A foundation for PROPs, algebras, and modules:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2015]
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| Series: | Mathematical surveys and monographs
volume 203 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Bibliografie Seite 303-305 |
| Physical Description: | xxxi, 311 Seiten Illustrationen |
| ISBN: | 9781470421977 |
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| 245 | 1 | 0 | |a A foundation for PROPs, algebras, and modules |c Donald Yau, Mark W. Johnson |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2015] | |
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Record in the Search Index
| _version_ | 1804175408246030336 |
|---|---|
| adam_text | Contents
Preface xv
Organizational Graphics xxii
Acknowledgments xxiii
List of Notations xxv
Part 1. Wheeled Graphs and Pasting Schemes 1
Chapter 1. Wheeled Graphs 3
1.1. Profiles 3
1.1.1. Colors 3
1.1.2. Profiles and Shuffles 3
1.1.3. Pairs of Profiles 5
1.2. Defining Wheeled Graphs 6
1.2.1. Basic Graphs 6
1.2.2. Structures on Graphs 9
1.3. Basic Examples of Wheeled Graphs 13
1.3.1. Corollas 13
1.3.2. Exceptional Edges 14
1.3.3. Exceptional Loops 14
1.3.4. Other Basic Graphs 15
1.4. Technical Variants of Wheeled Graphs 18
1.4.1. Pointed Graphs 18
1.4.2. Ordered Graphs 18
1.4.3. Multi-stage Graphs 19
1.4.4. Cyclic or Modular Graphs 19
Chapter 2. Special Sets of Graphs 21
2.1. Paths and Directed Paths 21
2.1.1. Paths 21
2.1.2. Directed Paths 22
2.2. Connected Graphs 24
2.2.1. Flag-Connected Graphs 24
2.2.2. Connected Graphs 24
2.3. Simply-Connected and Wheel-Free Graphs 26
2.3.1. Simply-Connected Graphs 26
2.3.2. Wheel-Free Graphs 26
2.4. Half-Graphs and Dioperadic Graphs 27
2.4.1. Half-Graphs 27
2.4.2. Basic Dioperadic Graphs 28
vii
viii
CONTENTS
2.5. Trees 30
2.5.1. Simple Trees 30
2.5.2. Level Trees 30
2.5.3. Special Trees 31
2.5.4. Wheeled and Truncated Trees 32
Chapter 3. Basic Operations on Wheeled Graphs 35
3.1. Relabeling 36
3.1.1. Input and Output Relabeling 36
3.1.2. Permuted Corollas 37
3.2. Disjoint Union 37
3.2.1. Disjoint Union of Wheeled Graphs 37
3.2.2. Disjoint Union of Corollas 38
3.2.3. Disjoint Union Decompositions of Wheeled Graphs 39
3.3. Grafting 39
3.3.1. Grafting of Ordinary Wheeled Graphs 39
3.3.2. Grafted Corollas 40
3.3.3. Partial Grafting 41
3.3.4. The Comp-i Operation 43
3.3.5. The j-comp-z Operation 44
3.4. Contraction 44
3.4.1. Contraction of Ordinary Wheeled Graphs 44
3.4.2. Contracted Corolla 45
Chapter 4. Graph Groupoids 47
4.1. Strict Isomorphisms 47
4.1.1. Strict Isomorphisms and Automorphisms 47
4.1.2. Decomposing the Strict Automorphism Group 50
4.2. Weak Isomorphisms 52
4.3. Graph Groupoids 54
4.3.1. Graph Groupoids Defined by Strict and Weak Isomorphisms 55
4.3.2. Examples of Graph Groupoids 55
Chapter 5. Graph Substitution 57
5.1. Graph Substitution in the Ordinary Case 57
5.2. Pre-Graphs and Pre-Substitution 59
5.2.1. Pre-Graphs 61
5.2.2. Pre-Substitution 61
5.2.3. Properties of Pre-substitution 63
5.3. Ambiguous Paths and Associated Graphs 64
5.3.1. Ambiguous Paths 64
5.3.2. The Associated Graph Construction 66
5.3.3. Compatibility of Associated Graphs with Pre-substitution 67
5.4. General Graph Substitution 70
5.4.1. Technical Variants of Graph Substitution 72
Chapter 6. Properties of Graph Substitution 75
6.1. Operations Coming from Graph Substitution 75
6.1.1. Characterizing Representable Graph Operations 76
6.1.2. Altering the Listing by Graph Substitution 76
CONTENTS ix
6.1.3. Disjoint Union from Graph Substitution 77
6.1.4. Grafting from Graph Substitution 77
6.1.5. Contraction from Graph Substitution 78
6.1.6. Shrinking Internal Edges Is Not a Representable Graph Operation 78
6.1.7. Input and Output Extensions 79
6.2. Properties Preserved by Graph Substitution 80
6.2.1. Connected Graphs 80
6.2.2. Wheel-free Graphs 81
6.2.3. Simply-Connected Graphs 82
6.3. Substitution Properties of Basic Operations 82
6.3.1. Iterated Relabelings 83
6.3.2. Union 83
6.3.3. Contraction 84
6.3.4. Grafting 87
6.4. Substitution Properties of Partial Grafting 89
6.4.1. Partial Grafting 89
6.4.2. Partial Grafting and Contraction 96
6.5. Substitution Properties for Trees and Basic Dioperadic Graphs 99
6.5.1. Simple Trees 100
6.5.2. Special Trees 101
6.5.3. Truncated and Contracted Trees 103
6.5.4. Basic Dioperadic Graphs 104
Chapter 7. Generators for Graphs 107
7.1. Graph Simplices and Generating Sets 109
7.1.1. Equivalent Graph Simplices 109
7.1.2. Relaxed Moves 110
7.1.3. Pointed Graph Simplices 111
7.1.4. Generating Set for a Graph Groupoid 111
7.2. Strong Generating Set for Wheeled Graphs 112
7.3. Strong Generating Set for Wheel-Free Graphs 115
7.4. Strong Generating Set for Level Trees 117
7.5. Strong Generating Set for Unit al Trees 119
7.6. Strong Generating Set for Wheeled Trees 120
7.6.1. Building from Level Trees to Wheeled Trees 120
7.6.2. Strong Generating Set for Wheeled Trees 122
7.7. Strong Generating Set for Simply-Connected Graphs 123
7.8. Strong Generating Set for Half-Graphs 124
7.9. Strong Generating Sets for Connected Wheel-Free Graphs 125
7.9.1. An Alternative Strong Generating Set for Connected Wheel-Free
Graphs 128
7.10. Strong Generating Set for Connected Wheeled Graphs 129
Chapter 8. Pasting Schemes 133
8.1. Definitions and First Examples 133
8.1.1. Definition of a Pasting Scheme 134
8.1.2. Partial Ordering on Pasting Schemes 134
8.1.3. Restriction on 5 135
8.1.4. First Examples of Pasting Schemes 135
X
CONTENTS
8.1.5. Unital Pasting Schemes 136
8.1.6. Key Examples of Pasting Schemes 136
8.1.7. Summary of Relationships Between Pasting Schemes 138
8.1.8. Virtual Pasting Schemes 138
8.1.9. Non-E Pasting Schemes 138
8.2. Free Product Decompositions of Pasting Schemes 139
8.2.1. Free Product 139
8.2.2. Generating Sets for Free Products 141
8.2.3. Separating Generating Sets for Free Products 141
8.3. Monogenic Pasting Schemes 144
Chapter 9. Well-Matched Pasting Schemes 147
9.1. Kontsevich Groupoid 147
9.1.1. Intersection 148
9.1.2. Orthogonality 148
9.1.3. Prime Graph Groupoids 149
9.1.4. Kontsevich Groupoid 149
9.2. Well-Matched Pasting Schemes 152
9.2.1. Well-Matched Examples 153
9.2.2. Examples That Are Not Well-Matched 154
Part 2. Generalized PROPs, Algebras, and Modules 157
Chapter 10. Generalized PROPs 159
10.1. Categorical Preliminaries 160
10.1.1. Monoidal Categories 160
10.1.2. Symmetric Monoidal Categories 162
10.1.3. Ordered and Unordered Tensor Products 164
10.1.4. Monads and Their Algebras 165
10.2. Pointed Extensions of Monads 169
10.2.1. Monad Replacement for Pointed Extensions of a Monad 171
10.3. Colored Objects, Bimodules, and Decorated Graphs 176
10.3.1. Diagram Categories 176
10.3.2. Colored Objects and Bimodules 177
10.3.3. Decorated Graphs 179
10.4. Generalized PROPs as Monadic Algebras 181
10.4.1. The Monad Associated to a Pasting Scheme 181
10.4.2. Generalized PROPs 184
10.4.3. Bi-equivariant Structure 185
10.4.4. Generalized PROPs for Unital Pasting Schemes 186
10.5. First Examples of Generalized PROPs 187
10.5.1. Unital Linear PROPs 188
10.5.2. Contraction PROPs 188
10.5.3. Horizontal and Vertical PROPs 188
10.5.4. Generalized PROPs over Monogenic Pasting Schemes 189
Chapter 11. Biased Characterizations of Generalized PROPs 191
11.1. Biased Definition Theorem 192
11.2. Biased Morphism Theorem 194
CONTENTS xi
11.3. Markl Non-Unital Operads as Tree-PROPs 195
11.3.1. Defining a Markl Non-Unital Operad 195
11.3.2. Interpreting the Axioms for a Markl Non-Unital Operad 196
11.3.3. Markl Non-Unital Operads Are Tree-PROPs 196
11.4. May Operads as UTree-PROPs 196
11.4.1. Defining a May Operad 196
11.4.2. Interpreting the Axioms for a May Operad 198
11.4.3. May Operads are UTree-PROPs 198
11.5. Dioperads as Gr^-PROPs 199
11.5.1. Defining a Dioperad 199
11.5.2. Interpreting the Axioms for a Dioperad 201
11.5.3. Dioperads are Gr^-PROPs 201
11.6. Half-PROPs as Gri-PROPs 202
11.6.1. Defining a Half-PROP 202
11.6.2. Interpreting the Axioms for a Half-PROP 203
11.6.3. Half-PROPs are Gr i-PROPs 203
2
11.7. Properads as GrJ-PROPs 203
11.7.1. Defining a Properad 203
11.7.2. An Alternate Definition of a Properad 207
11.7.3. Interpreting the Axioms for a Properad 208
11.7.4. Interpreting the Axioms for an Alternate Properad 208
11.7.5. Properads and Alternate Properads are Gr^-PROPs 208
11.8. PROPs as Grr-PROPs 209
11.8.1. Defining a PROP 209
11.8.2. Interpreting the Axioms for a PROP 212
11.8.3. PROPs are Gr^PROPs 212
11.9. Wheeled PROPs as Gr^-PROPs 213
11.9.1. Defining a Wheeled PROP 213
11.9.2. Interpreting the Axioms for a Wheeled PROP 215
11.9.3. Wheeled PROPs are GrP-PROPs 215
11.10. Wheeled Properads as Gr«P֊PROPs 215
11.10.1. Defining a Wheeled Properad 215
11.10.2. Interpreting the Axioms for Wheeled Properads 217
11.10.3. Wheeled Properads are GrP-PROPs 217
11.11. Wheeled Operads as Tree ^-PROPs 218
11.11.1. Defining a Wheeled Operad 218
11.11.2. Interpreting the Axioms for a Wheeled Operad 220
11.11.3. Wheeled Operads are Tree^-PROPs 221
Chapter 12. Functors of Generalized PROPs 223
12.1. Adjunction Induced by an Inclusion of Pasting Schemes 223
12.1.1. The Right Adjoint 224
12.1.2. Adjunction for a Well-Matched Pair 224
12.1.3. The General Left Adjoint 227
12.1.4. Examples of the Left Adjoint 231
12.2. Generalized PROPS under a Change of Base Category 233
12.2.1. Transferring Generalized PROPs 233
12.2.2. Changing Pasting Scheme and Base Category 235
Xll
CONTENTS
12.2.3. An Application to Homology 236
12.3. Notes 237
Chapter 13. Algebras over Generalized PROPs 239
13.1. Endomorphism Objects 240
13.1.1. Horn-Tensor Adjunction 240
13.1.2. Endomorphism Objects 241
13.1.3. Pasting Scheme Admitting an Endomorphism Object 242
13.1.4. Examples of Endomorphism Objects 243
13.1.5. Relative Endomorphism Object 245
13.2. Unbiased Algebras 249
13.2.1. Unbiased Definition of Algebras 250
13.2.2. Alternative Descriptions of Algebras 251
13.3. Algebras under Change of Pasting Scheme or Base Category 252
13.3.1. Change of Pasting Scheme 253
13.3.2. Change of Ç-PROP 254
13.3.3. Change of Base Category 254
13.4. Biased Algebras 257
13.4.1. Biased Algebra Theorem 257
13.4.2. Algebras over a Markl Non-Unital Operad 257
13.4.3. Algebras over a May Operad 258
13.4.4. Algebras over a Dioperad 258
13.4.5. Algebras over a Half-PROP 259
13.4.6. Algebras over a Properad 260
13.4.7. Algebras over a PROP 260
13.4.8. Algebras over a Wheeled PROP 261
13.4.9. Algebras over a Wheeled Properad 261
13.4.10. Algebras over a Wheeled Operad 261
13.5. Notes 262
Chapter 14. Alternative Descriptions of Generalized PROPs 263
14.1. Generalized PROPs as Operadic Algebras 263
14.1.1. The Operad Associated to a Pasting Scheme 264
14.1.2. The Colored Operad of £-PROPs 265
14.2. Generalized PROPs as Multicategorical Functors 266
14.2.1. Defining an Enriched Multicategory 266
14.2.2. Functors of Enriched Multicategories 268
14.2.3. Corepresenting Ç-PROPs 269
14.3. Notes 272
Chapter 15. Modules over Generalized PROPs 273
15.1. Pointed Decorated Graphs and a Monad Variation 274
15.1.1. Pointed Decorated Graphs 274
15.1.2. A Pointed Extension of the Monad Fg 274
15.2. Unbiased Modules over a {?-PROP 276
15.2.1. Unbiased Definition of Modules over a Ç-PROP 276
15.2.2. Bi-equivariant Structure of a Module 278
15.2.3. Graphical Interpretation 278
CONTENTS
xiii
15.3. Modules under Change of Pasting Scheme, Q~PROP, or Base
Category 280
15.3.1. Change of Pasting Scheme 280
15.3.2. Change of ¿7-PROP 280
15.3.3. Change of Base Category 281
15.4. Biased Characterizations of Modules 282
15.4.1. Biased Module Theorem 282
15.4.2. Modules over a Markl Non-Unital Operad 284
15.4.3. Modules over a May Operad 285
15.4.4. Modules over a Dioperad 285
15.4.5. Modules over a Half-PROP 286
15.4.6. Modules over a Properad 287
15.4.7. Modules over a PROP 287
15.4.8. Modules over a Wheeled PROP 288
15.4.9. Modules over a Wheeled Properad 289
15.4.10. Modules over a Wheeled Operad 290
Chapter 16. May Modules over Algebras over Operads 291
16.1. Preliminaries on Modules over an Algebra over an Operad 291
16.1.1. Groupoid-Indexed Colimit 292
16.1.2. A Pointed Monad Extension for Modules over an Algebra over a
Colored Operad 293
16.2. May Modules over an Operadic Algebra 297
16.2.1. Definition of a May Module 297
16.2.2. Alternative Description of a May Module 299
16.2.3. May Modules are Monadic Algebras 300
16.2.4. Modules Over Generalized PROPs are May Modules 300
Bibliography 303
Index 307
|
| any_adam_object | 1 |
| author | Yau, Donald 1977- Johnson, Mark W. 1973- |
| author_GND | (DE-588)141581875 (DE-588)1073989151 |
| author_facet | Yau, Donald 1977- Johnson, Mark W. 1973- |
| author_role | aut aut |
| author_sort | Yau, Donald 1977- |
| author_variant | d y dy m w j mw mwj |
| building | Verbundindex |
| bvnumber | BV043043335 |
| classification_rvk | SK 230 SK 320 |
| ctrlnum | (OCoLC)931760344 (DE-599)GBV813383455 |
| dewey-full | 512/.21 13 55 18 81 |
| dewey-hundreds | 500 - Natural sciences and mathematics 000 - Computer science, information, general works |
| dewey-ones | 512 - Algebra 013 - [Unassigned] 055 - Serials in Italian, Romanian & related languages 018 - [Unassigned] 081 - General collections in American English |
| dewey-raw | 512/.21 13 55 18 81 |
| dewey-search | 512/.21 13 55 18 81 |
| dewey-sort | 3512 221 |
| dewey-tens | 510 - Mathematics 010 - Bibliography 050 - General serial publications 080 - General collections |
| discipline | Geologie / Paläontologie Psychologie Anglistik / Amerikanistik Mathematik Philosophie |
| format | Book |
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| id | DE-604.BV043043335 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:15:49Z |
| institution | BVB |
| isbn | 9781470421977 |
| language | English |
| lccn | 2014049153 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028467896 |
| oclc_num | 931760344 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 DE-11 DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-20 DE-11 DE-19 DE-BY-UBM |
| physical | xxxi, 311 Seiten Illustrationen |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Yau, Donald 1977- Verfasser (DE-588)141581875 aut A foundation for PROPs, algebras, and modules Donald Yau, Mark W. Johnson Providence, Rhode Island American Mathematical Society [2015] © 2015 xxxi, 311 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs volume 203 Bibliografie Seite 303-305 Kategorie Mathematik (DE-588)4129930-9 gnd rswk-swf Permutationsgruppe (DE-588)4173833-0 gnd rswk-swf Permutationsgruppe (DE-588)4173833-0 s Kategorie Mathematik (DE-588)4129930-9 s DE-604 Johnson, Mark W. 1973- Verfasser (DE-588)1073989151 aut Erscheint auch als Online-Ausgabe 978-1-4704-2347-6 Mathematical surveys and monographs volume 203 (DE-604)BV000018014 203 DE-601 pdf/application http://www.gbv.de/dms/tib-ub-hannover/813383455.pdf Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028467896&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yau, Donald 1977- Johnson, Mark W. 1973- A foundation for PROPs, algebras, and modules Mathematical surveys and monographs Kategorie Mathematik (DE-588)4129930-9 gnd Permutationsgruppe (DE-588)4173833-0 gnd |
| subject_GND | (DE-588)4129930-9 (DE-588)4173833-0 |
| title | A foundation for PROPs, algebras, and modules |
| title_auth | A foundation for PROPs, algebras, and modules |
| title_exact_search | A foundation for PROPs, algebras, and modules |
| title_full | A foundation for PROPs, algebras, and modules Donald Yau, Mark W. Johnson |
| title_fullStr | A foundation for PROPs, algebras, and modules Donald Yau, Mark W. Johnson |
| title_full_unstemmed | A foundation for PROPs, algebras, and modules Donald Yau, Mark W. Johnson |
| title_short | A foundation for PROPs, algebras, and modules |
| title_sort | a foundation for props algebras and modules |
| topic | Kategorie Mathematik (DE-588)4129930-9 gnd Permutationsgruppe (DE-588)4173833-0 gnd |
| topic_facet | Kategorie Mathematik Permutationsgruppe |
| url | http://www.gbv.de/dms/tib-ub-hannover/813383455.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028467896&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT yaudonald afoundationforpropsalgebrasandmodules AT johnsonmarkw afoundationforpropsalgebrasandmodules |
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