; TeX output 1999.09.17:1336'7 YRXQ cmr12CHAPTER8&Nff cmbx12Tfos3olbox/^o cmr939(*7 &e408. %TOOLBO9XYSmN cmbx129.hmAdjointFunctorsandtheYonedaLemmaTheorem8.9.1.p@ cmti12(YoneffdaLemma)Let!!", cmsy10C+beacategory.GivenacovariantfunctorFc:URC!FSet,^and35anobjeffctg cmmi12A2C5.fiThenthemap^DË:URNat(Mor5"K cmsy8C(A;-33);F1)UR37!(A)(12cmmi8A)2F(A)is35bijeffctivewiththeinversemapi^>n9 |{Ycmr81 :URF1(A)3a7!h aY!2Nat(Mor5C(A;-33);F);wherffe35h2aϹ(B)(fG)UR=F1(f)(a).i<- cmcsc10Proof.@_FVorUR2Nat(Mor5C(A;-);F1)wrehaveamap(A)UR:MorOC(A;A)4!1F1(A),hence!:Zwithn9()UR:=(A)(1A)!isawrellde nedmap..FVor21ֹwrehavetocheckthath2awisanaturaltransformation.8GivrenfQ:URBX4!_7CFinC5.ThenthediagramQF1(B)yF1(Cܞ)┞32fdRO line10-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( ):܍Remark8.9.4.j6By4thepreviouscorollarytherepresenrtingobjectAisuniquelydetermineduptoisomorphismbrytheisomorphismclassofthefunctorMor Cd(A;-).RProblem8.9.1.nR1.BDetermineCLexplicitlyallnaturalendomorphismsfrom* msbm10Ga GtoGaw(asde nedinLemma2.3.5).2.8DeterminealladditivrenaturalendomorphismsofGaϹ.3.8DetermineallnaturaltransformationsfromGawtoGm l(seeLemma2.3.7).4.8DetermineallnaturalautomorphismsofGmĹ.Prop`osition8.9.5.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{)dcommrutes.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?suchthatthediagramKDF1G.(DS) D)32fdMά-H`fkğׁ @kğ @kğ @kğ @D>@D>RHF1(Cܞ)2Ǡ*Ffe3dǠ?`2F((gI{)dcommrutes.A*vpair*(G.(DS);ǹ)thatsatis estheaborve*conditionsiscalledauniversalmsolutionofthe(co-)univrersalproblemde nedbyFandDS.6Prop`osition8.9.14.LffetF;p: _C!.`DPbeleftadjointtoG: _D_!ЁC5.ThenF1(Cܞ)andOtheunitd=(Cܞ):CA$! G.F1(C)Oforma(cffo-)universalsolutionforthe(co-)universal35prffoblemde nedbyGcandCܞ.FurthermorffeQG.(DS)andthecounit(~=f (DS):F1G.(D)!3DformQauniversalsolution35fortheuniversalprffoblemde nedbyFdFandDS.-MH7 &e]9. %ADJOINT!FUNCTORSANDTHEYONED9ALEMMAQ5~45YProof.@_ByJTheorem8.9.10themorphisms:MorD!$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^Remark8.9.15.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`osition8.9.16.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.uIByTheorem8.9.5thereisafunctorFc:URC4!wfD?sucrhthatMor Cd(-;G.(-))P԰n:=Mor%5D,[(F1(-);-).ThesecondstatemenrtfollowsanalogouslyV.ocffxff ̟ff ̎ ̄cffRemark8.9.17.qN6OneacancrharacterizethepropSertiesthatG/:D)W4!C(resp.F:C5<4!DUV)1mrusthaveinordertopSossessaleft-(right-)adjointfunctor. Oneofthe(essenrtialpropSertiesforthisisthatGVpreserveslimits(hencedirectproSductsanddi erencekrernels).];7  -%n eufm10* msbm10#q% cmsy6"K cmsy8!!", cmsy10 ;cmmi62cmmi8g cmmi12|{Ycmr8- cmcsc10@ cmti12o cmr9N cmbx12Nff cmbx12XQ cmr12O line10p