CategoryTheoryFirstpublishedFriDec6,1996substantiverevisionFriOct3,2014

Categorytheoryhascometooccupyacentralpositionincontemporarymathematicsandtheoreticalcomputerscience,andisalsoappliedtomathematicalphysics.Roughly,itisageneralmathematicaltheoryofstructuresandofsystemsofstructures.Ascategorytheoryisstillevolving,itsfunctionsarecorrespondinglydeveloping,expandingandmultiplying.Atminimum,itisapowerfullanguage,orconceptualframework,allowingustoseetheuniversalcomponentsofafamilyofstructuresofagivenkind,andhowstructuresofdifferentkindsareinterrelated.Categorytheoryisbothaninterestingobjectofphilosophicalstudy,andapotentiallypowerfulformaltoolforphilosophicalinvestigationsofconceptssuchasspace,system,andeventruth.Itcanbeappliedtothestudyoflogicalsystemsinwhichcasecategorytheoryiscalledcategoricaldoctrinesatthesyntactic,prooftheoretic,andsemanticlevels.Categorytheoryisanalternativetosettheoryasafoundationformathematics.Assuch,itraisesmanyissuesaboutmathematicalontologyandepistemology.Categorytheorythusaffordsphilosophersandlogiciansmuchtouseandreflectupon.

1.GeneralDefinitions,ExamplesandApplications

2.BriefHistoricalSketch

3.PhilosophicalSignificance

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1.GeneralDefinitions,ExamplesandApplications1.1DefinitionsCategoriesarealgebraicstructureswithmanycomplementarynatures,e.g.,geometric,logical,computational,combinatorial,justasgroupsaremanyfacetedalgebraicstructures.Eilenberg&MacLane(1945)introducedcategoriesinapurelyauxiliaryfashion,aspreparationforwhattheycalledfunctorsandnaturaltransformations.Theverydefinitionofacategoryevolvedovertime,accordingtotheauthor'schosengoalsandmetamathematicalframework.Eilenberg&MacLaneatfirstgaveapurelyabstractdefinitionofacategory,alongthelinesoftheaxiomaticdefinitionofagroup.Others,startingwithGrothendieck(1957)andFreyd(1964),electedforreasonsofpracticalitytodefinecategoriesinsettheoreticterms.

Analternativeapproach,thatofLawvere(1963,1966),beginsbycharacterizingthecategoryofcategories,andthenstipulatesthatacategoryisanobjectofthatuniverse.Thisapproach,underactivedevelopmentbyvariousmathematicians,logiciansandmathematicalphysicists,leadtowhatarenowcalledhigherdimensionalcategories(Baez1997,Baez&Dolan1998a,Batanin1998,

Leinster2002,Hermidaetal.2000,2001,2002).Theverydefinitionofacategoryisnotwithoutphilosophicalimportance,sinceoneoftheobjectionstocategorytheoryasafoundationalframeworkistheclaimthatsincecategoriesaredefinedassets,categorytheorycannotprovideaphilosophicallyenlighteningfoundationformathematics.Wewillbrieflygooversomeofthesedefinitions,startingwithEilenberg's&MacLane's(1945)algebraicdefinition.However,beforegoinganyfurther,thefollowingdefinitionwillberequired.

Definition:Amappingewillbecalledanidentityifandonlyiftheexistenceofanyproducteoreimpliesthate=ande=Definition(Eilenberg&MacLane1945):AcategoryCisanaggregateObofabstractelements,calledtheobjectsofC,andabstractelementsMap,calledmappingsofthecategory.Themappingsaresubjecttothefollowingfiveaxioms:

(C1)Giventhreemappings , and ,thetripleproduct ( )isdefinedifandonlyif( ) isdefined.Wheneitherisdefined,theassociativelaw

( )=( )

holds.Thistripleproductiswritten .

(C2)Thetripleproduct isdefinedwheneverbothproducts and aredefined.

(C3)Foreachmapping,thereisatleastoneidentitye suchthate isdefined,andatleastoneidentitye suchthate isdefined.

(C4)Themappinge correspondingtoeachobjectXisanidentity.

(C5)ForeachidentityethereisauniqueobjectXofCsuchthate =e.

AsEilenberg&MacLanepromptlyremark,objectsplayasecondaryroleandcouldbeentirelyomittedfromthedefinition.Doingso,however,wouldmakethemanipulationoftheapplicationslessconvenient.Itispracticallysuitable,andperhapspsychologicallymoresimpletothinkintermsofmappingsandobjects.ThetermaggregateisusedbyEilenberg&MacLanethemselves,presumablysoastoremainneutralwithrespecttothebackgroundsettheoryonewantstoadopt.

Eilenberg&MacLanedefinedcategoriesin1945forreasonsofrigor.Astheynote:

Itshouldbeobservedfirstthatthewholeconceptofacategoryisessentiallyanauxiliaryoneourbasicconceptsareessentiallythoseofafunctorandofnaturaltransformation().Theideaofacategoryisrequiredonlybythepreceptthateveryfunctionshouldhaveadefiniteclassasdomainandadefiniteclassasrange,forthecategoriesareprovidedasthedomainsandrangesoffunctors.Thusonecoulddropthecategoryconceptaltogetherandadoptanevenmoreintuitivestandpoint,inwhichafunctorsuchasHomisnotdefinedoverthecategoryofallgroups,butforeachparticularpairofgroupswhichmaybegiven.Thestandpointwouldsufficeforapplications,inasmuchasnoneofourdevelopmentswillinvolveelaborateconstructionsonthecategoriesthemselves.(1945,chap.1,par.6,p.247)

Thingschangedinthefollowingtenyears,whencategoriesstartedtobeusedinhomologytheoryandhomologicalalgebra.MacLane,Buchsbaum,GrothendieckandHellerwereconsidering

1 2 3 3 2 1

3 2 1

3 2 1 3 2 1

3 2 1

3 2 1 3 2 2 1

1 1

2 2

X

X

categoriesinwhichthecollectionsofmorphismsbetweentwofixedobjectshaveanadditionalstructure.Morespecifically,givenanytwoobjectsXandYofacategoryC,thesetHom(X,Y)ofmorphismsfromXtoYformanabeliangroup.Furthermore,forreasonsrelatedtothewayshomologyandcohomologytheoriesarelinked,thedefinitionofacategoryhadtosatisfyanadditionalformalproperty(whichwewillleaveasideforthemoment):ithadtobeselfdual.Theserequirementsleadtothefollowingdefinition.

Definition:AcategoryCcanbedescribedasasetOb,whosemembersaretheobjectsofC,satisfyingthefollowingthreeconditions:

Morphism:ForeverypairX,Yofobjects,thereisasetHom(X,Y),calledthemorphismsfromXtoYinC.IffisamorphismfromXtoY,wewritef:XY.Identity:ForeveryobjectX,thereexistsamorphismid inHom(X,X),calledtheidentityonX.

Composition:ForeverytripleX,YandZofobjects,thereexistsapartialbinaryoperationfromHom(X,Y)Hom(Y,Z)toHom(X,Z),calledthecompositionofmorphismsinC.Iff:XYandg:YZ,thecompositionoffandgisnotated(gf):XZ.

Identity,morphisms,andcompositionsatisfytwoaxioms:

Associativity:Iff:XY,g:YZandh:ZW,thenh(gf)=(hg)f.Identity:Iff:XY,then(id f)=fand(fid )=f.

Thisisthedefinitiononefindsinmosttextbooksofcategorytheory.Assuchitexplicitlyreliesonasettheoreticalbackgroundandlanguage.Analternative,suggestedbyLawvereintheearlysixties,istodevelopanadequatelanguageandbackgroundframeworkforacategoryofcategories.Wewillnotpresenttheformalframeworkhere,foritwouldtakeustoofarfromourmainconcern,butthebasicideaistodefinewhatarecalledweakncategories(andweakcategories),andwhathadbeencalledcategorieswouldthenbecalledweak1categories(andsetswouldbeweak0categories).(See,forinstance,Baez1997,Makkai1998,Leinster2004,Baez&May2010,Simpson2011.)

Alsointhesixties,Lambekproposedtolookatcategoriesasdeductivesystems.Thisbeginswiththenotionofagraph,consistingoftwoclassesArrowsandObjects,andtwomappingsbetweenthem,s:ArrowsObjectsandt:ArrowsObjects,namelythesourceandthetargetmappings.Thearrowsareusuallycalledtheorientededgesandtheobjectsnodesorvertices.Followingthis,adeductivesystemisagraphwithaspecifiedarrow:

(R1)id :XX,

andabinaryoperationonarrows:

(R2)Givenf:XYandg:YZ,thecompositionoffandgis(gf):XZ.

Ofcourse,theobjectsofadeductivesystemarenormallythoughtofasformulas,thearrowsarethoughtofasproofsordeductions,andoperationsonarrowsarethoughtofasrulesofinference.Acategoryisthendefinedthus:

X

Y X

X

Definition(Lambek):Acategoryisadeductivesysteminwhichthefollowingequationsholdbetweenproofs:forallf:XY,g:YZandh:ZW,

(E1)fid =f,id f=f,h(gf)=(hg)f.

Thus,byimposinganadequateequivalencerelationuponproofs,anydeductivesystemcanbeturnedintoacategory.Itisthereforelegitimatetothinkofacategoryasanalgebraicencodingofadeductivesystem.Thisphenomenonisalreadywellknowntologicians,butprobablynottoitsfullestextent.AnexampleofsuchanalgebraicencodingistheLindenbaumTarskialgebra,aBooleanalgebracorrespondingtoclassicalpropositionallogic.SinceaBooleanalgebraisaposet,itisalsoacategory.(NoticealsothatBooleanalgebraswithappropriatehomomorphismsbetweenthemformanotherusefulcategoryinlogic.)Thusfarwehavemerelyachangeofvocabulary.Thingsbecomemoreinterestingwhenfirstorderandhigherorderlogicsareconsidered.TheLindenbaumTarskialgebraforthesesystems,whenproperlycarriedout,yieldscategories,sometimescalledconceptualcategoriesorsyntacticcategories(MacLane&Moerdijk1992,Makkai&Reyes1977,Pitts2000).

1.2ExamplesAlmosteveryknownexampleofamathematicalstructurewiththeappropriatestructurepreservingmapyieldsacategory.

1. ThecategorySetwithobjectssetsandmorphismstheusualfunctions.Therearevariantshere:onecanconsiderpartialfunctionsinstead,orinjectivefunctionsoragainsurjectivefunctions.Ineachcase,thecategorythusconstructedisdifferent

2. ThecategoryTopwithobjectstopologicalspacesandmorphismscontinuousfunctions.Again,onecouldrestrictmorphismstoopencontinuousfunctionsandobtainadifferentcategory.

3. ThecategoryhoTopwithobjectstopologicalspacesandmorphismsequivalenceclassesofhomotopicfunctions.Thiscategoryisnotonlyimportantinmathematicalpractice,itisatthecoreofalgebraictopology,butitisalsoafundamentalexampleofacategoryinwhichmorphismsarenotstructurepreservingfunctions.

4. ThecategoryVecwithobjectsvectorspacesandmorphismslinearmaps.

5. ThecategoryDiffwithobjectsdifferentialmanifoldsandmorphismssmoothmaps.

6. ThecategoriesPordandPoSetwithobjectspreordersandposets,respectively,andmorphismsmonotonefunctions.

7. ThecategoriesLatandBoolwithobjectslatticesandBooleanalgebras,respectively,andmorphismsstructurepreservinghomomorphisms,i.e.,(,,,)homomorphisms.

8. ThecategoryHeytwithobjectsHeytingalgebrasand(,,,,)homomorphisms.9. ThecategoryMonwithobjectsmonoidsandmorphismsmonoidhomomorphisms.

10. ThecategoryAbGrpwithobjectsabeliangroupsandmorphismsgrouphomomorphisms,i.e.(1,,?)homomorphisms

11. ThecategoryGrpwithobjectsgroupsandmorphismsgrouphomomorphisms,i.e.(1,,?)homomorphisms

12. ThecategoryRingswithobjectsrings(withunit)andmorphismsringhomomorphisms,i.e.

X Y

(0,1,+,)homomorphisms.

13. ThecategoryFieldswithobjectsfieldsandmorphismsfieldshomomorphisms,i.e.(0,1,+,)homomorphisms.

14. AnydeductivesystemTwithobjectsformulaeandmorphismsproofs.

Theseexamplesnicelyillustrateshowcategorytheorytreatsthenotionofstructureinauniformmanner.Notethatacategoryischaracterizedbyitsmorphisms,andnotbyitsobjects.Thusthecategoryoftopologicalspaceswithopenmapsdiffersfromthecategoryoftopologicalspaceswithcontinuousmapsor,moretothepoint,thecategoricalpropertiesofthelatterdifferfromthoseoftheformer.

Weshouldunderlineagainthefactthatnotallcategoriesaremadeofstructuredsetswithstructurepreservingmaps.Thusanypreorderedsetisacategory.Forgiventwoelementsp,qofapreorderedset,thereisamorphismf:pqifandonlyifpq.Henceapreorderedsetisacategoryinwhichthereisatmostonemorphismbetweenanytwoobjects.Anymonoid(andthusanygroup)canbeseenasacategory:inthiscasethecategoryhasonlyoneobject,anditsmorphismsaretheelementsofthemonoid.Compositionofmorphismscorrespondstomultiplicationofmonoidelements.Thatthemonoidaxiomscorrespondtothecategoryaxiomsiseasilyverified.

Hencethenotionofcategorygeneralizesthoseofpreorderandmonoid.Weshouldalsopointoutthatagroupoidhasaverysimpledefinitioninacategoricalcontext:itisacategoryinwhicheverymorphismisanisomorphism,thatisforanymorphismf:XY,thereisamorphismg:YXsuchthatfg=id andgf=id .

1.3FundamentalConceptsoftheTheoryCategorytheoryunifiesmathematicalstructuresintwodifferentways.First,aswehaveseen,almosteverysettheoreticallydefinedmathematicalstructurewiththeappropriatenotionofhomomorphismyieldsacategory.Thisisaunificationprovidedwithinasettheoreticalenvironment.Second,andperhapsevenmoreimportant,onceatypeofstructurehasbeendefined,itisimperativetodeterminehownewstructurescanbeconstructedoutofthegivenone.Forinstance,giventwosetsAandB,settheoryallowsustoconstructtheirCartesianproductAB.Itisalsoimperativetodeterminehowgivenstructurescanbedecomposedintomoreelementarysubstructures.Forexample,givenafiniteAbeliangroup,howcanitbedecomposedintoaproductofcertainofitssubgroups?Inbothcases,itisnecessarytoknowhowstructuresofacertainkindmaycombine.Thenatureofthesecombinationsmightappeartobeconsiderablydifferentwhenlookedatfromapurelysettheoreticalperspective.

Categorytheoryrevealsthatmanyoftheseconstructionsareinfactcertainobjectsinacategoryhavingauniversalproperty.Indeed,fromacategoricalpointofview,aCartesianproductinsettheory,adirectproductofgroups(Abelianorotherwise),aproductoftopologicalspaces,andaconjunctionofpropositionsinadeductivesystemareallinstancesofacategoricalproductcharacterizedbyauniversalproperty.Formally,aproductoftwoobjectsXandYinacategoryCisanobjectZofCtogetherwithtwomorphisms,calledtheprojections,p:ZXandq:ZYsuchthatandthisistheuniversalpropertyforallobjectsWwithmorphismsf:WXandg:WY,thereisauniquemorphismh:WZsuchthatph=fandqh=g.

X Y

NotethatwehavedefinedaproductforXandYandnottheproductforXandY.Indeed,productsandotherobjectswithauniversalpropertyaredefinedonlyuptoa(unique)isomorphism.Thusincategorytheory,thenatureoftheelementsconstitutingacertainconstructionisirrelevant.Whatmattersisthewayanobjectisrelatedtotheotherobjectsofthecategory,thatis,themorphismsgoinginandthemorphismsgoingout,or,putdifferently,howcertainstructurescanbemappedintoagivenobjectandhowagivenobjectcanmapitsstructureintootherstructuresofthesamekind.

Categorytheoryrevealshowdifferentkindsofstructuresarerelatedtooneanother.Forinstance,inalgebraictopology,topologicalspacesarerelatedtogroups(andmodules,rings,etc.)invariousways(suchashomology,cohomology,homotopy,Ktheory).Asnotedabove,groupswithgrouphomomorphismsconstituteacategory.Eilenberg&MacLaneinventedcategorytheorypreciselyinordertoclarifyandcomparetheseconnections.Whatmattersarethemorphismsbetweencategories,givenbyfunctors.Informally,functorsarestructurepreservingmapsbetweencategories.GiventwocategoriesCandD,afunctorFfromCtoDsendsobjectsofCtoobjectsofD,andmorphismsofCtomorphismsofD,insuchawaythatcompositionofmorphismsinCispreserved,i.e.,F(gf)=F(g)F(f),andidentitymorphismsarepreserved,i.e.,F(id )=id .Itimmediatelyfollowsthatafunctorpreservescommutativityofdiagramsbetweencategories.Homology,cohomology,homotopy,Ktheoryareallexampleoffunctors.

Amoredirectexampleisprovidedbythepowersetoperation,whichyieldstwofunctorsonthecategoryofsets,dependingonhowonedefinesitsactiononfunctions.ThusgivenasetX,(X)istheusualsetofsubsetsofX,andgivenafunctionf:XY,(f):(X)(Y)takesasubsetAofXandmapsittoB=f(A),theimageoffrestrictedtoAinX.Itiseasilyverifiedthatthisdefinesafunctorfromthecategoryofsetsintoitself.

Ingeneral,therearemanyfunctorsbetweentwogivencategories,andthequestionofhowtheyareconnectedsuggestsitself.Forinstance,givenacategoryC,thereisalwaystheidentityfunctorfromCtoCwhichsendseveryobject/morphismofCtoitself.Inparticular,thereistheidentityfunctoroverthecategoryofsets.

Now,theidentityfunctorisrelatedinanaturalmannertothepowersetfunctordescribedabove.Indeed,givenasetXanditspowerset(X),thereisafunctionh whichtakesanelementxofXandsendsittothesingletonset{x},asubsetofX,i.e.,anelementof(X).Thisfunctioninfactbelongstoafamilyoffunctionsindexedbytheobjectsofthecategoryofsets{h :Y(X)|YinOb(Set)}.Moreover,itsatisfiesthefollowingcommutativitycondition:givenanyfunctionf:XY,theidentityfunctoryieldsthesamefunctionId(f):Id(X)Id(Y).Thecommutativityconditionthusbecomes:h Id(f)=(f)h .Thusthefamilyoffunctionsh()relatesthetwofunctorsinanaturalmanner.Suchfamiliesofmorphismsarecallednaturaltransformationsbetweenfunctors.Similarly,naturaltransformationsbetweenmodelsofatheoryyieldtheusualhomomorphismsofstructuresinthetraditionalsettheoreticalframework.

Theabovenotions,whileimportant,arenotfundamentaltocategorytheory.Thelatterheadingarguablyincludethenotionsoflimit/colimitinturn,thesearespecialcasesofwhatiscertainlythecornerstoneofcategorytheory,theconceptofadjointfunctors,firstdefinedbyDanielKanin1956andpublishedin1958.

Adjointfunctorscanbethoughtofasbeingconceptualinverses.Thisisprobablybestillustrated

X

FX

X

Y

Y X

byanexample.LetU:GrpSetbetheforgetfulfunctor,thatis,thefunctorthatsendstoeachgroupGitsunderlyingsetofelementsU(G),andtoagrouphomomorphismf:GHtheunderlyingsetfunctionU(f):U(G)U(H).Inotherwords,Uforgetsaboutthegroupstructureandforgetsthefactthatmorphismsaregrouphomomorphisms.ThecategoriesGrpandSetarecertainlynotisomorphic,ascategories,tooneanother.(Asimpleargumentrunsasfollows:thecategoryGrphasazeroobject,whereasSetdoesnot.)Thus,wecertainlycannotfindaninverse,intheusualalgebraicsense,tothefunctorU.ButtherearemanynonisomorphicwaystodefineagroupstructureonagivensetX,andonemighthopethatamongtheseconstructionsatleastoneisfunctorialandsystematicallyrelatedtothefunctorU.Whatistheconceptualinversetotheoperationofforgettingallthegrouptheoreticalstructureandobtainingaset?Itistoconstructagroupfromasetsolelyonthebasisoftheconceptofgroupandnothingelse,i.e.,withnoextraneousrelationordata.Suchagroupisconstructedfreelythatis,withnorestrictionwhatsoeverexceptthoseimposedbytheaxiomsofthetheory.Inotherwords,allthatisrememberedintheprocessofconstructingagroupfromagivensetisthefactthattheresultingconstructionhastobeagroup.Suchaconstructionexistsitisfunctorialandityieldswhatarecalledfreegroups.Inotherwords,thereisafunctorF:SetGrp,whichtoanysetXassignsthefreegroupF(X)onX,andtoeachfunctionf:XY,thegrouphomomorphismF(f):F(X)F(Y),definedintheobviousmanner.Thesituationcanbedescribedthusly:wehavetwoconceptualcontexts,agrouptheoreticalcontextandasettheoreticalcontext,andtwofunctorsmovingsystematicallyfromonecontexttotheotherinoppositedirections.Oneofthesefunctorsiselementary,namelytheforgetfulfunctorU.Itisapparentlytrivialanduninformative.Theotherfunctorismathematicallysignificantandimportant.ThesurprisingfactisthatFisrelatedtoUbyasimpleruleand,insomesense,itarisesfromU.Oneofthestrikingfeaturesofadjointsituationsispreciselythefactthatfundamentalmathematicalandlogicalconstructionsariseoutofgivenandoftenelementaryfunctors.

ThefactthatUandFareconceptualinversesexpressesitselfformallyasfollows:applyingFfirstandthenUdoesnotyieldtheoriginalsetX,butthereisafundamentalrelationshipbetweenXandUF(X).Indeed,thereisafunction:XUF(X),calledtheunitoftheadjunction,thatsimplysendseachelementofXtoitselfinUF(X)andthisfunctionsatisfiesthefollowinguniversalproperty:givenanyfunctiong:XU(G),thereisauniquegrouphomomorphismh:F(X)GsuchthatU(h)=g.Inotherwords,UF(X)isthebestpossiblesolutiontotheproblemofinsertingelementsofXintoagroup(whatiscalledinsertionofgeneratorsinthemathematicaljargon).ComposingUandFintheoppositeorder,wegetamorphism:FU(G)G,calledthecounitoftheadjunction,satisfyingthefollowinguniversalproperty:foranygrouphomomorphismg:F(X)G,thereisauniquefunctionh:XU(G)suchthatF(h)=gFU(G)constitutesthebestpossiblesolutiontotheproblemoffindingarepresentationofGasaquotientofafreegroup.IfUandFweresimplealgebraicinversestooneanother,wewouldhavethefollowingidentity:UF=I andFU=I ,whereI denotestheidentityfunctoronSetandI theidentityfunctoronGrp.Aswehaveindicated,theseidentitiescertainlydonotholdinthiscase.However,someidentitiesdohold:theyarebestexpressedwiththehelpofthecommutativediagrams:

UU UFU F

F FUF

U F

U F

Set Grp Set

Grp

wherethediagonalarrowsdenotetheappropriateidentitynaturaltransformations.

Thisisbutonecaseofaverycommonsituation:everyfreeconstructioncanbedescribedasarisingfromanappropriateforgetfulfunctorbetweentwoadequatelychosencategories.Thenumberofmathematicalconstructionsthatcanbedescribedasadjointsissimplystunning.Althoughthedetailsofeachoneoftheseconstructionsvaryconsiderably,thefactthattheycanallbedescribedusingthesamelanguageillustratestheprofoundunityofmathematicalconceptsandmathematicalthinking.Beforewegivemoreexamples,aformalandabstractdefinitionofadjointfunctorsisinorder.

Definition:LetF:CDandG:DCbefunctorsgoinginoppositedirections.FisaleftadjointtoG(GarightadjointtoF),denotedbyFG,ifthereexistsnaturaltransformations:I GFand:FGI ,suchthatthecomposites

GG GFG

G G

and

FF FGF

F F

aretheidentitynaturaltransformations.(Fordifferentbutequivalentdefinitions,seeMacLane1971or1998,chap.IV.)

Herearesomeoftheimportantfactsregardingadjointfunctors.Firstly,adjointsareuniqueuptoisomorphismthatisanytwoleftadjointsFandF'ofafunctorGarenaturallyisomorphic.Secondly,thenotionofadjointnessisformallyequivalenttothenotionofauniversalmorphism(orconstruction)andtothatofrepresentablefunctor.(See,forinstanceMacLane1998,chap.IV.)Eachandeveryoneofthesenotionsexhibitanaspectofagivensituation.Thirdly,aleftadjointpreservesallthecolimitswhichexistinitsdomain,and,dually,arightadjointpreservesallthelimitswhichexistinitsdomain.

Wenowgivesomeexamplesofadjointsituationstoillustratethepervasivenessofthenotion.

1. Insteadofhavingaforgetfulfunctorgoingintothecategoryofsets,insomecasesonlyapartofthestructureisforgotten.Herearetwostandardexamples:

ThereisanobviousforgetfulfunctorU:AbGrpAbMonfromthecategoryofabeliangroupstothecategoryofabelianmonoids:Uforgetsabouttheinverseoperation.ThefunctorUhasaleftadjointF:AbMonAbGrpwhich,givenanabelianmonoidM,assignstoitthebestpossibleabeliangroupF(M)suchthatMcanbeembeddedinF(M)asasubmonoid.Forinstance,ifMis,thenF()is,thatis,itisisomorphicto.Similarly,thereisanobviousforgetfulfunctorU:HausTopfromthecategoryofHausdorfftopologicalspacestothecategoryoftopologicalspaceswhichforgetstheHausdorffcondition.Again,thereisafunctorF:TopHaussuchthatFU.GivenatopologicalspaceX,F(X)yieldsthebestHausdorffspaceconstructedfromX:itisthequotientofXbytheclosureofthediagonal XX,whichisanequivalence

C D

X

relation.Incontrastwiththepreviousexamplewherewehadanembedding,thistimewegetaquotientoftheoriginalstructure.

2. ConsidernowthecategoryofcompactHausdorffspaceskHausandtheforgetfulfunctorU:kHausTop,whichforgetsthecompactnesspropertyandtheseparationproperty.TheleftadjointtothisUistheStoneCechcompactification.

3. ThereisaforgetfulfunctorU:Mod AbGrpfromacategoryofRmodulestothecategoryofabeliangroups,whereRisacommutativeringwithunit.ThefunctorUforgetstheactionofRonagroupG.ThefunctorUhasbothaleftandarightadjoint.TheleftadjointisR:AbGrpMod whichsendsanabeliangroupGtothetensorproductRGandtherightadjointisgivenbythefunctorHom(R,):AbGrpMod whichassignstoanygroupGthemodulesoflinearmappingsHom(R,G).

4. ThecasewherethecategoriesCandDareposetsdeservesspecialattentionhere.AdjointfunctorsinthiscontextareusuallycalledGaloisconnections.LetCbeaposet.Considerthediagonalfunctor:CCC,with(X)=X,Xandforf:XY,(f)=f,f:X,XY,Y.Inthiscase,theleftadjointtoisthecoproduct,orthesup,andtherightadjointtoistheproduct,ortheinf.Theadjointsituationcanbedescribedinthefollowingspecialform:

XYZ XZ,YZ

ZXY ZY,ZX

wheretheverticaldoublearrowcanbeinterpretedasrulesofinferencegoinginbothdirections.

5. Implicationcanalsobeintroduced.Considerafunctorwithaparameter:(X):CC.ItcaneasilybeverifiedthatwhenCisaposet,thefunction(X)isorderpreservingandthereforeafunctor.Arightadjointto(X)isafunctorthatyieldsthelargestelementofCsuchthatitsinfimumwithXissmallerthanZ.ThiselementissometimescalledtherelativepseudocomplementofXor,morecommonly,theimplication.ItisdenotedbyXZorbyXZ.Theadjunctioncanbepresentedasfollows:

YXZ

YXZ

6. ThenegationoperatorXcanbeintroducedfromthelastadjunction.Indeed,letZbethebottomelementofthelattice.Then,sinceYXisalwaystrue,itfollowsthatYXisalsoalwaystrue.ButsinceXXisalwaysthecase,wegetatthenumeratorthatXX=.Hence,XisthelargestelementdisjointfromX.WecanthereforeputX= X.

7. Limits,colimits,andallthefundamentalconstructionsofcategorytheorycanbedescribedasadjoints.Thus,productsandcoproductsareadjoints,asareequalizers,coequalizers,pullbacksandpushouts,etc.Thisisoneofthereasonsadjointnessiscentraltocategorytheoryitself:becauseallthefundamentaloperationsofcategorytheoryarisefromadjointsituations.

8. Anequivalenceofcategoriesisaspecialcaseofadjointness.Indeed,ifintheabove

R

R

R

def

triangularidentitiesthearrows:I GFand:FGI arenaturalisomorphisms,thenthefunctorsFandGconstituteanequivalenceofcategories.Inpractice,itisthenotionofequivalenceofcategoriesthatmattersandnotthenotionofisomorphismofcategories.

Itiseasytoprovecertainfactsabouttheseoperationsdirectlyfromtheadjunctions.Consider,forinstance,implication.LetZ=X.ThenwegetatthenumeratorthatYXX,whichisalwaystrueinaposet(asiseasilyverified).Hence,YXXisalsotrueforallYandthisisonlypossibleifXX=,thetopelementofthelattice.Notonlycanlogicaloperationsbedescribedasadjoints,buttheynaturallyariseasadjointstobasicoperations.Infact,adjointscanbeusedtodefinevariousstructures,distributivelattices,Heytingalgebras,Booleanalgebras,etc.(SeeWood,2004.)Itshouldbeclearfromthesimpleforegoingexamplehowtheformalismofadjointnesscanbeusedtogivesyntacticpresentationsofvariouslogicaltheories.Furthermore,andthisisakeyelement,thestandarduniversalandexistentialquantifierscanbeshowntobearisingasadjointstotheoperationofsubstitution.Thus,quantifiersareonaparwiththeotherlogicaloperations,insharpcontrastwiththeotheralgebraicapproachestologic.(See,forinstanceAwodey1996orMacLane&Moerdijk1992.)Moregenerally,Lawvereshowedhowsyntaxandsemanticsarerelatedbyadjointfunctors.(SeeLawvere1969b.)

Dualitiesplayanimportantroleinmathematicsandtheycanbedescribedwiththehelpofequivalencesbetweencategories.Inotherwords,manyimportantmathematicaltheoremscanbetranslatedasstatementsabouttheexistenceofadjointfunctors,sometimessatisfyingadditionalproperties.Thisissometimestakenasexpressingtheconceptualcontentofthetheorem.Considerthefollowingfundamentalcase:letCbethecategorywhoseobjectsarethelocallycompactabeliangroupsandthemorphismsarethecontinuousgrouphomomorphisms.Then,thePontryagindualitytheoremamountstotheclaimthatthecategoryCisequivalenttothecategoryC,thatis,totheoppositecategory.Ofcourse,theprecisestatementrequiresthatwedescribethefunctorsF:CCandG:CCandprovethattheyconstituteanequivalenceofcategories.

AnotherwellknownandimportantdualitywasdiscoveredbyStoneinthethirtiesandnowbearshisname.Inonedirection,anarbitraryBooleanalgebrayieldsatopologicalspace,andintheotherdirection,froma(compactHausdorffandtotallydisconnected)topologicalspace,oneobtainsaBooleanalgebra.Moreover,thiscorrespondenceisfunctorial:anyBooleanhomomorphismissenttoacontinuousmapoftopologicalspaces,and,conversely,anycontinuousmapbetweenthespacesissenttoaBooleanhomomorphism.Inotherwords,thereisanequivalenceofcategoriesbetweenthecategoryofBooleanalgebrasandthedualofthecategoryofBooleanspaces(alsocalledStonespaces).(SeeJohnstone1982foranexcellentintroductionandmoredevelopments.)TheconnectionbetweenacategoryofalgebraicstructuresandtheoppositeofacategoryoftopologicalstructuresestablishedbyStone'stheoremconstitutesbutoneexampleofageneralphenomenonthatdidattractandstillattractsagreatdealofattentionfromcategorytheorists.Categoricalstudyofdualitytheoremsisstillaveryactiveandsignificantfield,andislargelyinspiredbyStone'sresult.(Forrecentapplicationsinlogic,see,forinstanceMakkai1987,Taylor2000,2002a,2002b,Caramello2011.)

2.BriefHistoricalSketchItisdifficulttodojusticetotheshortbutintricatehistoryofthefield.Inparticularitisnotpossibletomentionallthosewhohavecontributedtoitsrapiddevelopment.Withthiswordofcautionout

C D

oftheway,wewilllookatsomeofthemainhistoricalthreads.

Categories,functors,naturaltransformations,limitsandcolimitsappearedalmostoutofnowhereinapaperbyEilenberg&MacLane(1945)entitledGeneralTheoryofNaturalEquivalences.Wesayalmost,becausetheirearlierpaper(1942)containsspecificfunctorsandnaturaltransformationsatwork,limitedtogroups.Adesiretoclarifyandabstracttheir1942resultsledEilenberg&MacLanetodevisecategorytheory.Thecentralnotionatthetime,astheirtitleindicates,wasthatofnaturaltransformation.Inordertogiveageneraldefinitionofthelatter,theydefinedfunctor,borrowingthetermfromCarnap,andinordertodefinefunctor,theyborrowedthewordcategoryfromthephilosophyofAristotle,Kant,andC.S.Peirce,butredefiningitmathematically.

Aftertheir1945paper,itwasnotclearthattheconceptsofcategorytheorywouldamounttomorethanaconvenientlanguagethisindeedwasthestatusquoforaboutfifteenyears.CategorytheorywasemployedinthismannerbyEilenberg&Steenrod(1952),inaninfluentialbookonthefoundationsofalgebraictopology,andbyCartan&Eilenberg(1956),inagroundbreakingbookonhomologicalalgebra.(Curiously,althoughEilenberg&Steenroddefinedcategories,Cartan&Eilenbergsimplyassumedthem!)Thesebooksallowednewgenerationsofmathematicianstolearnalgebraictopologyandhomologicalalgebradirectlyinthecategoricallanguage,andtomasterthemethodofdiagrams.Indeed,withoutthemethodofdiagramchasing,manyresultsinthesetwobooksseeminconceivable,orattheveryleastwouldhaverequiredaconsiderablymoreintricatepresentation.

ThesituationchangedradicallywithGrothendieck's(1957)landmarkpaperentitledSurquelquespointsd'algbrehomologique,inwhichtheauthoremployedcategoriesintrinsicallytodefineandconstructmoregeneraltheorieswhichhe(Grothendieck1957)thenappliedtospecificfields,e.g.,toalgebraicgeometry.Kan(1958)showedthatadjointfunctorssubsumetheimportantconceptsoflimitsandcolimitsandcouldcapturefundamentalconceptsinotherareas(inhiscase,homotopytheory).

Atthispoint,categorytheorybecamemorethanaconvenientlanguage,byvirtueoftwodevelopments.

1. Employingtheaxiomaticmethodandthelanguageofcategories,Grothendieck(1957)definedinanabstractfashiontypesofcategories,e.g.,additiveandAbeliancategories,showedhowtoperformvariousconstructionsinthesecategories,andprovedvariousresultsaboutthem.Inanutshell,Grothendieckshowedhowtodeveloppartofhomologicalalgebrainanabstractsettingofthissort.Fromthenon,aspecificcategoryofstructures,e.g.,acategoryofsheavesoveratopologicalspaceX,couldbeseenasatokenofanabstractcategoryofacertaintype,e.g.,anAbeliancategory.Onecouldthereforeimmediatelyseehowthemethodsof,e.g.,homologicalalgebracouldbeappliedto,forinstance,algebraicgeometry.Furthermore,itmadesensetolookforothertypesofabstractcategories,onesthatwouldencapsulatethefundamentalandformalaspectsofvariousmathematicalfieldsinthesamewaythatAbeliancategoriesencapsulatedfundamentalaspectsofhomologicalalgebra.

2. ThanksinlargeparttotheeffortsofFreydandLawvere,categorytheoristsgraduallycametoseethepervasivenessoftheconceptofadjointfunctors.Notonlydoestheexistenceofadjointstogivenfunctorspermitdefinitionsofabstractcategories(andpresumablythosewhicharedefinedbysuchmeanshaveaprivilegedstatus)butaswementionedearlier,manyimportanttheoremsandeventheoriesinvariousfieldscanbeseenasequivalenttothe

existenceofspecificfunctorsbetweenparticularcategories.Bytheearly1970's,theconceptofadjointfunctorswasseenascentraltocategorytheory.

Withthesedevelopments,categorytheorybecameanautonomousfieldofresearch,andpurecategorytheorycouldbedeveloped.Andindeed,itdidgrowrapidlyasadiscipline,butalsoinitsapplications,mainlyinitssourcecontexts,namelyalgebraictopologyandhomologicalalgebra,butalsoinalgebraicgeometryand,aftertheappearanceofLawvere'sPh.Dthesis,inuniversalalgebra.Thisthesisalsoconstitutesalandmarkinthishistoryofthefield,forinitLawvereproposedthecategoryofcategoriesasafoundationforcategorytheory,settheoryand,thus,thewholeofmathematics,aswellasusingcategoriesforthestudyofthelogicalaspectsofmathematics.

Overthecourseofthe1960's,Lawvereoutlinedthebasicframeworkforanentirelyoriginalapproachtologicandthefoundationsofmathematics.Heachievedthefollowing:

Axiomatizedthecategoryofsets(Lawvere1964)andofcategories(Lawvere1966)

Gaveacategoricaldescriptionoftheoriesthatwasindependentofsyntacticalchoicesandsketchedhowcompletenesstheoremsforlogicalsystemscouldbeobtainedbycategoricalmethods(Lawvere1967)

CharacterizedCartesianclosedcategoriesandshowedtheirconnectionstologicalsystemsandvariouslogicalparadoxes(Lawvere1969)

Showedthatthequantifiersandthecomprehensionschemescouldbecapturedasadjointfunctorstogivenelementaryoperations(Lawvere1966,1969,1970,1971)

Arguedthatadjointfunctorsshouldgenerallyplayamajorfoundationalrolethroughthenotionofcategoricaldoctrines(Lawvere1969).

Meanwhile,Lambek(1968,1969,1972)describedcategoriesintermsofdeductivesystemsandemployedcategoricalmethodsforprooftheoreticalpurposes.

Allthisworkculminatedinanothernotion,thankstoGrothendieckandhisschool:thatofatopos.Eventhoughtoposesappearedinthe1960's,inthecontextofalgebraicgeometry,againfromthemindofGrothendieck,itwascertainlyLawvereandTierney's(1972)elementaryaxiomatizationofatoposwhichgaveimpetustoitsattainingfoundationalstatus.Veryroughly,anelementarytoposisacategorypossessingalogicalstructuresufficientlyrichtodevelopmostofordinarymathematics,thatis,mostofwhatistaughttomathematicsundergraduates.Assuch,anelementarytoposcanbethoughtofasacategoricaltheoryofsets.Butitisalsoageneralizedtopologicalspace,thusprovidingadirectconnectionbetweenlogicandgeometry.(Formoreonthehistoryofcategoricallogic,seeMarquis&Reyes2012,Bell2005.)

The1970ssawthedevelopmentandapplicationofthetoposconceptinmanydifferentdirections.Theveryfirstapplicationsoutsidealgebraicgeometrywereinsettheory,wherevariousindependenceresultswererecastintermsoftopos(Tierney1972,Bunge1974,butalsoBlass&Scedrov1989,Blass&Scedrov1992,Freyd1980,MacLane&Moerdijk1992,Scedrov1984).Connectionswithintuitionisticand,moregenerallyconstructivemathematicswerenotedearlyon,andtoposesarestillusedtoinvestigatemodelsofvariousaspectsofintuitionismandconstructivism(Lambek&Scott1986,MacLane&Moerdijk1992,VanderHoeven&Moerdijk1984a,1984b,1984c,Moerdijk1984,Moerdijk1995a,Moerdijk1998,Moerdijk&Palmgren1997,Moerdijk&Palmgren2002),Palmgren2012.Formoreonthehistoryoftopostheory,see

McLarty(1992).

Morerecently,topostheoryhasbeenemployedtoinvestigatevariousformsofconstructivemathematicsorsettheory(Joyal&Moerdijk1995,Taylor1996,Awodey2008),recursiveness,andmodelsofhigherordertypetheoriesgenerally.Theintroductionofthesocalledeffectivetoposandthesearchforaxiomsforsyntheticdomaintheoryareworthmentioning(Hyland1982,Hyland1988,1991,Hylandetal.1990,McLarty1992,Jacobs1999,VanOosten2008,VanOosten2002andthereferencestherein).Lawvere'searlymotivationwastoprovideanewfoundationfordifferentialgeometry,alivelyresearchareawhichisnowcalledsyntheticdifferentialgeometry(Lawvere2000,2002,Kock2006,Bell1988,1995,1998,2001,Moerdijk&Reyes1991).Thisisonlythetipoftheicebergtoposescouldprovetobeforthe21stcenturywhatLiegroupsweretothe20thcentury.

Fromthe1980stothepresent,categorytheoryhasfoundnewapplications.Intheoreticalcomputerscience,categorytheoryisnowfirmlyrooted,andcontributes,amongotherthings,tothedevelopmentofnewlogicalsystemsandtothesemanticsofprogramming.(Pitts2000,Plotkin2000,Scott2000,andthereferencestherein).Itsapplicationstomathematicsarebecomingmorediverse,eventouchingontheoreticalphysics,whichemployshigherdimensionalcategorytheorywhichistocategorytheorywhathigherdimensionalgeometryistoplanegeometrytostudythesocalledquantumgroupsandquantumfieldtheory(Majid1995,Baez&Dolan2001andotherpublicationsbytheseauthors).

3.PhilosophicalSignificanceCategorytheorychallengesphilosophersintwoways,whicharenotnecessarilymutuallyexclusive.Ontheonehand,itiscertainlythetaskofphilosophytoclarifythegeneralepistemologicalandontologicalstatusofcategoriesandcategoricalmethods,bothinthepracticeofmathematicsandinthefoundationallandscape.Ontheotherhand,philosophersandphilosophicallogicianscanemploycategorytheoryandcategoricallogictoexplorephilosophicalandlogicalproblems.Inowdiscussthesechallenges,briefly,inturn.

Categorytheoryisnowacommontoolinthemathematician'stoolboxthatmuchisclear.Itisalsoclearthatcategorytheoryorganizesandunifiesmuchofmathematics.(SeeforinstanceMacLane1971,1998orPedicchio&Tholen2004.)Noonewilldenythesesimplefacts.

Doingmathematicsinacategoricalframeworkisalmostalwaysradicallydifferentfromdoingitinasettheoreticalframework(theexceptionbeingworkingwiththeinternallanguageofaBooleantoposwheneverthetoposisnotBoolean,thenthemaindifferenceliesinthefactthatthelogicisintuitionistic).Hence,asisoftenthecasewhenadifferentconceptualframeworkisadopted,manybasicissuesregardingthenatureoftheobjectsstudied,thenatureoftheknowledgeinvolved,andthenatureofthemethodsusedhavetobereevaluated.Wewilltakeupthesethreeaspectsinturn.

Twofacetsofthenatureofmathematicalobjectswithinacategoricalframeworkhavetobeemphasized.First,objectsarealwaysgiveninacategory.Anobjectexistsinanddependsuponanambientcategory.Furthermore,anobjectischaracterizedbythemorphismsgoinginitand/orthemorphismscomingoutofit.Second,objectsarealwayscharacterizeduptoisomorphism(inthebestcases,uptoauniqueisomorphism).Thereisnosuchthing,forinstance,asthenatural

numbers.However,itcanbearguedthatthereissuchathingastheconceptofnaturalnumbers.Indeed,theconceptofnaturalnumberscanbegivenunambiguously,viatheDedekindPeanoLawvereaxioms,butwhatthisconceptreferstoinspecificcasesdependsonthecontextinwhichitisinterpreted,e.g.,thecategoryofsetsoratoposofsheavesoveratopologicalspace.Itishardtoresistthetemptationtothinkthatcategorytheoryembodiesaformofstructuralism,thatitdescribesmathematicalobjectsasstructuressincethelatter,presumably,arealwayscharacterizeduptoisomorphism.Thus,thekeyherehastodowiththekindofcriterionofidentityatworkwithinacategoricalframeworkandhowitresemblesanycriteriongivenforobjectswhicharethoughtofasformsingeneral.Oneofthestandardobjectionspresentedagainstthisviewisthatifobjectsarethoughtofasstructuresandonlyasabstractstructures,meaningherethattheyareseparatedfromanyspecificorconcreterepresentation,thenitisimpossibletolocatethemwithinthemathematicaluniverse.(SeeHellman2003forastandardformulationoftheobjection,McLarty1993,Awodey2004,Landry&Marquis2005,Shapiro2005,Landry2011,Linnebo&Pettigrew2011,McLarty2011forrelevantmaterialontheissue.)

Aslightlydifferentwaytomakesenseofthesituationistothinkofmathematicalobjectsastypesforwhichtherearetokensgivenindifferentcontexts.Thisisstrikinglydifferentfromthesituationonefindsinsettheory,inwhichmathematicalobjectsaredefineduniquelyandtheirreferenceisgivendirectly.Althoughonecanmakeroomfortypeswithinsettheoryviaequivalenceclassesorisomorphismtypesingeneral,thebasiccriterionofidentitywithinthatframeworkisgivenbytheaxiomofextensionalityandthus,ultimately,referenceismadetospecificsets.Furthermore,itcanbearguedthattherelationbetweenatypeanditstokenisnotrepresentedadequatelybythemembershiprelation.Atokendoesnotbelongtoatype,itisnotanelementofatype,butratheritisaninstanceofit.Inacategoricalframework,onealwaysreferstoatokenofatype,andwhatthetheorycharacterizesdirectlyisthetype,notthetokens.Inthisframework,onedoesnothavetolocateatype,buttokensofitare,atleastinmathematics,epistemologicallyrequired.Thisissimplythereflectionoftheinteractionbetweentheabstractandtheconcreteintheepistemologicalsense(andnottheontologicalsenseoftheselatterexpressions.)(SeeEllerman1988,Marquis2000,Marquis2006,Marquis2013.)

Thehistoryofcategorytheoryoffersarichsourceofinformationtoexploreandtakeintoaccountforanhistoricallysensitiveepistemologyofmathematics.Itishardtoimagine,forinstance,howalgebraicgeometryandalgebraictopologycouldhavebecomewhattheyarenowwithoutcategoricaltools.(See,forinstance,Carter2008,Corfield2003,Krmer2007,Marquis2009,McLarty1994,McLarty2006.)Categorytheoryhasleadtoreconceptualizationsofvariousareasofmathematicsbasedonpurelyabstractfoundations.Moreover,whendevelopedinacategoricalframework,traditionalboundariesbetweendisciplinesareshatteredandreconfiguredtomentionbutoneimportantexample,topostheoryprovidesadirectbridgebetweenalgebraicgeometryandlogic,tothepointwherecertainresultsinalgebraicgeometryaredirectlytranslatedintologicandviceversa.Certainconceptsthatweregeometricalinoriginaremoreclearlyseenaslogical(forexample,thenotionofcoherenttopos).Algebraictopologyalsolurksinthebackground.Onadifferentbutimportantfront,itcanbearguedthatthedistinctionbetweenmathematicsandmetamathematicscannotbearticulatedinthewayithasbeen.Alltheseissueshavetobereconsideredandreevaluated.

Movingclosertomathematicalpractice,categorytheoryallowedforthedevelopmentofmethodsthathavechangedandcontinuetochangethefaceofmathematics.Itcouldbearguedthatcategorytheoryrepresentstheculminationofoneofdeepestandmostpowerfultendenciesintwentieth

centurymathematicalthought:thesearchforthemostgeneralandabstractingredientsinagivensituation.Categorytheoryis,inthissense,thelegitimateheiroftheDedekindHilbertNoetherBourbakitradition,withitsemphasisontheaxiomaticmethodandalgebraicstructures.Whenusedtocharacterizeaspecificmathematicaldomain,categorytheoryrevealstheframeuponwhichthatareaisbuilt,theoverallstructurepresidingtoitsstability,strengthandcoherence.Thestructureofthisspecificarea,inasense,mightnotneedtorestonanything,thatis,onsomesolidsoil,foritmightverywellbejustonepartofalargernetworkthatiswithoutanyArchimedeanpoint,asiffloatinginspace.Touseawellknownmetaphor:fromacategoricalpointofview,Neurath'sshiphasbecomeaspaceship.

Still,itremainstobeseenwhethercategorytheoryshouldbeonthesameplane,sotospeak,assettheory,whetheritshouldbetakenasaseriousalternativetosettheoryasafoundationformathematics,orwhetheritisfoundationalinadifferentsensealtogether.(Thatthisveryquestionappliesevenmoreforcefullytotopostheorywillnotdetainus.)

Lawverefromearlyonpromotedtheideathatacategoryofcategoriescouldbeusedasafoundationalframework.(SeeLawvere1964,1966.)Thisproposalnowrestsinpartonthedevelopmentofhigherdimensionalcategories,alsocalledweakncategories.(See,forinstanceMakkai1998.)Theadventoftopostheoryintheseventiesbroughtnewpossibilities.MacLanehassuggestedthatcertaintoposesbeconsideredasagenuinefoundationformathematics.(SeeMacLane1986.)Lambekproposedthesocalledfreetoposasthebestpossibleframework,inthesensethatmathematicianswithdifferentphilosophicaloutlooksmightnonethelessagreetoadoptit.(SeeCouture&Lambek1991,1992,Lambek1994.)Hehasrecentlyarguedthatthereisnotoposthatcanthoroughlysatisfyaclassicalmathematician.(SeeLambek2004.)(Formoreonthevariousfoundationalviewsamongcategorytheorists,seeLandry&Marquis2005.)

Argumentshavebeenadvancedforandagainstcategorytheoryasafoundationalframework.(Blass1984surveystherelationshipsbetweencategorytheoryandsettheory.Feferman1977,Bell1981,andHellman2003argueagainstcategorytheory.SeeMarquis1995foraquickoverviewandproposalandMcLarty2004andAwodey2004forrepliestoHellman2003.)Thismatterisfurthercomplicatedbythefactthatthefoundationsofcategorytheoryitselfhaveyettobeclarified.Fortheremaybemanydifferentwaystothinkofauniverseofhigherdimensionalcategoriesasafoundationsformathematics.Anadequatelanguageforsuchauniversestillhastobepresentedtogetherwithdefiniteaxiomsformathematics.(SeeMakkai1998forashortdescriptionofsuchalanguage.AdifferentapproachbasedonhomotopytheorybutwithclosedconnectionswithhigherdimensionalcategorieshasbeenproposedbyVoevodskyetal.andisbeingvigorouslypursued.SeethebookHomotopyTypeTheory,byAwodeyetal.2013.)

Itisanestablishedfactthatcategorytheoryisemployedtostudylogicandphilosophy.Indeed,categoricallogic,thestudyoflogicbycategoricalmeans,hasbeenunderwayforabout30yearsnowandstillvigorous.Someofthephilosophicallyrelevantresultsobtainedincategoricallogicare:

Thehierarchyofcategoricaldoctrines:regularcategories,coherentcategories,HeytingcategoriesandBooleancategoriesallthesecorrespondtowelldefinedlogicalsystems,togetherwithdeductivesystemsandcompletenesstheoremstheysuggestthatlogicalnotions,includingquantifiers,arisenaturallyinaspecificorderandarenothaphazardlyorganized

Joyal'sgeneralizationofKripkeBethsemanticsforintuitionisticlogictosheafsemantics

(Lambek&Scott1986,MacLane&Moerdijk1992)

Coherentandgeometriclogic,socalled,whosepracticalandconceptualsignificancehasyettobeexplored(Makkai&Reyes1977,MacLane&Moerdiejk1992,Johnstone2002,Caramello2011b,2012a)

Thenotionsofgenericmodelandclassifyingtoposofatheory(Makkai&Reyes1977,Boileau&Joyal1981,Bell1988,MacLane&Moerdijk1992,Johnstone2002,Caramello2012b)

Thenotionofstrongconceptualcompletenessandtheassociatedtheorems(Makkai&Reyes1977,Butz&Moerdijk1999,Makkai1981,Pitts1989,Johnstone2002)

Geometricproofsoftheindependenceofthecontinuumhypothesisandotherstrongaxiomsofsettheory(Tierney1972,Bunge1974,Freyd1980,1987,Blass&Scedrov1983,1989,1992,MacLane&Moerdijk1992)

Modelsanddevelopmentofconstructivemathematics(seebibliographybelow)

Syntheticdifferentialgeometry,analternativetostandardandnonstandardanalysis(Kock1981,Bell1998,2001,2006)

Theconstructionofthesocalledeffectivetopos,inwhicheveryfunctiononthenaturalnumbersisrecursive(McLarty1992,Hyland1982,1991,VanOosten2002,VanOosten2008)

Categoricalmodelsoflinearlogic,modallogic,fuzzysets,andgeneralhigherordertypetheories(Reyes1991,Reyes&Zawadoski1993,Reyes&Zolfaghari1991,1996,Makkai&Reyes1995,Ghilardi&Zawadowski2002,Rodabaugh&Klement2003,Jacobs1999,Taylor1999,Johnstone2002,Blute&Scott2004,Awodey&Warren2009,Awodeyet.al.2013)

Agraphicalsyntaxcalledsketches(Barr&Wells1985,1999,Makkai1997a,1997b,1997c,Johnstone2002).

Quantumlogic,thefoundationsofquantumphysicsandquantumfieldtheory(Abramsky&Duncan2006,Heunenet.al.2009,Baez&Stay2010,Baez&Lauda2011,Coecke2011,Isham2011,Dring2011).

Categoricaltoolsinlogicofferconsiderableflexibility,asisillustratedbythefactthatalmostallthesurprisingresultsofconstructiveandintuitionisticmathematicscanbemodeledinapropercategoricalsetting.Atthesametime,thestandardsettheoreticnotions,e.g.Tarski'ssemantics,havefoundnaturalgeneralizationsincategories.Thus,categoricallogichasrootsinlogicasitwasdevelopedinthetwentiethcentury,whileatthesametimeprovidingapowerfulandnovelframeworkwithnumerouslinkstootherpartsofmathematics.

Categorytheoryalsobearsonmoregeneralphilosophicalquestions.Fromtheforegoingdisussion,itshouldbeobviousthatcategorytheoryandcategoricallogicoughttohaveanimpactonalmostallissuesarisinginphilosophyoflogic:fromthenatureofidentitycriteriatothequestionofalternativelogics,categorytheoryalwaysshedsanewlightonthesetopics.Similarremarkscanbemadewhenweturntoontology,inparticularformalontology:thepart/wholerelation,boundariesofsystems,ideasofspace,etc.Ellerman(1988)hasbravelyattemptedtoshowthatcategorytheoryconstitutesatheoryofuniversals,onehavingpropertiesradicallydifferentfromsettheory,whichisalsoseenasatheoryofuniversals.Movingfromontologytocognitivescience,MacNamara&

Reyes(1994)havetriedtoemploycategoricallogictoprovideadifferentlogicofreference.Inparticular,theyhaveattemptedtoclarifytherelationshipsbetweencountnounsandmassterms.Otherresearchersareusingcategorytheorytostudycomplexsystems,cognitiveneuralnetworks,andanalogies.(See,forinstance,Ehresmann&Vanbremeersch1987,2007,Healy2000,Healy&Caudell2006,ArziGonczarowski1999,Brown&Porter2006.)Finally,philosophersofsciencehaveturnedtocategorytheorytoshedanewlightonissuesrelatedtostructuralisminscience.(See,forinstance,Brading&Landry2006,Bain2013,Lam&Wthrichforthcoming.)

Categorytheoryoffersthusmanyphilosophicalchallenges,challengeswhichwillhopefullybetakenupinyearstocome.

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