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MONOIDAL 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 URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. 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