveys and monographs
Volume 132 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | ix, 441 Seiten Diagramme |
| ISBN: | 9780821839225 0821839225 |
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MARC
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Record in the Search Index
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| adam_text | MATHEMATICAL SURVEYS AND MONOGRAPHS VOLUME 132 PARAMETRIZED HOMOTOPY
THEORY J. P. MAY J. SIGURDSSON AMERICAN MATHEMATICAL SOCIETY CONTENTS
PROLOGUE 1 PART I. POINT-SET TOPOLOGY, CHANGE FUNCTORS, AND PROPER
ACTIONS 11 INTRODUCTION 13 CHAPTER 1. THE POINT-SET TOPOLOGY OF
PARAMETRIZED SPACES 15 INTRODUCTION 15 1.1. CONVENIENT CATEGORIES OF
TOPOLOGICAL SPACES 15 1.2. TOPOLOGICALLY BICOMPLETE CATEGORIES AND
EX-OBJECTS 16 1.3. CONVENIENT CATEGORIES OF EX-SPACES 19 1.4. CONVENIENT
CATEGORIES OF EX-G-SPACES 22 1.5. PHILOSOPHICAL COMMENTS ON THE
POINT-SET TOPOLOGY 23 1.6. TECHNICAL POINT-SET TOPOLOGICAL LEMMAS 24
1.7. APPENDIX: NONASSOCIATIVITY OF SMASH PRODUCTS IN C^OP, 26 CHAPTER 2.
CHANGE FUNCTORS AND COMPATIBILITY RELATIONS 29 INTRODUCTION 29 2.1. THE
BASE CHANGE FUNCTORS/I,/*, AND/* 30 2.2. COMPATIBILITY RELATIONS 32 2.3.
CHANGE OF GROUP AND RESTRICTION TO FIBERS 35 2.4. NORMAL SUBGROUPS AND
QUOTIENT GROUPS 37 2.5. THE CLOSED SYMMETRIC MONOIDAL CATEGORY OF
RETRACTS 40 CHAPTER 3. PROPER ACTIONS, EQUIVARIANT BUNDLES AND
FIBRATIONS 43 INTRODUCTION 43 3.1. PROPER ACTIONS OF LOCALLY COMPACT
GROUPS 43 3.2. PROPER ACTIONS AND EQUIVARIANT BUNDLES 47 3.3. THE BUNDLE
CONSTRUCTION 48 3.4. SPACES OF THE HOMOTOPY TYPES OF G-CW COMPLEXES 51
3.5. SOME CLASSICAL THEOREMS ABOUT FIBRATIONS 53 3.6. QUASIFIBRATIONS 54
PART II. MODEL CATEGORIES AND PARAMETRIZED SPACES 57 INTRODUCTION 59
CHAPTER 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES 61 INTRODUCTION 61
4.1. MODEL THEORETIC PHILOSOPHY: H, Q, AND M-MODEL STRUCTURES 62 4.2.
STRONG HUREWICZ COFIBRATIONS AND FIBRATIONS 63 VI CONTENTS 4.3. TOWARDS
CLASSICAL MODEL STRUCTURES IN TOPOLOGICAL CATEGORIES 66 4.4. CLASSICAL
MODEL STRUCTURES IN GENERAL AND IN JFF AND % 69 4.5. COMPACTLY GENERATED
G-TYPE MODEL STRUCTURES 72 CHAPTER 5. WELL-GROUNDED TOPOLOGICAL MODEL
CATEGORIES 77 INTRODUCTION 77 5.1. OVER AND UNDER MODEL STRUCTURES 78
5.2. THE SPECIALIZATION TO OVER AND UNDER CATEGORIES OF SPACES 82 5.3.
WELL-GROUNDED TOPOLOGICALLY BICOMPLETE CATEGORIES 85 5.4. WELL-GROUNDED
CATEGORIES OF WEAK EQUIVALENCES 87 5.5. WELL-GROUNDED COMPACTLY
GENERATED MODEL STRUCTURES 90 5.6. PROPERTIES OF WELL-GROUNDED MODEL
CATEGORIES 91 CHAPTER 6. THE G/-MODEL STRUCTURE ON JIF B 97 INTRODUCTION
97 6.1. SOME OF THE DANGERS IN THE PARAMETRIZED WORLD 98 6.2. THE QF
MODEL STRUCTURE ON THE CATEGORY JF/B 100 6.3. STATEMENTS AND PROOFS OF
THE THICKENING LEMMAS 102 6.4. THE COMPATIBILITY CONDITION FOR THE
G/-MODEL STRUCTURE 105 6.5. THE QUASIFIBRATION AND RIGHT PROPERNESS
PROPERTIES 107 CHAPTER 7. EQUIVARIANT QF-TYPE MODEL STRUCTURES 109
INTRODUCTION 109 7.1. FAMILIES AND NON-COMPACT LIE GROUPS 110 7.2. THE
EQUIVARIANT Q AND (//-MODEL STRUCTURES 111 7.3. EXTERNAL SMASH PRODUCT
AND BASE CHANGE ADJUNCTIONS 115 7.4. CHANGE OF GROUP ADJUNCTIONS 118
7.5. FIBER ADJUNCTIONS AND BROWN REPRESENTABILITY 122 CHAPTER 8.
EX-FIBRATIONS AND EX-QUASIFIBRATIONS 127 8.1. EX-FIBRATIONS 128 8.2.
PRESERVATION PROPERTIES OF EX-FIBRATIONS 129 8.3. THE EX-FIBRANT
APPROXIMATION FUNCTOR 131 8.4. PRESERVATION PROPERTIES OF EX-FIBRANT
APPROXIMATION 133 8.5. QUASIFIBRANT EX-SPACES AND EX-QUASIFIBRATIONS 135
CHAPTER 9. THE EQUIVALENCE BETWEEN HO GX B AND HGW B 137 INTRODUCTION
137 9.1. THE EQUIVALENCE OF HO GJ(T B AND HGW B 138 9.2. DERIVED
FUNCTORS ON HOMOTOPY CATEGORIES 139 9.3. THE FUNCTORS /* AND F B ON
HOMOTOPY CATEGORIES 140 9.4. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS
AND BASE CHANGE 142 PART III. PARAMETRIZED EQUIVARIANT STABLE HOMOTOPY
THEORY 147 INTRODUCTION 149 CHAPTER 10. ENRICHED CATEGORIES AND
G-CATEGORIES 151 INTRODUCTION 151 10.1. PARAMETRIZED ENRICHED CATEGORIES
151 10.2. EQUIVARIANT PARAMETRIZED ENRICHED CATEGORIES 153 CONTENTS VII
10.3. G-TOPOLOGICAL MODEL G-CATEGORIES 155 CHAPTER 11. THE CATEGORY OF
ORTHOGONAL G-SPECTRA OVER B 159 INTRODUCTION 159 11.1. THE CATEGORY OF
J^C-SPACES OVER B 159 11.2. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B
163 11.3. ORTHOGONAL G-SPECTRA AS DIAGRAM EX-G-SPACES 166 11.4. THE BASE
CHANGE FUNCTORS/*,/T, AND/* 167 11.5. CHANGE OF GROUPS AND RESTRICTION
TO FIBERS 170 11.6. SOME PROBLEMS CONCERNING NON-COMPACT LIE GROUPS 172
CHAPTER 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA 175 INTRODUCTION
175 12.1. THE LEVEL MODEL STRUCTURE ON G^ B 176 12.2. SOME QUILLEN
ADJOINT PAIRS RELATING LEVEL MODEL STRUCTURES 179 12.3. THE STABLE MODEL
STRUCTURE ON G5?B 180 12.4. COFIBER SEQUENCES AND ^-ISOMORPHISMS 183
12.5. PROOFS OF THE MODEL AXIOMS 186 12.6. SOME QUILLEN ADJOINT PAIRS
RELATING STABLE MODEL STRUCTURES 190 CHAPTER 13. ADJUNCTIONS AND
COMPATIBILITY RELATIONS 195 INTRODUCTION 195 13.1. BROWN
REPRESENTABILITY AND THE FUNCTORS /* AND F B 196 13.2. THE CATEGORY GS B
OF EXCELLENT PRESPECTRA OVER B 200 13.3. THE LEVEL EX-FIBRANT
APPROXIMATION FUNCTOR P ON PRESPECTRA 202 13.4. THE AUXILIARY
APPROXIMATION FUNCTORS K AND E 205 13.5. THE EQUIVALENCE BETWEEN HO G&
B AND HGS B 207 13.6. DERIVED FUNCTORS ON HOMOTOPY CATEGORIES 208 13.7.
COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 209 CHAPTER
14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS 215
INTRODUCTION 215 14.1. PARAMETRIZED MODULE G-SPECTRA 215 14.2. CHANGE OF
UNIVERSE 219 14.3. RESTRICTION TO SUBGROUPS 223 14.4. NORMAL SUBGROUPS
AND QUOTIENT GROUPS 226 PART IV. PARAMETRIZED DUALITY THEORY 229
INTRODUCTION 231 CHAPTER 15. FIBERWISE DUALITY AND TRANSFER MAPS 233
INTRODUCTION 233 15.1. THE FIBERWISE DUALITY THEOREM 234 15.2. DUALITY
AND TRACE MAPS IN SYMMETRIC MONOIDAL CATEGORIES 236 15.3. TRANSFER MAPS
OF HUREWICZ FIBRATIONS 238 15.4. THE BUNDLE CONSTRUCTION ON PARAMETRIZED
SPECTRA 240 15.5. II-FREE PARAMETRIZED F-SPECTRA 242 15.6. THE FIBERWISE
TRANSFER FOR (II; F)-BUNDLES 244 CHAPTER 16. CLOSED SYMMETRIC
BICATEGORIES 247 VIII CONTENTS INTRODUCTION 247 16.1. RECOLLECTIONS
ABOUT BICATEGORIES / 248 16.2. THE DEFINITION OF SYMMETRIC BICATEGORIES
Y 249 16.3. THE DEFINITION OF CLOSED SYMMETRIC BICATEGORIES 252 16.4.
DUALITY IN CLOSED SYMMETRIC BICATEGORIES 255 16.5. COMPOSITES AND
NATURALITY OF DUALITIES 259 16.6. A QUICK REVIEW OF TRIANGULATED
CATEGORIES 261 16.7. COMPATIBLY TRIANGULATED SYMMETRIC BICATEGORIES 262
16.8. DUALITY IN TRIANGULATED SYMMETRIC BICATEGORIES 266 CHAPTER 17. THE
CLOSED SYMMETRIC BICATEGORY OF PARAMETRIZED SPECTRA 269 INTRODUCTION 269
17.1. THE DEFINITION OF THE BICATEGORY SX 269 17.2. BASE CHANGE SPECTRA
273 17.3. DUALITY OF BASE CHANGE SPECTRA 277 17.4. USING SX TO ENCODE
RELATIONS BETWEEN HOG^ S AND HOG^ 278 17.5. SKETCH PROOFS OF THE
COMPATIBLE TRIANGULATION AXIOMS 280 CHAPTER 18. COSTENOBLE-WANER DUALITY
285 INTRODUCTION 285 18.1. THE TWO NOTIONS OF DUALITY IN HO G5F B 286
18.2. COSTENOBLE-WANER DUALIZABILITY OF FINITE CELL SPECTRA 288 18.3.
COSTENOBLE-WANER F-DUALITY 290 18.4. PRELIMINARIES ON UNREDUCED RELATIVE
MAPPING CONES 292 18.5. F-DUALITY OF G-ENRS 295 18.6. PARAMETRIZED
ATIYAH DUALITY FOR CLOSED MANIFOLDS 296 18.7. PARAMETRIZED ATIYAH
DUALITY FOR MANIFOLDS WITH BOUNDARY 300 18.8. THE PROOF OF THE
COSTENOBLE-WANER DUALITY THEOREM 302 CHAPTER 19. FIBERWISE
COSTENOBLE-WANER DUALITY 311 INTRODUCTION 311 19.1. COSTENOBLE-WANER
DUALITY AND HOMOTOPICAL POINCARE DUALITY 312 19.2. THE BICATEGORIES SX B
314 19.3. COMPARISONS OF BICATEGORIES 316 19.4. THE BUNDLE CONSTRUCTION
PSEUDO-FUNCTOR 319 19.5. THE FIBERWISE COSTENOBLE-WANER DUALITY THEOREM
320 19.6. FIBERWISE POINCARE DUALITY 324 19.7. THE ADAMS ISOMORPHISM 326
19.8. SOME BACKGROUND AND COMPARISONS 328 PART V. HOMOLOGY AND
COHOMOLOGY, THORN SPECTRA, AND ADDENDA 333 INTRODUCTION 335 CHAPTER 20.
PARAMETRIZED HOMOLOGY AND COHOMOLOGY THEORIES 337 INTRODUCTION 337 20.1.
AXIOMS FOR PARAMETRIZED HOMOLOGY AND COHOMOLOGY THEORIES 338 20.2.
REPRESENTED HOMOLOGY AND COHOMOLOGY THEORIES 341 20.3. COEFFICIENT
SYSTEMS AND RESTRICTION MAPS 343 20.4. THE SERRE SPECTRAL SEQUENCE 344
CONTENTS IX 20.5. POINCARE DUALITY AND THE THORN ISOMORPHISM 347 20.6.
RELATIVE POINCARE DUALITY 350 20.7. PRODUCTS IN PARAMETRIZED HOMOLOGY
AND COHOMOLOGY 350 20.8. THE REPRESENTABILITY OF HOMOLOGY THEORIES 353
CHAPTER 21. EQUIVARIANT PARAMETRIZED HOMOLOGY AND COHOMOLOGY 357
INTRODUCTION 357 21.1. EQUIVARIANT HOMOLOGY AND COHOMOLOGY THEORIES 358
21.2. REPRESENTED EQUIVARIANT THEORIES 360 21.3. CHANGE OF BASE AND
EQUIVARIANT COFFICIENT SYSTEMS 361 21.4. DUALITY THEOREMS AND
ORIENTATIONS 363 21.5. PRODUCTS AND THE REPRESENTABILITY OF HOMOLOGY 366
21.6. FIBERWISE PARAMETRIZED HOMOLOGY AND COHOMOLOGY 367 21.7. FIBERWISE
POINCARE DUALITY AND ORIENTATIONS 369 CHAPTER 22. TWISTED THEORIES AND
SPECTRAL SEQUENCES 373 INTRODUCTION 373 22.1. TWISTED HOMOLOGY AND
COHOMOLOGY THEORIES 374 22.2. AUTOMORPHISM MONOIDS OF SPECTRA AND GL (K)
375 22.3. TWISTED IF-THEORY 378 22.4. THE SIMPLICIAL SPECTRAL SEQUENCE
380 22.5. CECH TYPE SPECTRAL SEQUENCES 384 22.6. THE TWISTED
ROTHENBERG-STEENROD SPECTRAL SEQUENCE 386 22.7. THE PARAMETRIZED
KIINNETH SPECTRAL SEQUENCE 388 CHAPTER 23. PARAMETRIZED FSP S AND
GENERALIZED THORN SPECTRA 393 INTRODUCTION 393 23.1. ^-FUNCTORS WITH
PRODUCTS IN SYMMETRIC MONOIDAL CATEGORIES 395 23.2. THE SPECIALIZATION
OF F-FP S TO SPACES AND EX-SPACES 397 23.3. GROUP, MONOID, AND MODULE
FCP S; EXAMPLES 399 23.4. THE TWO-SIDED BAR CONSTRUCTION ON FCP S 402
23.5. EXAMPLES: ITERATED THORN SPECTRA 403 23.6. / C -FCP S AND
JZF-SPACES 405 23.7. UNIVERSAL SPHERICAL FIBRATION SPECTRA 407 23.8.
SOME HISTORICAL BACKGROUND 408 CHAPTER 24. EPILOGUE: CELLULAR PHILOSOPHY
AND ALTERNATIVE APPROACHES 411 INTRODUCTION 411 24.1. CW SPACES OVER B
412 24.2. CW SPECTRA AND STABLE HOMOTOPY CATEGORIES 415 24.3. STRUCTURED
SPECTRA AND WELL-GROUNDED MODEL CATEGORIES 418 24.4. THE STABLE CATEGORY
OF PARAMETRIZED SPECTRA 420 24.5. TOWARDS PARAMETRIZED 5 3-MODULES 423
BIBLIOGRAPHY 425 INDEX 433 INDEX OF NOTATION 439
|
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| id | DE-604.BV025506194 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:35:31Z |
| institution | BVB |
| isbn | 9780821839225 0821839225 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020117015 |
| oclc_num | 634913167 |
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| owner_facet | DE-11 DE-19 DE-BY-UBM DE-355 DE-BY-UBR |
| physical | ix, 441 Seiten Diagramme |
| psigel | ebook |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | May, Jon Peter 1939- Verfasser (DE-588)122391810 aut Parametrized homotopy theory J. P. May ; J. Sigurdsson Providence, Rhode Island American Mathematical Society [2006] © 2006 ix, 441 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 132 Homotopietheorie (DE-588)4128142-1 gnd rswk-swf Homologietheorie (DE-588)4141714-8 gnd rswk-swf Homotopietheorie (DE-588)4128142-1 s DE-604 Homologietheorie (DE-588)4141714-8 s Jóhann Sigurðsson 1975- Verfasser (DE-588)138295166 aut Erscheint auch als Online-Ausgabe 978-1-4704-1359-0 Mathematical surveys and monographs Volume 132 (DE-604)BV000018014 132 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020117015&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | May, Jon Peter 1939- Jóhann Sigurðsson 1975- Parametrized homotopy theory Mathematical surveys and monographs Homotopietheorie (DE-588)4128142-1 gnd Homologietheorie (DE-588)4141714-8 gnd |
| subject_GND | (DE-588)4128142-1 (DE-588)4141714-8 |
| title | Parametrized homotopy theory |
| title_auth | Parametrized homotopy theory |
| title_exact_search | Parametrized homotopy theory |
| title_full | Parametrized homotopy theory J. P. May ; J. Sigurdsson |
| title_fullStr | Parametrized homotopy theory J. P. May ; J. Sigurdsson |
| title_full_unstemmed | Parametrized homotopy theory J. P. May ; J. Sigurdsson |
| title_short | Parametrized homotopy theory |
| title_sort | parametrized homotopy theory |
| topic | Homotopietheorie (DE-588)4128142-1 gnd Homologietheorie (DE-588)4141714-8 gnd |
| topic_facet | Homotopietheorie Homologietheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020117015&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT mayjonpeter parametrizedhomotopytheory AT johannsigurðsson parametrizedhomotopytheory |