Operads of wiring diagrams

Preface......Page 7
Contents......Page 8
 1.1 What Are Wiring Diagrams?......Page 11
 1.2 Purposes of this Monograph......Page 13
 1.4 Chapter Summaries......Page 15
 1.5 References for the Main Results and Examples......Page 18
Part I Wiring Diagrams......Page 20
  2.1 Colored Operads......Page 22
  2.2 Defining Wiring Diagrams......Page 32
  2.3 Operad Structure......Page 39
  2.4 Summary of this Chapter......Page 46
 3 Generators and Relations......Page 47
  3.1 Generating Wiring Diagrams......Page 48
  3.2 Internal Wasted Wires......Page 52
  3.3 Elementary Relations......Page 54
  3.4 Summary of this Chapter......Page 69
 4 Decomposition of Wiring Diagrams......Page 70
  4.1 Factoring Wiring Diagrams......Page 71
  4.2 Unary Wiring Diagrams......Page 76
  4.3 Unary Wiring Diagrams with No Loop Elements......Page 81
  4.4 Summary of this Chapter......Page 94
 5 Finite Presentation......Page 95
  5.1 Stratified Presentation......Page 96
  5.2 Finite Presentation for Wiring Diagrams......Page 100
  5.3 Finite Presentation for Normal Wiring Diagrams......Page 107
  5.4 Finite Presentation for Strict Wiring Diagrams......Page 108
  5.5 Summary of this Chapter......Page 111
 6 Finite Presentation for Algebras over Wiring Diagrams......Page 112
  6.1 Operad Algebras......Page 113
  6.2 Algebras over the Operad of Wiring Diagrams......Page 117
  6.3 Finite Presentation for the Propagator Algebra......Page 126
  6.4 Algebras over the Operad of Normal Wiring Diagrams......Page 145
  6.5 Finite Presentation for the Algebra of Discrete Systems......Page 146
  6.7 Finite Presentation for the Algebra of Open Dynamical Systems......Page 154
  6.8 Summary of this Chapter......Page 160
Part II Undirected Wiring Diagrams......Page 161
  7.1 Defining Undirected Wiring Diagrams......Page 162
  7.2 Pushouts......Page 167
  7.3 Operad Structure......Page 168
  7.4 Summary of this Chapter......Page 177
  8.1 Generating Undirected Wiring Diagrams......Page 178
  8.2 Elementary Relations......Page 181
  8.3 Wasted Cables......Page 189
  8.4 Summary of this Chapter......Page 192
  9.1 A Motivating Example......Page 193
  9.2 Factoring Undirected Wiring Diagrams......Page 197
  9.3 The Inner Undirected Wiring Diagram......Page 199
  9.4 The Outer Undirected Wiring Diagram......Page 202
  9.5 Iterated Splits......Page 206
  9.6 Iterated Loops......Page 209
  9.7 Summary of this Chapter......Page 211
  10.1 Stratified Presentation......Page 212
  10.2 Elementary Equivalences......Page 215
  10.3 Summary of this Chapter......Page 220
 11 Algebras of Undirected Wiring Diagrams......Page 221
  11.1 Finite Presentation for Algebras......Page 222
  11.2 Finite Presentation for the Relational Algebra......Page 229
  11.3 Spivak's Conjecture: Rigidity of the Relational Algebra......Page 232
  11.4 Finite Presentation for the Typed Relational Algebra......Page 236
  11.5 Summary of this Chapter......Page 239
Part III Maps Between Operads of Wiring Diagrams......Page 240
 12 A Map from Normal to Undirected Wiring Diagrams......Page 242
  12.1 Operad Maps......Page 243
  12.2 Normal to Undirected Wiring Diagrams......Page 246
  12.3 Examples of the Operad Map......Page 252
  12.4 Image of the Operad Map......Page 256
  12.5 Map from Strict to Undirected Wiring Diagrams......Page 262
  12.6 Summary of this Chapter......Page 266
 13 A Map from Wiring Diagrams to Undirected Wiring Diagrams......Page 267
  13.1 Wiring Diagrams to Undirected Wiring Diagrams......Page 268
  13.2 Examples of the Operad Map......Page 274
  13.3 Surjectivity of the Operad Map......Page 276
  13.4 Summary of this Chapter......Page 283
 14 Problems......Page 284
 A.1 Category Theory......Page 287
 A.3 Props......Page 288
 A.4 Applications of Compositional Structures......Page 289
B List of Notation......Page 291
References......Page 294
Index......Page 297