atedbyamodelforsyntacticcon dtoJohnCReynoldsontheOcasionofhisthBirthday ThisauthorwassupportedbyNSFgrantCCRThisauthorgratefullyac etaltroductionThispaperisacompanionpapertoSyntacticContro ID: 820996
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ElectronicNotesinTheoreticalComputerScie
ElectronicNotesinTheoreticalComputerScience URL http wwwelseviernllocateentcsvolumehtml pages J FUniversityofPennsylvania Philadelphia USA W OHearnacuseUniversity NewYork USAA J PatedbyamodelforsyntacticcondtoJohnC ReynoldsontheOcasionofhisthBirthdayThisauthorwassupportedbyNSFgrantCCR ThisauthorgratefullyacetaltroductionThispaperisacompanionpapertoSyntacticControlofInterfer inthisvInthispaperweintroduceageneralcategoricaltoanalysethepropertiesofthemodeloftheSCIRtypesystemgivenin Thispaperispurelycategoricalitcanbereadindependentlyof asacategorytheoreticpaperThebireectivityconcepthashwiderapplicabilitbutthispaperconcentratesonourleadingexampleenfrom althoughwewilldescribeitagainherewewillnotexplainitssignicanceThecentralsurprisingcategorytheoreticfeatureofthemodelofSCIRenin istheconceptofa bireective subcategoryywhicemeanasubcategorywithinclusionhavingbothleftandrightadjointwiththosetsequalandsatisfyinganevidentcoherenceconditionrelatingtheunitandcounitIn theoneandonlynontrivialbireectivesubcategoryoftheticcategoryisthesubcategoryofobjectsFormanycategorieshasandthecategoryofcpo sthereisnonontrivialsucsubcategoryandinfactweproethatanellpointedcategoryhasnotrivialsuchsubcategoryInthispaperwharacterizebireectivesubcategoriesofacategoryasequivttosplitidempotentnaturaltransformationsfromtheidenfunctorontoitselfTheconstructionimplicitinthisresultusesalimitinthecategorycalledanSowedescribethenotion
ofidenetheconstructionandproeourresult
ofidenetheconstructionandproeourresultIntheparticularcasethatisapresheafcategory weproemorethatanybireectivesubcategoryustitselfbeapresheafcategory andwegiveanexplicitdescriptionofaforwhicThesemanticcategory of isamildvtofapresheafandforourpurposessatisesthesameconditionsSoalthoughwstudypresheafcategoriesinthispaperitisroutinetoverifythatouranalysisallextendsto SpecicallyifisasmallmonoidalcategorythenisthefreemonoidalcocompletionofWithalittlemorestructureonwhicecallstructurewecanconstructanidempotentnaturaltransformationfromidtoitselfandhenceasplitonefromidtoitselfthusyieldingabireectivesubcategoryof oranexamplein thecategoryofworldsisasmallmonoidalcategorywithdiagonalstructureandgeneralizingmildlyfromtothecategoryofdomainsourconstructionyieldsthemonoidalstructureonthesemancategory of anditsrestrictiontothepassiveobjectseusediagonalstructuretodeduceseveralresultsabouttheinteractionofthebireectivesubcategorybothadjunctionsbeteenthemaremonoidaladjunctions hasniteproductsgivenbytherestrictionofthetensorproductoniscontainedinthecategoryofcommecomonoids andisanexponentialidealofTheseresultsarecentraltotheanalysisof orthispaperwedonotmakeheavyuseofcategoriesbeyondtheiretaldenition astandardreferencetoananalysisofthedenitionis byKellyandStreetThesemanticcategoryInthissectionwerecallthesemanticdenitionsof De nitionThecategoryofwThecategoryhasasobjectstablesetswithamorphismfRfromenbyafunctionandanequivalencerelationhthat andwithcompos
itiongSfRgivenbythefunctionandtherel
itiongSfRgivenbythefunctionandtherelationwherexTyfxSfyPropositionFiniteprductofsetsgivesasymmetricmonoidalstructurwithunittheterminalobjeProoforfRandgSthetensorfRgSisgivenbgRwherexyySyThecanonicalisomorphismsaregivenbythoseofniteproductswithtotalrelationsThesingletonsetisterminalinwiththeuniqueenbytheuniquefunctionandtheequalityrelationDe nitiondenethestatechangecytheidenyfunctionandtheidenyonNotethatisanidempotentnaturaltransformationonidewriteandsimilarlyforIn thetypetheorySCIRismodelledinthesemanticcisthecategoryofpossiblybottomlesscompleteposetsandconuousfunctionsDe nitioneobjectsiscalledidThefullsubcategoryof isgivythepassiveobjectsThefullinclusioniswrittenDe nitionDeneamonoidalstructureon asfollowsFfgthetensorproductisgivenbabwherefXgYandforabagTheunitistheterminalobjectof Throughthecourseofthispaperweproethefollowingpropertiesofthecategoryofpassiveobjectsusedin etalPropositionThefullsubisbothrctiveandcctiveinthesemanticcmorover therctorandcctorcPropositionThechasniteprProposition ThesymmetricmonoidalstructureonestrictstothecartesianstructureonPropositionBoththeinclusionandtherctorctorarestrsymmetricmonoidalfunctors i e theypreservethemonoidalstructurProposition Pisanexponentialidealof i e given theexponentialobjeAPliesinOfcourseonecouldproetheseresultsdirectlyratherthanbyappealtotheabstracttheorywedevelophereeritseemslikelythatothermodelsofsyntacticcontrolofinterferencewillbedevelo
pedinfuturesoratherthanhavingtoproesuch
pedinfuturesoratherthanhavingtoproesuchresultseverytimeonediscoersanewmodelitseemsusefultohaeageneralresultfromwhichonecandeducethemMoreoerourgeneralresultsprovidenecessaryandsucienconditionsforthenaturallevelofgeneralityoftheargumentssotheysetparameterstothesearchformodelsthatsatisfythepropertieswestudyRemarkRobinCocettandRobertSeelyhaepointedoutpersonalunicationthatasecondtensorcanbedenedonthecategoryorldsonobjectsityieldsthedisjointunionofthesetsandonmorphismsyieldsthesumofthefunctionpartsandthe join oftheequivpartsThesecondtensoralsoliftstothesemanticcategory whictogetherwiththebireectivesubcategoryprovidesanexampleofawemodelofnegationfreelinearlogic withandbothgivythebireectorThisconstructioncannotbenontriviallygeneralizedtomodelfulllinearlogicforifthesemanticcategorywautonomoustheesubcategorywhichisboththecategoryofalgebrasforandthecategoryofcoalgebrasforwouldbebothcartesianclosedandcocartesianclosedandhencedegenerateInthissectionwedenethenotionofbireectivesubcategoryandcizebireectivesubcategoriesinagivencategoryAftergivingafewexampleseusethischaracterizationtoshowthatanybireectivesubcategoryofapresheafcategoryisitselfapresheafcategorySoinparticularthecategoryofpassiveobjectsoftheprevioussectionisapresheafcategoryInfactitwsfromouranalysisthatitisthetrivialbireectivesubcategoryof De nitionsubcategoryofacategoryisasubcategorywithinclusionthathasleftandrightadjointsequalsaetalwithJSAJSAutingwhereistheunitofadjunctionisthecounito
fPropositionnybirctivesubofacisfull c
fPropositionnybirctivesubofacisfull closeundersubctformation andclosedunderquotientformation ProofWithnotationasinDenitionforSoforanfJwiththetranspositionoforsubobjectformationletbeamonomorphismeshowthatidThisisequivtto usingthecoherenceconditionmAJwiththetranspositionofClosureunderquotientformationisedduallyInthisandsubsequentsectionsendonaturaltransformationswhosecomponentsareallsplitidempotentsplayacentralroleWecallsuchanaturaltransformationaotentnaturaltrTheoremGivenac togiveabirctivesubgoryoftogiveasplitidempotentnaturaltransformationonInordertoproethisweneedtheconstructionofabireectivesubcategoryfromasplitidempotentnaturaltransformationonidThisisgivenbalimitinthecategorycalledanDe nitionbeacategoryandf beacellinitTheistheuniversalcellhthatidSpellingthisouthastopropertiesigivhthatisanidenycellthereexistsauniquehthath XkWkcommetaliigivkkbothidentitiesandthereexistsauniquecellhthattiersarelimitsincategoriesasexplainedinKelly sarticle Proof ofTheorem idbeasplitidempotennaturaltransformationtheidentierofandidthesplittingofThenidBytheunivyoftheidenonehasauniquefunctorTheadjunctionisgivenbSABAJBSA BwithunitApplyingthesameargumenttooneobtainsorthereversedirectionesthedesiredsplitidempotentItiseasytoverifytheseconstructionsaremutuallyinTheoremallowsonetoreplaceananalysisofbireectivesubcategoriesythatofsplitidempotentnaturaltransformationswhichisofteneasierExampleThecat
egoryofnitesemilatticesisbireectiveint
egoryofnitesemilatticesisbireectiveinthecategoryofnitecommesemigroupsFirstnotethatanyonegeneratorniteesemigrouphasexactlyoneidempotenWiththeadditivnotationletbegeneratedbwithrelationikThereisauniquewithh ksaSinceonehasnkxieisanidempotentForunicitifisalsoanidempotenGivenanitecommesemigroupbetheuniqueidempotentinthenitesubsemigroupofgeneratedbThefunctionisanendomorphismonsincegivxyisanidempotentinandhenceytheuniquenessTheuniquenessalsoimpliesthatisnaturalinsplitswiththeretractwhichisasemilatticewiththeorderSimilarlysemilatticesinthecategoryoftorsioncommesemigroupsformabireectivesubcategoryOnemaalsoreplacesemigroupsbymonoidsExampleThecategoryofsetsandrelationsisbireectiveinSProctheinteractioncategoryofsynchronousprocesses PropositionanobjectSProcisapair ofsetswithanonemptprexclosedsubsetof amorphismfromisastrongbisimilarclassof labeledtransitionsystemswhosetracesareconintheobvioussense thecompositeP Bisgivenby synchronization atieforacthereisatransitionifandonlyifthereexistswithaanda andnallytheidenetalisgivenbythe labeledtransitionsystemwhosetracesareenanobjectletthesubsetofenbythestringsoflengthatmostThereisatrivialonestep labeledtransitionsystemwith start a b end ab aSAjAbotharebisimilartothetransitionsystem truncated toatmostonestepSoisnaturalinSProcThisalsosplitsgivingtheretractwhereustheemptystringThestatementatthebeginningholdssincethefullsubcategoryofSProcenbythosetheidentra
nsitionsystemispreciselyExamplebeatop
nsitionsystemispreciselyExamplebeatoposandRelthecategoryofrelationsinieRelhasthesameobjectsasandaRelmorphismisanequivalenceclassofmonicpairswiththecompositiongivenbypullbacksandtheepimonofactorizationLetbeacollectionofobjectsthatincludestheterminalobjectThenRelthefullsubcategoryofRelgivenbytheobjectsinisbireectiveinRelifandonlyifasafullsubcategoryisreectiveinandclosedunderpoerobjectformationThisrequiresanontrivialproofbuttheproofrestsonthefactthatcertainsplitidempotennaturaltransformationsonidarenecessarilyequivalencerelationsFsubcategoriesofatoposclosedunderpoerobjectformationarestudiedindetailbreyd withanapplicationinlogiceplantoprepareasequelgivingfulldetailsoftheseapplicationsandotherharacterizationsofbireectivitenowreturntoourleadingexampleenacategoryacoidentierinisanidentierinrevthecellsinthedenitionForourmainexampleofacoidenExample beacategoryandletididbeanidempotentnaturaltransformationThenthecoidenisgivenbyfactoringythecongruencewhereforfgoseethisrstobservethatisacongruenceonitisobviouslyanalenceoneachhomsetAB itrespectscompositioninbecauseisnaturalwifisidentiedwiththeidenyandthenisidentiedwithwhichisidentiedwithsoidSowemaydescribethecoidenObObABABifandonlyifenanyfunctorbeteensmallcategoriesonehasafunctor iscompleteandcocomplete hasleftandrightadjointsgivenbyleftandrightKanextensionIfthecoidentierofanaturaltransformationbeteenfunctorswhosevalueisequalonobjectsitfollowsfromtheuniversalpropertythat
isfullyfaithfulexhibiting asequivtto
isfullyfaithfulexhibiting asequivttoafullsubcategoryof ThatetalsubcategoryisgivenbythosehthatidwhereisthenaturaltransformationPropositionGivenaceeveryidempotentsplitsandaalidempidid thefullinclusionasabirctivesubgoryof TheadjointoftothesplittingofProofThisfollowsbyusing tosendcolimitsintolimitshencecoidentierstoidentiersandbyapplyingtheconstructionofTheoremNotethat alwyssplitsDe nition Afullyfaithfulfunctorifthe CisfaithfulSpellingthisoutafullsubcategoryifforanyparallelpairofdistinctmapsfgthereexistsanobjectandamaphthataredistinctForexampletheunitcategoryisgeneratinginandinandthearrowcategoryisgeneratingAcategorywithaterminalobjectiswellpiftheinclusionisgeneratingPropositionGivenageneratingfunctor anyendonaturansformationonisuniquelydeterminedbyitsrestrictiontoProofstidimpliesforeacandynaturalitSotherearenomoreendonaturaltransformationsonidthanthereareonCorollaryorawellp thereisnonontrivialidemptentnaturaltransformationonProofTheonlynaturaltransformationontheinclusionofistheidenonRemarkInourcategoryofworldsDenitionisnotageneratorasonecannotdistinguishtomorphismswhichdi!eronlyintheiralencerelationpartsbuttheoneobjectsubcategoryofythetoelementsetisApplyingtheaboepropositionthereareatmostsixnaturaltransformationsonidofwhichfourcanbeidempotenByexaminingeachonecanconcludeidistheonlyidempotennaturaltransformationonidotherthantheidenemayuseRemarktodeducethatoursemanticcategory hasonlyonenontrivialbiree
ctivesubcategorywhichisofcoursethesubc
ctivesubcategorywhichisofcoursethesubcategoryofpassiveobjectsInfactweshowastrongerresulttogiveansplitidempotentnaturaltransformationonidistogiveanidempotnaturaltransformationonidThisgivesaconersetoPropositionetalincasethebasecategoryisTheliftingofthisresultfromroutinewegiveitforforeaseofexpositionPropositionorasmallc togiveanidempotentnaturansformationonistogiveoneonProofenanidempotentnaturaltransformationiditextendsto yhommingwgivenanyidempotennaturaltransformationidythefactthatevisacolimitofrepresenisfullydeterminedbitsbehaviouronrepresenuseverysucarisesfromauniqueidExampleenamonoidwithzeroelementforallisbireectiveinthecategoryofsetscorrespondingtotheidempotentonidregardedasaoneobjectcategoryRemarkyseemstobethedistinctivecategoricalpropertthatdi!erentiatesthemodelbasedonthecategoryfromotherextantexamplesoffunctorcategorysemanticsForexampleOlesoriginallyusedafullonobjectssubcategorythemapsfRbeingthosewheretherestrictionoftoanalenceclassisbijectiveClearlythisrulesoutthestatechangeconstraintendomorphismsandsothereisonlyonenaturalidempotentonidnamelytheidenAsaresultthefunctorcategoryusedbyOlespossessesnonontrivialbireectivesubcategoriesDiagonalCategoriesInthissectionwedenestructureonasymmetricmonoidalcategoryAdiagonalstructureconsistsofthedataandsomeoftheaxiomsrequiredtoforcethemonoidalstructuretobeniteproductstructureOfcoursethecategoryofwhasdiagonalstructureasdoesanycategorywithniteproductsromdiagonalstructureonecanobtaina
nidempotentnaturaltransformationthatina
nidempotentnaturaltransformationthatinaprecisesensemeasurestheextenttowhichthediagonalstructurefailstobeniteproductstructureThisidempotentallowsustodeneabireectivesubcategoryofthepresheafcategoryasintheprevioussectionandthediagonalstructurefurtherallowsustodeduceresultssuchasthatthemonoidalstructureonthepresheafcategoryrestrictstoniteproductstructureonthebireectivesubcategoryandthattheadjunctionbecomesamonoidaladjunctionDe nitioncategoryisasymmetricmonoidalcategorywhoseunitistheterminalobjectoftogetherwithanaturaltransformationwithcomponencalledthemorphismonsucetal A AAAA A AA AAAA AA A ARAAc AAAB AB A BRABABItisroutinetoverifythatinadiagonalcategorythemapsformanidempotentnaturaltransformationfromidtoidOurleadingexampleofdiagonalstructureisasfolloExampledeneythediagonaltogetherwiththetotalrelationExampleConsiderthesymmetricmonoidalclosedcategorypointedsetsthemonoidalstructurebeingsmashproductAisacategorywithzeromorphismsConsiderancategorywithniteproductsThentheniteproductsdenethesymmetricmonoidalstructureandwemaydenetobethezeromorphismSpecicexamplesofsuchcategoriesarethecategoriesofmonoidsofpointedsetsandofcpo swithbottomandbottompreservingmapsExampleycategorywithniteproductsItiseasytoseethatinExamplesandthestructureisnotthatofniteproductssincethediagram A AAAdoesnotcommuteObservethatinExampleisthestatectidempotenAlsoinExampleandthethreespecicexamplesinistheonlyno
ntrivialidempotentnaturaltransformationo
ntrivialidempotentnaturaltransformationontheidenyfunctorbyPropositionPropositionThedataforadiagonalcgoryformniteprductstrucetaleifandonlyif A AAidRAProofInanycategorywithniteproductsthecompositeofthediagonalwiththeprojectionmustbetheidenGivenAgdenetobe C CCItisroutinetovusingtheequationandtheterminalobjectconditionthattheappropriatetodiagramscommyissimilarusingthethirdofthethreediagonalcommPropositioneadiagonalcgory Thenthefrgoryonthatforesthediagonaldataoftobeniteprducts isgivenbythectierofthenaturaltransformationdeterminedbyProofThisfollowsfromPropositionbecausesendingthediagonaldatatoniteproductstructurenecessitatestheidenticationofwithidandsuchidenticationwiththeadditionofnofurtherobjectsorwsyieldsaniteproductstructureExampleApplyingtheconstructionofPropositiontoExampleyieldsthecategoryofcountablesetsorExampletheconstructionofyieldsacategoryequivttotheunitcategoryTheconstructionoftheofworldsfromthecategoryofcountablesetsgeneralizeseasilytoaconstructiononanysmallcategorywithnitelimitsOnestillacquiresadiagonalcategoryandfollowingthatconstructionbythatofPropositionreturnstheoriginalcategoryoendthissectionwedigressbrieytoobservethatforgeneralreasonsmonoidalstructureongivingrisetoanidempotentnaturaltransformationonidisrestrictedTheargumentgoesasfolloProposition oltzKellyandLairGivenamonoidalcwithunit anyidempextendstoanidempotentnaturaltrmorisinjeProoffortherightidenonehasamonoidhomomorphismII
CCidCidCf fCCrC CIf CI
CCidCidCf fCCrC CIf CIwithrightinerseIISoisamonomorphismThiseasilyrestrictstoidempotenSonaturaltransformationsonidlimitthepossiblemonoidalstructuresInparticularetalRemarkorthecategorythereareexactlytoidempotentnaturaltransformationsididRemarkSoforanymonoidalstructureonwhoseunithasanidempotentonittheunitiseithertheterminalobjectortheinitialobjectsinceotherwisetherewouldbemorethantoendomorphismsontheunitInthissectionwetakeasmalldiagonalcategoryconstructthepresheafcategoryonitandapplytheconstructionofPropositiontotheidempotentoobtainabireectivesubcategoryof Thepresheafcategoryisthefreemonoidalcocompletionofeusethisfacttogetherwiththediagonalstructureontodeducetherelationshipbeteentheinducedmonoidalstructureon andniteproductsinthebireectivsubcategoryItfollowsthatthelatterisafullsubcategoryofthecategoryofemonoidson eonlyrefertoasourbasecategoryinthissectionwhereasforourleadingexamplethebaseisthecategoryofdomainsAllourresultshereextendto infacttheyextendtoanycartesianclosedcompleteandcocompletecategoryifwestartwithasmalldiagonalEvsmalldiagonalcategorycanbeseentriviallyasasmalldiagonalsowecandeduceresultsforourleadingexampleimmedieexpressourresultsonlyintermsofandordinarycategoriesmerelyforeaseofexpositionTheorem ImandKelly easmallsymmetricmonoidalgory Then thefresymmetricmonoidalcompletionofwithsymmetricmonoidalstructuregivenbyleftKanextensionCCY C CopSetCY CYYSpellingthisout iscocompleteandcolimitsAn
explicitformulaforisX YCXthecoequaliz
explicitformulaforisX YCXthecoequalizerinX Y Xu XY CX X YCXoftheevidenomapsXYYxywxgywanduvxywxyRemarkThemonoidalstructureonthesemanticcategory givetalinDenitionagreeswiththemonoidalstructuredeterminedbyPropositionandtheextensionofTheoremtoratherthanwassumeisadiagonalcategoryewritefortheidempotenbethecoidentierdeterminedbExampleConsiderthefollowingdiagramY CChY CSaehaeseenbyPropositionthat isfullyfaithfulwithleftandtadjointequalgivenbysendingtothesplittingofsendsthemonoidalstructureoftoniteproductstructurePropositionThecategory iscartesianclosedandcocompleteSobytheuniversalpropertyof wehaPropositionGiventhediagramabtoniteductsonPropositionoranyfgProofThefamiliesf gandf gbothformnaturaltransformationsfromtoitselfThesetonaturaltransformationsareequalifandonlyiftheirrestrictionsCCareequal thisisimmediatefromthedenitionofasaleftKanextensionSoitsucestoproethatforeacandforXYthemapsZZXZXandZareequal butthatholdsbyaroutinecalculationusingthethirdcommyinthedenitionofdiagonalcategoryewriteforthefullinclusion PropositionoranyfgProofehaidandidandweshowrstthatliesin id butthisfollowsimmediatelyfromPropositionSoSinceandthetadjoinehaSJfSJgJSJfJSJgJgPuttingthistogetherwehaTheoremThefullinclusionsendsniteprductsinthemonoidalstructureof andhasleftandrightadjointsendingtothesplittingof sendingthesymmetricmonoidalstructuretoniteducts So bemonoidaladjunctions etalCorollary
isafullsubgoryofthecgoryofctivecomono
isafullsubgoryofthecgoryofctivecomonoidsinProofrestrictstoniteproductson eachobjectof possessesauniquecommecomonoidstructurefaithfulnessisobIntheparticularcaseofExamplecalculationoftheformulaforealsthatispreciselythecategoryofcommecomonoidsinthissectionweaddressclosedstructureNoneofourresultsherestrictlyrequiresthefactthatwehaeabireectivesubcategory infacttheydonotrequirewehaepresheaeseitherHoertheleadingexampleisasthroughthecourseofthepapertheinclusionofto Recallfromtheprevioussectionthatgivenasmallsymmetricmonoidalcategorythecategory isthefreesymmetricmonoidalcocompletionofInfactmoreistrue issymmetricmonoidalclosedThatresulttogetherwithallouranalysisoftheprevioussectionextendstosmallsymmetricmonoidalcategoriesproallypresentableasacsee Thecategoryofdomainsissuchacategorysoforgeneralreasons issymmetricmonoidalclosedoproetheresultsofthissectionweconsideramoregeneralsituationPropositionesymmetricmonoidalclosed withafullinclusionwithleftadjointeservingsymmetricmonoidalstructuruptocentisomorphism ThenisanexponentialidealofProofItsucestoshowthatforanAJXliesintheimageofoseethatapplyYonedatothefollowingsequenceofnaturalisomorphimsforanAJXAJXFAXFCCJFAXByasimilarcalculationonecanshowthatgivenanyfullcoreectivsubcategoryofasymmetricmonoidalclosedcategoryhthatclosedunderthemonoidalstructureofthenissymmetricmonoidalclosedThisallowsustodeduceTheoremeafullrctiveandcctivesubgoryofsymmetricmonoidalclose andassumeisclosedunderthemonoidalstr
uceofandtheleftadjointpreservesthesymme
uceofandtheleftadjointpreservesthesymmetricmonoidalstructurissymmetricmonoidalclosedandisinfactanexponentialidealofPuttingthistogetherwithearlierresultswemayconcludeetalCorollarynysmalldiagonalcesabirctivesub suchthatisaprafcgory hencartesiancloseandanexponentialidealin withboththeinclusionandadjointprservingthesymmetricmonoidalstructurSAbramsky SGa andRNagarajanteractionCategoriesandtheoundationsofTypedConcurrentProgrammingdingsofNAdStudyInstitute InternationalSummerSchool Marktob RFBlute JRBCocett andRAGSeely and StorageastensorialstrengthToappearinMathematicalStructuresinComputerScienceBDaOnclosedcategoriesoffunctorsInSMacLane editor ortsoftheMidwestCategorySeminarolumeofeNotesinMathematicspagesSpringerVerlag FFoltz GMKelly andCLairAlgebraiccategorieswithfewmonoidalbiclosedstructuresornoneJournalofPureandApplie PreydTheaxiomofcJournalofPureandApplie GBImandGMKellyAuniversalpropertyoftheconolutionmonoidalJournalofPureandApplie GMKellyStructuresdenedbynitelimitsintheenrichedcontextICahiersdeTopetGeomDi GMKellyElementaryobservationsoncategoricallimitsBulletinofthealianMathematicalSo GMKellyandRStreetReviewoftheelementsofcategoriesInADoldandBEckmann editors gorySeminarSydney olume ofeNotesinMathematics pagesSpringerVerlag PWOHearn AJPer MTama andRTtSyntacticConofInterferenceRevisitedI