BRICSLS-98-2C.Butz:RegularCategoriesandRegularLogic - PDF document

BRICSLS-98-2C.Butz:RegularCategoriesandRegularLogic BRICS BasicResearchinComputerScienceRegularCategoriesandRegularLogicCarstenButzBRICSLectureSeriesLS-98-2 ISSN1395-2048October1998 1998,BRICS,DepartmentofComputerScienceUniversityofAarhus.Allrightsreserved.Reproductionofallorpartofthisworkispermittedforeducationalorresearchuseonconditionthatthiscopyrightnoticeisincludedinanycopy.SeebackinnerpageforalistofrecentBRICSLectureSeriespublica-tions.Copiesmaybeobtainedbycontacting:BRICSDepartmentofComputerScienceUniversityofAarhusNyMunkegade,building540DK–8000AarhusCTelephone:+4589423360Telefax:+4589423255Internet:BRICS@brics.dkBRICSpublicationsareingeneralaccessiblethroughtheWorldWideWebandanonymousFTPthroughtheseURLs:Thisdocumentinsubdirectory RegularCategoriesandRegularLogic CarstenButz CarstenButzDepartmentofComputerScienceUniversityofAarhusNyMunkegadeDK-8000AarhusC,DenmarkOctober1998 asicesearchomputerCentreoftheDanishNationalResearchFoundation. NoteshandedouttostudentsattendingthecourseonCategoryTheoryattheDepartmentofComputerScienceinAarhus,Spring1998.ThesenotesweresupposedtogivemoredetailedinformationabouttherelationshipbetweenregularcategoriesandregularlogicthaniscontainedinJaapvanOosten'sscriptoncategorytheory(BRICSLecturesSeriesLS-95-1).Regularlogicistherecalledcoherentlogic.IwouldliketothankJaapvanOostenforsomecommentsonthesenotes. ContentsPrefacev1Prologue12RegularCategories23RegularLogic114ASoundCalculus185TheInternalLogicofaRegularCategory226TheGenericModelofaRegularTheory257Epilogue31 RegularCategoriesandRegularLogic 1PrologueInthesenoteswedescribeindetailtherelationbetweenregularcategoriesandregularlogic,thelatterbeingthefragmentofrstorderlogicthatcanexpressstatementsoftheform)),wherearebuiltupusingatomicformulae,thetruthconstant,binarymeetsandexistentialquantication.Aregularcategoryisacategorywithallnitelimitsinwhicheveryarrowcanbefactoredasaregularepimorphismfollowedbyamonomorphism.Intuitively,theobjectarisinginthisfactorisationistheofthemap.Aregularfunctorbetweensuchcategoriesisafunctorthatpreservesallthisstructure.ThisgivesthecategoryRegCatsmallregularcategories.Insucharegularcategorywecaninterpretsignaturesandextendsuchinterpretationstoallregularformulae,i.e.,tothoseformulaebuiltupusingatomicformulae,binarymeetsandexistentialquantication.Thenaninter-pretationisamodelofasequent))iftheinterpretationof)factorsthroughtheinterpretationof)(necessarilyasamonomor-phism).Withtheappropriatenotionofmapsbetweenmodelsthisgivesforeachtheory(i.e.,foreachsetofsequents)andforeachregularcategoryacategoryMod )ofmodelsof.Thisconstructionisnaturalinbecausearegularfunctorpreservesmodels,sothatwehaveafunctor,foreachtheoryMod RegCatCatfromsmallregularcategoriestosmallcategories.Animportantpointisthatthisfunctorisrepresentable(inaweaksense)byasmallregularcategory),thatis,thereareequivalencessuchthatforeachfunctorC!DthediagramMod Mod T;F =// RegCat Mod (D)=// RegCatcommutes.Therepresentingobjectisconstructedwiththehelpofasmallcalculusforregularlogic,soundforinterpretationsinregularcategories.Inparticular,applyingtheaboveequivalencetotheidentityfunctor,thisrep-resentingcategorycontainsamodel,genericinthesensethat RegularCategoriesandRegularLogic asequentissatisedbyifandonlyifitisderivablefrominthecalcu-lus.Hencewehaveacompletenesstheoremforregularlogicwithrespecttoregularcategories.Ourtreatmentofthematerialisfairlydetailed,butwehopethatthereaderdoesnotgetbored.Weincludedallthoseminorfacts,almosttrivial,thatonereallyhastocheck.Forthelogicianamongthereadersweshouldstressthattheresultsofthisnoteareanoftherelationshipbetweenlogicandcategorytheory.Theypresentthekindofresultsonecanget.2RegularCategoriesbeacategory,anobjectofsubobjectisanequivalenceclassofmonomorphismswhereareisomorphicover,i.e.,thereshouldbeanisomorphismsothat commutes.Wewritetoindicatethatisasubobjectof,leavingoftenunspeciedalthoughitisanimportantpartofthedata.Sub(denotestheclassofsubobjectsof,beingpartiallyorderedbyandonlyiffactorsthrough.Thispartiallyorderedclasshasalargestelement,representedbytheidentitymapidDenition2.1Wesaythatwell-poweredSub(isasetforallobjectsSupposenowthathaspullbacks.ThenforalltheclassSub()isameet-semilattice,themeetofbeingrepresentedbythecompositionarisinginthepullbacksquare _ B_ A// X Ofcourse,regularlogiciscompletewithrespecttomodelsintheregularcategoryofsets,thankstoGodel'scompletenesstheoremforrst-orderlogic.Butsimilarresultsasinthisnoteholdforotherpairsoflogics/categories,whereonecannotappealtoknownresultsoflogic. 2.RegularCategories Moreover,foreacharrow,pullbackalonginducesameet-preservingmap:Sub(Sub((Sometimesonewritesinsteadof.)Theobjectiscalledtheinverseimageof.Ifiswell-poweredtheresultingfunctorSub((orthefunctorSub(tothecategoryofmeet-semilattices)iscalledthesubobjectfunctorDenition2.2Acategoryiscalledifithasallnitelimits,ifco-equalisersofkernelpairsexist,andifregularepimorphismsarestableunderpullbacks.Herethekernelpair()ofanarrowconsistsofthetwoprojectionsTheimportantpointtonoteaboutregularcategoriesisthatanyar-rowcanbefactoredasaregularepimorphismfollowedbyamonomorphism.Moreover,thisfactorisationisunique(uptoisomorphism).Beforeprov-ingthiswecollectsomeminorfactsaboutregularepimorphismsinsuchacategory:Lemma2.3Letbearegularcategory.(i)Anyregularepimorphismisthecoequaliserofitskernelpair.(ii)Aregularepimorphismwhichismonoisanisomorphism.(iii)Thecompositeoftworegularepimorphismsisagainaregularepimor-(iv)Ifarearrowssuchthatbothareregularepimorphisms,soisProof.Fortherstparttakesomecoequaliser g/ Be// ,constructthekernelpairoftogetf;g BEB2/ 1/ Be// E Wenotethattheliteratureknowsmanydierent`denitions'ofregularcategories.Buttheyallgivethesameclassofcategoriesprovidedtherequestfornitelimitsis RegularCategoriesandRegularLogic anddenotebythecoequaliserof.Sinceisauniquemapsatisfying.Sincecoequalisesitsatises,sothatwendamapsuchthatClearly,areinversesofeachother.ForthesecondpartnotethatsinceisthecoequaliserofitskernelpairandsinceismonothatTherefore,theidentitymapidcoequalisesthemandisinfactasplitmonomorphismandanepimorphism,henceanisomorphism.Nowletbetworegularepimorphisms.Werstlookatthepullbackdiagram  EE0B/  Be BE0E//   EE0E/  Ee0 Be// Ee0// toseethatthecanonicalmapisepi,asitisthecompositeoftwo(regular)epimorphisms.Nextweusethediagram BE0BeE0e/// q1 q2 EE0Ep1 p2 Be// Ee0// toshowthatisthecoequaliserofthetwoparallelarrows(projections).Notethat.Moreover,)isthekernelpairofand(q;q)isthekernelpairofissuchthatcoequalisesthekernelpairofsowendauniquesatisfying.Usingthatisepiandcommutativityofthediagramaboveweseethat,sothatthereisauniquearrowthatsatises.Thus,isthecoequaliserofFinally,letbetwoarrowssuchthatbothareregularepimorphisms.Writeforthekernelpairforthekernelpairof.(Notethat,asabove, 2.RegularCategories )andsimilarforthesecondprojections.)Ifamapsuchthatcoequalises,sothatwendasatisfyinghgf.Sinceisepi,Proposition2.4Inaregularcategoryeacharrowcanbefactoredasaregularepimorphismfollowedbyamonomorphism.Moreover,foreachcom-mutativediagram e Ym X0g// witharegularepimorphismandamonomorphismthereexistsa(unique)makingbothtrianglescommute.Inparticular,theregularepi{monofactorisationisuniqueuptoisomorphism.Proof.Forthefactorisationletbearbitraryandcoequaliserofthekernelpair.Inparticular,thereexistsauniquearrowsuchthat.Wehavetoshowthatismono.bethekernelpairof.Sincethereexistsauniquearrowsuchthatasin p2/ b&MMMMMMMMMMXf// e????????YEYEq1/ q2/ AsintheproofofLemma2.3weobservethatisanepimorphism.Butbecause,whichimpliesthatisamonomorphism:Indeed,fortwoparallelarrowsg;hwiththearrowg;hexists,andg;hg;hToshowthesecondpartconsiderthekernelpair()of.Sincemfpmfpismonowededuce,andthereexistsauniquemapsuchthat.Thenmdebecauseisepi. RegularCategoriesandRegularLogic beanarrowinaregularcategory.Themonomorphismarisinginthefactorisationofiscalledthe(direct)imagedenotedIm().SometimeswealsosaythatistheimageofTheimageisonlyuniqueuptoisomorphism,butdeterminesauniquesub-objectofwhichisdenoted).WeoftenconfuseIm()andtheobjectMoregenerally,forasubobjectwedene):=Im(whichgivesawell-denedmap:Sub(Sub(Lemma2.5Letbeanarrow.(i)Themapismonotoneandleft-adjointtothepullbackfunctorthatis,:Sub((ii)Ifisanotherarrowthen:Sub(Sub(Proof.FormonotonicityoftakeinSub().Wefactoriserstandthen(!)togetthediagram YB?OO // 9fB?OO B0// ?OO J*0GG ?OO Thefactorisationisaregularepi{monofactorisationofrepresentsForadjointnesswetakesubobjectsandlookatthesolidarrowsin Yf�1B?OO / B?OO A3FF/// ={{{{{9fA��}}}}3 FF 6 2.RegularCategories thenthedashedmapexistsandtheoutersquarecommutes,hencethedashedarrowexists(beingamonomorphismis),and.Conversely,ifwecanjustfactorisethemaptogettheimageof,inparticular,thisimagethenfactorsthroughFinally,theidentityholdssinceitjuststateshowleft-adjointscompose.AsaconsequenceweobservethatpreservesallmeetswhichexistinSub().Nextweprovetheso-calledFrobeniusidentityLemma2.6Letbearegularcategory,anarrowandtwosubobjects.Then,assubobjectsofProof.Noterstthatisobtainedasthepullbackof,whichfactorsasasinthetoppartof 9fX// YA(55kkkkkkkkkk/// 9fA(55kkkkkkkkA^f�1B?OO / 9fA^B?OO // ?OO Thus,pullingbackwerstget(asasubobjectofandthen.Themapisaregularepimorphism,sinceitisthepullbackofone,soistheregularepi{monofactorisationofandwegettheequalityofthelemma.Nextweshowhowwecancodeanarrowbyitsgraphsubobjectof.Wedeneastheimageofthemap(observethatthecanonicalmap)isanisomorphism).Letdenotetheprojectionsrespectively. RegularCategoriesandRegularLogic Lemma2.7Letbeasubobjectof.ThenLogically,thelemmasaysthat,forapredicateoftheinterpretationofProof.Againthisisamatteroflookingattherightdiagram,whereallsquaresarepullbacksquares,constructedfromrighttoleft(weendupwiththemonomorphismsincethecompositehorizontalmaptheidentity): f)// XYX/ YO XA?OO // �1XA^graph(f)?OO // �1XA?OO / ?OO Themap)isaregularepimorphism,sowegettheimagefactorisationofastheimagefactorisationofthemapcomposedwith)(hereweuseLemma2.3(iii)).Inparticular,theidentityofthelemmaholds.Wenotethatwecanrecoverfromitsgraph:Asubobjectmaybeseenasarelationbetween`elements'of.Calltotal(whichmeansintuitivelythatthesetofallsuchthatthereexistssomewithxRyequals);andfunctionalifthecanonicalarrowfactorsthroughtheinclusionid.(Sincerepresentsthe`object'oftriples(x;y)suchxRyxRythismeansintuitivelythatfromxRyxRyshouldbeabletodeduceThegraphofanarrowisatotalandfunctionalrelationon(ExerciseE.3).Lemma2.8Foreachtotalandfunctionalrelationthereexistsauniquearrowsuchthat=graph( 2.RegularCategories Proof.istotalthecompositeisaregularepimor-phismwhichisthecoequaliserofitskernelpair()constructedin _ p2' p1 'OOOOOOYXR/  R_r P=XYX
Y(QQQQQQQRXY//  XYY1;3/ 1;2 XYX Y5555555555555555Rr// XYX/ (Thedashedfactorisationexistssinceisfunctional.)Thetwowaysfromarejusttheprojectionontothesecondcoordinate,hencewededucecoequalisessothatthereexistsauniquesatisfyingToshowthat=graph()welookatthefollowingdiagramwheretherightpartisjustthedenitionofgraph( $$$ XYX/ X// hX;fi& f4iiiiiiiiiiiiiiiiiiiiiiiiUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUf)// XYYO X Thecompositeisaregularepimorphismandwecanobtain)aswellastheregularepi{monofactorisationofthecomposite.ButthismapistheuniquesuchmapwhencomposedwithgivesandwhencomposedwithgivesSothismaphastobethemono,theimageofwhichisAnotherimportantfactonehastoknowisthebehaviourofimagesofmapsthatarerelatedbyapullbacksquare: RegularCategoriesandRegularLogic Lemma2.9Let f0 Xf Zg/ beapullbacksquare.Then:Sub(Sub(Proof.Forasubobjectwelookatthefollowingcube,wherethefrontisourpullbacksquare,theleftsideisapullbacksquareobtainedfrompullingback,thebottomfactistheimagefactorisationsof,whiletherightsideandthebacksideareagaintwopullbacksquares: // ?????????f�19gA  ?????????ZYXf0 / Xf A // ??????????9gAo?????????Zg// )thefactorisationregularepi{monofactorisationof,sowegettheequalityasstatedinthelemma.CallafunctorC!Dbetweenregularcategoriesregularifitpreservesnitelimitsandcoequalisersofkernelpairs(thelattermakessensesincepreservesnitelimits).WedenotebyRegCatthecategorywithobjectsthesmallregularcategoriesandarrowsregularfunctors.Sinceinacategorywithpullbacksanarrowismonoifandonlyif id f f/  3.RegularLogic isapullbacksquare,anyregularfunctorC!Dinduces,foreachobject,amap(orjust:Subwhichpreservesnitemeetsandthetopelement.Inparticular,itismono-tone.Moreover,ifandifbecausepreservespullbacks.OnewaytosumthisupistosaythatinducesanaturaltransformationInaddition,preservesbydenitionaswellimages:Ifisgiven)).Forthisjustrememberthatwasdenedasthecoequaliserofthekernelpairof.Fortherecord:Lemma2.10Aregularfunctorbetweenregularcategoriespreservesbothinverseimagesand(direct)images.3RegularLogicRegularlogicisroughlyspeakingthe{fragmentofmany-sortedrst-orderlogic.Althoughwedonotprovethisherewementionthatthereisnodierencebetweentheintuitionisticandtheclassicalversion.A(typed)signatureconsistsofasetofbasicsorts(basictypes)sort ,andofsetsconst ,funct andrel oftypedconstants,functionandrelationsymbols.WewriteexpressionslikeX;f


Y;R


toindicatethetypingofthesesymbols.(Notethat


isjustaformalexpressionthatshouldsupportourintuition!)Weusuallyabbreviate


Givenasignature,the)(orbetter),wherethesuperscriptstandsforregular)consistsofthesignature,foreachsort Therearesomeconventionsonhowtodenotelogics.Heretherstsubscriptstandsforthefactthatthereareonlyniteconjunctions,i.e.,conjunctionsoversetsofcardinalitylessthen.Thesecondstandsforthefactthatthereareonlyniteblocksofquantiers(inourcase,ofexistentialquantiers). RegularCategoriesandRegularLogic acountablelistofvariables(wewritetoindicatethatisavariableoftype,aswell,hasitsobviousmeaning),andthesetsof(typed)termsandformulaedenedasfollows:(T1)isatermoftype,providedthatisavariableoftype(T2)Similarly,aconstantisatermoftype(T3)If;:::;taretermsoftype;:::;Xrespectivelyandif


isafunctionsymbolthen;:::;t)isatermoftype(F1)Ifaretermsoftype(orbetter:)isaformula.(F2)If;:::;taretermsoftype;:::;Xrespectively,andif


isarelationsymbol,then;:::;t)isaformula.(thelogicalconstant`true')isaformula.(F4)Ifareformulaesoareavariableofsometype).Foraformula,FV()denotesthethesetoffreevariablesoftheory(formulatedin))isasetofsequentswhereare(regular)formulae.ThelatterisshorthandedtoBelowwewilldeneinterpretationsofalanguage)inaregularcategory.Thenwillbeamodelofassubobjectsof


(andthesetofvariablesoccurringfreeineither).Inthisisequivalenttosaythatisamodelof),andindeed,thisistheintuitiononeshouldhaveaboutasequentRemark3.1Inwhatfollowswewillalwaysassumethataregularcategorycomesequippedwithachoiceofaterminalobject,ofbinaryproductsandofequalisers.Usingbracketingfromlefttoright,asequence


means(


.Ofcourse,tomakesuchchoiceswehavetousetheAxiomofChoice.Tobeprecise,aninterpretation)inaregularcategoryconsists|anobjectforeachbasicsortsort S; 3.RegularLogic |anarrowforeachconstant|anarrow


foreachfunctionsymbol


infunct |andasubobject


foreachrelationsymbol


Wewritefortheproduct


.Thisinterpretationofthesignatureisextendedtoalltermsandallformulae.Toatermoftypewithfreevariablesamongweassignanarrow,toaformulawithfreevariablesamongweassignasubobjectasfollows:(T1)Ifisavariableoftypeisthecomposite.(Hereweareveryspecic:isinterpretedbytheidentity,theprojectionisneededtohandlethe`dummy'variablesoccurringin.Notethat,byassumption,thefreevariablesoftheterm(i.e.,thevariable)arecontainedinthelist(T2)Ifisaconstantthenisthecomposite(T3)Let


beafunctionsymbol,termsoftype.Byinduction,thetermsareinterpretedbyarrows;:::;t))isinterpretedbythecomposite;:::;t;:::;t X(M)f(M)/ istheequaliserof t2(z)(M)/ ;:::;tisthesubobjectofdenedbythepullback  ;:::;t R(M)// �jg;andnally,,whereistheprojectionAninterpretationiscalledamodelofasequent)ifassubobjectsof,where RegularCategoriesandRegularLogic isthetuple(orset)ofthosevariablesoccurringfreeeitherinorin.Theinterpretationisamodelofatheory)ifitisamodelofeachsequentinExample3.2bethesignaturewithsortsX;Y;Zandthreefunctionsymbols.Aninterpretationinaregularcategoryisamodelof))ifandonlyif.Indeed,isthesubobjectdeterminedbytheequaliserofthetwoarrows g(M)f(M)/ ;and))iInthenextsectionwewillgiveasound(andcomplete)calculusforregularlogic.Herewehavetoprovetwotechnicallemmaswhichgiveinformationaboutdummyvariablesandaboutsubstitution.Lemma3.3Letbeaformulawithfreevariablesamongthetuple.Writemoreoverfortheprojectionto.ThenProof.WerstnotethatfortermsoftypeoneprovesbyinductionthatThenweproceedagainbyinduction,thistimeon.Thecasetrivial,while)followssincewehavethediagram  Y(M)Z(M) / Y(M) fzjR(t)g(M)//  Z(M)ht(M)i / 1R// whereallsquaresarepullbacks.Thecaseholds(byinduction)becausepreservesbinarymeets.Forwecalculatey;x;(bydenition)(byinduction)(byLemma2.9) 3.RegularLogic whereweusedthepullbacksquare XZ Y(M)Z(M)Z X(M)Z(M)q// Thecaseholdssincetakingtheequaliseroftwoarrowscommuteswithtakingproducts.Asaconsequenceoftheproofwecanalwayswriteinsteadofsincethesearrowsareuniquelydeterminedprovidedweknowwhattheyareinthecase=FV(Lemma3.4Letbeaformulawithfreevariablesamongz;yandletatermoftype(withfreevariablesamong)whichissubstitutablefor,i.e.,aftersubstitutionnofreevariableofbecomesboundin;thatis,thereisapullbacksquare // Z(M)hb(M);Z(M)i f(z)j g(M)// Proof.Weprovethisbyinductionon,butrstwenotethatforatermoftypewithfreevariablesamong(b=y;(Theproofisbyinductionaswell.)ThenthecasefollowsfromExerciseE.1,whilethecases,andareeithertrivialoreasy.Finally,suppose).Byassumption(issubstitutablein)wehaveFV().Theproofofthiscaseiscontainedinthefollowingcube,whereweassumedforsimplicitythatistheemptysequence.Herethefrontsquareisjustthepullbacksquare,asistheleftside.Thenthemapx;yx;yisfactoredasaregularepimorphismfollowed RegularCategoriesandRegularLogic byamonomorphism,withtheimagebydenition.Thisyieldsthebottomofthecube.Finally,therightsideandthebacksideareobtainedaspullbacksoftheregularepi{monofactorisationalongthemapx;b // ???????????????????(b(M))�1fyg(M)  ????????????????????X(M)h;b(M)i / 1b(M) x;y // ???????????????????fyg(M)???????????????????X(M)Y(M)// Wededucethatthefactorisationx;bx;yistheregularepi{monofactorisationofx;b1,sothatwegettheequalityf
j9x;bx;yWewanttomaketheclassofmodelsofatheoryinaregularcategoryintoanothercategoryMod ).Abetweentwomodelsisafamilyofmapssort whichcommutewiththeinterpretationsofthebasicoperationsinourlanguage,i.e.,for 3.RegularLogic aconstant,afunctionsymbolthediagrams c(N)!CCCCCCCCCX(M)hX X(M)f(M)/ 


 Y(M)hY X(N)X(N)f(N)/ commute;whileforarelationsymbolthecomposite(


shouldfactorthroughtheinclusionByinductiononeprovesthatforallterms)oftype hZ Y(M)hY Z(N)t(N)/ commutes(beingtheproduct


);andforeachformulathecompositefactorsthroughC!Dbearegularfunctorbetweenregularcategories.Ifisamodelofatheory(formulatedin))wegetaninterpretation)of)inasfollows:Wedene|forasortinsort |foraconstant):1|foranarrowinsort |andforarelationsymbol,Astraightforwardinductionshowsthatforalltermsoftypewithfreevariablesamongandnally,againbyinduction,oneprovesthatforallregularformulaetheunderlyinglanguage Strictlyspeaking,doesnotpreservethechosenterminalobjects,thebinaryproductsandtheequalisers.But,forexample,thereisacanonicalisomorphism(1)1.Wesuppressedmentioningexplicitlytheseisomorphisms,butforbeingprecisetheyshouldbepresent.Thepointisthattheydonoharmbecausetheyareunique. RegularCategoriesandRegularLogic Itfollowsthat)isamodelof,sinceforasequenthavepreservestheorderofsubobjects.Moreover,ifisamapinMod )wecanapplypointwisetogetamapsort betweenthemodels)and).Fortherecord:Lemma3.5AregularfunctorC!Dbetweenregularcategoriesinduces,foreachregulartheory,afunctorMod :Mod Mod InparticularwehaveafunctorRegCatCat,fromsmallregularcategoriestosmallcategories.Ontheotherside,ifisaxedmodelofinaregularcategorygetforeachregularcategoryafunctorRegCatMod whichonobjectssendsafunctorE!Dtothemodel)innaturaltransformationissenttothemapbetweenmodelssort Moreover,ifD!CisaregularfunctorthediagramoffunctorsRegCat F(�) Mod (D)FT RegCat Mod commutes.4ASoundCalculusHerewexalanguage)anddeneasequenceofentailmentrelations(ordeductionrelations)betweenformulae,therelationsbeingindexedbynitesetsof(typed)variables.Thesequentisonlydenedifboththefreevariablesofarecontainedinthe.Writingdownexpressionslikethatitisalwaysassumedthatthissideconditionisfullled. 4.ASoundCalculus Thereasonforthesubscriptsisasfollows:Eveninthesimplestcasewhereisaclosedregularformulathereisanessentialdierencebetweenthetwointerpretationsf
j1andsomeinterpretationinaregularcategory).Fromtherstwecanalwaysgetthesecondbypullbackalong!1,buttheseconddoesnotdeterminetherst.Thisisonlythecaseif!isaregularepimorphism,inwhichcasef
j.Thus,ifwewouldhaverestrictedourselvestointerpretationsinwhicheachsortisinterpretedbyanobjectwith`globalsupport'(i.e.,objectssuchthat!isaregularepimorphism),thenwecouldgetridofthesesubscripts.Weremarkaswellthathasglobalsupportifandonlyif,i.e.,ifitistrueinWegrouptheaxiomsandrulesofinferenceforourentailmentrelationsasfollows:1.Structuralrules p`Fr;p`Fq p`Fygq;'(y)`F (y) whereisavariable,isatermoftype,andissubstitutableonbothsides,thatis,nofreevariableofbecomesboundin)and)aftersubstitution.Aswell,implicitlyitisassumedthatFV(2.Logicalrules2.2if;andifboth Inintuitionisticlogicbeinginhabitedisapositivewaytoexpressnon-emptiness. RegularCategoriesandRegularLogic 2.3iffygp;andconverselyififygpthen9y (y)`Fp.3.Rulesforequality3.4foreachfunctionsymbolinthelanguage.(Herestandsofcoursefor3.5foreachrelationsymbolinthelanguage.Wewrite.GivenatheorywewriteT;',wheredenotesentailmentinthecalculusaboveextendedbytheaxiomsFV((FV(Togivethewholeamoresymmetricoutlookwewriteaswellforthefactthat(modulo.ThenstandsforthefactthatmodulotheformulaisprovablyequivalenttotheformulaLemma4.1Letbeatheory,andamodelofinaregularcategory.IfT;'then,assubobjectsProof.Weprovethislemmabyinductionoveraderivationof.Incaseofanaxiomthisispartofthedenitionofbeingamodel.Thecaseoftheaxiom1.1istrivial,while2.1holdssince�jgFortheinductionstepsrule1.2issoundsincetheorderonistran-sitive,while1.3issoundbecausepullbackalongtheprojectioninducesamonotonemapSub(Sub((NotethatLemma3.3entersheredescribing 4.ASoundCalculus Rule2.2issoundbecauseSub(while2.3issoundsinceifandonlyif,fortheprojection(hereweuseLemma3.3again).Soundnessofrule1.4followsfromLemma3.4.Itremainstoprovesoundnessoftherulesforequality.Rule3.1issoundistheequaliserofid,which;rule3.2issoundbecausetheequaliseroffactorsthrough(infact,isequalto)theequaliserofTheinterpretationofisthepullback _p E2;3_i2;3 E1;2i1;2/ X(M)X(M)X(M)1// 2/ 3 2 thecanonicalmapfactorsthroughtheequaliserof,theinterpretationofRule3.4issoundsincethediagonal,theinterpre-tationof,equalises))= X(M)X(M)1// f2/ 2 1 YX(M)O'44iiiiiiiiiiiiiiiiiiii// Finally,forrule3.5,theleftside)isthepullback _p X(M)_
 R(M)X(M) r(M)/ X(M)X(M)1 R(M)r(M)/ X(M): RegularCategoriesandRegularLogic Itfollowsthatfactorsthrough,inparticularthrough,theinterpretationofLetusmentionthatclassicalrst-orderlogicisconservativeoverregularlogic(providedoneallowsemptydomainsaswell),sothatintuitivelyallclausesofthefollowinglemmaaretrue.Weleavetheproofasanexercise,butmentiontheconnectionbetween(i)andLemma2.6.Lemma4.2provideddoesnotoccurfreein5TheInternalLogicofaRegularCategorybearegularcategory.Toitweassociateasignatureandlanguageasfollows:hasasbasicsortstheobjectsofthecategory.Againwexaterminalobjectandforeachnitelistofobjectsofonespeciedproductofthisnitesetofobjects.Thenoursignaturehasforeacharrowoneconstantsymbol,foreacharrowonefunctionsymbol,andforeachsubobjectonerelationsymbol.Obviously,thelanguage)hasacanonicalinterpretation.Wedenetobethetheoryofthisinterpretation,i.e.,thesetofallsequentsinthelanguage)whicharetrueundertheinterpretation.Insteadofonewritesusuallyjust.Inwhatfollows,wedonotdistinguishbetweenasymbol,sayafunctionsymbol,inourlanguageanditsinterpretationinthecategoryWewillrstshowhowdierentcategoricalnotionsarecapturedbytheinternallanguageofthecategory.Lemma5.1Letbearegularcategory.(i)Letbetwoarrowsand.Thenifandonlyif,whereisafreevariableoftype(ii)Anarrowismonoifandonlyif(iii)Thearrowisaregularepimorphismifandonlyif.(Oneshouldreadthisas` 5.TheInternalLogicofaRegularCategory Proof.TherstpartisasinExample3.2.Forthecharacterisationofmonomorphismswelookat XX1/ 2/ Xm thatis,thetoplineisanequaliserdiagramandistheequaliserof.Ifisamonomorphismthentheinclusionequalises,sofactorsthrough.Conversely,ifwehavethisfactorisationthenfortwoparallelarrowsf;gsuchthemapf;gfactorsthrough,inparticularthroughtheinterpretationof.Hencef;gf;gisamonomorphism.Forthelastpartwenoterstthatgraph(istheequaliser(ExerciseE.4),theinterpretationof.ThenwecanapplyLemma2.3tothediagram f*UUUUUUUUUUUUUUUUUUUUUUUUf)// &MMMMMMMMMMMXY2 toseethatisaregularepimorphismigraph(is,whichholdsifandonlyifThefollowinglemmacharacterisesnitelimitsinaregularcategorywiththehelpoftheinternallogic.Lemma5.2Letbeagainaregularcategory.(i)Anobjectisterminali(ii)Twomapsaproductofifandonlyif(iii)Inadiagram Xf/ g/ suchthat,theobjectistheequaliserofifandonlyifisamonoand.Thiscanbeexpressedusingthepredicateassociatedtothesubobject RegularCategoriesandRegularLogic Proof.Foranobjectifandonlyif
:isanisomorphism,whichisequivalenttosaythat!1isamonomorphism.Combiningthiswiththefactthat)ifandonlyif!1isaregularepimorphismwegettherstpartofthelemma.Forthesecondpartnoterstthattheinterpretationof)istheequaliseroff;gf;Gasin ZZ1/ 2/ f;g Then,wehaveseenargumentslikethisbefore,factorsthroughthediagonal(theinterpretationof)ifandonlyiff;gisaMoreover,fromthefollowingdiagramofpullbacks,constructedbyrstpullingbacktherightverticalmapalong1andthenpullingbacktheresultingmap,wededucethatgraph(graph(),theinterpretationofistheimageofthemap   // XZ / Z g)// _ x;z;y _ graph( _ g)_ ZY// graph( XZY/ Thereforef;gisaregularepimorphismifandonlyif f;g %KKKKKKKKKKKXZY Finally,theequaliserof,andbecausethereisafactorisation fxj=g(C)// Xf/ g/ ZOOOO e3gggggggggggggggggggggggggggggg 6.TheGenericModelofaRegularTheory isamonoifandonlyifisamonomorphism(whichisequivalenttobeinganisomorphismsincethismapisaregularepimorphism),ifandonlyifthemonomorphismisaregularepimorphism.ThesetwoconditionsareclearlyequivalenttobeingthecoequaliserofInthepreviouslemmaswesawthatwecancharacteriseallthoseprop-ertiesofacategorywhichmakeitaregularcategory.Thereforeitshouldbenosurprisethatwecandescriberegularfunctorsintermsoflogicaswell.C!Disafunctorbetweenregularcategories(notnecessarilyreg-ular)wegetaninterpretationofthe(functionalpartofthe)signatureasfollows:),forsort ),foranarrowinProposition5.3ThefunctorisregularifandonlyifTobeprecisewehavetorestrictourselvestothosesequentswhichinvolveonlyfunctionsymbols.Proof.OnedirectionisjustpartofLemma3.5.Fortheother,ifisaterminalobjectthen).Then),i.e.,usingtheinternallogicofweseethatisaterminalobject(byLemma5.2again).Inthesamewayoneprovespreservesproductsandequalisers,whichcanbedenedusingthefunctionalpartoftheinternallanguageofacategory.Aswell,preservestheimagefactorisationofamap,henceinparticularthoseregularepimorphismsarisingasthecoequaliserofthekernelpairofanarrow(andthiskernelpair,beingapullback,ispreservedaswell).6TheGenericModelofaRegularTheoryTheaimofthissectionistoshowthatthereisaregularcategory)andequivalencesofcategoriesMod RegCat RegularCategoriesandRegularLogic naturalin.(ThisjustmeansthatthefunctorMod RegCatCatrepresentableinaweaksense.)Inparticular,)willcontainaconserva-tivemodelof,amodelinwhichprovabilityinthecalculusforregularlogicandsatisabilitycoincide.Wex,andconstruct)asasortofLindenbaum{Tarski-category:Objectsareequivalenceclassesofpairs(X;p))whereisanitelistofsorts,andisaregularformula.(Hereiscalledthecontext.)Onlyformulaeinthesamecontextcanbeequivalent.ThenX;p))and(X;p))areequivalentifwhereisasetoffreshvariablesoftype.Wedenoteequivalenceclassesbyandassumethatthecontextisunderstood.(Notethatthisnotionsomehowbindsthevariablesoccurringinisanequivalenceclassofregularformulae{in{context(Y; ))whereisprovablyfunctional:(Intuitivelythissaysthat)isthegraphofafunction.)Again,twosuchformulae{in{contextareequivalentiftheyareprovablyequivalent.Herewedenoteequivalenceclassesbyorsimply|Thecompositionoftwoarrowsg!fisgivenbytheequivalenceclassoftheformula(Weleaveitasanexercisetoshowthatthisisindeedwell-dened.)SummingupthisconstructionwegetasmallcategoryLemma6.1Thecategoryhasnitelimits:(i)Theobjectf
j�g(theequivalenceclassoftheformula{in{context)istheterminalobjectin 6.TheGenericModelofaRegularTheory (ii)Theproductofisgivenbytheobjectwithprojectiontotheequivalenceclass,andsimilarfortheotherprojection.(Hereweusethecommaintoseparatesourceandtarget!)(iii)Theequaliseroftwoparallelarrowsistheobject,withinclusionthe(iv)Giventwoarrowsg!fg!ftheirpullbackistheobjectwithcanonicalprojections.Wenotethatinourcategory)objectsliketheterminalobjectortheproductoftwogivenobjectsareactuallyunique,andnotjustuniqueuptoisomorphism.Thereasonistheequivalencerelationweincorporatedinthedenitionoftheobjectsof).Strictlyspeaking,thiswasnotnecessary(seeExerciseE.13),butitfacilitatesourproofs.Proof.WeargueinformallyandmakeextensiveuseofthepropertiesoftheprovabilityrelationObviously,givenanarbitraryobjectthereisatleastonearrowg!f�
jgnamelytheoneinducedbytheformula{in{context(Iftherearetwoofthem,say,thenbydenitionofbeingarrows,)and).Thelatterisequivalenttosothatbymonotonicityofthedeductionrelation.Byasimilarargumentfortheotherdirectionweconcludethatmoduloisprovablyequivalenttosothattheinducedarrowsareidentical.Forproductsx,andthetwoprojections f1g whereforexampleisgivenby.(Weleaveittothereadertocheckthatbothareindeedarrows).)Ifg!farearrowsin)wedene RegularCategoriesandRegularLogic andclaimthatinducestheuniquearrowsuchClearly,))since.Thenshowsthatistotal.Itisfunctionalbecausewededuce(modulo)from)rst)andthen,andsimilarfor.SummingupweseethatisindeedamapThecompositeisgivenbywithfreevariablesand.Sincewecangetridoftheexistentialquantier,andtheformulaisequivalent(modulo)to)).Because)wededucethatgfisinducedbyaformulaprovablyequiv-alentto,andthisarrowthusequals.Byasimilarargumenttheothertrianglecommutes.Itremainstoshowuniqueness:Thisweprovebyshowingthatif.Theassumptionsforthersttrianglesaythatmodulotheformula)isequivalenttothatis,)andsimilarfor.Thus,againmodulo)isequivalentto)),whichinturnisequivalentto)becauseisfunctionalandtheimageof,i.e.,thetuple()suchthat)isprovablyunique.Weleavethecaseofequalisersasanexercise(ExerciseE.10)andnotethatthedescriptionofpullbacksfollowsfromtheconstructionofpullbacksusingproductsandequalisers(seeExerciseE.2).Theproofofthefollowinglemmaisleftasanexercise:Lemma6.2Anarrowg!f(i)amonomorphismifandonlyif(ii)andaregularepimorphismifandonlyif(iii)Amapisamonomorphismifandonlyif 6.TheGenericModelofaRegularTheory Proposition6.3isaregularcategory.Proof.�FromLemma6.1weknowthat)hasnitelimits.Inapullback fg fxjpgf'g fyjqgf g/ themapisinducedbySupposeisaregularepimorphism,i.e.,).Thenfrom)wededuce),whichisequivalentto)).Modulowecandeducefurther(usingtheassumption))).ByLemma4.2thisisprovablyequivalentto))(weinter-changedtheexistentialquantiers),aswewanted.Soregularepimorphismsarestableunderpullbacks.Thecategory)containsanaturalinterpretationoftheunderlyingwhereissomevariableoftypef�
jg!foreachconstantintheunderlyinglanguage.x;yforeachfunctionsymbolinfunct .WenotethatthisisindeedanarrowinourcategorybyExerciseE.12.Furthermoreweuse


whichiseasilyseentobeasubobjectofAneasyinductionshowsthatforterms)oftypez;yanarrow;andforregularformulae)that RegularCategoriesandRegularLogic Itfollowsthatisamodelof:Ifisasequentinisamonomorphismfrom,sothatindeed.Themodelhastheadditionalpropertythatitisconservative,i.e.,forallsequentsIndeed,ifisamodelofthenin)thereisamonomorphismg!f,sobyLemma6.2(iii),.Fortherecord:Proposition6.4Thecanonicalinterpretationintheregularcategoryisaconservativemodelof.Inparticular,thecalculusgivenaboveiscom-pletewithrespecttointerpretationsin(small)regularcategories.WearenowreadytodenethefunctorsinvolvedintheequivalenceMod RegCat),naturalinThefunctorRegCatMod )isthefunctortheendofSection3,whichsendsafunctortothemodel,andanaturaltransformationtothefamilysort Thefunctor:Mod RegCat)sendsamodeltothefunctorg7!fg!fg7!`theuniquearrowsuchthatgraph((ThemapinthearrowpartexistsbyLemma2.8.Uniquenessofthismapensuresthatwereallygotafunctor.)Soundnessofthecalculusandthefactthatbeingamodelisdenedusingtheinternallogicofprovesthatisaregularfunctor.Amorphismbetweenmodelsgivesrisetoafamilyofmaps(seethediscussionattheendofSection3)whichisnaturalbecauseifwehaveamapthen,sincebothsquaresbelowcommute, 7.Epilogue theouterdoesaswellandthatisjustthenaturalitysquare: hfxjpg/ fxjpg(N) fyy)g(M)hfyy)g/ _ fyy)g(N)_ fyjqg(M)hfyjqg/ Thisfunctorisagainnaturalin:IfD!CisaregularfunctorthenMod (D)FD/ FT RegCat Mod (C)FC/ RegCatcommutes.Weconcludewiththefollowingtheorem:Theorem6.5ThefunctorsinduceanequivalenceofcategoriesMod RegCatnaturalin.Uptoequivalenceanysmallregularcategoryarisesthiswayasthe`classifyingcategory,ofaregulartheorysinceProof.Itisstraightforwardfromtheexplicitdenitionthatbothwaysroundareisomorphictotheidentity.ThesecondpartfollowsfromProposition5.3.7EpilogueInthesenoteswesawacloseconnectionbetweenafragmentofrst-orderlogicandaparticularclassofcategories.Thematerialofthesenotesistreatedmoreorlessdetailed(more`lessdetailed')in[12,2,4].Similarresultsholdforthefollowingpairs: RegularCategoriesandRegularLogic geometric(coherent)logicgeometriccategoriesintuitionisticrst{orderlogicHeytingcategoriesclassicalrst{orderlogicBooleancategorieshigher{orderlogic(elementary)toposestyped{calculuscartesianclosedcategories-partofMartin-Loftypetheorieslocallycartesianclosedcategories.Goodreferencesareforgeometriclogic[8],forrst-orderlogic[4],forhigher{orderlogicandforthe{calculus[7],andforMartinLoftypetheoriesthepaper[11].ThereisanaturalGrothendiecktopologyonaregularcategory,andthesheaedYonedaembeddingintothesheaftoposoverthissitepreservesandre ectsalltheregularstructure(andmore).Thisexplainssimilarresultsasinthisnotefor(innitary)geometriclogicandGrothendiecktoposes,theclassicaltreatmentofwhichiscontainedin[9].Thesituationforrst{orderlogicandGrothendiecktoposesisnotasgood,butstilltherearestrongresults(likecompleteness,etc),seethereferences[6,3,10].TheexistenceofclassifyingtoposesforgeometrictheoriesisalreadyimplicitinthethesisofM.Hakim[5].SheshowedamongotherthingsthattheZariskitoposclassieslocalrings,andtheassociatedetaletoposclassieshenselianlocalrings.Finally,thereisacompletelydierentapproachtocategoricallogic,basedsketches.Roughlyspeaking,asketchisacategoryequippedwithtwoclassesofdiagrams.Amodelofsuchasketchisafunctorintosomecategorysendingtherstclassofarrowstolimit-diagrams,andthesecondtocolimit-diagrams.Thus,asketchissomethinglikeatheory,andrestrictionsontheclassofdiagramsallowed(nite,orniteandonlylimit-diagrams,...)specifyinwhichlanguagethistheoryisformulated.Hereonecouldstartreadingin[2,1].ExercisesLetbeacategorywithnitelimitsand Xp1/ p2/ anequaliserdiagram.Foranarbitraryobjectwegettheequaliserofthetwoparallel Geometriclogicisobtainedbyreplacingregularformulaebythosebuiltfromatomicformulae,thelogicalconstantsand,thebinaryoperationsandandexistentialquantication 7.Epilogue arrows p2idZ/ .ShowthatfortheprojectionShowthatthepullbackandg:canbecon-structedasthepullbackofalongthediagonal
asin  XYfg Z
Z// (Here
istheuniquemapProvethatthegraphofanarrowgivesatotalandfunctionalrelationon(subobjectof)Showthatforanarrowthemonomorphismgraph(istheequaliserofthetwoparallelarrowsProvethatthecategoryofgroupsisregular.Canyoudothesameforthecategoryofrings?Anabeliangrouptorsion-freeifforallnaturalnumbers1andallelements=0implies=0.Showthatthecategoryoftorsion-freeabeliangroups(whichisafullsub-categoryofthecategoryofabeliangroups)isregular.ShowthatTop,thecategoryoftopologicalspaces,hasallnitelimitsandallcoequalisers.Givenanexampleofaregularepimorphismthatisnotstableunderpullbacks.ConcludethatTopisnotregular.Letbea(small)regularcategory,a(small)category.Showthatthefunctorcategory[]isregular.Letbearegularcategory.ProvethattheslicecategoryisagainregularforeachobjectE.10CompleteintheproofofLemma6.1thedescriptionofequalisers.E.11ProvidetheproofofLemma6.2. RegularCategoriesandRegularLogic E.12Verifythatifisafunctionsymbolinsomelanguagethenx;yisamorphismin)from.(Heresomexedtheoryformulatedintheunderlyinglanguage.)E.13Deneacategory)similaras),butusingasobjectsformulae{in{contextsinsteadofequivalenceclassesthereof.Provethat)isequivalent).Canyouuseforarrowsaswellformulae{in{contextsinsteadofequiva-lenceclasses?Inthenextcoupleofexerciseswedevelopsomeforcingsemanticsforregularlogic.Wextheinternallogic)ofaregularcategory.Forasortgeneralisedelementatstageisanarrow.Foraformula)withfreevariableandageneralisedelementwesaythatforces)(insymbols:factorsthrough,i.e.,ifinSub().Thisdenitionextendsimmediatelytoformulaewithmorefreevariables.ForanarrowwewriteforthegeneralisedelementatstageE.14Provethefollowingtwopropertiesoftheforcingrelation:(i)(Monotonicity.)If)thenforany,also(ii)(Localcharacter.)Ifisaregularepimorphismandthen).[Hint:Showrstthatpulledbackalongisanisomorphismanddeducethenthatpulledbackalongisalreadyanisomorphism.]E.15Showthattheforcingrelationinaregularcategoryobeysthefollowingrules:alwaysholds.)ifandonlyif)andy;)ifandonlyifthereexistsaregularepimorphismandageneralisedelementsuchthat;(iv)Supposethatandaretermsoftypewithfreevariable.Showthat)ifandonlyifE.16Extendtheforcingrelationtosequents)ifforallarrows,if)then(i)Showthat)ifandonlyifforallandallgeneralisedelements(ii)Concludethatforasequentiforalland,if)then REFERENCES [1]M.BarrandC.Wells.CategoryTheoryforComputingScience.PrenticeHall,NewYork1990.[2]F.Borceux.HandbookofCategoricalAlgebra.Vol.1,2and3.CambridgeUniversityPress,Cambridge1994.[3]C.ButzandP.T.Johnstone.Classifyingtoposesforrst-ordertheories.Ann.PureAppl.Logic,91:33-58,1998.[4]P.FreydandA.Scedrov.Categories,Allegories.North{Holland,Ams-terdam1990.[5]M.Hakim.ToposAnnelesetSchemasRelatifs.Springer{Verlag,Berlin[6]P.Johnstone.Openmapsoftoposes.ManuscriptaMath.31:1980,no.1-3,217{247.[7]J.LambekandP.J.Scott.IntroductiontoHigherOrderCategoricalLogic.CambridgeUniversityPress,Cambridge1986.[8]S.MacLaneandI.Moerdijk.SheavesinGeometryandLogic.Verlag,NewYork1992.[9]M.MakkaiandG.Reyes.FirstOrderCategoricalLogic.Verlag,Berlin1977.(LectureNotesinMathematics611.)[10]E.Palmgren.Constructivesheafsemantics.Math.LogicQuart.no.3,321{327.[11]R.A.G.Seely.LocallyCartesianclosedcategoriesandtypetheory.Math.Proc.CambridgePhilos.Soc.95:1984,no.1,33{48.[12]J.vanOosten.BasicCategoryTheory.BRICSLectureSeries,LS-95-1,January1995. RecentBRICSLectureSeriesPublicationsLS-98-2CarstenButz.RegularCategoriesandRegularLogic.OctoberLS-98-1UlrichKohlenbach.ProofInterpretations.June1998.LS-97-1JanChomickiandDavidToman.TemporalLogicinInforma-tionSystems.November1997.viii+42pp.FullversionappearsinChomickiandSaake,editors,LogicsforDatabaseandInfor-mationSystems,3:31–70,KluwerAcademicPublishers,1998.LS-96-6TorbenBrauner.IntroductiontoLinearLogic.December1996.iiiv+55pp.LS-96-5DevdattP.Dubhashi.WhatCan'tYouDoWithLP?1996.viii+23pp.LS-96-4SvenSkyum.ANon-LinearLowerBoundforMonotoneCircuitSize.December1996.viii+14pp.LS-96-3KristofferH.Rose.ExplicitSubstitution–Tutorial&SurveySeptember1996.v+150pp.LS-96-2SusanneAlbers.CompetitiveOnlineAlgorithms.September1996.iix+57pp.LS-96-1LarsArge.External-MemoryAlgorithmswithApplicationsinGe-ographicInformationSystems.September1996.iix+53pp.LS-95-5DevdattP.Dubhashi.ComplexityofLogicalTheories.September1995.x+46pp.LS-95-4DanyBreslauerandDevdattP.Dubhashi.CombinatoricsforComputerScientists.August1995.viii+184pp.LS-95-3MichaelI.Schwartzbach.PolymorphicTypeInference.June1995.viii+24pp.LS-95-2SvenSkyum.IntroductiontoParallelAlgorithms.June1995.viii+17pp.SecondEdition.LS-95-1JaapvanOosten.BasicCategoryTheory.January1995.vi+75