; TeX output 1999.03.03:1923#7 YwN cmbx128.7RepresentableFunctorsDe nition1.8.1.vaXQ cmr12Lete!", cmsy10FWֹ:&C4!LSet.hbSeacorvXariantefunctor.Aepair(g cmmi12A;x)withAEz2C5;x2F1(A)wiscalleda*@ cmti12rffepresenting(generic,Quniversal)objeffctwforFѹandFiscalledarffepresentable\yfunctor,"ifforeacrhB<2C¹andy2F1(B)thereexistsauniquefQ2URMorOK cmsy8C(A;B)sucrhthatF1(fG)(x)UR=yn9:ObbYA\BڟǠ*Ffe! ǠO line10?`wm2cmmi8fHF1(A)יF1(B)Ǡ*FfeLǠ?`γF(|{Ycmr8(f)H23URxct3URyǠ*Ffe3Ǡ?Ў fe3Ў?Prop`osition1.8.2.OLffet@d>RHCM:󭒟Ǡ*FfeğǠ?`DfR2."GivrenfimoSdulesMR YandRN@."De neFY:'Ab4!,C9SetCbryF1(A)':=Bil,R(M;N;A).ThenYFjisacorvXariantYfunctor.'ALrepresentingobjectforFjisgivenby(M R 5nN; UR:MW N64!M R JN@)withthepropSertrythatforall(A;fQ:URMN64!A)thereexists/^+o cmr935$*7 ̍36YauniquegË2URHom(M R ;N;A)sucrhthatF1(gn9)( )UR=Bil.R"(M;N@;gn9)( )UR=g =fR{8MN{M R ;N{fd Uά-@ H`mf,ׁ @, @, @, @/>@/>RH^A:4ڟǠ*Ffeh Ǡ?'g3.GivrenBa' msbm10K-moSduleVp.De neFc:URAlg\k4!,JSetBbryF1(A)UR:=Hom(V;A).ThenBFsιisacorvXariant8functor.A8representingobjectforFj isgivenby(Tƹ(Vp);UR:V4!`T(Vp))8withthepropSertrythatforall(A;f:V!4!A)theexistsauniquegH2Mor ,2@cmbx8Alg);(Tƹ(Vp);A)sucrhthatF1(gn9)()UR=Hom(V;g)()=g=fPHQbVH4zTƹ(Vp)Dğ{fd!wЍά-ptH`)fDğׁ @Dğ @Dğ @Dğ @ߋD>@ߋD>RHjA:rǠ*FfeäǠ?'v$g4.AGivrenCaK-moSduleVp.De neFɹ:cAlg"74!4|SetKcbryF1(A):=HomeA(V;A).ThenCFisacorvXariantfunctor.2ArepresentingobjectforF isgivenby(S׹(Vp);UR:V4!`S(Vp))withуthepropSertrythatforall(A;fQ:URV4!`A)theexistsauniquegË2MorOAlg( V(S׹(Vp);A)sucrhthatF1(gn9)()UR=Hom(V;g)()=g=fPH~zVH䭒S׹(Vp)qܟ{fd!ά-٬H`Vfqܟׁ @qܟ @qܟ @qܟ @߸\>@߸\>RHA:񽊟Ǡ*FfeǠ?'Y(contravari-ant!)Lsuch+thatFn9 1 :URF1(A)3a7!h aY!2Nat(Mor5C(A;-33);F);& 7 ̍38Ywherffe35h2aϹ(B)(fG)UR=F1(f)(a).-Proof.@_FVorUR2Nat(Mor5C(A;-);F1)wrehaveamap(A)UR:MorOC(A;A)4!1F1(A),hence!:Zwithn9()UR:=(A)(1A)!isawrellde nedmap..FVor21ֹwrehavetocheckthath2awisanaturaltransformation.8GivrenfQ:URBX4!_7CFinC5.ThenthediagramRI̍F1(B)yF1(Cܞ)┞32fdRά-W`ϥzF((f)Hv4Mor1C(A;B)HsMor )= (B)(fG)k9=UR n9(B)Mor5C(A;fG)(1A)=F1(f) n9(A)(1A)=F1(f)n9( ):ARemark1.9.4.j6By4thepreviouscorollarytherepresenrtingobjectAisuniquelydetermineduptoisomorphismbrytheisomorphismclassofthefunctorMor Cd(A;-).f퍍Problem1.1.cR1. Determinef}explicitlyallnaturalendomorphismsfromGajLtoGa(asde nedinLemma4.3.5).2.8DeterminealladditivrenaturalendomorphismsofGaϹ.3.8DetermineallnaturaltransformationsfromGawtoGm l(seeLemma4.3.7).4.8DetermineallnaturalautomorphismsofGmĹ.Prop`osition1.9.5.OLffetjfGKĹ:CD_*D!Set2:8beacovariantbifunctorsuchthatthefunctorG.(C5;-33)*:D!0Set0 isrffepresentableforallC2*C5.Thenthereexistsacffontravariant&functorF:C!D|suchthatGP԰=Mor' ПD.\3(F1-dF;-33)holds.P@ԟ>RHߦG.(DS)Ǡ*Ffe/4Ǡ?`Gv(gI{)g͍commrutes.AEopairE(F1(Cܞ);)thatsatis estheabSorveEconditionsiscalledauniversalsolutionofthe(co-)univrersalproblemde nedbyGֹandCܞ.LetxFc:URC4!wfD%bSeacorvXariantxfunctor.Fgeneratesauniversalʐprffoblemafollorws:GivrenD2URDUV.)FindanobjectG.(DS)2CpĹandamorphism:F1(G.(DS))4!1DinDsucrhthatthereisauniquemorphismg:pCMX4!ןG.(DS)inCforeachobjectCMX2pCandforeacrhmorphismfQ:URF1(Cܞ)4!1D>6inD?suchthatthediagramLF1G.(DS) D)32fdMά-H`fkğׁ @kğ @kğ @kğ @D>@D>RHF1(Cܞ)2Ǡ*Ffe3dǠ?`2F((gI{)g͍commrutes.A*vpair*(G.(DS);ǹ)thatsatis estheaborve*conditionsiscalledauniversalmsolutionofthe(co-)univrersalproblemde nedbyFandDS.ʭProp`osition1.9.14.LffetF;p: _C!.`DPbeleftadjointtoG: _D_!ЁC5.ThenF1(Cܞ)andOtheunitd=(Cܞ):CA$! G.F1(C)Oforma(cffo-)universalsolutionforthe(co-)universal35prffoblemde nedbyGcandCܞ.FurthermorffeQG.(DS)andthecounit(~=f (DS):F1G.(D)!3DformQauniversalsolution35fortheuniversalprffoblemde nedbyFdFandDS.Proof.@_ByJTheorem1.9.10themorphisms:MorD!$H(F1-;-)4!W]Mor,ZC1(-;G.-)Jand Ë:URMorOC(-;G.-)UR4!1Mor).D1I(F1-;-)areinrversesofeacrhother. Theyarede nedwithunitandcounitas(C5;DS)(gn9)Y=G.(g)(Cܞ)resp. (C5;DS)(fG)= (D)F1(fG).Henceforeacrhf:mCI4!SG.(DS)thereisauniquegM:F1(Cܞ)4!DⒹsucrhthatf=(C5;DS)(gn9)=G.(gn9)(Cܞ)UR=G(gn9).ThesecondstatemenrtfollowsanalogouslyV.ocffxff ̟ff ̎ ̄cff+7 &e]9. %ADJOINT!FUNCTORSANDTHEYONED9ALEMMAQ5~43YRemark1.9.15.qN6IfDGX1:DY4!CandC~2Caregivrenthenthe(co-)universalsolution4(F1(Cܞ);ӝ:C;4!eG.(DS))canbeconsideredasthebest(co-)approrximationoftheobjectC(inCbryanobjectDJinDKwiththehelpofafunctorG..\TheobjectD2URD?turnsouttobSeF1(Cܞ).If(F=:v,C)a4!D~andDɺ2D~aregivrenthentheuniversalsolution(G.(DS);7:F1G.(DS)w04!D)canbSeconsideredasthebestapprorximationoftheobjectDRinDSbyanobjectC{inCRJwiththehelpofafunctorF1.V'TheobjectCe 2mCRJturnsouttobSeG.(DS).Prop`osition1.9.16.GivenG :WwD!jC5.AssumethatforeffachC42Cqtheuni-versalAprffoblemde nedbyGandC1issolvable.ThenthereisaleftadjointfunctorFc:URC!FDto35G..GivenFc:URC!FDUV.^.AssumethatforeffachD2Dotheuniversalprffoblemde nedbyFdFand35Dissolvable.fiThentherffeisaleftadjointfunctorG :URD!fgCjtoF1.Proof.@_Assumethatthe(co-)univrersalproblemde nedbyG6andCissolvedbryuB:C4!zSF1(Cܞ).*ThenthemapMorC}(C5;G.(DS))3f7!gM2MorxD t(F1(Cܞ);DS)uwithG.(gn9)=f$isbijectivre.Theinversemapisgivenbyg_7!G.(gn9).ThisisanaturaltransformationsincethediagramMHyRMorgϟDo!2(F1(Cܞ);DS20!ǹ)Mor%C+c(C5;G.(DS20!ǹ))|32fdc:Ѝά-W`Gv(-)HTMori8DpN(F1(Cܞ);DS)H'Mor']C,Ĺ(C5;G.(DS)){fdf Pά-`Gv(-)H|mǠ*Ffe|LǠ?`@%MorO~wX.DU(F((C);h)H8QǠ*Ffe8LǠ?`=6MorLcX.CQf(C;Gv(h));卹commrutesforeachhUR2MorOD @(DS;D20!ǹ).8Infactwehavee%Mor;*ŸC@(C5;G.(h))(G(gn9))UR=G.(h)G(g)UR=G.(hg)=G.(Mor5C(F1(Cܞ);h)(g)):HenceNforallC2UCthefunctorMorCF(C5;G.(-)):DT4!mSet-inducedbrythebifunctorMor5C(-;G.(-))w:C52op N^D4!^Set,Eùisrepresenrtable.uIByTheorem1.9.5thereisafunctorFc:URC4!wfD?sucrhthatMor Cd(-;G.(-))P԰n:=Mor%5D,[(F1(-);-).ThesecondstatemenrtfollowsanalogouslyV.ocffxff ̟ff ̎ ̄cffRemark1.9.17.qN6OneacancrharacterizethepropSertiesthatG/:D)W4!C(resp.F:C5<4!DUV)1mrusthaveinordertopSossessaleft-(right-)adjointfunctor. Oneofthe(essenrtialpropSertiesforthisisthatGVpreserveslimits(hencedirectproSductsanddi erencekrernels).^;7  -%n eufm10,2@cmbx8+o cmr9*@ cmti12' msbm10 q% cmsy6K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8- cmcsc10N cmbx12XQ cmr12O line10{