Table Of ContentLecture Notes in Mathematics 2192
Donald Yau
Operads
of Wiring
Diagrams
Lecture Notes in Mathematics 2192
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Donald Yau
Operads of Wiring Diagrams
123
DonaldYau
DepartmentofMathematics
TheOhioStateUniversityatNewark
Ohio,USA
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-95000-6 ISBN978-3-319-95001-3 (eBook)
https://doi.org/10.1007/978-3-319-95001-3
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To EunSooand Jacqueline
Preface
Wiring diagrams and undirected wiring diagrams are graphical languages for
describing interconnected processes and their compositions. These objects have
enormouspotentialsforapplicationsinmanydifferentdisciplines,includingcom-
puter science, cognitive neuroscience, dynamical systems, network theory, and
circuit diagrams. It is known that the collection of wiring diagrams is an operad
andlikewiseforundirectedwiringdiagrams.
This monograph is a comprehensive study of the combinatorial structure of
various operads of wiring diagrams and undirected wiring diagrams. Our first
main objective is to provea finite presentationtheorem for each operadof wiring
diagrams, describing each one in terms of just a few operadic generators and a
smallnumberofgeneratingrelations.Forexample,theoperadofwiringdiagrams
has8generatorsand28generatingrelations,whiletheoperadofundirectedwiring
diagramshas6generatorsand17generatingrelations.
Our second main objective is to prove a corresponding finite presentation
theorem for algebras over each operad of wiring diagrams. As applications we
provide finite presentations for the propagator algebra, the algebra of discrete
systems,thealgebraofopendynamicalsystems,andthe(typed)relationalalgebra.
WealsoprovideapartialverificationofSpivak’sconjectureregardingthequotient-
freenessoftherelationalalgebra.
Ourthirdmainobjectiveistoconstructexplicitoperadmapsamongtheseveral
operadsofwiringdiagrams.Inparticular,thereisasurjectiveoperadmapfromthe
operad of all wiring diagrams, including delay nodes, to the operad of undirected
wiringdiagrams.
This monograph is intended for graduate students, mathematicians, scientists,
and engineers interested in operads and wiring diagrams. Assuming no prior
knowledge of categories, operads, and wiring diagrams, this monograph is self-
containedandcanbeusedasasupplementinagraduatecourseandforindependent
study. There are over 100 graphical illustrations and a chapter with a list of
problems.
Newark,OH,USA DonaldYau
vii
Contents
1 Introduction................................................................. 1
1.1 WhatAreWiringDiagrams? ........................................ 1
1.2 PurposesofthisMonograph......................................... 3
1.3 AudienceandFeatures............................................... 5
1.4 ChapterSummaries .................................................. 5
1.5 ReferencesfortheMainResultsandExamples .................... 8
PartI WiringDiagrams
2 WiringDiagrams ........................................................... 13
2.1 ColoredOperads...................................................... 13
2.2 DefiningWiringDiagrams........................................... 23
2.3 OperadStructure ..................................................... 30
2.4 SummaryofthisChapter ............................................ 37
3 GeneratorsandRelations.................................................. 39
3.1 GeneratingWiringDiagrams........................................ 40
3.2 InternalWastedWires................................................ 44
3.3 ElementaryRelations ................................................ 46
3.4 SummaryofthisChapter ............................................ 61
4 DecompositionofWiringDiagrams...................................... 63
4.1 FactoringWiringDiagrams.......................................... 64
4.2 UnaryWiringDiagrams.............................................. 69
4.3 UnaryWiringDiagramswithNoLoopElements .................. 74
4.4 SummaryofthisChapter ............................................ 87
5 FinitePresentation ......................................................... 89
5.1 StratifiedPresentation................................................ 90
5.2 FinitePresentationforWiringDiagrams............................ 94
5.3 FinitePresentationforNormalWiringDiagrams................... 101
5.4 FinitePresentationforStrictWiringDiagrams..................... 102
5.5 SummaryofthisChapter ............................................ 105
ix
x Contents
6 FinitePresentationforAlgebrasoverWiringDiagrams............... 107
6.1 OperadAlgebras...................................................... 108
6.2 AlgebrasovertheOperadofWiringDiagrams..................... 112
6.3 FinitePresentationforthePropagatorAlgebra..................... 121
6.4 AlgebrasovertheOperadofNormalWiringDiagrams............ 140
6.5 FinitePresentationfortheAlgebraofDiscreteSystems........... 141
6.6 AlgebrasovertheOperadofStrictWiringDiagrams .............. 149
6.7 FinitePresentationfortheAlgebraofOpenDynamical
Systems............................................................... 149
6.8 SummaryofthisChapter ............................................ 155
PartII UndirectedWiringDiagrams
7 UndirectedWiringDiagrams ............................................. 159
7.1 DefiningUndirectedWiringDiagrams.............................. 159
7.2 Pushouts .............................................................. 164
7.3 OperadStructure ..................................................... 165
7.4 SummaryofthisChapter ............................................ 174
8 GeneratorsandRelations.................................................. 175
8.1 GeneratingUndirectedWiringDiagrams........................... 175
8.2 ElementaryRelations ................................................ 178
8.3 WastedCables........................................................ 186
8.4 SummaryofthisChapter ............................................ 189
9 DecompositionofUndirectedWiringDiagrams ........................ 191
9.1 AMotivatingExample............................................... 191
9.2 FactoringUndirectedWiringDiagrams............................. 195
9.3 TheInnerUndirectedWiringDiagram.............................. 197
9.4 TheOuterUndirectedWiringDiagram ............................. 200
9.5 IteratedSplits......................................................... 204
9.6 IteratedLoops ........................................................ 207
9.7 SummaryofthisChapter ............................................ 209
10 FinitePresentationforUndirectedWiringDiagrams.................. 211
10.1 StratifiedPresentation................................................ 211
10.2 ElementaryEquivalences............................................ 214
10.3 SummaryofthisChapter ............................................ 219
11 AlgebrasofUndirectedWiringDiagrams............................... 221
11.1 FinitePresentationforAlgebras..................................... 222
11.2 FinitePresentationfortheRelationalAlgebra...................... 229
11.3 Spivak’sConjecture:RigidityoftheRelationalAlgebra........... 232
11.4 FinitePresentationfortheTypedRelationalAlgebra .............. 236
11.5 SummaryofthisChapter ............................................ 239
Contents xi
PartIII MapsBetweenOperadsofWiringDiagrams
12 AMapfromNormaltoUndirectedWiringDiagrams ................. 243
12.1 OperadMaps ......................................................... 244
12.2 NormaltoUndirectedWiringDiagrams............................ 247
12.3 ExamplesoftheOperadMap........................................ 253
12.4 ImageoftheOperadMap............................................ 257
12.5 MapfromStricttoUndirectedWiringDiagrams................... 263
12.6 SummaryofthisChapter ............................................ 267
13 AMapfromWiringDiagramstoUndirectedWiringDiagrams...... 269
13.1 WiringDiagramstoUndirectedWiringDiagrams ................. 270
13.2 ExamplesoftheOperadMap........................................ 276
13.3 SurjectivityoftheOperadMap...................................... 278
13.4 SummaryofthisChapter ............................................ 285
14 Problems..................................................................... 287
A FurtherReading............................................................ 291
B ListofNotation ............................................................. 295
References......................................................................... 299
Index............................................................................... 303
