Categorical structure of closure operators: with applications to topology, algebra and discrete mathematics
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
1995
|
| Schriftenreihe: | Mathematics and its applications
346 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 356 S. graph. Darst. |
| ISBN: | 0792337727 |
Internformat
MARC
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| 245 | 1 | 0 | |a Categorical structure of closure operators |b with applications to topology, algebra and discrete mathematics |c by D. Dikranjan and W. Tholen |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1995 | |
| 300 | |a XVII, 356 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 346 | |
| 650 | 7 | |a Catégories (mathématiques) |2 ram | |
| 650 | 4 | |a Correspondance Pumplün-Röhr | |
| 650 | 4 | |a Lemme Frolik | |
| 650 | 4 | |a Opérateur fermeture | |
| 650 | 4 | |a Réflecteur Barr | |
| 650 | 4 | |a Théorème Clementino | |
| 650 | 4 | |a Categories (Mathematics) | |
| 650 | 4 | |a Closure operators | |
| 650 | 0 | 7 | |a Kategorientheorie |0 (DE-588)4120552-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Abgeschlossener linearer Operator |0 (DE-588)4141055-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Abgeschlossener linearer Operator |0 (DE-588)4141055-5 |D s |
| 689 | 0 | 1 | |a Kategorientheorie |0 (DE-588)4120552-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Tholen, W. |e Verfasser |4 aut | |
| 830 | 0 | |a Mathematics and its applications |v 346 |w (DE-604)BV008163334 |9 346 | |
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Datensatz im Suchindex
| _version_ | 1804125069031505920 |
|---|---|
| adam_text | Table of Contents
Preface xi
Introduction xjjj
1. Preliminaries on Subobjects, Images, and Inverse Images
1.1 .M subobjects 1
1.2 Inverse images are A4 pullbacks 3
1.3 Review of pairs of adjoint maps 3
1.4 Adjointness of image and inverse image 5
1.5 The right M factorization of a morphism 6
1.6 Constructing images from right A4 factorizations 7
1.7 Stability properties of X subobjects 9
1.8 A4 subobjects of A4 subobjects 12
1.9 When the subobjects form a large complete lattice 14
1.10 The right .M factorization of a sink 16
1.11 The last word on least and last subobjects 19
Exercises 20
Notes 23
2. Basic Properties of Closure Operators
2.1 The categorical setting 24
2.2 The local definition of closure operator 25
2.3 Closed and dense subobjects 26
2.4 Idempotent and weakly hereditary closure operators 27
2.5 Minimal and hereditary closure operators 30
2.6 Grounded and additive closure operators 34
2.7 Productive closure operators 36
2.8 Restriction of closure operators to full subcategories 38
Exercises 39
Notes 43
3. Examples of Closure Operators
3.1 Kuratowski closure operator, Cech closure operator 44
3.2 Filter convergence spaces and Katetov closure operator 45
3.3 Sequential closure, 6 closure, ^ closure, E closure 47
3.4 Preradicals of i? modules and Abelian groups 51
3.5 Groups, rings and fields 55
3.6 Graphs and partially ordered sets 56
3.7 Directed complete posets and Scott closure 61
Exercises 64
Notes 71
4. Operations on Closure Operators
4.1 The lattice structure of all closure operators 72
4.2 Composition of closure operators 73
4.3 Cocomposition of closure operators 75
4.4 Closedness and density for (co)composites 78
4.5 Properties stable under meet or join 79
4.6 Idempotent hull and weakly hereditary core 81
4.7 Indiscrete operator, proper closure operators 85
4.8 Additive core 88
4.9 Fully additive core 90
4.10 Minimal core and hereditary hull 94
4.11 Productivity of idempotent closure operators 98
Exercises 102
Notes 108
5. Closure Operators, Functors, Factorization Systems
5.1 Pointed endofunctors and prereflections 109
5.2 Closure operators are prereflections 112
5.3 Factorization systems 116
5.4 Recognizing classes of C dense and C closed subobjects 120
5.5 Closure operators versus .M preradicals 125
5.6 .M preradicals versus £ prereflections 130
5.7 (C, £ ) continuous functors 135
5.8 Lifting closure operators along Ai fibrations 140
5.9 Application to topological groups 144
5.10 Closure operators and CS valued functors 147
5.11 Closure structured categories, uniform spaces 151
5.12 Pointed modifications of closure operators 154
5.13 Closure operators and adjoint functors 158
5.14 External closure operators 164
Exercises 168
Notes 176
6. Regular Closure Operators
6.1 .4 epimorphisms and ,4 regular monomorphisms 177
6.2 .4 epi closure and .4 regular closure 180
6.3 Computing the .4 regular closure for reflective A 184
6.4 The magic cube 187
6.5 Frolik s Lemma 192
6.6 The strong modification of a closure operator 195
6.7 Regular closure in pointed and additive categories 198
6.8 Clementino s Theorem 203
6.9 Regular closure for topological spaces 209
6.10 Pointed topological spaces 213
Exercises 219
Notes 224
7. Subcategories Defined by Closure Operators
7.1 The Salbany correspondence 225
7.2 Two diagonal theorems 228
7.3 Essentially equivalent closure operators 231
7.4 Regular hull and essentially strong closure operators 232
7.5 Characterization of additive regular closure operators 235
7.6 The Pumplun Rohrl correspondence 238
7.7 The maximal epi preserving extension 242
7.8 Nabla subcategories 247
7.9 Companions of A(C) in topological categories 250
Exercises 253
Notes 258
8. Epimorphisms and Cowellpoweredness
8.1 Categorical preliminaries 259
8.2 Reflectivity and cowellpoweredness 262
8.3 Epimorphisms in subcategories of Top a first summary 265
8.4 Projective closure operators and the categories Haus(7 ) 267
8.5 Cowellpowered subcategories of Top 273
8.6 Non cowellpowered subcategories of Top 275
8.7 Quasi uniform spaces 280
8.8 Topological groups 284
8.9 Epimorphisms and cowellpoweredness in algebra 287
8.10 The Frobenius closure operator of fields 292
Exercises 295
Notes 304
9. Dense Maps and Pullback Stability
9.1 Hereditariness revisited 305
9.2 Initial and open morphisms 307
9.3 Modal closure operators 311
9.4 Barr s reflector 314
9.5 Total density 320
Exercises 326
Notes 329
Bibliography 331
Index of Definitions 345
Notation Index 351
Tables of Results 357
|
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| dewey-full | 511.3/3 |
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/3 |
| dewey-search | 511.3/3 |
| dewey-sort | 3511.3 13 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010597296 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:55:41Z |
| institution | BVB |
| isbn | 0792337727 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007067414 |
| oclc_num | 33104039 |
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| owner | DE-12 DE-11 |
| owner_facet | DE-12 DE-11 |
| physical | XVII, 356 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Dikranjan, Dikran N. Verfasser aut Categorical structure of closure operators with applications to topology, algebra and discrete mathematics by D. Dikranjan and W. Tholen Dordrecht [u.a.] Kluwer 1995 XVII, 356 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 346 Catégories (mathématiques) ram Correspondance Pumplün-Röhr Lemme Frolik Opérateur fermeture Réflecteur Barr Théorème Clementino Categories (Mathematics) Closure operators Kategorientheorie (DE-588)4120552-2 gnd rswk-swf Abgeschlossener linearer Operator (DE-588)4141055-5 gnd rswk-swf Abgeschlossener linearer Operator (DE-588)4141055-5 s Kategorientheorie (DE-588)4120552-2 s DE-604 Tholen, W. Verfasser aut Mathematics and its applications 346 (DE-604)BV008163334 346 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007067414&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dikranjan, Dikran N. Tholen, W. Categorical structure of closure operators with applications to topology, algebra and discrete mathematics Mathematics and its applications Catégories (mathématiques) ram Correspondance Pumplün-Röhr Lemme Frolik Opérateur fermeture Réflecteur Barr Théorème Clementino Categories (Mathematics) Closure operators Kategorientheorie (DE-588)4120552-2 gnd Abgeschlossener linearer Operator (DE-588)4141055-5 gnd |
| subject_GND | (DE-588)4120552-2 (DE-588)4141055-5 |
| title | Categorical structure of closure operators with applications to topology, algebra and discrete mathematics |
| title_auth | Categorical structure of closure operators with applications to topology, algebra and discrete mathematics |
| title_exact_search | Categorical structure of closure operators with applications to topology, algebra and discrete mathematics |
| title_full | Categorical structure of closure operators with applications to topology, algebra and discrete mathematics by D. Dikranjan and W. Tholen |
| title_fullStr | Categorical structure of closure operators with applications to topology, algebra and discrete mathematics by D. Dikranjan and W. Tholen |
| title_full_unstemmed | Categorical structure of closure operators with applications to topology, algebra and discrete mathematics by D. Dikranjan and W. Tholen |
| title_short | Categorical structure of closure operators |
| title_sort | categorical structure of closure operators with applications to topology algebra and discrete mathematics |
| title_sub | with applications to topology, algebra and discrete mathematics |
| topic | Catégories (mathématiques) ram Correspondance Pumplün-Röhr Lemme Frolik Opérateur fermeture Réflecteur Barr Théorème Clementino Categories (Mathematics) Closure operators Kategorientheorie (DE-588)4120552-2 gnd Abgeschlossener linearer Operator (DE-588)4141055-5 gnd |
| topic_facet | Catégories (mathématiques) Correspondance Pumplün-Röhr Lemme Frolik Opérateur fermeture Réflecteur Barr Théorème Clementino Categories (Mathematics) Closure operators Kategorientheorie Abgeschlossener linearer Operator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007067414&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
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