Table Of ContentMONOIDAL TOPOLOGY
MonoidalTopologydescribesanactiveresearchareathat,aftervariouspast
proposalsonhowtoaxiomatize“spaces”intermsofconvergence,beganto
emergeatthebeginningofthemillennium.ItcombinesBarr’srelational
presentationoftopologicalspacesintermsofultrafilterconvergencewith
Lawvere’sinterpretationofmetricspacesassmallcategoriesenrichedoverthe
extendedrealhalf-line.Hence,equippedwithaquantaleV (replacingthereals)
andamonadT(replacingtheultrafiltermonad)laxlyextendedfromsetmapsto
V-valuedrelations,thebookdevelopsacategoricaltheoryof(T,V)-algebras
thatisinspiredsimultaneouslybyitsmetricandtopologicalroots.Thebook
highlightsinparticularthedistinguishedroleofequationallydefinedstructures
withinthegivenlax-algebraiccontextandpresentsnumerousnewresults
rangingfromtopologyandapproachtheorytodomaintheory.Allthenecessary
pre-requisitesinorderandcategorytheoryarepresentedinthebook.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplication
inmathematicsormathematicalscienceandforwhichadetaileddevelopmentof
theabstracttheoryislessimportantthanathoroughandconcreteexplorationof
theimplicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedas
exercisesattheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
106 A.MarkoeAnalyticTomography
107 P.A.MartinMultipleScattering
108 R.A.BrualdiCombinatorialMatrixClasses
109 J.M.BorweinandJ.D.VanderwerffConvexFunctions
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmannetal.TheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLévyNoise
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributedParameter
Systems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopologicalGraph
Theory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,ComputerScience,
andEngineering
135 V.BerthéandM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Kristály,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,Geometry,
andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland
DifferentialEquations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethodfor
MaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.Cabrera-GarcíaandÁ.RodríguezPalaciosNon-AssociativeNormedAlgebrasI:The
Vidav–PalmerandGelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Monoidal Topology
A Categorical Approach to Order, Metric,
and Topology
Editedby
DIRK HOFMANN
UniversidadedeAveiro,Portugal
GAVIN J. SEAL
SwissFederalInstituteofTechnology
WALTER THOLEN
YorkUniversity,Toronto
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107063945
©CambridgeUniversityPress2014
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2014
PrintedintheUnitedKingdombyCPIGroupLtd,CroydonCR04YY
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationdata
Monoidaltopology:acategoricalapproachtoorder,metric,andtopology/editedby
DirkHofmann,UniversidadedeAveiro,GavinJ.Seal,SwissFederalInstituteofTechnology,
WalterTholen,YorkUniversity,Toronto.
pages cm.–(Encyclopediaofmathematicsanditsapplications)
ISBN978-1-107-06394-5(hardback)
1. Topologicalsemigroups. 2. Grouptheory. I. Hofmann,Dirk,1970–
II. Seal,GavinJ. III. Tholen,W.(Walter),1947–
QA387.M65 2014
514(cid:2).32–dc23
2013046221
ISBN978-1-107-06394-5Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
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ToHorstHerrlich
Summary of contents
Preface pagexv
I Introduction 1
RobertLowenandWalterTholen
II Monoidalstructures 18
GavinJ.SealandWalterTholen
III Laxalgebras 145
DirkHofmann,GavinJ.Seal,andWalterTholen
IV Kleislimonoids 284
DirkHofmann,RobertLowen,RoryLucyshyn-Wright,
andGavinJ.Seal
V Laxalgebrasasspaces 375
MariaManuelClementino,EvaColebunders,andWalterTholen
Bibliography 467
Selectedcategories 480
Selectedfunctors 484
Selectedsymbols 487
Index 491
Contents
Preface pagexv
I Introduction 1
I.1 Theubiquityofmonoidsandtheiractions 1
I.1.1 Monoidsandtheiractionsinalgebra 2
I.1.2 Ordersandmetricsasmonoidsandlaxalgebras 3
I.1.3 Topologicalandapproachspacesasmonoidsand
laxalgebras 5
I.1.4 Thecaseforconvergence 7
I.1.5 FilterconvergenceandKleislimonoids 9
I.2 Spacesascategories,andcategoriesofspaces 10
I.2.1 Ordinarysmallcategories 10
I.2.2 Consideringaspaceasacategory 11
I.2.3 Movingtothelargecategoryofallspaces 13
I.3 Chapterhighlightsanddependencies 14
II Monoidalstructures 18
II.1 Orderedsets 18
II.1.1 TheCartesianstructureofsetsanditsmonoids 18
II.1.2 Thecompositionalstructureofrelations 19
II.1.3 Orders 21
II.1.4 Modules 22
II.1.5 Adjunctions 22
II.1.6 Closureoperationsandclosurespaces 24
II.1.7 Completeness 25
II.1.8 Adjointnesscriteria 27
II.1.9 Semilattices,lattices,frames,andtopologicalspaces 28
II.1.10 Quantales 30
