; TeX output 1999.09.17:13327 YRXQ cmr12CHAPTER8&Nff cmbx12Tfos3olbox]o cmr96*7 &e3. %NA:TURAL!TRANSF9ORMATIONS=f7YN cmbx123.NaturalTransformationsDe nition8.3.1.vaùLetP !", cmsy10FB:j1Cf4!$D۹andG _:Cf4!$D۹bSetrwoPfunctors. jxA,@ cmti12naturffalDtransformationoɹorafunctorialmorphismg cmmi12'UR:Fc4!BG%isoafamilyofmorphismsf'(A)UR:F1(A)4!1G.(A)jA2C5g꨹sucrhthatthediagramZCvF1(B)G.(B)pt32fd/pO line10- Vp'(B)HDF1(A)HG.(A)D{fd0Ѝά-Pt'(A)HҟǠ*FfeǠ?`FF1(fG)HǠ*FfeğǠ?`DG.(fG)%TcommrutesforallfQ:URA4!1BinC5,i.e.8G.(fG)'(A)UR='(B)F1(f).NLemma8.3.2.g5QGivencffovariantfunctorsF=IdjB.2@cmbx8Set&N:Set 4!5A_SetN'andG\3=Mor5Set#t(Mor5Set(;A);A)UR:Set!+OFSetBk^for35asetA.fiThen'UR:Fc!B"Gcwith8H'(B)UR:BX3b7!(Mor5Set#t(B;A)3fQ7!fG(b)2A)2G.(B)is35anaturffaltransformation.- cmcsc10Proof.@_GivrengË:URBX4!_7Cܞ.8ThenthefollowingdiagramcommutesY荍H^BHoMorSetU(Mor5Set#t(B;A);A)kD{fd*Fά-Psq'(B)^͎C MorSetY(Mor5Set#t(C5;A);A)k432fd*Ѝά-s@'(Cܞ)Hc<Ǡ*FfecoǠ? XgHث2Ǡ*FfedǠ?`ݐMorSet|(Mor5Set#t(gn9;A);A),since8ʍLu'(Cܞ)F1(gn9)(b)(fG)UR='(C)gn9(b)(fG)=fgn9(b)='(B)(b)(fgn9),g=UR['(B)(b)Mor5Set#t(gn9;A)](fG)=[Mor5Set(Mor5Set(gn9;A);A)'(A)(b)](fG):!' %cffxff ̟ff ̎ ̄cffLemma8.3.3.g5QLffetfQ:URA!BbeamorphisminC5.ThenMor5!K cmsy8C(f;-33)UR:MorOC(B;-)!Mor5C(A;-33)_givenbyMor\C}(f;Cܞ):MorC,¹(B;Cܞ)3g?7!gn9f2MorC,¹(A;Cܞ)_isanaturffaltrffansformation35ofcovariantfunctors.LffetJWf':(A!fB]beJWamorphisminC5.ThenMorTC(-35;fG)(:Mor%C#(-;A)!fMor*C0 y(-;B)givenbyMor=C͹(C5;fG)UR:MorOC(C;A)3gË7!fGg2MorOC(C5;B)isanaturffaltransformationof35cffontravariantfunctors.7 &e88. %TOOLBO9XYProof.@_LethUR:C14!Cܞ20bSeamorphisminC5.8ThenthediagramsR3X~GMor(DC(B;Cܞ20׹)fMor cCa"(A;Cܞ20׹)32fd0Cά-knÞ0|{Ycmr8MorǟX."q% cmsy6CQ(2cmmi8fh;C-:0B})HYdMoraC (B;Cܞ)H$Mor ZC?(A;Cܞ){fd30ά-`MorOX.C'5(fh;C)HǠ*FfeE4Ǡ?`pxMorX.C(Bd;h)HǠ*FfeǠ?`f4Mor'˟X.C,U(A;h)|andJaMor^CS(C5;A)BMor ?Cd(C5;B)d32fd3ά-W`BMorӛKX.Cr(C;f)H~Mor*Cr(Cܞ20;A)HMor <CS(Cܞ20;B)ST{fd0Cά-n­^MorX.C(C-:0B};f)HҟǠ*FfeǠ?`pˢMor$9X.C(h;A)H&RǠ*FfeYǠ?` Mor'dX.C,<%(h;Bd)commrute.w0Gcffxff ̟ff ̎ ̄cffYRemark8.3.4.j6The$compSositionoftrwo$naturaltransformationsisagainanat-uraltransformation.8Theidenrtityid LF(A)UR:=1F((A)C@isalsoanaturaltransformation.De nition8.3.5.vaùAmznaturalmtransformation'48:FeI4!G#ʹismcalledanaturffaliso-morphismifthereexistsanaturaltransformation T:G[I4!FJsucrhthat'ʒ =id oGand 1n'B=idpF".Thenaturaltransformation չisuniquelydeterminedbry'.WVewrite'21ι:=UR n9.A|functor}F.issaidtobSeisomorphictoafunctorG3Kifthereexistsanaturalisomorphism'UR:Fc4!BG..Problem8.3.1.nR1.ҖLets:F1;G:=C4!HRDȐbSefunctors.Shorwthatanaturaltrans-formation:'UR:Fc4!BGMhisanaturalisomorphismifandonlyif'(A)isanisomorphismforallobjectsAUR2C5.2.Let5A(ArB;pA;pBN>)bSetheproductofAandBGinC5.ThenthereisanaturalisomorphismߍmrMor(-;AB)PUR԰n:=Mor%5C*(-;A)Morय़CNd(-;B):^3.yLetC^ bSeacategorywith niteproducts.yFVoreacrhobjectAinC^ showthatthereexistsamorphismA Q:sA4!'AAsatisfyingp1A=s1A=p2A.nShorwthatthisde nesanaturaltransformation.8Whatarethefunctors?4. wlLet,C]abSeacategorywith niteproducts. wlShorwthatthereisabifunctor-- :{CmEC/ 4!pC~sucrhIthat(--)(A;B)istheobjectofaproSductofAandB.|WVedenote4elemenrtsintheimageofthisfunctorbyA8BX:=UR(-{-)(A;B)4andsimilarlyfgn9.5. DWiththenotationoftheprecedingproblemshorwthatthereisanaturaltransformation (A;B;Cܞ):(A&B)CPh4԰=I!A(BCܞ).[Shorwthatthediagram 7 &e3. %NA:TURAL!TRANSF9ORMATIONS=f9Y(cffoherence35orcffonstraints)B4卍((AB)Cܞ)Dl(A(BECܞ))DgW:2fdEHά-̯rA (A;Bd;C)1`8,A((BECܞ)DS)|:2fdEHά-̯!)< (A;BdC;D(AE)B)A(E^B)L:2fdC ά-̯} (A;Er;Bd)NABꃀ(A)1l Ql攴Ql?^QlQz̟0Qz̟0sꃀp 1(Bd)L L攴L?^L#0#0+De nition8.3.6.vaùLetdCandD%bSecategories.0A]corvXariantdfunctorFc:URC4!wfDiscalled9aneffquivalence]ofcffategories9ifthereexistsacorvXariant9functorGY:D74!Cnandnaturalisomorphisms'UR:G.FPc԰K=1 IdIC%Mand Ë:F1GP ԰$h=)IdzfD"ɹ.A:conrtravXariant@functorFc:URC4!wfD)iscalledadualityofcffategories@ifthereexistsa DconrtravXariantfunctorG':CDO4!Cyandnaturalisomorphisms':G.FPu ԰=ZIdҗC*Mand Ë:URF1GP ԰$h=)IdzfD"ɹ.A_$category_GC|issaidtobSeeffquivalenttoacategoryDifthereexistsanequivXalenceF¹:nC!4!$DUV.eAcategoryCŹissaidtobSedualtoacategoryDNifthereexistsadualitryFc:URC4!wfDUV.Problem8.3.2.nR1.8ShorwthatthedualcategoryC52op isdualtothecategoryC5.2.yLetMDUbSeacategorydualtothecategoryC5.ShorwthatDUisequivXalenttothedualcategoryC52op R.3.-Let