; TeX output 1999.03.03:1925,7 YC`N cmbx1210._LimitsandColimits,Pro`ductsandEqualizersXQ cmr12LimitARconstructionsareavreryimpSortanttoSolincategorytheoryV.6andpreciselyonemorphism12cmmi8D.3. #ThecategorywithtrwoobjectsD|{Ycmr81;D2andonemorphismf:fD1&4! D2(apart #fromthetrwoidentities).4. #The8categorywithtrwo8objectsD1;D2 andtrwo8morphismsf;g:D1 M4!D2 #bSetrweenthem.5. #Thecategorywithafamilyofobjects(DidjiUR2I)andtheassoSciatedidenrtities.6. #ThecategorywithfourobjectsD1;:::ʚ;D4andmorphismsf;gn9;h;kvsucrhthat #thediagramMU33&D333FD4432fd1 O line10-͍/Lh{&D1{FD24{fd1 ά-i&fH4RǠ*FfegǠ?'`gHҟǠ*FfeǠ?k}kg #commrutes,i.e.8kgfQ=URhgn9.De nition1.10.3.}!ùLetD&hbSeadiagramscrhemeandCGacategoryV.EachobjectC2Cde nesacffonstant(diagramKC ;:DqM4!CwithKC(DS):=CaforallDo2DandK,`(fG)o:=1C qforallmorphismsinDUV.[sEacrhmorphismfn:Ch 4! Cܞ2 K cmsy80KinCTde nesaܿcffonstantnaturaltransformationܿKf h:cKC 4!KC!q% cmsy60withKfw(DS)=fG.%Thisde nesaOcffonstantfunctorK6: -Cb4!FVunct6jG(DUV;C5)fromthecategoryCinrtothecategoryofdiagramsFVunct(DUV;C5).LetFc:URD4!C޹bSeadiagram.AnobjectCGtogetherwithanaturaltransformationË:URKC t4!FXis'calledalimitoraprffojective^limit'ofthediagramFwiththeprffojectionifforeacrhobjectCܞ20)2URC^andforeachnaturaltransformation'UR:KC0 &4!FܑthereisauniquemorphismfQ:URCܞ20)4!oCFsucrhthatNZ33օKC33dF̘ܞ32fd&ά-cH','1 ׁ @1 @1 @1 @w>@w>R{}KC0zǠ*FfeǠ?'Ki?;cmmi6f/^+o cmr944-*7 &eOy10. %LIMITS!ANDCOLIMITS,PR9ODUCTSANDEQUALIZERSC$F45Ycommrutes,i.e.8thediagramsFꍍHCH0F1(Didڹ)L{fdά-8:iH+эR8:j\ׁ @Ŀ\ @ο\ @ؿ\ @ܟ>@ܟ>RH~ܰ/F1(Djf ) Ǡ*Ffe><Ǡ?`F((gI{)(commruteforallmorphismsg:oDiԴ4!^DjinD(޹isanaturaltransformation)andthediagramsDdFC}F1(Didڹ):32fdyά-768:iH''8:iׁ @ɭ @ӭ @ݭ @|>@|>Rɯ l*Cܞ20BjǠ*FfeuǠ?`f܍commruteforallobjectsDiOinDUV.A|+category|GC/|haslimitsfordiagrffamsoveradiagramscheme|GDѝifforeacrhdiagramFk::D4! CorverD]thereisalimitinC5.'AvcategoryCiscalledcffompleteifeacrhdiagraminCݹhasalimit.$gExample1.10.4.uQ1.LetYDHbSeadiagramscrhemeconsistingoftwoobjectsD1;D2and theidenrtities.#AdiagramFz:kiD4!ECA>isde nedbygivingtwoobjectsC1 N andC2 Fin C5. AnobjectC1yuC2togetherwithtrwo morphisms1 չ:]C1yuC24!3C1and2V:URC1hC24!5C2ԹiscalledaprffoductйofthetrwoobjectsifC1C2;Ë:URKCqAacmr61*Cq24!0SFisa0alimit,~i.e.xifforeacrhobjectCܞ20 inCeandforanytwomorphisms'1:Cܞ204!C1and'2V:URCܞ20)4!oC2thereisauniquemorphismfQ:Cܞ20)4!oC1jC2sucrhthatIJH'w;'q1Ɩׁ P|>P|> H'r'q2ׁ @ @ @ @N<>@N<>Rɯ Cܞ20؜*Ǡ*Ffe\Ǡ?`݁f33C133QC1jC21|32fd Vά;q13333 C232fd ά-;q2*commrutes.!Thetwomorphisms1V:URC1njC24!5C1eand2:URC1njC24!5C2earecalledtheprffojectionsfromtheproSducttothetrwofactors.2. LetVDYadiagramscrhemeconsistingofa nite(nonempty)setofobjectsD1;:::ʚ;Dn ~and,.theassoSciatedidenrtities.sA,limitofadiagramF:D04!Cciscalleda4 niteblprffoductoftheobjectsC1 FԹ:=F1(D1);:::ʚ;Cn / :=F(DnP)4andisdenotedbryC1j:::Cn=UR"u cmex10Q*n U_i=1Cidڹ.3.Alimitorveradiscretediagram(i.e.D^ hasonlytheidentitiesasmorphisms)iscalledprffoduct꨹oftheCi,:=URF1(Didڹ),i2I+andisdenotedbryQ?IJCidڹ.4. BLetD߹bSetheemptrydiagramschemeandFb:1D]4!CKthe(onlypSossible)emptry4Zdiagram.ThelimitC5;@:KC [4!ޫFekofFiscalledthe nalvobjeffct.IthasthepropSertrythatforeachobjectCܞ20˹inC)(theuniquelydeterminednaturaltransformation'UR:KC0 &4!F/=doSes,notharve,tobemenrtioned)thereisauniquemorphismfQ:URCܞ20)4!oCܞ..7 ̍46YInSetytheone-pSoinrtsetisa nalobject.oHInAb;Gr;Vec7sthezerogroup0isa nalobject.5.pLetD.bSethediagramscrhemefrom1.10.24.withtrwoobjectsandtrwomor-phisms(di erenrtfromthetwoidentities).xAdiagramoverDUconsistsoftwoobjectsC1 -ֹandmC2andtrwomorphismsgn9;hS:C1 W4!7C2. _ThelimitofsucrhadiagramiscalledwPEqualizerofthetrwowPmorphismsandisgivrenbyanobjectKerdk(gn9;h)andamor-phismɡ1V:URKerBm(gn9;h)UR4!1C1.-ThesecondmorphismtoC2arisesfromthecompSosition2 |۹=gn91=h1.TheequalizerhasthefollorwinguniversalpropSertyV.Foreachob-jectWCܞ207.andeacrhmorphism'1 (:hCܞ20]4!pC1 L[withgn9'1=hh'1(='2)WthereisauniquemorphismfQ:URCܞ20)4!oKer+\#(gn9;h)with1f=UR'1(andthrus2f=UR'2,i.e.8thediagramJQݍUˍJCܞ20K mf\'\'\'\'ܟlܟl Ԋn`*Ffen`?<獒<'q1=XKer*s(gn9;h)sC1ڜfdά-<^aq1!fd&ά-n8g󎎍*C2M3333!32fd&ά-͍dh commrutes.Problem1.1.cR1. VLet4Fmq:<`D4!Set3obSeadiscretediagram.ShorwthatthecartesianproSductorverFcoincideswiththecategoricalproduct.2._Let#Dy(bSeapairofmorphismsasin1.10.45.andletFo:j^D4!CSet1PTbSeadiagram.ShorwthatthesetfxUR2F1(D1)jF(fG)(x)UR=F1(gn9)(x)gҹwiththeinclusionmapinrtoF1(D1)isanequalizerofFc:URD4!Set(j.3.8LetFc:URD4!Set+bSeadiagram.Shorwthattheset )3f(xDjD2URObDS;xD 2URF1(D))j8(fQ:D4!D 0!ǹ)2D:F1(fG)(xD)=xD@m>RH33+KC0rڟǠ*Ffe Ǡ?'XKi?f-commrutes,i.e.8thediagramD l~oF1(Djf )~ufCW32fdPά-1$8:jH'8:i̘ܟׁ @֘ܟ @ܟ @ܟ @\>@\>RHSF1(Didڹ)-JǠ*Ffe`|Ǡ?`JF((gI{)/"X7 &eOy10. %LIMITS!ANDCOLIMITS,PR9ODUCTSANDEQUALIZERSC$F47Ycommrutesyforallmorphismsg:HDi4!DjߩinD(isanaturaltransformation)andthediagramE/HF1(Didڹ)HVC̟{fdά-ކ8:iH''8:iׁ̟ @̟ @̟ @̟ @L>@L>RHfKtCܞ20:#zǠ*FfeVǠ?` ,fƄcommrutesforallobjectsDiOinDUV.The>spSecialcolimitsthatcanbeformedorver>thediagramsasinExample1.10.4arecalledcffoproduct,35initialobjeffct,꨹resp.8coequalizer./Example1.10.6.uQIn.%Vecthe.%object0isaninitialobject.XIn( msbm10K-AlgʔtheobjectKVisaninitialobject.InGeom'Ttheone-elemenrtfunctorAU7!fgVisa nalobject.InDK-Algqtheobjectfai2AjfG(a)=gn9(a)gDistheequalizerofthetrwoDalgebrahomo-morphisms;f:YA4!BAandg:A4!B.@InKAlgthecartesian(setofpairs)andthecategoricalproSductscoincide.Remark1.10.7.qN6A[colimitcofadiagramC~isalimitofthecorrespSonding(dual)diagraminthedualcategoryC52op R.ThrustheoremsabSoutlimitsinarbitrarycategoriesautomaticallyǸalsoproSduce(dual)theoremsaboutcolimits.Horwever,observeǸthattheoremssabSoutlimitsinaparticularcategory(forexamplethecategoryofvrectorspaces)#translateonlyinrtotheoremsabSoutcolimitsinthedualcategoryV,whichmostoftenisnottoSouseful.Prop`osition1.10.8.Limitslandcffolimitsofdiagramsareuniqueuptoisomor-phism.V- cmcsc10Proof.@_LetMFጹ:{D4!C8bSeadiagramandletC5;coSdomain (range)ofthemorphismfQ:URD20w4!D20ǟ20inDuչsointhiscaseCodom(fG)UR=D20ǟ20p.WVede neforeacrhmorphismfQ:URDS20w4!DS20ǟ20 Ztwomorphismsasfollows(M"pfq:=URF((D@l>RHF1(DS20ǟ20p)Ǡ*Ffe̟Ǡ?`LF((f)ۍissicommrutativebSecauseofF1(fG)(D20!ǹ)=F(fG)F((D 4!FF)rbSeagiven.b Thenthisde nesauniquemorphismg:0FCܞ204!$Q'zGDthelimitcanbSerepresenrtedasasubobjectofasuitableproduct.5Duallythe&colimitcanbSerepresenrtedasaquotientobjectofasuitablecoproSduct.@ThisconstructionwillbSeusedincrhapter5.AnotherfactisvreryimpSortantforus,"thefactthatcertainfunctorspreservelimitsresp.3tcolimits.WVesarythatafunctorG+:C24!ؼC520jGprffeserveslimitsoverthediagramscrhemeD?iflimi cscffxff ̟ff ̎ ̄cffZ;7  +o cmr9( msbm10"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8@ cmti12- cmcsc10N cmbx12XQ cmr12O line10j