; TeX output 1999.03.03:19127 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,and3. #a4familyfMor5C(A;B)`MorΟC(B;Cܞ)3(f;gn9)7!gfd2MorRCt(A;Cܞ)jA;B;CV2 #Ob1&C5g꨹ofmapscalledcffompositions.Cݹiscalledacffategory꨹ifthefollorwingaxiomsholdforC1. #AssoSciativreLaw: #8A;B;C5;D2URObC5;fQ2URMorOC(A;B);gË2URMorOC(B;Cܞ);hUR2MorOC(C5;DS):>7h(gn9fG)UR=(hg)fG;2. #IdenrtityLaw: #8A^V2ObC912cmmi8A<:2MorSC"(A;A)I8B;C:2^VObC5;I8fU2^VMorSC"(A;B);8g̏2 #Mor5YC:D(C5;A)UR:31AgË=URgѹandJ1fG1A 36=f:Examples1.1.2.th1.8ThecategoryofsetsSetӋ.2.The categoriesofRJ-moSdulesR-Mo`dn:,Ikg-vrectorspacesk-Vec[^ork-Mo`dn:,IgroupsGr ,aabSelian>groupsAb{,monoidsMonL,commrutative>monoidscMon$L,ringsRi|, eldsFldr,topSologicalspacesTopn5.Since^moSdulesarehighlyimportanrtforallwhatfollows,{werecallthede nitionandsomebasicpropSerties.De nitionandRemark1.1.3.ßLetR޹bSearing(alwraysassociativewithunit).ADleftRJ-moffduleR 'M`xisan(additivrelywritten)abSeliangroupMtogetherwithanopSerationRM63UR(rr;m)7!rm2M+sucrhthat1. #(rSs)mUR=r(sm),2. #(r6+s)mUR=rSm+sm,3. #rS(m+m209)UR=rm+rm209,4. #1mUR=mforallrr;sUR2RJ,m;m20#2M@.EacrhabSeliangroupisa( msbm10Z-moduleinauniquewrayV.],o cmr91*7 &e21. %TOOLBO9XYAhomomorphism*ofleftRJ-moffdulesfga:bR M`F4!5R N8isagrouphomomorphismsucrhthatfG(rSm)UR=rfG(m).Analogouslywrede nerightRJ-moSdulesMR ;andtheirhomomorphisms.WVeudenotebryHomUR"I(:M;:N@)thesetofhomomorphismsofleftRJ-moSdulesR MYandRN@.PSimilarlyHomlR#_(M:;N:)denotesthesetofhomomorphismsofrighrtRJ-moSdulesMR ;andNR.8BothsetsareabSeliangroupsbry(f+gn9)(m)UR:=fG(m)+gn9(m).fFVorarbitrarycategorieswreadoptmanyofthecustomarynotations.ʴNotation1.1.4.p>6fQ2URMorOC(A;B)willbSewrittenasf:URA4!1BorArf ㎍4!B.8Aꔹiscalledthedomain,BtherffangeoffG.TheDcffompositionoftrwomorphismsfu:;vA4!yBJandg:B|4!+Ciswrittenasgn9fQ:URA4!1CForasgf:URA4!1Cܞ.fDe nitionandRemark1.1.5.ßAmorphismf鴹:A4!\BHiscalledanisomor-phismaTifthereexistsamorphismg:PBV4!3AinCsucrhthatfGg=P1B andgn9fgO=1A.TheYmorphismgisuniquelydeterminedbryfsincegn920Ĺ=URgn920:8(M;T1)4!'(M;T2)isbijectivreandcontinuous.%Theinversemap,hhowever,isnotcontinuous,hhencef׹isnoisomorphism(homeomorphism).ManrywellknownconceptscanbSede nedforarbitrarycategories.WVearegoingtoapplysomeofthem.8Herearetrwoexamples.De nition1.1.7.vaù1. ]?Amorphismrf0:@1A4!B#d> H'Egڤׁ @ڤ @ڤ @ڤ @!$>@!$>RA!hAB432fdpά!7pX.;cmmi6ApB32fd@ά-!7pX.BHoǠ*FfeآDǠ?`T(fh;gI{)ecommrutes.AnobjectE:2h#C?Riscalleda nalǝobjeffctifforevreryobjectT 2h#C?RthereexistsauniquemorphismeUR:T4!eE(i.e.8MornݟCܜ(T;E)consistsofexactlyoneelemenrt).AcategoryCwhicrhhasaproSductforanytwoobjectsAandBandwhichhasa nalobjectiscalledacategorywith niteproSducts.I8Remark1.1.9.j6IfSEtheproSduct(AuqB;pA;pBN>)SEoftrwoSEobjectsAandBKinCzexiststhenitisuniqueuptoisomorphism.Ifthe nalobjectEinCݹexiststhenitisuniqueuptoisomorphism.Problem1.1.cRLetCXbSeacategorywith niteproducts.gQGivreade nitionofaproSduct*ofafamilyA1;:::ʚ;An nz(nUR0).,Shorw*thatproductsofsucrhfamiliesexistinC5.De nitionandRemark1.1.10._LetnCabSeacategoryV.1ThenC52opDwiththefol-lorwing}DdataOb'C52op 뤹:=URObC5,%Mor"Cmrop&>(A;B)UR:=MorOC(B;A),%and}Df8op VgË:=g9r9fCde nesanewcategoryV,thedual35cffategory꨹toC5.Remark1.1.11.qN6Anry#~notionexpressedincategoricalterms(withobjects,1mor-phisms,?andtheircompSosition)hasadualnotion,i.e. }thegivrennotioninthedualcategoryV.MonomorphismsHfGinthedualcategoryC52op fareepimorphismsintheoriginalcat-egoryCandconrverselyV.H9A nalobjectsIIinthedualcategoryC52op isaninitial7objeffctintheoriginalcategoryC5.De nition1.1.12.}!ùThecffoproductoftrwoobjectsinthecategoryCչisde nedtobSeaproductoftheobjectsinthedualcategoryC52op R.Remark1.1.13.qN6EquivXalenrt"totheprecedingde nitionisthefollowingde ni-tion.GivrenA;BX2URC5.!AnobjectAqB?inCW̹togetherwithmorphismsjA 36:A4!1AqBandsjB :URBX4!_7AqBX4!Byissa(categorical)coproSductofAandBifforevreryobjectT 2kTC˹andevrerypairofmorphismsfS:A4!5T\andgٍ:BZ4!;T\thereexistsauniquemorphism[f;gn9]UR:AqBX4!_7TnsucrhthatthediagramLƅH_T`f#dׁ @#d @#d @#d @i>@i>RH'Eg!$ׁ !$ !$ !$ ڤ>ڤ> bYAbYLAqB4{fdά-ejX.AbYbYpBzԟ{fdPYάeښjX.BHoǠ*FfeآDǠ?`T[߱8fh;gI{]7 &e41. %TOOLBO9XYcommrutes.ThecategoryCissaidtoharve nite2cffoproductsifC52op0isacategorywith niteproSducts.8Inparticularcoproductsareuniqueuptoisomorphism.ꍍU2.FunctorsDe nition1.2.1.vaùLetCݹandD?bSecategories.8LetFconsistof 1. #amapObFC3URA7!F1(A)2ObDUV;2. #afamilyofmaps@<)_fFA;B]:URMorOC(A;B)UR3fQ7!FA;B(fG)2MorODڲ(F1(A);F(B))jA;BX2URC5g[or1"fFA;B]:URMorOC(A;B)UR3fQ7!FA;B(fG)2MorODڲ(F1(B);F(A))jA;BX2URC5g] Fiscalledacffovariant[contravariant$D]functorif1. #FA;A<(1A)UR=1F((A)C@forallAUR2ObC5,2. #FA;C(gn9fG)UR=FBd;CJJ(g)FA;B(fG)forallA;B;C12URObC5: #[FA;C(gn9fG)UR=FA;B(f)FBd;CJJ(gn9)forallA;B;C12URObC5].Notation:8WVewrite ʍ~\AUR2C~insteadofXAUR2ObC~}fQ2URC~insteadof2fQ2URMorOC(A;B);F1(fG)~insteadofFA;B(fG).#󍍍Examples1.2.2.h1. Id:URSet4!+fSet2. #FVorget:8RJ-Mo`dÌ4!02kSet3. #FVorget:8Ri_4!'Ab4. #FVorget:8Ab?4!+Gr5. #P晹:Set4!,ySet?\;P(M@):=BpSorwersetofM@.>P(fG)(X):=f21 {(X)Bforf2:M*4! #N;XFURN+isaconrtravXariantfunctor.6. #Qҹ:Set 4!.SetA;Q(M@):=pSorwersetofM@.dQ(fG)(X):=f(X)forfѹ:MѶ4! #N;XFURM+isacorvXariantfunctor.şLemma1.2.3.y5Q1.XLffet35XF2URC5.fiThen@<AOb~C3URA7!MorOC(XJg;A)2ObSet) Mor> CDMɹ(A;B)UR3fQ7!M@orCm(XJg;fG)2MorO12@cmbx8Set&(Mor5C(XJg;A);Mor5CU(X;B)); #withMorԁC (XJg;fG) :MorRC q(X;A)3gB7!fGg2MorRC q(XJg;B)orMorԁC (X;fG)(gn9) = #fGgnis35acffovariantfunctorMori2C ʹ(XJg;-33).2. #Lffet35XF2URCܞ.fiThenOb+QC3URA7!MorOC(A;X)2ObSet,MorACGN(A;B)UR3fQ7!MorOC(f;X)2MorOSet&(Mor5C(B;X);Mor5C(A;X)) #withHMor~ݟC윹(f;X)UR:MorOC(B;X)3gË7!gn9fQ2MorOC(A;X)HorMor~ݟC윹(f;X)(gn9)UR=gf #is35acffontravariant35functorMori2C(-;X)./7 &e3. %NA:TURAL!TRANSF9ORMATIONS=f5Y- cmcsc10Proof.@_1. oMorC$(XJg;1A)(gn9)=1Ag'Ϲ=g=id:(gn9);R[MorXC#(XJg;fG)Mor5CU(X;gn9)(h)=fGgn9hUR=MorOC(XJg;fgn9)(h).2.8analogouslyV.Jcffxff ̟ff ̎ ̄cffRemark1.2.4.j6TheprecedinglemmashorwsthatMor&ğC(-;-)isafunctorinbSothargumenrts.1A functorNintwoargumentsiscalledabifunctor.1WVecanregardsthebifunctorMor Cd(-;-)asacorvXariantfunctorb,$OMorZLC (-;-)UR:C5 op @C4!wfSet(`I:TheuseofthedualcategoryremorvesthefactthatthebifunctorMorC`(-;-)iscon-trarvXariantinthe rstvariable.ObrviouslyDthecompSositionoftwofunctorsisagainafunctorandthiscompSositionisassoSciativre.8FVurthermoreforeachcategoryCݹthereisanidentityfunctorIdC.FVunctors-oftheformMorcʟCщ(XJg;-)resp.Mor/C(-;X)arecalledrffepresentablefunctors(corvXariant^resp.contravariant)^andXiscalledtherffepresenting@Hobject^(seealsosection1.8). ߍ3.NaturalTransformationsDe nition1.3.1.vaùLetPFB:j1Cf4!$D۹andG _:Cf4!$D۹bSetrwoPfunctors. jxAnaturffalDtransformationoɹorafunctorialmorphism'UR:Fc4!BG%isoafamilyofmorphismsf'(A)UR:F1(A)4!1G.(A)jA2C5g꨹sucrhthatthediagramSm vF1(B)G.(B)pt32fd/pά- Vp'(B)HDF1(A)HG.(A)D{fd0Ѝά-Pt'(A)HҟǠ*FfeǠ?`FF1(fG)HǠ*FfeğǠ?`DG.(fG)̍commrutesforallfQ:URA4!1BinC5,i.e.8G.(fG)'(A)UR='(B)F1(f).Lemma1.3.2.g5QGivencffovariantfunctorsF=IdjBSet&N:Set 4!5A_SetN'andG\3=Mor5Set#t(Mor5Set(;A);A)UR:Set!+OFSetBk^for35asetA.fiThen'UR:Fc!B"Gcwithb,H'(B)UR:BX3b7!(Mor5Set#t(B;A)3fQ7!fG(b)2A)2G.(B)is35anaturffaltransformation.Proof.@_GivrengË:URBX4!_7Cܞ.8ThenthefollowingdiagramcommutesR`H^BHoMorSetU(Mor5Set#t(B;A);A)kD{fd*Fά-Psq'(B)^͎C MorSetY(Mor5Set#t(C5;A);A)k432fd*Ѝά-s@'(Cܞ)Hc<Ǡ*FfecoǠ? XgHث2Ǡ*FfedǠ?`ݐMorSet|(Mor5Set#t(gn9;A);A)?~7 &e61. %TOOLBO9XYsince1ʍLu'(Cܞ)F1(gn9)(b)(fG)UR='(C)gn9(b)(fG)=fgn9(b)='(B)(b)(fgn9),g=UR['(B)(b)Mor5Set#t(gn9;A)](fG)=[Mor5Set(Mor5Set(gn9;A);A)'(A)(b)](fG): %cffxff ̟ff ̎ ̄cffbLemma1.3.3.g5QLffetfQ:URA!BbeamorphisminC5.ThenMor5C(f;-33)UR:MorOC(B;-)!Mor5C(A;-33)_givenbyMor\C}(f;Cܞ):MorC,¹(B;Cܞ)3g?7!gn9f2MorC,¹(A;Cܞ)_isanaturffaltrffansformation35ofcovariantfunctors.LffetJWf':(A!fB]beJWamorphisminC5.ThenMorTC(-35;fG)(:Mor%C#(-;A)!fMor*C0 y(-;B)givenbyMor=C͹(C5;fG)UR:MorOC(C;A)3gË7!fGg2MorOC(C5;B)isanaturffaltransformationof35cffontravariantfunctors.bProof.@_LethUR:C14!Cܞ20bSeamorphisminC5.8ThenthediagramsME~GMor(DC(B;Cܞ20׹)fMor cCa"(A;Cܞ20׹)32fd0Cά-knÞ0MorǟX.!q% cmsy6CQ(fh;C-:0B})HYdMoraC (B;Cܞ)H$Mor ZC?(A;Cܞ){fd30ά-`MorOX.C'5(fh;C)HǠ*FfeE4Ǡ?`pxMorX.C(Bd;h)HǠ*FfeǠ?`f4Mor'˟X.C,U(A;h)>andFaMor^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$9X.C(h;A)H&RǠ*FfeYǠ?` Mor'dX.C,<%(h;Bd)c፹commrute.w0Gcffxff ̟ff ̎ ̄cffbRemark1.3.4.j6The$compSositionoftrwo$naturaltransformationsisagainanat-uraltransformation.8Theidenrtityid LF(A)UR:=1F((A)C@isalsoanaturaltransformation.De nition1.3.5.vaùAmznaturalmtransformation'48:FeI4!G#ʹismcalledanaturffaliso-morphismifthereexistsanaturaltransformation T:G[I4!FJsucrhthat'ʒ =id oGand 1n'B=idpF".Thenaturaltransformation չisuniquelydeterminedbry'.WVewrite'21ι:=UR n9.A|functor}F.issaidtobSeisomorphictoafunctorG3Kifthereexistsanaturalisomorphism'UR:Fc4!BG..Problem1.2.cR1.ILetEF1;G5:C<4!DbSefunctors.Shorwthatanaturaltransfor-mationiI',۹:F]4!TGwisanaturalisomorphismifandonlyif'(A)isanisomorphismforallobjectsAUR2C5.2.Let5A(ArB;pA;pBN>)bSetheproductofAandBGinC5.Thenthereisanaturalisomorphism̍mrMor(-;AB)PUR԰n:=Mor%5C*(-;A)Morय़CNd(-;B):N7 &e3. %NA:TURAL!TRANSF9ORMATIONS=f7Y3.yLetC^ bSeacategorywith niteproducts.yFVoreacrhobjectAinC^ showthatthereexistsamorphismA Q:sA4!'AAsatisfyingp1A=s1A=p2A.nShorwthatthisde nesanaturaltransformation.8Whatarethefunctors?4. wlLet,C]abSeacategorywith niteproducts. wlShorwthatthereisabifunctor-- :{CmEC/ 4!pC~sucrhIthat(--)(A;B)istheobjectofaproSductofAandB.|WVedenote4elemenrtsintheimageofthisfunctorbyA8BX:=UR(-{-)(A;B)4andsimilarlyfgn9.5. DWiththenotationoftheprecedingproblemshorwthatthereisanaturaltransformation (A;B;Cܞ):(A&B)CPh4԰=I!A(BCܞ).[Shorwthatthediagram(cffoherence35orcffonstraints)Iw((AB)Cܞ)Dl(A(BECܞ))DgW:2fdEHά-̯rA (A;Bd;C)1`8,A((BECܞ)DS)|:2fdEHά-̯!)< (A;BdC;D(AE)B)A(E^B)L:2fdC ά-̯} (A;Er;Bd)NABꃀ(A)1l Ql攴Ql?^QlQz̟0Qz̟0sꃀp 1(Bd)L L攴L?^L#0#0+FDe nition1.3.6.vaùLetdCandD%bSecategories.0A]corvXariantdfunctorFc:URC4!wfDiscalled9aneffquivalence]ofcffategories9ifthereexistsacorvXariant9functorGY:D74!Cnandnaturalisomorphisms'UR:G.FPc԰K=1 IdIC%Mand Ë:F1GP ԰$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!$DUV.eAcategoryCŹissaidtobSedualtoacategoryDNifthereexistsadualitryFc:URC4!wfDUV.FProblem1.3.cR1.8ShorwthatthedualcategoryC52op isdualtothecategoryC5.2.yLetMDUbSeacategorydualtothecategoryC5.ShorwthatDUisequivXalenttothedualcategoryC52op R.3.-Let@/>RHA4ڟǠ*Ffeh Ǡ? gecommrutes.TheelementsofM R ^ENEarecalledtensors,K;theelementsoftheformm n꨹arecalleddeffcomposable35tensors.Warning: IfDyrouwanttode neahomomorphismfQ:URM2 RN64!AwithatensorproSduct[rasdomainyroumustde neitbygivinganRJ-bilinearmapde nedonM$N@.%Lemma1.4.3.g5QA-\tensor-]prffoduct(Mސ R ?N; )de nedbyMR andRNnAisuniqueup35toauniqueisomorphism.Proof.@_Let(M R ;N; )and(M% msam10R ;N;)bSetensorproducts.8ThenH2YMN`> ^ҁ ^ׁ ^܁ ^ ^ ^ ^ x^ sќdžsќdžH-»Ɩׁ P|>P|> H` ׁ @ @ @ @N<>@N<>RH- @ҁ H@ׁ H@܁ H@ H@ H@ H%@ H/@ H3džH3džj33LM R ;N33MR ;N}T32fdϠά-:}Zh3333 32fdAά-:Սk33?M R ;N337MR ;N-zl32fdϠά-:-whWimpliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cffBecause}ofthisfactwrewillhenceforthtalkabSoutthetensorproductofMaandNorverRJ.Prop`osition1.4.4.O(Rules!wofcomputationinatensorproSduct)zGLffet(MP RIN; )bffe35thetensorproduct.fiThenwehaveforallr2URRJ,m;m20#2M@,n;n20#2N r>7 &ea>4. %TENSOR!PR9ODUCTS9Y1. #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 rSn·(observeinpffarticular,[that t:M^N!3M Nkis·not #injeffctive35ingeneral),5. #ifJ5f:MN!#AisanRJ-bilineffarmapandg":M R DN!#Aistheinducffed #homomorphism,35thenPgn9(m n)UR=fG(m;n):ۍProof.@_1. +LetlB:= hm niM R NPdenotelthesubgroupofM R NgeneratedԞbrythedecompSosabletensorsmr n.2LetԞj%:URBX4!_7M RfRNbetheemrbeddinghomomorphism.]WVe getaninducedmap 20#:URM5QN64!B.InthefollorwingdiagramOlXY[MNYHXB{fdά-; 20{󎎍{POM R ;ND̟{fdά-#趪j33HXB33POM R ;ND̞32fdά-^c趪jH披E\ 20ׁ @ @ @ @ >@ >RH:Ǡ*Ffe lǠ?褍A id BHzǠ*FfeìǠ?hv,jpH pLׁ L L L D̟>D̟> Zwrebhaveid\BB 20?V=q 209,pwithp,j 20?V=qp = 20_existsbsince 20isRJ-bilinear.Becauseyofjpv Թ=jgIv 20 = =idxM" X.RN/* y޹wregetjpԹ=idxM" 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 ̎ ̄cffۍRemark1.4.5.j6TVoconstructtensorproSducts,,wreusethenotionofafreemodule.LetLXϹbSeasetandRԖbearing.)AnRJ-moduleRJXϹtogetherwithamapUR:XF4!RJXiscalledafrffeeR-modulegeneratedbyX,ifforevreryRJ-moSduleMwandforeverymap@&f.չ:XY4!ؼM thereexistsauniquehomomorphismofRJ-moSdulesgU:RJXY4!ؼMsucrhthatthediagramE[3bYgTXbY*RJX{fd$pά- ѯH`efׁ @ @ @ @?,>@?,>RH(MDZǠ*FfewǠ? * gycommrutes.FVreeRJ-moSdulesexistandcanbeconstructedasRJX:=f :X4!5 RJjforalmostallxUR2XFչ: (x)=0g.ۍProp`osition1.4.6.OGivenIRJ-moffdulesMR andRN@.gThentherffeexistsatensorprffoduct35(M R ;N; ).Proof.@_De ne:M R N:=0ZfMN@g=UXwhereZfMN@gisafreeZ-moSduleorverMN+(thefreeabSeliangroup)andUisgeneratedbry }7 &e101. %TOOLBO9XY #(m+m209;n)(m;n)(m209;n) #(m;m+n209)(m;n)(m;n209) #(mrr;n)(m;rSn)s2forallr2URRJ,m;m20#2M@,n;n20#2N@.8ConsiderJ:ٍH@MNHZfMN@gj!t{fd!;ά- xHHeM R ;N/{fdЍά- ׳HHVh=URZfMN@g=UHqA` l\uPv\+P\ׁP\tP\+P\P\sP\+P\P\rP\+P\P0$P0$qH 5+ʬQtQQ+QrQQ턴Q턴sHǠ*FfeDǠ? geLetUt íbSegivren.yDThenthereisaunique 2Hom(ZfM4B^N@g;A)Utsuchthat = n9.Since IisRJ-bilinearwreget((mN5+m209;n)(m;n)(m209n))1= n9(mN5+m20;n) n9(m;n)Z (m209;n) =0andsimilarly((m;nZ+n209)(m;n)(m;n209)) =0and((mrr;n)~(m;rSn))h=0.So&wreget(U@)h=0.This&impliesthatthereisauniqueg=2Hom(M7m R N;A)Zsucrhthatgn9˹=(homomorphismtheorem).6Let :=P.Then isbilinearsince(m4+m209) n$/=4(m+m20;n)$/=ǹ((m4+m20;n))$/=ǹ((mfE+m209;n)(m;n)(m209;n)+(m;n)+(m209;n))*[=ǹ((m;n)fE+(m209;n))*[==;v(m;n)+(m209;n)UR=m;v n+m20 n.&The8othertrwo8propSertiesareobtainedinananalogouswrayV.WVe>=harvetoshowthat(M` R>N; )isatensorproSduct.gTheaborvediagramshowsthat~foreacrhabSeliangroupAandforeachRJ-bilinearmap N:gMY )N4!(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鄍Prop`ositionandDe nition1.4.7.ajGiven35twohomomorphismsednfQ2URHomٟR$l(M:;M@ 0:)35andgË2URHomٟR(:N;:N@ 0):Then35therffeisauniquehomomorphism}Hf R ;gË2URHom(M RN;M@ 0 RN@ 0)such35thatf R ;gn9(m n)UR=fG(m) g(n),35i.e.fithefollowingdiagrffamcommutesM%卒UM@20N@20卒QM@20 R ;N@20]432fdpά- W` {$0MN{M R ;Nȋ{fdά-` H꒟Ǡ*FfeğǠ?`ffgH`Ǡ*FfeDǠ?`Ef R ;g?Proof.@_ (fgn9)isbilinear.@cffxff ̟ff ̎ ̄cff鄍Notation1.4.8.p>6WVe_roftenwritefB+ R N\:=f R1N xand_rM; RgT:=1M ~ Rgn9.WVeharvethefollowingruleofcomputation:Gf R ;gË=UR(f RN@ 0)(M Rgn9)UR=(M@ 0 Rgn9)(f RN@)sincefgË=UR(fN@20)(Mgn9)UR=(M@20gn9)(fN@). 7 &ea>4. %TENSOR!PR9ODUCTS 11YProp`osition1.4.9.OThe35followingde necffovariantfunctorss21. #-'e N6:URMo`dÌ-"Rn!*[Ab&";2. #M - 3/:URRJ-LMo`d# !4AbF};3. #-'e - 3/:URMo`dÌ-"RRJ-LMo`d# !4AbF}.q(Proof.@_(f  gn9)(fG20Bg204. %TENSOR!PR9ODUCTS 13Y4.3@WVriteW(A;B)UR:A- BX4!_7B#3 AforW(A;B)UR:a- bUR7!b- a.3@Shorwthat0isanaturaltransformation(bSetrweenwhichfunctors?).8ShowthatM%H=#(A B) CH0(BE A) CQܟ{fd,cά-`Ftr(X&;Bd) 1HH(#BE (A Cܞ)\{fd,cά- 7` H^:Ǡ*Ffe^lǠ?]CTU HIv:Ǡ*FfeIlǠ?`N[1 r(A;C)=#A (BE Cܞ)0(BE Cܞ) AQܞ32fd,cά-z r(A;Bd C)(#BE (CF A)\32fd,cά-Í 7` 8鍹commrutesforallA;B;C12URK-Mo`d#CandthateW(B;A)(A;B)UR=id A BforallA,BinK-Mo`dX.5.8FindanexampleofM@,N62URK-Mo`dX-#CKsucrhthatM K cN6P԰= N K cM@.z;7  12@cmbx8-%n eufm10,o cmr9+@ cmti12*O+msbm6)ppmsbm8( msbm10% msam10"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12|{Ycmr8- cmcsc10N cmbx12Nff cmbx12XQ cmr12O line10