Infinity properads and infinity wheeled properads:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham [u.a.]
Springer
2015
|
| Series: | Lecture notes in mathematics
2147 |
| Online Access: | 01 Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 351 - 354 |
| Physical Description: | XV, 358 S. graph. Darst. 235 mm x 155 mm, 0 g |
| ISBN: | 3319205463 9783319205465 |
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MARC
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| 245 | 1 | 0 | |a Infinity properads and infinity wheeled properads |c Philip Hackney ; Marcy Robertson ; Donald Yau |
| 264 | 1 | |a Cham [u.a.] |b Springer |c 2015 | |
| 300 | |a XV, 358 S. |b graph. Darst. |c 235 mm x 155 mm, 0 g | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 2147 | |
| 500 | |a Literaturverz. S. 351 - 354 | ||
| 700 | 1 | |a Robertson, Marcy |e Verfasser |0 (DE-588)1077452802 |4 aut | |
| 700 | 1 | |a Yau, Donald |d 1977- |e Verfasser |0 (DE-588)141581875 |4 aut | |
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Titel: Infinity properads and infinity wheeled properads
Autor: Hackney, Philip
Jahr: 2015
Contents 1 Introduction. 1 1.1 (Wheeled) Properads as Generalized Categories. 1 1.2 Infinity Categories and Infinity Operads. 4 1.3 Infinity (Wheeled) Properads. 6 1.4 Chapter Summaries. 7 Part I Infinity Properads 2 Graphs. 13 2.1 Wheeled Graphs. 14 2.1.1 Profiles of Colors. 14 2.1.2 Generalized Graphs. 15 2.1.3 Structures on Generalized Graphs. 16 2.2 Examples of Graphs. 17 2.3 Connected Graphs. 22 2.3.1 Paths. 23
2.3.2 Connected Graphs. 24 2.3.3 Wheel-Free Graphs. 25 2.4 Graph Substitution. 28 2.4.1 Properties of Graph Substitution. 28 2.4.2 Examples. 30 2.5 Closest Neighbors. 32 2.5.1 Motivating Examples. 32 2.5.2 Closest Neighbors. 33 2.5.3 Inner Properadic Factorization. 34 2.6 Almost Isolated Vertices. 36 2.6.1 Definition and Examples. 36 2.6.2 Extremal Paths. 39 2.6.3 Existence of Almost Isolated Vertices. 41 2.6.4 Outer Properadic Factorization. 42 xi
Contents 2.7 Deletable Vertices and Internal Edges. 2.7.1 Deletable Vertices . 2.7.2 Internal Edges. 2.8 Disconnectable Edges and Loops. 2.8.1 Disconnectable Edges. 2.8.2 Loops. 3 Properads. 3.1 Biased Properads. 3.1.1 17-Bimodules and Colored Objects. 3.1.2 Biased Definition of a Properad. 3.1.3 Algebras over a Properad. 3.2 Unbiased Properads. 3.2.1 Monads and Their Algebras. 3.2.2 Decorated Graphs. 3.2.3 Free Properad Monad.
3.2.4 Maps from Free Properads. 4 Symmetric Monoidal Closed Structure on Properads. 4.1 Symmetric Monoidal Product. 4.1.1 Smash Product. 4.1.2 Tensor Product. 4.1.3 Distributivity. 4.1.4 Symmetric Monoidal Structure. 4.2 Symmetric Monoidal Product of Free Properads. 4.2.1 Motivation . 4.2.2 The First Two Relations. 4.2.3 Generating Distributivity. 4.2.4 Consequences of Theorem 4.14. 4.3 Proof of Theorem 4.14. 4.3.1 Reduction of Proof. 4.3.2 Induction Step .
4.4 Internal Horn of Special Properads. 4.4.1 Natural Transformation. 4.4.2 Internal Horn. 4.4.3 Symmetric Monoidal Closed Structure. 5 Graphical Properads. 5.1 Properads Generated by Connected Wheel-Free Graphs. 5.1.1 1 -Colored Graphs. 5.1.2 Generating Set. 5.1.3 Size of Graphical Properads. 5.2 Symmetric Monoidal Product. 5.2.1 Symmetric Monoidal Product of Graphical Properads . 5.2.2 The Connected Wheel-Free Graphs. 5.2.3 The Smash Product. 44 45 48 50 50 53 55 56 56 58 60 61 62 63 64 66 69 70 70 71 72 75 76 76 77 81 82 83 83 84 92 92 96 96 99 100 100 101 103 106 106 107 108
Contents xiii 5.2.4 Generating Distributivity. 109 5.2.5 Generating Distributivity in Action. 112 5.3 Maps of Graphical Properads. 113 5.3.1 Maps Between Graphical Properads. 114 5.3.2 Non-determination by Edge Sets . 115 5.3.3 Non-injections. 117 5.4 Maps with Simply Connected Targets. 120 5.4.1 Simply Connected Targets . 121 5.4.2 Injectivity on Inputs and Outputs. 121 5.4.3 Non-injections. 122 5.4.4 Unique Determination by Edge Sets. 123 6 Properadic Graphical Category. 125 6.1 Coface and Codegeneracy Maps. 126 6.1.1 Inner Coface Maps .
126 6.1.2 Outer Coface Maps. 130 6.1.3 Codegeneracy Maps. 132 6.2 Graphical Identities. 134 6.2.1 Identities of Codegeneracy Maps. 134 6.2.2 Identities Involving Both Codegeneracy and Coface Maps. 135 6.2.3 Identities of Coface Maps. 138 6.3 Graphical Category. 145 6.3.1 Input and Output Relabeling. 145 6.3.2 Subgraphs. 146 6.3.3 Images. 150 6.3.4 Graphical Maps. 153 6.3.5 Graphical Category. 155 6.3.6 Factorization of Graphical Maps. 156 6.4 A Generalized Reedy Structure on
T. 160 7 Properadic Graphical Sets and Infinity Properads . 165 7.1 oo-Properads. 166 7.1.1 Graphical Sets. 166 7.1.2 Why a Graphical Set Resembles a Properad. 168 7.1.3 Properadic Nerve. 171 7.1.4 Symmetric Monoidal Closed Structure on Graphical Sets. 173 7.1.5 Faces and Horns. 174 7.1.6 oo-Properads. 178 7.2 Properadic Segal Maps. 179 7.2.1 Motivation of the Properadic Segal Maps. 179 7.2.2 Outer Coface Maps from Corollas. 182 7.2.3 Compatibility over Ordinary Edges. 184 7.2.4 Properadic Segal Maps. 184
XIV Contents 7.2.5 Alternative Description of the Properadic Segal Maps . 187 7.2.6 Properadic Nerves Satisfy the Segal Condition. 188 7.2.7 Properadic Nerve Is Fully Faithful. 189 7.2.8 The Segal Condition Implies Strict oo-Properad. 190 7.3 Characterization of Strict oo-Properads. 194 7.3.1 Properad Associated to a Strict oo-Properad. 195 7.3.2 Strict oo-Properads Are Properadic Nerves. 202 7.3.3 Fundamental Properad. 207 8 Fundamental Properads of Infinity Properads . 209 8.1 Homotopy in a Graphical Set. 210 8.1.1 Motivation from Fundamental Categories. 210 8.1.2 Homotopy of 1-Dimensional Elements. 212 8.1.3 Homotopy Relations Are Equivalence Relations. 218 8.1.4 Same Equivalence Relation.
226 8.2 Properad Associated to an oo-Properad. 233 8.2.1 Properadic Composition of 1 -Dimensional Elements. 233 8.2.2 Properad of Homotopy Classes. 245 Part II Infinity Wheeled Properads 9 Wheeled Properads and Graphical Wheeled Properads . 251 9.1 Biased and Unbiased Wheeled Properads. 252 9.1.1 Biased Wheeled Properads. 252 9.1.2 Unbiased Wheeled Properads. 255 9.1.3 Symmetric Monoidal Structure. 256 9.2 Graphical Wheeled Properads. 257 9.2.1 Wheeled Properads Generated by Connected Graphs . 257 9.2.2 Size of Graphical Wheeled Properads. 260 9.2.3 Maps Between Graphical Wheeled Properads. 261 9.3 Wheeled Properadic Coface Maps . 262 9.3.1 Motivation for Coface Maps. 263
9.3.2 Coface and Codegeneracy Maps. 264 9.3.3 Graphical Identities. 269 9.3.4 Codimensional 2 Property. 271 9.4 Wheeled Properadic Graphical Category. 278 9.4.1 Subgraphs. 278 9.4.2 Images. 281 9.4.3 Graphical Maps. 283 9.4.4 Graphical Category. 285 9.4.5 Factorization of Graphical Maps. 287 10 Infinity Wheeled Properads. 293 10.1 oo-Wheeled Properads. 294 10.1.1 Wheeled Properadic Graphical Sets and Nerve. 294 10.1.2 Symmetric Monoidal Closed Structure. 296
Contents XV 10.1.3 Faces and Horns . 297 10.1.4 oo-Wheeled Properads . 302 10.2 Characterization of Strict oo-Wheeled Properads. 303 10.2.1 Outer Coface Maps from Corollas. 304 10.2.2 Wheeled Properadic Segal Maps . 304 10.2.3 Wheeled Properadic Nerves Satisfy the Segal Condition. 308 10.2.4 Wheeled Properadic Nerve Is Fully Faithful. 308 10.2.5 The Segal Condition Implies Strict oo-Wheeled Properad. 309 10.2.6 Wheeled Properad Associated to a Strict oo-Wheeled Properad. 313 10.2.7 Strict oo-Wheeled Properads Are Nerves. 320 10.3 Fundamental Wheeled Properads of oo-Wheeled Properads. 324 10.3.1 Homotopy of 1-Dimensional Elements. 324 10.3.2 Dioperadic Compositions of 1-Dimensional Elements . 326 10.3.3 Contractions of 1-Dimensional Elements. 329 10.3.4 Wheeled Properad of Homotopy Classes. 334 10.3.5 Fundamental Wheeled Properad. 338 11 What’s Next? . 341 11.1 Homotopy Theory of Infinity Properads. 341 11.2 String Topology Infinity Properad. 342 11.3 Operadic Approach. 342 11.4 Strong Homotopy Properads. 343 11.5 Deformation Theory. 344 11.6 Weber Theory. 344 Notation . 347 References. 351 Index 355 |
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| author | Hackney, Philip Robertson, Marcy Yau, Donald 1977- |
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| discipline | Mathematik |
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| spelling | Hackney, Philip Verfasser (DE-588)1077452519 aut Infinity properads and infinity wheeled properads Philip Hackney ; Marcy Robertson ; Donald Yau Cham [u.a.] Springer 2015 XV, 358 S. graph. Darst. 235 mm x 155 mm, 0 g txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2147 Literaturverz. S. 351 - 354 Robertson, Marcy Verfasser (DE-588)1077452802 aut Yau, Donald 1977- Verfasser (DE-588)141581875 aut Erscheint auch als Online-Ausgabe 978-3-319-20547-2 Lecture notes in mathematics 2147 (DE-604)BV000676446 2147 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=5266770&prov=M&dok_var=1&dok_ext=htm 01 Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028354109&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hackney, Philip Robertson, Marcy Yau, Donald 1977- Infinity properads and infinity wheeled properads Lecture notes in mathematics |
| title | Infinity properads and infinity wheeled properads |
| title_auth | Infinity properads and infinity wheeled properads |
| title_exact_search | Infinity properads and infinity wheeled properads |
| title_full | Infinity properads and infinity wheeled properads Philip Hackney ; Marcy Robertson ; Donald Yau |
| title_fullStr | Infinity properads and infinity wheeled properads Philip Hackney ; Marcy Robertson ; Donald Yau |
| title_full_unstemmed | Infinity properads and infinity wheeled properads Philip Hackney ; Marcy Robertson ; Donald Yau |
| title_short | Infinity properads and infinity wheeled properads |
| title_sort | infinity properads and infinity wheeled properads |
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