; TeX output 1998.02.24:19167 YRXQ cmr12CHAPTER1&Nff cmbx12Tfos3olbox"N cmbx121.ijCategoriesDe nition1.1.1.vaùLet!", cmsy10Cݹconsistof 1. #aclassObFCݹwhoseelemenrtsg cmmi12A;B;C5;:::2URObCarecalled+@ cmti12objeffcts, 2. #a4familyfMor5 K cmsy8C(A;B)jA;B2wOb";C5g4ofmrutuallydisjointsetswhoseelements #f;gn9;:::2URMorOC(A;B)arecalledmorphisms,and 3. #a4familyfMor5C(A;B)`Mor͟C(B;Cܞ)3(f;gn9)7!gfd2MorRCs(A;Cܞ)jA;B;CU2 #Ob1&C5g꨹ofmapscalledcffompositions.Cݹiscalledacffategory꨹ifthefollorwingaxiomsholdforC 1. #AssoSciativreLaw: #8A;B;C5;D2URObC5;fQ2URMorOC(A;B);gË2URMorOC(B;Cܞ);hUR2MorOC(C5;DS):>7h(gn9fG)UR=(hg)fG; 2. #IdenrtityLaw: #8A^U2ObC912cmmi8A<92MorRC"(A;A)I8B;C:2^UObC5;I8fT2^UMorRC"(A;B);8g̎2 #Mor5YC:D(C5;A)UR:31AgË=URgѹandJ1fG1A 36=f:퍍Examples1.1.2.th1.8ThecategoryofsetsSetӋ.2.The categoriesofRJ-moSdulesR-Mo`dn:,Ikg-vrectorspacesk-Vec[^ork-Mo`dn:,IgroupsGr ,aabSelian>groupsAb|,monoidsMonL,commrutative>monoidscMon$L,ringsRi}, eldsFldr,topSologicalspacesTopn5.Since^moSdulesarehighlyimportanrtforallwhatfollows,{werecallthede nitionandsomebasicpropSerties.De nitionandRemark1.1.3.ßLetR޹bSearing(alwraysassociativewithunit).ADleftRJ-moffduleR &M`wisan(additivrelywritten)abSeliangroupMtogetherwithanopSerationRM63UR(rr;m)7!rm2M+sucrhthat 1. #(rSs)mUR=r(sm), 2. #(r6+s)mUR=rSm+sm, 3. #rS(m+m209)UR=rm+rm209, 4. #1mUR=mforallrr;sUR2RJ,m;m20#2M@.EacrhabSeliangroupisa( msbm10Z-moduleinauniquewrayV.],o cmr91*7 &e21. %TOOLBO9XYAhomomorphism*ofleftRJ-moffdulesfg`:aR M`Ey!3R N8isagrouphomomorphismsucrhthatfG(rSm)UR=rfG(m).Analogouslywrede nerightRJ-moSdulesMR ;andtheirhomomorphisms.WVeudenotebryHomUR"I(:M;:N@)thesetofhomomorphismsofleftRJ-moSdulesR MYandRN@.PSimilarlyHomlR#_(M:;N:)denotesthesetofhomomorphismsofrighrtRJ-moSdulesMR ;andNR.8BothsetsareabSeliangroupsbry(f+gn9)(m)UR:=fG(m)+gn9(m).FVorarbitrarycategorieswreadoptmanyofthecustomarynotations.hNotation1.1.4.p>6fQ2URMorOC(A;B)willbSewrittenasf:URAn!1BorAhfJn!B.8Aꔹiscalledthedomain,BtherffangeoffG.TheCcffompositionoftrwomorphismsft:;uAU!wBIandg:B{!+}Ciswrittenasgn9fQ:URAn!1CForasgf:URAn!1Cܞ.De nitionandRemark1.1.5.ßAmorphismf鴹:A@!\BHiscalledanisomor-phismaSifthereexistsamorphismg:OBU!1AinCsucrhthatfGg=O1B andgn9fgN=1A.TheYmorphismgisuniquelydeterminedbryfsincegn920Ĺ=URgn920:8(M;T1)RX!'(M;T2)isbijectivreandcontinuous.%Theinversemap,hhowever,isnotcontinuous,hhencef׹isnoisomorphism(homeomorphism).ManrywellknownconceptscanbSede nedforarbitrarycategories.WVearegoingtoapplysomeofthem.8Herearetrwoexamples.De nition1.1.7.vaù1.) sA[zmorphism[f:ɺAE!B߹iscalledamonomorphismif8C12URObC5;8gn9;hUR2MorOC(C5;A):rj^fGgË=URfh꨹=)g=URh (f2isleftcancellable)aW:2.)`A morphism f:>AW!Bعiscalledanepimorphismif8C2ObC5; 8gn9;h2Mor5C(B;Cܞ)UR:f4gn9fQ=URhf2=)@gË=h (f2isrighrtcancellable)iv:De nition1.1.8.vaùGivrenkA;Bm620C5.)AnobjectACBeqinkC}togetherwithmor-phisms!pA E:aAGBNgg!UAandpB :aABNgg!UBiscalleda(categorical)prffoductofA 7 &eA1. %CA:TEGORIES3YandMBifforevreryobjectT2C=andeverypairofmorphismsfDĹ:T!Aandgj:T!nBthereexistsauniquemorphism(f;gn9)UR:T!eABsucrhthatthediagramSdbZ_T`fiׁ O line10i i i #d>#d> H'Egڤׁ @ڤ @ڤ @ڤ @!$>@!$>RA!hAB432fdpά!7pX.;cmmi6ApB32fd@ά-!7pX.BHoǠ*FfeآDǠ?`T(fh;gI{)&commrutes.AnobjectE:2h#C?Qiscalleda nalǜobjeffctifforevreryobjectT 2h#C?QthereexistsauniquemorphismeUR:T!eE(i.e.8MornݟCܜ(T;E)consistsofexactlyoneelemenrt).AcategoryCwhicrhhasaproSductforanytwoobjectsAandBandwhichhasa nalobjectiscalledacategorywith niteproSducts.ÍRemark1.1.9.j6IfSEtheproSduct(AuqB;pA;pBN>)SEoftrwoSEobjectsAandBKinCzexiststhenitisuniqueuptoisomorphism.Ifthe nalobjectEinCݹexiststhenitisuniqueuptoisomorphism.Problems1.1.10.zLet)C]^bSeacategorywith niteproducts.#aGivreade nitionofaproSductofafamilyA1;:::ʚ;An (nl0).ShorwthatproductsofsucrhfamiliesexistinC5.De nitionandRemark1.1.11._LetmCabSeacategoryV.1ThenC52opDwiththefol-lorwing}EdataOb'C52op 뤹:=URObC5,&Mor#Cmrop&>(A;B)UR:=MorOC(B;A),&and}Ef9op WgË:=g9s:fDde nesanewcategoryV,thedual35cffategory꨹toC5.Remark1.1.12.qN6Anry#~notionexpressedincategoricalterms(withobjects,1mor-phisms,>andtheircompSosition)hasadualnotion,i.e. |thegivrennotioninthedualcategoryV.MonomorphismsIfHinthedualcategoryC52op fareepimorphismsintheoriginalcat-egoryCandconrverselyV.H9A nalobjectsIHinthedualcategoryC52op isaninitial7objeffctintheoriginalcategoryC5.De nition1.1.13.}!ùThecffoproductoftrwoobjectsinthecategoryCչisde nedtobSeaproductoftheobjectsinthedualcategoryC52op R.Remark1.1.14.qN6EquivXalenrt"totheprecedingde nitionisthefollowingde ni-tion.GivrenA;BX2URC5.!AnobjectAqB?inCW̹togetherwithmorphismsjA 36:An!1AqBandsjB :URBX !_7AqBX !Byissa(categorical)coproSductofAandBifforevreryobjectT 2kTC˹andevrerypairofmorphismsfS:A!5T\andgٍ:BZ!;T\thereexistsauniquet7 &e41. %TOOLBO9XYmorphism[f;gn9]UR:AqBX !_7TnsucrhthatthediagramMȍH_T`f#dׁ @#d @#d @#d @i>@i>RH'Eg!$ׁ !$ !$ !$ ڤ>ڤ> bZAbZLAqB4{fdά-ejX.AbZbZpBzԟ{fdPYάeښjX.BHoǠ*FfeآDǠ?`T[߱8fh;gI{]Ĩcommrutes.ThecategoryCissaidtoharve nite2cffoproductsifC52op/isacategorywith niteproSducts.8Inparticularcoproductsareuniqueuptoisomorphism.GԍU2.FunctorsDe nition1.2.1.vaùLetCݹandD?bSecategories.8LetFconsistofS 1. #amapObFC3URA7!F1(A)2ObDUV; 2. #afamilyofmaps)_fFA;B]:URMorOC(A;B)UR3fQ7!FA;B(fG)2MorODڲ(F1(A);F(B))jA;BX2URC5g덍[or1"fFA;B]:URMorOC(A;B)UR3fQ7!FA;B(fG)2MorODڲ(F1(B);F(A))jA;BX2URC5g]QuFiscalledacffovariant[contravariant$D]functorifS 1. #FA;A<(1A)UR=1F((A)C@forallAUR2ObC5, 2. #FA;C(gn9fG)UR=FBd;CJJ(g)FA;B(fG)forallA;B;C12URObC5: #[FA;C(gn9fG)UR=FA;B(f)FBd;CJJ(gn9)forallA;B;C12URObC5].Notation:8WVewriteSʍ~[AUR2C~insteadofVAUR2ObC~|fQ2URC~insteadof1fQ2URMorOC(A;B);F1(fG)~insteadofFA;B(fG)."UExamples1.2.2.1. Id:URSet!+fSet 2. #FVorget:8RJ-Mo`dÌ !02kSet 3. #FVorget:8Ri_y!'Ab 4. #FVorget:8Ab?Y!+Gr 5. #P)]:,SetBq[!0SetCq;P(M@),:=pSorwersetofM@.wP(fG)(X),:=f21 {(X)forftƹ:,M #"=!1gN;XFURN+isaconrtravXariantfunctor. 6. #QC:Seti !29SetE;Q(M@):=hpSorwersetofM@. gQ(fG)(X)C:=f(X)hforf'B:CM #"=!1gN;XFURM+isacorvXariantfunctor.Lemma1.2.3.xUU1.XLffet35XF2URC5.fiThenĨAOb~C3URA7!MorOC(XJg;A)2ObSet덍) Mor> CDMɹ(A;B)UR3fQ7!M@orCm(XJg;fG)2MorO12@cmbx8Set&(Mor5C(XJg;A);Mor5CU(X;B));Qu #withMorԁC (XJg;fG) :MorRC q(X;A)3gB7!fGg2MorRC q(XJg;B)orMorԁC (X;fG)(gn9) = #fGgnis35acffovariantfunctorMori2C ʹ(XJg;-33).-:7 &e3. %NA:TURAL!TRANSF9ORMATIONS=f5Y 2. #Lffet35XF2URCܞ.fiThenǍOb+QC3URA7!MorOC(A;X)2ObSet),MorACGN(A;B)UR3fQ7!MorOC(f;X)2MorOSet&(Mor5C(B;X);Mor5C(A;X)) #withHMor~ݟC윹(f;X)UR:MorOC(B;X)3gË7!gn9fQ2MorOC(A;X)HorMor~ݟC윹(f;X)(gn9)UR=gf #is35acffontravariant35functorMori2C(-;X).+- cmcsc10Proof.@_1. oMorC$(XJg;1A)(gn9)=1Ag'ι=g=id9(gn9);RZMorWC#(XJg;fG)Mor5CU(X;gn9)(h)=fGgn9hUR=MorOC(XJg;fgn9)(h).2.8analogouslyV.Jcffxff ̟ff ̎ ̄cffRemark1.2.4.j6TheprecedinglemmashorwsthatMor&ğC(-;-)isafunctorinbSothargumenrts.1A functorMintwoargumentsiscalledabifunctor.1WVecanregardsthebifunctorMor Cd(-;-)asacorvXariantfunctorǍ$OMorZLC (-;-)UR:C5 op @C"!wfSet(`I:TheuseofthedualcategoryremorvesthefactthatthebifunctorMorC`~(-;-)iscon-trarvXariantinthe rstvariable.ObrviouslyDthecompSositionoftwofunctorsisagainafunctorandthiscompSositionisassoSciativre.8FVurthermoreforeachcategoryCݹthereisanidentityfunctorIdC.FVunctors-oftheformMorcʟCщ(XJg;-)resp.Mor/C(-;X)arecalledrffepresentablefunctors(corvXariant^resp.contravariant)^andXiscalledtherffepresenting@Hobject^(seealsosection1.8). 3.NaturalTransformationsDe nition1.3.1.vaùLetPFA:j0Ce 6!"D۹andG ^:Ce 6!"D۹bSetrwoPfunctors. jwAnaturffalEtransformationoɹorafunctorialmorphism'UR:Fc!BG%isoafamilyofmorphismsf'(A)UR:F1(A)n!1G.(A)jA2C5g꨹sucrhthatthediagramSvF1(B)G.(B)pt32fd/pά- Vp'(B)HDF1(A)HG.(A)D{fd0Ѝά-Pt'(A)HҟǠ*FfeǠ?`FF1(fG)HǠ*FfeğǠ?`DG.(fG)FgcommrutesforallfQ:URAn!1BinC5,i.e.8G.(fG)'(A)UR='(B)F1(f).Lemma1.3.2.g5QGivencffovariantfunctorsF=IdjBSet&N:Set 4"X!5A_SetN'andG\3=Mor5Set#t(Mor5Set(;A);A)UR:Set!+OFSetBk^for35asetA.fiThen'UR:Fc!B"GcwithǍH'(B)UR:BX3b7!(Mor5Set#t(B;A)3fQ7!fG(b)2A)2G.(B)is35anaturffaltransformation.=B7 &e61. %TOOLBO9XYProof.@_GivrengË:URBX !_7Cܞ.8ThenthefollowingdiagramcommutesQFH^BHoMorSetU(Mor5Set#t(B;A);A)kD{fd*Fά-Psq'(B)^͏C MorSetY(Mor5Set#t(C5;A);A)k432fd*Ѝά-s@'(Cܞ)Hc<Ǡ*FfecoǠ? XgHث2Ǡ*FfedǠ?`ݐMorSet|(Mor5Set#t(gn9;A);A)rsinceeʍLt'(Cܞ)F1(gn9)(b)(fG)UR='(C)gn9(b)(fG)=fgn9(b)='(B)(b)(fgn9),g=UR['(B)(b)Mor5Set#t(gn9;A)](fG)=[Mor5Set(Mor5Set(gn9;A);A)'(A)(b)](fG):ȍ %cffxff ̟ff ̎ ̄cff!Lemma1.3.3.g5QLffet8fQ:URA!Bbe8amorphisminC5.ThenMormCۼ(f;-33)UR:MorOC(B;-)fk!Mor&C,_{(A;-33)givenbyMorC"(f;Cܞ)UR:MorOC(B;Cܞ)3gË7!gn9fQ2MorOC(A;Cܞ)isanaturffaltrffansformation35ofcovariantfunctors.LffetJWf':(A!fB]beJWamorphisminC5.ThenMorTC(-35;fG)(:Mor%C#(-;A)!fMor*C0 y(-;B)givenbyMor=C͹(C5;fG)UR:MorOC(C;A)3gË7!fGg2MorOC(C5;B)isanaturffaltransformationof35cffontravariantfunctors.!Proof.@_LethUR:C1K{!Cܞ20bSeamorphisminC5.8ThenthediagramsMHy~GMor(DC(B;Cܞ20׹)gMor dCa#(A;Cܞ20׹)32fd0Cά-knÞ1MorȟX.!q% cmsy6CR(fh;C-:0B})HYdMoraC (B;Cܞ)H$Mor ZC?(A;Cܞ){fd30ά-`MorOX.C'5(fh;C)HǠ*FfeE4Ǡ?`pxMorX.C(Bd;h)HǠ*FfeǠ?`f4Mor'˟X.C,U(A;h)9andF aMor^CS(C5;A)BMor ?Cd(C5;B)d32fd3ά-W`BMorӛKX.Cr(C;f)H~Mor*Cs(Cܞ20;A)HMor <CS(Cܞ20;B)ST{fd0Cά-n­_MorX.C݀(C-:0B};f)HҟǠ*FfeǠ?`pˢMor$9X.C(h;A)H&RǠ*FfeYǠ?` Mor'dX.C,<%(h;Bd)commrute.w0Gcffxff ̟ff ̎ ̄cff!Remark1.3.4.j6The$compSositionoftrwo$naturaltransformationsisagainanat-uraltransformation.8Theidenrtityid LF(A)UR:=1F((A)C@isalsoanaturaltransformation.De nition1.3.5.vaùAmznaturalmtransformation'47:FeH~! G#ɹismcalledanaturffaliso-morphismifthereexistsanaturaltransformation S:G[Ht!FJsucrhthat'ʑ =id oGand 1n'B=idpF".Thenaturaltransformation Թisuniquelydeterminedbry'.WVewrite'21ι:=UR n9.A|functor}F-issaidtobSeisomorphictoafunctorG3Jifthereexistsanaturalisomorphism'UR:Fc!BG..Lܠ7 &e3. %NA:TURAL!TRANSF9ORMATIONS=f7YProblems1.3.6.sX1.!Let.hF1;G~ҹ:ȤC{d!^ DbSefunctors.Shorwthatanaturaltrans-formation;'UR:Fc!BGMiisanaturalisomorphismifandonlyif'(A)isanisomorphismforallobjectsAUR2C5.2.Let5A(ArB;pA;pBN>)bSetheproductofAandBGinC5.ThenthereisanaturalisomorphismꍍmrMor(-;AB)PUR԰n9=Mor%5C*(-;A)Morय़CNd(-;B):23.yLetC^ bSeacategorywith niteproducts.yFVoreacrhobjectAinC^ showthatthereexistsamorphismA Q:sAW!%AAsatisfyingp1A=s1A=p2A.nShorwthatthisde nesanaturaltransformation.8Whatarethefunctors?4. wkLet+C]`bSeacategorywith niteproducts. wkShorwthatthereisabifunctor-- :{CmEC/ H!nC~sucrhIthat(--)(A;B)istheobjectofaproSductofAandB.|WVedenote4elemenrtsintheimageofthisfunctorbyA8BX:=UR(-{-)(A;B)4andsimilarlyfgn9.5. CWiththenotationoftheprecedingproblemshorwthatthereisanaturaltransformation (A;B;Cܞ):(A&B)CPh4԰=I!A(BCܞ).[Shorwthatthediagram(cffoherence35orcffonstraints)Dύ((AB)Cܞ)Dl(A(BECܞ))DgW:2fdEHά-̯rA (A;Bd;C)1`8,A((BECܞ)DS)|:2fdEHά-̯!)< (A;BdC;D(AE)B)A(E^B)L:2fdC ά-̯} (A;Er;Bd)NABꃀ(A)1l Ql攴Ql?^QlQz̟0Qz̟0sꃀp 1(Bd)L L攴L?^L#0#0+De nition1.3.7.vaùLetdCandD%bSecategories.0A^corvXariantdfunctorFc:URC"!wfDiscalled8aneffquivalence]ofcffategories8ifthereexistsacorvXariant8functorGY:D6!Cmandnaturalisomorphisms'UR:G.FPc԰J=1 IdIC%Mand Ë:F1GP ԰$g=)IdzfD"ɹ.A;conrtravXariant@functorFc:URC"!wfD)iscalledadualityofcffategories@ifthereexistsa DconrtravXariantfunctorG':CDO!Cyandnaturalisomorphisms':G.FPu ԰=ZIdҗC*Mand Ë:URF1GP ԰$g=)IdzfD"ɹ.A_%category_HC}issaidtobSeeffquivalenttoacategoryDifthereexistsanequivXalenceF¹:nC!;q!$DUV.eAcategoryCŹissaidtobSedualtoacategoryDNifthereexistsadualitryFc:URC"!wfDUV.Problems1.3.8.sX1.8ShorwthatthedualcategoryC52op isdualtothecategoryC5.^ʠ7 &e81. %TOOLBO9XY2.yLetMDUbSeacategorydualtothecategoryC5.ShorwthatDUisequivXalenttothedualcategoryC52op R.3.-Let@/>RHA4ڟǠ*Ffeh Ǡ? gdcommrutes.TheelementsofM R ^ENEarecalledtensors,K;theelementsoftheformm n꨹arecalleddeffcomposable35tensors.Warning: IfEyrouwanttode neahomomorphismfQ:URM2 RN6!AwithatensorproSduct[sasdomainyroumustde neitbygivinganRJ-bilinearmapde nedonM %N@.G΍Lemma1.4.3.g5QA-\tensor-^prffoduct(Mސ R ?N; )de nedbyMR andRNnBisuniqueup35toauniqueisomorphism.Proof.@_Let(M R ;N; )and(M% msam10R ;N;)bSetensorproducts.8ThenHZMN`> ^ҁ ^ׁ ^܁ ^ ^ ^ ^ x^ sќdžsќdžH-»Ɩׁ P|>P|> H` ׁ @ @ @ @N<>@N<>RH- @ҁ H@ׁ H@܁ H@ H@ H@ H%@ H/@ H3džH3džj34LM R ;N34MR ;N}T32fdϠά-:}Zh3434 32fdAά-:Սk34?M R ;N347MR ;N-zl32fdϠά-:-wh p7 &ea>4. %TENSOR!PR9ODUCTS9Yimpliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cffRBecause}ofthisfactwrewillhenceforthtalkabSoutthetensorproductofMaandNorverRJ.ፍProp`osition1.4.4.O(Rules!wofcomputationinatensorproSduct)zHLffet(MP RJN; )bffe35thetensorproduct.fiThenwehaveforallr2URRJ,m;m20#2M@,n;n20#2N4w 1. #M R ;N6=URf"u cmex10P imi nij35mi,2M;ni2N@g; 2. #(m+m209) nUR=m n+m20x n; 3. #m (n+n209)UR=m n+m n20; 4. #mrqB nt=m rSnΆ(observeinpffarticular,Zthat t:M^N?!1M NjisΆnot #injeffctive35ingeneral), 5. #ifJ4f:MN'7!!AisanRJ-bilineffarmapandg!:M R DN'7!!Aistheinducffed #homomorphism,35thenPgn9(m n)UR=fG(m;n):ፍProof.@_1. *LetkB:= hm niM R NOdenotekthesubgroupofM R NgeneratedԞbrythedecompSosabletensorsmr n.2LetԞj%:URBX !_7M RfRNbetheemrbeddinghomomorphism.^WVe getaninducedmap 20#:URM5QN6!B.InthefollorwingdiagramP$Z[MNZHXB{fdά-; 20{􎎍{POM R ;ND̟{fdά-#趫j34HXB34POM R ;ND̞32fdά-^c趫jH披E\ 20ׁ @ @ @ @ >@ >RH:Ǡ*Ffe lǠ?褍A id BHzǠ*FfeìǠ?hv,jpH pLׁ L L L D̟>D̟> $&wrebhaveid\BB 20?U=q 209,pwithp,j 20?U=qp = 20_existsbsince 20isRJ-bilinear.Becauseyofjpu ӹ=jgHu 20 = =idwM" X.RN/) y޹wregetjpӹ=idwM" X.RN-+,ݬhenceytheemrbSeddingj{issurjectiveandthustheidentityV.2.8(m+m209) nUR= (m+m20;n)UR= (m;n)+ (m209;n)UR=m n+m20x n.3.8and4.analogouslyV.5.8ispreciselythede nitionoftheinducedhomomorphism.dǷcffxff ̟ff ̎ ̄cffፍRemark1.4.5.j6TVoconstructtensorproSducts,,wreusethenotionofafreemodule.LetSXֹbSeasetandRbearing.&AnRJ-moduleRJXֹtogetherwithamapm:X!wRJXwѹisNcalledafrffeeNR-moduleFgeneratedbyNX,7ifforevreryRJ-moSduleM2andforevreryhwmapfsu:+vX6!aM[thereexistsauniquehomomorphismofRJ-moSdulesg:RJX!nM+sucrhthatthediagramFbZgUXbZ+RJX{fd$pά- ѯH`efׁ @ @ @ @?,>@?,>RH(MDZǠ*FfewǠ? * gcommrutes. 7 &e101. %TOOLBO9XYFVreeRJ-moSdulesexistandcanbeconstructedasRJX:=f :X!5 RJjforalmostallxUR2XFչ: (x)=0g.認Prop`osition1.4.6.OGivenIRJ-moffdulesMR andRN@.gThentherffeexistsatensorprffoduct35(M R ;N; ).Proof.@_De ne:M R N:=/ZfMN@g=UXwhereZfMN@gisafreeZ-moSduleorverMN+(thefreeabSeliangroup)andUisgeneratedbry~ʍ #(m+m209;n)(m;n)(m209;n) #(m;m+n209)(m;n)(m;n209) #(mrr;n)(m;rSn)forallr2URRJ,m;m20#2M@,n;n20#2N@.8ConsiderVR H@MNHZfMN@gj!t{fd!;ά- xHHeM R ;N/{fdЍά- ׳HHVi=URZfMN@g=UHqA` l\uPv\+P\ׁP\tP\+P\P\sP\+P\P\rP\+P\P0$P0$qH 5+ʬQtQQ+QrQQ턴Q턴sHǠ*FfeDǠ? gLetUt íbSegivren.yDThenthereisaunique 2Hom(ZfM4B^N@g;A)Utsuchthat = n9.Since IisRJ-bilinearwreget((mN4+m209;n)(m;n)(m209n))1= n9(mN4+m20;n) n9(m;n)Z (m209;n) =0andsimilarly((m;nZ+n209)(m;n)(m;n209)) =0and((mrr;n)}(m;rSn))g=0.So&wreget(U@)g=0.This&impliesthatthereisauniqueg<2Hom(M7l R N;A)Zsucrhthatgn9ʹ=(homomorphismtheorem).6Let :=O.Then isbilinearsince(m3+m209) n$.=3(m+m20;n)$.=ǹ((m3+m20;n))$.=ǹ((mfE+m209;n)(m;n)(m209;n)+(m;n)+(m209;n))*[=ǹ((m;n)fE+(m209;n))*[=>;w(m;n)+(m209;n)UR=m;w n+m20 n.&The8othertrwo8propSertiesareobtainedinananalogouswrayV.WVe>=harvetoshowthat(M` R>N; )isatensorproSduct.gTheaborvediagramshowsthat}foreacrhabSeliangroupAandforeachRJ-bilinearmap M:gMY )N!(AthereisVag{52 Hom(M5 R N;A)sucrhthatgbX  = n9.|GivenVh2Hom(M5 R N;A)withh UR= n9.8ThenhloUR= .8Thisimpliesh=UR=gohencegË=h.(Մcffxff ̟ff ̎ ̄cff認Prop`ositionandDe nition1.4.7.ajGiven35twohomomorphismsdnfQ2URHomٟR$l(M:;M@ 0:)35andgË2URHomٟR(:N;:N@ 0):Then35therffeisauniquehomomorphism}Hf R ;gË2URHom(M RN;M@ 0 RN@ 0) {7 &ea>4. %TENSOR!PR9ODUCTS 11Ysuch35thatf R ;gn9(m n)UR=fG(m) g(n),35i.e.fithefollowingdiagrffamcommutesOJ*卒UM@20N@20卒QM@20 R ;N@20]432fdpά- W` {$1MN{M R ;Nȋ{fdά-` H꒟Ǡ*FfeğǠ?`ffgH`Ǡ*FfeDǠ?`Ef R ;gProof.@_ (fgn9)isbilinear.@cffxff ̟ff ̎ ̄cffNotation1.4.8.p>6WVe_qoftenwritefB* R N\:=f R1N xand_qM; RgT:=1M ~ Rgn9.WVeharvethefollowingruleofcomputation:=jGf R ;gË=UR(f RN@ 0)(M Rgn9)UR=(M@ 0 Rgn9)(f RN@)sincefgË=UR(fN@20)(Mgn9)UR=(M@20gn9)(fN@).Prop`osition1.4.9.OThe35followingde necffovariantfunctors 1. #-'e N6:URMo`dÌ-"Rn!*[Ab&"; 2. #M - 3/:URRJ-LMo`d# %vv!4AbF}; 3. #-'e - 3/:URMo`dÌ-"RRJ-LMo`d# %vv!4AbF}.Proof.@_(f gn9)(fG20Ag20rHk}Aϳ,>UH`rׁ̟ ̟ ̟ ̟ 댟>댟> eLet@+AandB1bSeK-algebras.9kA@homomorphismofalgebrffas@+f.߹:Ak!MBis@+aK-linearmapsucrhthatthefollowingdiagramscommute:LWABE32fd6^ά- fZCHA AZBE B{fd@ά-if fH4RǠ*FfegǠ?JrX.AHҟǠ*FfeǠ?rX.Band?pj⍒%)K'X.Aγׁ ɳ ij  d>d> H'tX.Bׁ A A A AD>AD>U AŚB:Rd32fd(Ѝά- EfRemark1.5.2.j6EvreryK-algebraAisaringwiththemultiplicationӍxAA2 pURn!1A A2jrpURn!A:eTheunitelemenrtisn9(1),where1istheunitelementofK.ObrviouslyMthecompSositionoftwohomomorphismsofalgebrasisagainahomo-morphismofalgebras.FVurthermoretheidenrtitymapisahomomorphismofalge-bras.iHencetheK-algebrasformacategoryK-Algǹ.ThecategoryofcommrutativeK-algebraswillbSedenotedbryK-cAlgǹ.Problems1.5.3.sX1.8ShorwthatEndgK!3)(Vp)isaK-algebra.2.mShorwthat(A;r:A* Az!EkA;(:Kz!EkA)isaK-algebraifandonlyifAxwith themrultiplicationAgA2o p1ʼ!{A A23Irp1ʼ!A andtheunitn9(1)isaringandj:1K!n߹Cenrt)y(A)isaringhomomorphisminrtothecenterofA.3.;LetV6bSeaK-module.;ShorwthatD(Vp)3:=KjV6withthemrultiplication(r1;v1)(r2;v2)UR:=(r1r2;r1v2j+r2v1)isacommrutativeK-algebra.c7 &eT:5. %ALGEBRAS15YLemma1.5.4.g5QLffetoAandB bealgebras. ThenA B isoanalgebrawiththemultiplicffation35(a1j b1)(a2 b2)UR:=a1a2j b1b2.\RProof.@_Certainly thealgebrapropSertiescaneasilybecrhecked byasimplecal-culationwithelemenrts.8WVepreferforlaterapplicationsadiagrammaticproSof.LetprA T4:vPA@ A!-ApandrB Ď:BF @BV*!3B/vdenotethemrultiplicationsofthetrwo;algebras.nThenthenewmrultiplicationisrA B:=M(rA  + rBN>)(1A E 1BN>)M:AP BV A B(!A B.where/:sB As!yA B.isthesymmetrymapfromTheorem1.4.15.8Norwthefollowingdiagramscommutepz%A BE A B A BpzKA A BE B A B4mifd(ά-jF1 I{ 1-:Aacmr63pzpz`)A BE A B4mifd)7ά-jF6»r r 1-:2Z%A BE A A B BZKA A A BE B B4{fd(ά-MU|D1 X.BI;A A} 1-:3ZZ`)A A BE B4{fd)7ά-23ir 1 r 1m5H@ `G fe@S4`?, 1-:3* I{ 1m5HEŸ`G fex`?͍+t1 X.BI B;A 1m5H}0Ÿ`G fe}c`?ht1 I{ 1ꍑX8A A BE A B BH"E1 I{ 1-:3P`Z`d`n`x`}Tx}Tx*m5?P61-:3* I{ 14zn@4n@4n@4n@4n@ԟԟ?PH"1-:2* I{ 1-:2ԟwHԟwHԟwHԟwHԟwH4H4j)A BE A B!A A BE Bl 32fdX ά-n1 I{ 1xXA BFD32fdXPά-0>FLr rH@ Ǡ*Ffe@S4Ǡ?p1 r rHEŸǠ*FfexǠ?p+t1 r 1 rH}0ŸǠ*Ffe}cǠ?ptr reIn>.theleftuppSerrectangleofthediagramthequadranglecommrutesbythepropSertiesofNthetensorproSductandthetrwoNtrianglescommrutebyinnerpropSertiesofn9.eTherighrt0WuppSerandleftlowerrectanglescommutesinceisanaturaltransformationandtherighrtlowerrectanglecommutesbytheassoSciativityofthealgebrasAandB.FVurthermorewreusethehomomorphismË=URA B:Kn!1K"v K1K{!A BCinthefollorwingcommutativediagramyp WK A BPX԰ ?=A BPX԰ ?=A BE KpA BE K Kqdmifd'8ά-jܠpp`A BE A B?mifdЍά-iQ>1-:2* I{ m5.ڍ1sठzn@H}ठn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@H टn@Hटn@Hटn@H'टn@H1टn@H;टn@HEटn@HOटn@HYटn@Hcटn@Hmटn@HndƑHndƑjm5 zn@Hn@H—䟄n@H̗䟉n@H֗䟎n@H䟓n@H䟘n@H䟝n@H$䟟H$䟟jBA K BE KB`A A BE B? RfdЍά-\퍒;1 I{ 1 m5 br*Ffe?H$1 I{ 1m5 򟟴*Ffe$?1 I{ 1 ǠG fe$Ǡ?Gr rm5 9)dzn@HC)dn@HM)dn@HW)dn@Ha)dn@Hk)dn@Hu)dn@H)dn@HdHdjB7K K A BBuJK A K BT Rfdά-͎U1 I{ 1m5 (r*Ffe(?aA BE A BuA A BE BT32fdά-nU1 I{ 1vA B:t32fdt0ά-0;r r (rǠG fe(Ǡ?ևōGI{  1-:2 ǠG fe'$Ǡ?]tdI{ 1  1e %cffxff ̟ff ̎ ̄cff\RDe nition1.5.5.vaùLetKbSeacommrutativering.?gLetV3beaK-module.?gASK-algebraK0Tƹ(Vp)togetherwithahomomorphismofK-moSdules:V!AT(Vp)K0iscalledU7 &e161. %TOOLBO9XYatensor8algebrffaoverV5-ifforeacrhK-algebraAandforeachhomomorphismofK-moSdules_fD:V !OAthereexistsauniquehomomorphismofK-algebrasgj:Tƹ(Vp)!nA꨹sucrhthatthediagramJHQbVH4zTƹ(Vp)Dğ{fd!wЍά- H` }fDğׁ @Dğ @Dğ @Dğ @ߋD>@ߋD>RH\0ArǠ*FfeäǠ? v$g4commrutes.Note:"IfW,yrouwanttode neahomomorphismgË:URTƹ(Vp)n!1AW,withatensoralgebraasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVp.Lemma1.5.6.g5QAmtensorualgebrffa(Tƹ(Vp);)de nedbyVisuniqueuptoauniqueisomorphism.Proof.@_Let(Tƹ(Vp);)and(T20o(Vp);209)bSetensoralgebrasorverV.8ThenL]PbZ_V9<ҁ <ׁ <܁ < < < < w< s<džs<džH <20✟ׁ ✟ ✟ ✟ >> H9S\ׁ @S\ @S\ @S\ @ܟ>@ܟ>RH V<20苼ҁ H򋼟ׁ H܁ H H H H$ H. H3džH3džjURTƹ(Vp)mvTƟ20o(Vp)s32fd@`ά-:{hӌ32fdQά-:ՍkCTƹ(Vp)>6TƟ20o(Vp)#L32fd@`ά-:,h'impliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cffProp`osition1.5.7.O(Rulesofcomputationinatensoralgebra)ZLffet(Tƹ(Vp);)bethe35tensoralgebrffaoverVp.fiThenwehave 1. #:Vl!Tƹ(Vp)uisinjeffctive(sowemayidentifytheelements(vn9)andvforall #vË2URVp), 2. #Tƹ(Vp)UR=fP n;'ߍRk!iIviq1 :):::vinjLNiWy=(i1;:::ʚ;inP)35multiindexoflengthnng;> 3. #ifZfQ:URVX-!AisahomomorphismofK-moffdules,vAisaK-algebra,vandgË:URTƹ(Vp) #"!1GA35istheinducffed35homomorphismofK-algebrffas,then⍑i gn9(Xun;'ߍRk!iUVviq1 :):::vin)UR=X/n;'ߍRk!ifG(viq1):::f(vin):%Proof.@_1.UseFtheemrbSeddinghomomorphismj`":OVO iJ!D(Vp),whereD(Vp)isde nedAasin1.5.3.3.>toconstructgX6:Tƹ(Vp)!DS(V)AsucrhthatgTX=jӹ.>SinceAjҹisinjectivresois.2.$Let_BX:=URfP n;'ߍRk!iIviq1 /:::)/vinjLNiWy=(i1;:::ʚ;inP)mrultiindexoflengthoBng.Obrviously>Bis(ythesubalgebraofTƹ(Vp)generatedbrytheelementsofVp.TLetjk]:BYs!1Tƹ(V)bSetheemrbSeddinghomomorphism.p)Then/:V3 M*![Tƹ(Vp)factorsthroughalinearmap7 &eT:5. %ALGEBRAS17Y20#:URV M!`B.8InthefollorwingdiagramRRbZVbZY@Bt{fd*Fά-{G 20HHEjTƹ(Vp)U{fd!wЍά-#=jY@BEjTƹ(Vp)U32fd!wЍά-^c=jH 20tׁ @t @t @t @>@>RH"Ǡ*FfeTǠ?褍QidBH bǠ*Ffe ԔǠ?hjpH p4ׁ 4 4 ݜ4 U>U> 䍹wreXhaveid"Bp20ݪ=q209.pwithpj20ݪ=qp=20&9existsXsince20isahomomorphismofK-moSdules.6BecauseofjpUR=jKo20#==id T.:(V)#4K¹wregetjpUR=id T.:(V)!4M,hencetheemrbSeddingj{issurjectiveandthusj{istheidentityV.3.8ispreciselythede nitionoftheinducedhomomorphism.dǷcffxff ̟ff ̎ ̄cffProp`osition1.5.8.OGiven,aK-moffduleVp. PThenthereexistsatensoralgebra(Tƹ(Vp);).Proof.@_De ne$TƟ2nJ(Vp)/:=Vn 3:::o 3VT=V2 nto$bSethen-foldtensorproductofVp.8De neTƟ20aʹ(V)UR:=KandTƟ21aʹ(Vp):=V.8WVede ne8pTƹ(Vp)UR:=M i0wvT i(Vp)=KVG(V Vp)(V V Vp):::uD:"ThecompSonenrtsTƟ2nJ(Vp)ofTƹ(V)arecalledhomoffgeneous35components.ThecanonicalisomorphismsTƟ2m (Vp)] TƟ2nJ(V)PUR԰n9=TƟ2m+nkR(V)takrenasmultiplication$ʍ rUR:TƟ2m (Vp) TƟ2nJ(V)URn!1TƟ2m+nkR(V)sYrUR:Tƹ(Vp) T(Vp)URn!1T(V)andUtheemrbSeddingy: `K=TƟ20aʹ(Vp)$!0MTƹ(V)UinducethestructureofaK-algebraonTƹ(Vp).8FVurthermorewrehavetheembSeddingUR:V M!`TƟ21aʹ(Vp)Tƹ(V).WVeharvetoshowthat(Tƹ(Vp);)isatensoralgebra.bLetfhɹ: V: !AbSeaho-momorphismofK-moSdules.EacrhelementinTƹ(Vp)isasumofdecompSosableten-sors6v1 :::LD vnP. De negz: Tƹ(Vp)&F!3A6brygn9(v1 :::LD vnP) :=fG(v1):::ʜf(vn)(and(gC{:BTƟ20aʹ(Vp)!A)=(:K!A)). cByinductiononeseesthatgg>isahomomorphismofalgebras. oSince(gJ :TƟ21aʹ(Vp)r!)=(f#:V xW !mA)wregetg?==fG. S0IfHh:Tƹ(Vp)!lAisahomomorphismofalgebraswithh==fwregeth(v1j ::: vnP)UR=h(v1):::ʜh(vn)=fG(v1):::ʜf(vn)hencehUR=gn9.QWcffxff ̟ff ̎ ̄cffProp`osition1.5.9.OThejcffonstructionoftensoralgebrasTƹ(Vp)de nesafunctorTG:QK-35Mo`d%(Y[!:K-35AlgthatisleftadjointtotheunderlyingfunctorUe:K-35Algfk!K-35Mo`do.Proof.@_FVollorwsfromtheuniversalpropSertyand1.11.6.uMcffxff ̟ff ̎ ̄cffProblems1.5.10.z1.~LetWOXHҹbSeasetandV:=BKXbSethefreeK-moduleorverX.Shorw?thatXF``!V M!`Tƹ(Vp)de nesafrffee9algebra?޹overX,bi.e.foreveryK-algebra 7 &e181. %TOOLBO9XYA4bandevrerymapfQ:URXF``!AthereisauniquehomomorphismofK-algebrasgË:Tƹ(Vp)!nA꨹sucrhthatthediagramH8ȍHXHTƹ(Vp)&<{fd ဍά-H`Xfׁ @ɏ @ӏ @ݏ @l>@l>RHXAۚǠ*Ffe̟Ǡ? Lg6commrutes.WVe:V 9!jyS(Vp),Fsucrhthat(vn9)w(v20@߸\>RHHA񽊟Ǡ*FfeǠ? > H9S\ׁ @S\ @S\ @S\ @ܟ>@ܟ>RH V<20苼ҁ H򋼟ׁ H܁ H H H H$ H. H3džH3džjUiRS׹(Vp)uSן20(Vp)sY\32fdά-:{hy\32fdRά-:ՍkS׹(Vp)>5Sן20(Vp)#32fdά-:,hf6impliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cffProp`osition1.5.13.(Rulespofcomputationinasymmetricalgebra)Lffet(S׹(Vp);)bffe35thesymmetricalgebraoverVp.fiThenwehaves2 1. #:V@d !JS׹(Vp)r(isinjeffctive(wewillidentifytheelements(vn9)andvaforall #vË2URVp), 2. #S׹(Vp)UR=fP n;'ߍRk!iIviq1 :):::vinjLNiWy=(i1;:::ʚ;inP)35multiindexoflengthnng;> 3. #ifkfֹ:sV G v!AisahomomorphismofK-moffdulessatisfyingfG(vn9)f(vn920Bӹis5thesubalgebraofS׹(Vp)generatedbrytheelementsofVp.PLetj ::Bp@!_S׹(V)bSetheIemrbSeddinghomomorphism.Then:VB& [!iS׹(Vp)Ifactorsthroughalinearmap20#:URV M!`B.8InthefollorwingdiagramOebZVbZY@Bt{fd*Fά-{G 20HHjS׹(Vp)U{fd!ά-#zjY@BjS׹(Vp)U32fd!ά-^czjH 20tׁ @t @t @t @>@>RH"Ǡ*FfeTǠ?褍QidBH bǠ*Ffe ԔǠ?hjpH p4ׁ 4 4 ݜ4 U>U> +wrelhaveid77Bs20=2u209,pwithpj20=2up=20:̹existslsince20isahomomorphismofK-moSdulessatisfying209(vn9)-20(vn920@߸\>RHHA񽊟Ǡ*FfeǠ? propSertryjfollowssincethede ningconditionfG(vn9)f(vn920@>RHpAǠ*Ffe;Ǡ? dgecommrutes.ThealgebraK[X]:=S׹(KX)iscalledthepffolynomial@O$>RH ATRǠ*Ffe񇄟Ǡ? :g3commrutes.ThemrultiplicationinE(Vp)isusuallydenotedbyu^vn9.Note:=If0yrouwanttode neahomomorphismg{: _E(Vp)&!4KA0withanexterioralgebraqasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVsatisfyingfG(vn9)22V=UR0forallv;v202URVp.*+Problems1.5.19.z1.LetfQ:URV M!`AbSealinearmapsatisfyingfG(vn9)22V=0forallvË2URVp.8ThenfG(vn9)f(v20Let2bSeinrvertibleinK(e.g.Ka eldofcrharacteristic6=2).Letf˹:VD<]!AbSe1alinearmapsatisfyingfG(vn9)f(v20> H9S\ׁ @S\ @S\ @S\ @ܟ>@ܟ>RH V<20苼ҁ H򋼟ׁ H܁ H H H H$ H. H3džH3džjTXE(Vp){E20P(Vp)t+̞32fdOά-:{hK̞32fdQ ά-:ՍkE(Vp)>";E20P(Vp)$Q32fdOά-:,himpliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cffፍProp`osition1.5.21.(Rules`ofcomputationinanexterioralgebra)sLffet(E(Vp);)bffe35theexterioralgebraoverVp.fiThenwehaveMX 1. #:VT !E(Vp)^isinjeffctive(wewillidentifytheelements(vn9)andvforall #vË2URVp), 2. #E(Vp)UR=fP n;'ߍRk!iIviq1 :)^:::^vinjLNiWy=(i1;:::ʚ;inP)35multiindexoflengthnng;> 3. #ifkfֹ:sV G v!AisahomomorphismofK-moffdulessatisfyingfG(vn9)f(vn920BisthesubalgebraofE(Vp)generatedbrytheelementsofVp.LetjI:B7Qb!5E(V)bSetheemrbSeddinghomomorphism.JThen:V 7!)E(Vp)factorsthroughalinearmap20#:URV M!`B.8InthefollorwingdiagramS捍bZVbZY@Bt{fd*Fά-{G 20HHpE(Vp)U{fd ά-#jY@BpE(Vp)U32fd ά-^cjH 20tׁ @t @t @t @>@>RH"Ǡ*FfeTǠ?褍QidBH bǠ*Ffe ԔǠ?hjpH p4ׁ 4 4 ݜ4 U>U> dxwrelhaveid77Bs20=2u209,pwithpj20=2up=20:̹existslsince20isahomomorphismofA7K-moSdulessatisfying209(vn9)20(vn920 cmmi10n oiG^"*.2. aShorw(thatthesymmetricgroupSn gxopSerates(fromtheleft)onTƟ2nJ(Vp)byn9(v1j ::: vnP)UR=vI{1 ;;(1) ::: vI{1 ;;(n)F/withË2Sn andvi,2Vp.؍3./Atensor`ά-WZwA A MZ?A M,{fd'8ά-ipid HJǠ*Ffe|Ǡ?э;r idH tJǠ*Ffe |Ǡ?'YmkandG⍒nMP6԰=@K M⍒oA M{fd@ά-i?~I{ idH7M zǠ*Ffe Ǡ?',Hk}lid"ҁ H"ׁ H"܁ H" H" H" H" H" HdžHdžjv7 &e241. %TOOLBO9XYcommrute.LetlA PMPandlANbSelA-modulesandletfQ:URM6!NPbeaK-linearmap./Themapf2iscalledahomomorphism35ofmoffdules꨹ifthediagramQߍ+A NqN,32fd@`ά-W =X.NZ0A MZMĺ{fd>`ά-iwX.MH Ǡ*Ffe=<Ǡ?`!1 fH Ǡ*Ffe /<Ǡ?`f2ҍcommrutes.ThepleftA-moSdulesandtheirhomomorphismsformthecffategoryA MofA-moffdules.]%Problems1.5.25.zShorwthatanabSeliangroupM޹isaleftmoduleorvertheringAifandonlyifMisaK-moSduleandanA-moduleinthesenseofDe nition1.5.24.-6._CoalgebrasDe nition1.6.1.vaùA&K-cffoalgebra&isaK-moSduleCtogetherwithacffomultiplica-tion꨹ordiagonalUR:C1K{!CF CFthatiscoassoSciativre:RV?.CF C9CF C Cܞ32fd)@ά-aid< Z6CZ1.CF C|{fdAά-̍JNHǠ*Ffe,Ǡ?`PHǠ*Ffe,Ǡ?э} id_andacffounitoraugmentationUR:C1K{!K:ZoCZ}FCF C{fdY?ά-̍&HuǠ*FfeğǠ?`~HǠ*FfeDǠ?эid I "zCF C"ȎK CP1԰J׹=ܙCP1԰J׹=CF K:t32fd&fpά-aP idHk}idҁ Hׁ H܁ H H H H H H妄džH妄džjAK-coalgebraCFiscffocommutative꨹ifthefollorwingdiagramcommutesNپCF CCF Cψl32fdά-č9bZ-C``ϳ,ׁ ʳ, ų, , >> H`댟ׁ A댟 A댟 A댟 A̟>A̟>U7 &e 6. %CO9ALGEBRAS/25YLet|CYandDЋbSeK-coalgebras.A|homomorphismofcffoalgebras|fd:NeC+D!DisaK-linearmapsucrhthatthefollowingdiagramscommute:M~CF CcD6 Dm,32fdyά- աf fbZ'CbZyDr̟{fd5氍ά-iۍfH%JǠ*FfeX|Ǡ?XX.CHʟǠ*FfeǠ?|X.D؍andFf_H""%)K-PX.Cdׁ AÐd AȐd A͐d Aγ>Aγ>UH-tX.DDׁ D D D >> bZCbZ+DS:Ž{fd(ά-iEfRemark1.6.2.j6ObrviouslyhQthecompSositionoftwohomomorphismsofcoalgebrasisagainahomomorphismofcoalgebras.'FVurthermoretheidenrtitymapisahomo-morphism8ofcoalgebras. "{HencetheK-coalgebrasformacategoryK-Coalg!.ThecategoryofcoScommrutativeK-coalgebraswillbedenotedbryK-cCoalg'.Problems1.6.3.sX1.ShorwJthatV cVp2 isacoalgebraforevery nitedimensionalvrectorspaceV>1overa eldKifthecomultiplicationisde nedby(vC ՝vn92.=)@:=P* n U_ i=1v v2n9RAi vi vn92 wherefvidgandfv2n9RAi.=garedualbasesofVresp.Vp2\t.2.ShorwthatthefreeK-moSdulesKX #withthebasisXandthecomrultiplication(x)UR=x x꨹isacoalgebra.8Whatisthecounit?Isthecounitunique?3.&ShorwthatK<,VQwith(1)UR=1<, 1,c(vn9)UR=ve <,1+1 v"ʹde nesacoalgebra.4.Let^CandDbSecoalgebras.ThenC.- QDisacoalgebrawiththecomrultiplicationC DG:=UR(1C  c& 1D)(C D)UR:Cы DH{ C Dk!C D=andcounit"UR="C DG:CF Dk!K K1K{!K.8(TheproSofisanalogoustotheproofofLemma1.5.4.)TVo?describSethecomrultiplicationofaK-coalgebraintermsofelementsweintro-duceanotation rstinrtroSducedbySweedlersimilartothenotationr(a^ b)X=abusedforalgebras.8Insteadof(c)UR=Pci c20RAiOwrewrite'(c)UR=Xc(1)$ c(2) \|:wObservrevXthatonlythecompleteexpressionontherighthandsidemakessense,Dnotthe4compSonenrtsc(1)=orc(2)whicrharenotconsideredasfamiliesofelementsofCܞ.>ThisnotationtalonedoSesnothelpmruchtinthecalculationswrehavetopSerformlateron.SowreintroSduceamoregeneralnotation.De nition1.6.4.vaù(SwreedlerTNotation)LetMW8bSeanarbitraryK-moduleandCbSeaK-coalgebra.8Thenthereisabijectionbetrweenallmrultilinearmaps0fQ:URCF:::C1K{!M<7 &e261. %TOOLBO9XYandalllinearmapsmfG 0k:URCF ::: C1K{!M:1JThesemapsareassoSciatedtoeacrhotherbytheformula}jfG(c1;:::ʚ;cnP)UR=f 08(c1j ::: cnP):FVorcUR2CFwrede neэXiMfG(c(1) \|;:::ʚ;c(n) Dȹ)UR:=f 08( n1̹(c));Mwherec2n1ӹdenotesthen1-foldcapplicationof,forexample2n1="5( 1 :::uH 1)( 1).Inparticularwreobtainforthebilinearmap UR:CFC13(c;d)7!c d2CF C>X2oc(1)$ c(2)ι=UR(c);andforthemrultilinearmap 22V:URCFCC1K{!C C CoVXjITc(1)$ c(2) c(3)ι=UR( 1)(c)=(1 )(c):[Withthisnotationonevreri eseasilyDeXX c(1)$ ::: (c(i) R) ::: c(n)=URXc(1)$ ::: c(n+1)pandʍ0͟P>xc(1)$ ::: (c(i) R) ::: c(n)ݭ=URPc(1)$ ::: 1 ::: c(n1)ݭ=URPc(1)$ ::: c(n1)This4notationanditsapplicationtomrultilinearmapswillalsobSeusedinmoregeneralconrtextslikecomoSdules.lProp`osition1.6.5.OLffet.NC beacoalgebraandAanalgebra.rThenthecompositionfgË:=URrA(f gn9)C de nes35amultiplicffation}IC{Homa(C5;A) Hom$1(C;A)UR3f gË7!fgË2Hom(C5;A);such.TthatHom(C5;A)bffecomes.Tanalgebrffa.WTheunitelementisgivenbyK&Y3 97!(cUR7!n9( (c)))2Hom(C5;A).Proof.@_The`mrultiplicationofHom(C5;A)obviouslyisabilinearmap. Themul-tiplicationOisassoSciativresince(fgn9)hbY=rA((rA(f gn9)C) h)C =bYrA(rA 1)((f lgn9) h)(CC 1)C t=URrA(1 rA)(f (g h))(1 C)C t=URrA(f (rA(g h)C))C <=fab(g<h).MFVurthermore!itisunitarywithunit1HomD(C;A)+_l=AC >sinceAC fQ=URrA(AC fG)C t=URrA(A { 1A)(1K fG)(C 1C)C t=URf,|and}similarlyfAC t=URfG.d cffxff ̟ff ̎ ̄cffDe nition1.6.6.vaùThe*$mrultiplicationUR:Hom(C5;A)!o Hom(C;A)URn!1Hom-=(C;A)iscalledcffonvolution.Corollary1.6.7.sWLffet~C[zbeaK-coalgebra. I_ThenCܞ2 X=zHom5K&pŹ(C5;K)isanK-algebrffa.67 &e 6. %CO9ALGEBRAS/27YProof.@_UsethatKitselfisaK-algebra.ìecffxff ̟ff ̎ ̄cffRemark1.6.8.j6IfwrewritetheevXaluationasCܞ2bj Cu3"a c7!ha;ci2Kthenan,elemenrta2Cܞ2 ιis,completelydeterminedbythevXaluesofha;ciforallc2Cܞ.nSotheproSductofaandbinCܞ2 Jisuniquelydeterminedbrytheformula]iHhab;ciUR=ha b;(c)iUR=Xa(c(1) \|)b(c(2)):]TheunitelemenrtofCܞ2 JisUR2Cܞ2.Lemma1.6.9.g5QLffetKbea eldandAbea nitedimensionalK-algebra.ThenA2V=URHom۟K$ (A;K)35isaK-cffoalgebra.Proof.@_De nethecomrultiplicationonCܞ2 JbyuG~;UR:A 2 r-:pVj!(A A) 2Vcanb-:1p Î!"tA j A :nTheBcanonicalmapcan:URA2 A2V .!5(A A)2fFisBinrvertible,$sinceAis nitedimensional.By&ZadiagrammaticproSoforbrycalculationwithelementsitiseasytoshowthatA2bSecomesaK-coalgebra./ׄcffxff ̟ff ̎ ̄cffRemark1.6.10.qN6If(Kisanarbitrarycommrutative(ring,uthenA2V=URHom۟K$ (A;K)isaK-coalgebraifAisa nitelygeneratedprojectivreK-moSdule.Problems1.6.11.zFind"sucienrtconditionsforanalgebraAresp.0acoalgebraC(sucrhL`thatHom(A;Cܞ)bSecomesacoalgebrawithco-convolutionascomultiplication.De nition1.6.12.}!ùLetCbSeaK-coalgebra.AleftXCܞ-cffomoduleisaK-moduleM+togetherwithahomomorphismM B:URM6!CF M@,sucrhthatthediagramsQ;aCF MeCF C M32fd&ά-a>id˾ ZMZCF M\{fd>y@ά--(HǠ*Ffe,Ǡ?k}HǠ*Ffe,Ǡ?э} idtandF=bZMŸǠ*FfeHǠ?k}"CF M"VK MP6԰=@M:32fd:]ά-aF idHk}؄4idҁ Hׁ H܁ HĹ Hι Hع H⹴ H카 HFdžHFdžj commrute.7 &e281. %TOOLBO9XYLet~џ2C iMand~џ2CNbSe~Cܞ-comodulesandletf:QMf!uNbeaK-linearmap.\Themapf2iscalledahomomorphism35ofcffomodules꨹ifthediagramR$NCF N>,32fd@Z@ά-EX.NZ MZ^CF M.{fd>y@ά-zX.MHDleftCܞ-comoSdulesandtheirhomomorphismsformthecffategory2C YMofcffomod-ules.LethNLbSeanarbitraryK-moduleandMLbeaCܞ-comodule. "ThenthereisabijectionbSetrweenallmultilinearmapsܰLfQ:URCF:::M6!NandalllinearmapsK`fG 0k:URCF ::: M6!N:i}ThesemapsareassoSciatedtoeacrhotherbytheformulaq,fG(c1;:::ʚ;cnP;m)UR=f 08(c1j ::: cnR m):FVormUR2M+wrede nein邟X>fG(m(1) \|;:::ʚ;m(n) D;m(M") h)UR:=f 08(s2 n(m));vOwheres22n *denotesthen-foldapplicationofs2,i.e.82n pԹ=UR(1 ::: 1 )(1 ).Inparticularwreobtainforthebilinearmap UR:CFM6!C MˢX Jm(1)$ m(M")u=URs2(m);andforthemrultilinearmap 22V:URCFCM6!C C MPUȟXcm(1)$ m(2) m(M")u=UR(1 s2)(c)UR=( 1)s2(m):WProblems1.6.13.zShorwthata nitedimensionalvectorspaceVqisacomoSduleorver]thecoalgebraV1 8Vp2 ѹasde nedinproblem1.6.3.1withthecoactions2(vn9):=Pv v2n9RAi vi,2UR(VG Vp2\t) VwhereꨟPSv2n9RAi viOisthedualbasisofVinVp2  Vp.h(Theorem1.6.14.wX(Fundamental~ThefforemforComodules)LetKbea eld.ILetMKbffe aleftCܞ-comoduleandletmUR2MKbe given.Thenthereexistsa nitedimensionalsubffcoalgebragCܞ20ضCanda nitedimensionalCܞ20-cffomodulegM@20 withm2M@20Mw7 &e 6. %CO9ALGEBRAS/29Ywherffe35M@20doURMtisaK-submodule,suchthatthediagramMHWMϮCF M32fd>y@ά-͍/ n;M@20/ uCܞ20U M@20L{fd:?ά- L̟-:0HǠ*Ffe$̟Ǡ?H㚟Ǡ*Ffe̟Ǡ?ۍcffommutes.uCorollary1.6.15.z1.~Each;>elementcd42Cof;>acffoalgebra;>iscffontained;>ina nitedimensional35subffcoalgebraofCܞ.2. wEachoelementmu2M$Sofoacffomoduleoiscffontainedoina nitedimensionalsubffcomodule35ofM@.Corollary1.6.16.z1. ^Eachl nitedimensionalsubspffacelVofacffoalgebralCI$iscffontained35ina nitedimensionalsubffcoalgebra35Cܞ20 ofCܞ.2.ÊEach nitedimensionalsubspffaceVDofacffomoduleMyiscffontainedina nitedimensional35subffcomoduleM@20BRofM@.Corollary1.6.17.z1.FEachQcffoalgebraisaunionof nitedimensionalsubcoalge-brffas.2.fiEach35cffomoduleisaunionof nitedimensionalsubcomodules.Proof.@_(oftheThefforem)AWVecanassumethatmUR6=0AforelsewrecanuseM@20do=UR0andCܞ20)=UR0.Undertherepresenrtationsofs2(m)UR2CG M8as nitesumsofdecompSosabletensorspicrkone!Os2(m)UR= sX ㇍Si=1ci mi5ofshortestlengths.WThenthefamilies(cidjiw=1;:::ʚ;s)and(mijiw=1;:::ʚ;s)arelinearlyindepSendenrt.8Choosecoecienrtscij 62URCFsuchthat {N(cjf )UR= wtX ㇍Si=1ci cijJ; 8j%=1;:::ʚ;s;bryrsuitablyextendingthelinearlyindepSendentfamily(cidji<̹=1;:::ʚ;s)rtoalinearlyindepSendenrtfamily(cidjiUR=1;:::ʚ;t)andtURs.WVeɧ rstshorwthatwecanchoSoset=s.ByɧcoassociativitrywehaveP*tTs U_tTi=1!ciV As2(midڹ)t=P*]s U_]jv=1"a(cjf ) mjں=tP*]s U_]jv=1P*/ t U_/ i=1?Mcit cij h~ mjf .3\Since|theciVandthemjareclinearlyindepSendenrtwecancomparecoSecientsandgetzs2(midڹ)UR= sX ㇍jv=1cij mjf ; 8i=1;:::ʚ;s(1)7 &e301. %TOOLBO9XYand0UR=P*s U_jv=1!Bcij mjPfori>s.8Thelaststatemenrtimplies駍{Ucij 6=UR0; 8i>s;j%=1;:::ʚ;s:HencewregettUR=s꨹andf{N(cjf )UR= sX ㇍Si=1ci cijJ; 8j%=1;:::ʚ;s: ΍De neu^ nitedimensionalsubspacesCܞ20 =,hcijJji;j=1;:::ʚ;siCQandu^M@20 I=hmidjin=1;:::ʚ;siM@.Then&]bry(1)weget:nM@20 } Z!Cܞ20,r M@20.WVeshowthatmL2M@20,and|thattherestrictionoftoCܞ20&givresalinearmapL:Cܞ20 2!^Cܞ20 4  Cܞ20so6ythattherequiredpropSertiesofthetheoremaresatis ed.UFirstobservrethatm_=P"(cidڹ)mi,2URM@20Źandcj\=P"(ci)cij 62Cܞ20׹.8UsingcoassoSciativitryweget!Q$P*]Ϭn U_]Ϭi;jv=1tS~ci (cijJ) mj=URP*s U_k6;jv=1(B(ck#) ck6j D mj=URP*s U_i;j$;k6=1- ci cik  ck6j D mjphence+ (cijJ)UR= sX'؍?k6=1cik  ck6j :(2)ς %cffxff ̟ff ̎ ̄cffЍRemark1.6.18.qN6WVegivreasketchofasecondproSofwhichissomewhatmoretecrhnical.ͩSinceCisaK-coalgebra,WthedualCܞ2Eisanalgebra.ThecomoSdulestructureE:bMF ,!5C {Mҹleads`toamoSdulestructurebry=(ev / 1)(1{ s2):Cܞ2 F M!Cܞ2 M C M&" ?!$M@. ConsiderlthesubmoSduleN&":=>Cܞ2m.ThenNis nitedimensional, 6sincec2m@=P*Hn U_Hi=1Chc2;cidimizZforallc2^D2@Cܞ2 "whereP*-n U_-i=1!ci, mi=s2(m).Observrea*thatCܞ2misasubspaceofthespacegeneratedbythemidڹ.hButitdoSesnotdepSendIonthecrhoiceofthemidڹ.FVurthermoreifwetakes2(m)N=PLci /mi#withIashortest6represenrtationthenthemiaareinCܞ2msincec2ms4=Phc2;cidimi=miafor6c2anelemenrtofadualbasisofthecidڹ.NisaCܞ-comoSdulesinces2(c2m)#=PShc2;cidis2(mi)#=PShc2;ci(1)AVici(2)@ bmi 2CF Cܞ2m.Norw9weconstructasubScoalgebraDXǹofC׹suchthatNFisaDS-comodulewiththeinduced%7coaction.LetD :=Nj ҆N@2.By1.6.13NfisacomoSduleorver%7thecoalgebraN jN@2.Constructalinearmap5Z:Ds!CKbryn n2 ^7!Pn(1) \|hn2;n(N")i.Byde nitionofthedualbasiswrehavenUR=Pnidhn2RAi;ni.8Thusweget1ʍ)#( )D(n n2)گ=UR( )(Pn n2RAij ni n2)گ=URPn(1) \|hn2RAi;n(N")i ni(1)AVhn2;ni(N")iگ=URPn(1)$ ni(1)AVhn2;ni(N")ihn2RAi;n(N")iگ=URPn(1)$ n(2) \|hn2;n(N")iUR=PC(n(1) \|)hn2;n(N")iگ=URC(n n2):7 &ebH7. %BIALGEBRAS 31YFVurthermore~"C(n n2)="(Pn(1) \|hn2;n(N")i=hn2;P"(n(1) \|)n(N")i=hn2;ni="(n n2).Hence UR:Dk!Cԩisahomomorphismofcoalgebras,(DKis nitedimensionalandtheimageCܞ20T:=(DS)isa nitedimensionalsubcoalgebraofCܞ."ClearlyNisalsoaCܞ20׹-comoSdule,sinceitisaD-comodule.FinallyǧwreshowthattheDS-comoduleǧstructureonNifliftedtotheCܞ-comoSdulestructurecoincideswiththeonede nedonM@.8WVeharve&ʍ8C(c2m)J0t> HGyGI{ gׁ Al Aq Av AwT>AwT>UAP0Bo BE BNL32fd.ά- YZIBE BZuBӔ{fd.ά-̍PrH""3K` ׁ AĞ Aɞ AΞ A4>A4>UH]Dԟׁ ԟ ԟ ԟ >> j⍒,FKj⍒fK9 {fd*ά--J6idHIB'35 tׁ A: t A? t AD t AE0>AE0>UH]Dd$RiǠWiǠ\iǠaiǠbTׂbTׂcommrute,i.e.7Handarehomomorphismsofalgebrasresp.randUarehomomor-phismsofcoalgebras.2.ɱGivrenpCbialgebrasAandB.Ap mapf:8ARF!B IispCcalledahomomorphismofbialgebrffas꨹ifitisahomomorphismofalgebrasandahomomorphismofcoalgebras.3.8ThecategoryofbialgebrasisdenotedbryK-Bialg"S.䍍Problems1.7.2.sX1.Let߈(B;r;n9)bSeanalgebraand(B;;")bSeacoalgebra.ThefollorwingareequivXalent:ٍa) #(B;r;n9;;")isabialgebra.*b) #UR:BX !_7BE Band":BX !K꨹arehomomorphismsofK-algebras.xc) #rUR:BE BX !_7BandË:Kn!1BarehomomorphismsofK-coalgebras.2.#LetBF¹bSea nitedimensionalbialgebraorver eldK.ShorwthatthedualspaceB2 Eisabialgebra. 7 &e321. %TOOLBO9XYOneofthemostimpSortanrtpropertiesofbialgebrasBisthatthetensorproductorverKoftrwoB-moSdulesortrwoB-comoSdulesisagainaB-moSdule.lProp`osition1.7.3.o1.s+LffetLB8Rbeabialgebra.4qLetM0andNbffeleftB-modules. #Then35M K cNtisaB-moffdulebythemap$TBE M N2G 1p6 !@B B M N261 I{ 1p]!"ϊB M B Nh՚ J6 !@M N:¤ 2. #Lffet B2beabialgebra.LetMandNbffeleftB-comodules.ThenM5 K }Nisa #B-cffomodule35bythemapԍ$TM N2L p6 !@BE M BE N261 I{ 1p]!"ϊB B M N2Gr 1p6 !@B M N: 3. #K35isaB-moffdulebythemapBE KPUR԰n9=B2 "pX E!K.D捍 4. #K35isaB-cffomodule35bythemapKh KJUR !BPX԰ ?=BE K.lProof.@_WVegivreadiagrammaticproSoffor1.8Theassociativitrylawisgivenby︍vBE B M Nv&BE B B M NYsH2fd#ά-o0W1  1 1vvgfBE B M B N0sH2fd#:ά-onxQ1 1 I{ 1vvBE M NdpsH2fd0ά-o>͍Y1  9BE B B B M N_BE B B M B N`A2fd(eά-n1 1 1 I{ 1NBE B M Nz!@A2fd[ά-7͍uK1 1  Ԡ9BE B B B M NԠ_BE B M B B N`:2fd(eά-έzzЀS1 1 I{(Bd B;M") 1ԠԠNBE M B Nz!@:2fd[ά-0͍uK1  1  BE M N׎BE B M NMP32fd;Ѝά-0]  1 1%BE M B N32fd;kά-n1 I{ 1"M NL32fd0aά- s*徟Ǡ|z@fe+Ǡ?ې/pr 1 1sT>ՠ@fepՠ?␍9 1 1 1 1sDՠ@feDՠ?␍Ip 1 1 1 1s~ՠ@feՠ?␍0 1 1T>Π@fepΠ?91 I{ 1 1 1DΠ@feDΠ?Ip1 I{(Bd;B M") 1 1~Π@feΠ?01 I{ 1T>Ǡ@fepǠ?Ԑ9r r 1 1DǠ@feDǠ?ԐIpr 1 r 1~Ǡ@feǠ?'0 Theunitlarwisthecommutativityof︍v"T,yM NP6԰=@K M Nv"t!BE M NˊLsH2fdDvpά-o>͍LI{ 1 1"`K K M N"ZBE B M N߽ A2fd$+@ά-7͍I{  1 1ԩ"SM NP6԰=@K M K Nԩ"ZBE M B NB:2fd `ά-0͍|I{ 1  1shꟙՠ@feՠ?מ⍍NN=hΠ@feΠ?N1 I{ 1s2Sꟙՠ@fe2ՠ?␍79 1 12SΠ@fe2Π?791 I{ 12SǠ@fe2Ǡ?'79 sdҊΠM@feeΠ?Y=M NjX1ۿ`X?`X`X?`X`XɄ?`Xӄ`X݄?`X焜`X񄜟?`X`X?`XXzTherDcorrespSondingpropertiesforcomodulesfollorwsfromthedualizeddiagrams.ThemoSduleandcomodulepropertiesofKareeasilycrhecked.\cffxff ̟ff ̎ ̄cff!٠7 &eI8. %REPRESENT:ABLE!FUNCTORSANDTHEYONED9ALEMMA=}33YDe nition1.7.4.vaù1.Let2(B;r;n9;;)bSeabialgebra.LetAbealeftB-modulewith9structuremapUR:B l}An!1A..Let9furthermore(A;rA;A)9bSeanalgebrasucrhthatqNrA O2andAarehomomorphismsofB-moSdules.Then(A;rA;A;)qNiscalledaB-moffdule35algebra.2.Let(B;r;n9;;)bSeabialgebra.LetCobealeftB-modulewithstructuremapUR:B {C1K{!Cܞ.SLetUfurthermore(C5;C;"C)UbSeacoalgebrasucrhthatC uand"CarehomomorphismsofB-moSdules.8Then(C5;C;"C;)iscalledaB-moffdule35coalgebra.3.%Let9(B;r;n9;;)bSeabialgebra.%LetAbealeftB-comodulewithstructuremapICi:ZA!AB A.TLetfurthermore(A;rA;A)ICbSeanalgebrasucrhthatrA ''andA yare"homomorphismsofB-comoSdules.Then(A;rA;A;s2)"iscalledaB-cffomodulealgebrffa.4.Let2(B;r;n9;;)bSeabialgebra.LetC>bealeftB-comodulewithstructuremapi:7Cd~`!BZ Cܞ.Letfurthermore(C5;C;"C)bSeacoalgebrasucrhthatC (&and"C G are'thomomorphismsofB-comoSdules.FThen(C5;C;"C;s2)'tiscalledaB-cffomodulecffoalgebra.5Remark1.7.5.j6Ifg(C5;C;"C)gisaK-coalgebraand(C5;)isaB-moSdule,Wthen(C5;C;"C;)isaB-moSdulecoalgebrai isahomomorphismofK-coalgebras.If(A;rA;A)isaK-algebraand(A;s2)isaB-comoSdule,Xthen(A;rA;A;s2)isaB-comoSdulealgebrai ]ڹisahomomorphismofK-algebras.SimilarstatemenrtformoSdulealgebrasorcomodulecoalgebrasdonothold. ݍ?E8.UERepresentableFunctorsandtheYonedaLemmaDe nition1.8.1.vaùLeteFWչ:&C!JSet.hbSeacorvXariantefunctor.Aepair(A;x)withAEz2C5;x2F1(A)wiscalledarffepresenting(generic,Puniversal)objeffctwforFйandFiscalledarffepresentable\yfunctor,"ifforeacrhB<2C¹andy2F1(B)thereexistsauniquefQ2URMorOC(A;B)sucrhthatF1(fG)(x)UR=yn9:LtbZA\BڟǠ*Ffe! Ǡ?`wmfHF1(A)יF1(B)Ǡ*FfeLǠ?`γF((f)H23URxdt3URyǠ*Ffe3Ǡ?Ў fe3Ў?eTheorem1.8.2.p(YoneffdaLemma)LetC,beacategory.GivenacovariantfunctorFc:URCn!FSet,^and35anobjeffctA2C5.fiThenthemap^DË:URNat(Mor5C(A;-33);F1)UR37!(A)(1A)2F(A)is35bijeffctivewiththeinversemapi^>n9 1 :URF1(A)3a7!h aY!2Nat(Mor5C(A;-33);F);wherffe35h2aϹ(B)(fG)UR=F1(f)(a)."7 &e341. %TOOLBO9XYProof.@_FVorUR2Nat(Mor5C(A;-);F1)wrehaveamap(A)UR:MorOC(A;A)n!1F1(A),hence!:Zwithn9()UR:=(A)(1A)!isawrellde nedmap..FVor21ֹwrehavetocheckthath2awisanaturaltransformation.8GivrenfQ:URBX !_7CFinC5.ThenthediagramWYyF1(B)yF1(Cܞ)┞32fdRά-W`ϥ{F((f)Hv5Mor2C(A;B)HsMor )= (B)(fG)k9=UR n9(B)Mor5C(A;fG)(1A)=F1(f) n9(A)(1A)=F1(f)n9( ):UTRemark1.8.5.j6By4thepreviouscorollarytherepresenrtingobjectAisuniquelydetermineduptoisomorphismbrytheisomorphismclassofthefunctorMor Cd(A;-).Problems1.8.6.sX1.#4DetermineexplicitlyallnaturalendomorphismsfromGastoGaw(asde nedinLemma4.3.5).2.8DeterminealladditivrenaturalendomorphismsofGaϹ.3.8DetermineallnaturaltransformationsfromGawtoGm l(seeLemma4.3.7).4.8DetermineallnaturalautomorphismsofGmĹ.8 ;7  #12@cmbx8-%n eufm10,o cmr9+@ cmti12*O+msbm6)ppmsbm8( msbm10% msam10"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8- cmcsc10N cmbx12Nff cmbx12XQ cmr12 b> cmmi10O line10u cmex10@N