Homotopy theory of higher categories: [from segal categories to n-categories and beyond]
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2012
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | New Mathematical Monographs
19 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Enth. Literaturverz. und Index |
| Beschreibung: | XVIII, 634 S. graph. Darst. |
| ISBN: | 9780521516952 |
Internformat
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| 245 | 1 | 0 | |a Homotopy theory of higher categories |b [from segal categories to n-categories and beyond] |c Carlos Simpson |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2012 | |
| 300 | |a XVIII, 634 S. |b graph. Darst. | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a New Mathematical Monographs |v 19 | |
| 500 | |a Enth. Literaturverz. und Index | ||
| 650 | 4 | |a Homotopy theory | |
| 650 | 4 | |a Categories (Mathematics) | |
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Datensatz im Suchindex
| _version_ | 1804148823954554880 |
|---|---|
| adam_text | Titel: Homotopy theory of higher categories
Autor: Simpson, Carlos
Jahr: 2012
Contents
Preface page xi
Acknowledgements xvii
PART I HIGHER CATEGORIES 1
1 History and motivation 3
2 Strict /»-categories 21
2.1 Godement relations: the Eckmann-Hilton argument 23
2.2 Strict n-groupoids 25
2.3 The need for weak composition 38
2.4 Realization functors 39
2.5 n-groupoids with one object 40
2.6 The case of the standard realization 41
2.7 Nonexistence of strict 3-groupoids of 3-type S2 43
3 Fundamental elements of «-categories 51
3.1 A globular theory 51
3.2 Identities 54
3.3 Composition, equivalence and truncation 54
3.4 Enriched categories 57
3.5 The (/1 + l)-category of n-categories 58
3.6 Poincaré n-groupoids 60
3.7 Interiors 61
3.8 The case« = oo 62
4 Operadk approaches 65
4.1 May s delooping machine 65
4.2 Baez-Dolan s definition 66
4.3 Batanin s definition 69
vi Contents
4.4 Trimble s deiinition and Cheng s comparison 73
4.5 Weak units 75
4.6 Other notions 78
5 Simplicial approaches 81
5.1 Strict simplicial categories 81
5.2 Segal s delooping machine 83
5.3 Segal categories 86
5.4 Rezk categories 91
5.5 Quasicategories 93
5.6 Going between Segal categories and n-categories 96
6 Weak enrichment over a cartesian model category:
an introduction 98
6.1 Simplicial objects in M 98
6.2 Diagrams over A* 99
6.3 Hypotheses on M 100
6.4 Precategories 101
6.5 Unitality 102
6.6 Rectification of Ax-diagrams 104
6.7 Enforcing the Segal condition 105
6.8 Products, intervals and the model structure 107
PART II CATEGORICALPRELIMINARIES 109
Model categories 111
7.1 Lifting properties 112
7.2 Quillen s axioms 113
7.3 Left propemess 116
7.4 The Kan-Quillen model category of simplicial sets 119
7.5 Homotopy Uftings and extensions 121
7.6 Model structures on diagram categories 124
7.7 Cartesian model categories 129
7.8 Internal Hom 132
7.9 Enriched categories 135
Cell complexes in locally presentable categories 144
8.1 Locally presentable categories 146
8.2 The srnall object argument 151
8.3 More on cell complexes 154
8.4 Conbrantly generated, combinatorial and tractable
model categories 168
10
11
12
Contents vii
8.5 Smith s recognition principle 171
8.6 Injective cofibrations in diagram categories 177
8.7 Pseudo-generating sets 183
Direct left Bousfield localization 192
9.1 Projection to a subcategory of local objects 192
9.2 Weak monadic projection 199
9.3 New weak equivalences 208
9.4 Invariance properties 211
9.5 New fibrations 216
9.6 Pushouts by new trivial cofibrations 218
9.7 The model category structure 220
9.8 Transfer along a left Quillen functor 222
PARTffl GENERATORS AND RELATIONS 225
Precategories 227
10.1 Enriched precategories with a fixed set of objects 227
10.2 The Segal conditions 229
10.3 Varying the set of objects 230
10.4 The category of precategories 232
10.5 Basic examples 233
10.6 Limits, colimits and local presentability 236
10.7 Interpretations as presheaf categories 242
Algebraic theories in model categories 251
11.1 Diagrams over the categories 6 (n) 252
11.2 Imposing the product condition 257
11.3 Algebraic diagram theories 263
11.4 Unitality 266
11.5 Unital algebraic diagram theories 272
11.6 The generation Operation 273
11.7 Reedy structures 274
Weak equivalences 275
12.1 Local weak equivalences 275
12.2 Unitalization adjunctions 280
12.3 The Reedy structure 282
12.4 Global weak equivalences 289
12.5 Categories enriched over ho( M) 292
12.6 Change of enrichment category 294
viii Contents
13 Cofibrations 297
13.1 Skeleta and coskeleta 297
13.2 Some natural precategories 302
13.3 Projective cofibrations 304
13.4 Injective cofibrations 307
13.5 A pushout expression for the skeleta 308
13.6 Reedy cofibrations 310
13.7 Relationship between the classes of cofibrations 323
14 Calculus of generators and relations 326
14.1 The T precategories 326
14.2 Some trivial cofibrations 329
14.3 Pushout by isotrivial cofibrations 332
14.4 An elementary generation step Gen 340
14.5 Fixing the fibrant condition locally 343
14.6 Combining generation Steps 344
14.7 Functoriality of the generation process 345
14.8 Example: generators and relations for 1-categories 347
15 Generators and relations for Segal categories 350
15.1 Segal categories 350
15.2 The Poincare-Segal groupoid 352
15.3 Looping and delooping 355
15.4 The calculus 359
15.5 Computing the loop space 370
15.6 Example: jr3(S2) 378
PARTIV THE MODEL STRUCTURE 383
16 Sequentially free precategories 385
16.1 Imposing the Segal condition on T 385
16.2 Sequentially free precategories in general 387
17 Products 397
17.1 Products of sequentially free precategories 397
17.2 Products of general precategories 408
17.3 The role of unitality, degeneracies and higher coherences 416
17.4 Why we can t truncate A. 419
18 Intervab 421
18.1 Contractible objects and intervals in M 422
18.2 Intervals for ^-enriched precategories 424
Contents ix
19
20
21
22
18.3 The versahty property 429
18.4 Contractibility of intervals for JC -precategories 432
18.5 Construction of a left Quillen functor K ?*¦ M 433
18.6 Contractibility in general 435
18.7 Pushout of trivial cofibrations 437
18.8 A versahty property 442
The model category of M -enriched precategories 444
19.1 A Standard factorization 444
19.2 The model structures 446
19.3 The cartesian property 449
19.4 Properties of fibrant objects 450
19.5 The model category of strict M -enriched categories 450
PARTV HIGHER CATEGORY THEORY 453
Iterated higher categories 455
20.1 Initialization 456
20.2 Notation for n-categories 457
20.3 Truncation and equivalences 466
20.4 Homotopy types and higher groupoids 469
20.5 The (n + l)-category nCAT 477
Higher categorical techniques 480
21.1 The opposite category 481
21.2 Equivalent objects 481
21.3 Homotopies and the homotopy 2-category 484
21.4 Constructions with T 489
21.5 Acyclicity of inversion 495
21.6 Localization and interior 499
21.7 Limits 506
21.8 Coümits 510
21.9 Invariance properties 511
21.10 Limits of diagrams 516
Iimits of weak enriched categories 527
22.1 Cartesian families 528
22.2 TiieYonedaembeddings 531
22.3 Universe considerations 536
22.4 Diagrams in quasifibrant precategories 538
22.5 Extension properties 543
22.6 Limits of weak enriched categories 553
x Contents
22.7 Cardinality 563
22.8 Splitting idempotents 567
22.9 Colimits of weak enriched categories 576
22.10 Fiber products and amalgamated sums 592
23 Stabilization 596
23.1 Minimal dimension 598
23.2 The stabilization hypothesis 608
23.3 Suspension and free monoidal categories 611
Epilogue 616
References 618
Index 630
|
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| author | Simpson, Carlos 1962- |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512.62 |
| dewey-search | 512.62 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| indexdate | 2024-07-10T00:13:16Z |
| institution | BVB |
| isbn | 9780521516952 |
| language | English |
| lccn | 2011026520 |
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| spelling | Simpson, Carlos 1962- Verfasser (DE-588)1017429251 aut Homotopy theory of higher categories [from segal categories to n-categories and beyond] Carlos Simpson 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2012 XVIII, 634 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier New Mathematical Monographs 19 Enth. Literaturverz. und Index Homotopy theory Categories (Mathematics) Homotopietheorie (DE-588)4128142-1 gnd rswk-swf Kategorientheorie (DE-588)4120552-2 gnd rswk-swf Homotopietheorie (DE-588)4128142-1 s Kategorientheorie (DE-588)4120552-2 s DE-604 New Mathematical Monographs 19 (DE-604)BV035420183 19 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024738594&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Simpson, Carlos 1962- Homotopy theory of higher categories [from segal categories to n-categories and beyond] New Mathematical Monographs Homotopy theory Categories (Mathematics) Homotopietheorie (DE-588)4128142-1 gnd Kategorientheorie (DE-588)4120552-2 gnd |
| subject_GND | (DE-588)4128142-1 (DE-588)4120552-2 |
| title | Homotopy theory of higher categories [from segal categories to n-categories and beyond] |
| title_auth | Homotopy theory of higher categories [from segal categories to n-categories and beyond] |
| title_exact_search | Homotopy theory of higher categories [from segal categories to n-categories and beyond] |
| title_full | Homotopy theory of higher categories [from segal categories to n-categories and beyond] Carlos Simpson |
| title_fullStr | Homotopy theory of higher categories [from segal categories to n-categories and beyond] Carlos Simpson |
| title_full_unstemmed | Homotopy theory of higher categories [from segal categories to n-categories and beyond] Carlos Simpson |
| title_short | Homotopy theory of higher categories |
| title_sort | homotopy theory of higher categories from segal categories to n categories and beyond |
| title_sub | [from segal categories to n-categories and beyond] |
| topic | Homotopy theory Categories (Mathematics) Homotopietheorie (DE-588)4128142-1 gnd Kategorientheorie (DE-588)4120552-2 gnd |
| topic_facet | Homotopy theory Categories (Mathematics) Homotopietheorie Kategorientheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024738594&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT simpsoncarlos homotopytheoryofhighercategoriesfromsegalcategoriestoncategoriesandbeyond |