; TeX output 1999.11.18:1033-7 Y"@AV dcr12APPENDIXߢ8(rxOff dcbx12T|os+olbox/2 dcr945.*7 468. %TOOLBOcfponstantfunctorK8: CCx+!pFeunct6kԹ(DUV;C5)fromthecategoryCsinwtothecategoryofdiagramslFeunct(DUV;C5).LetgFc:URD[!CbSeadiagram.AnobjectCtogetherwithanaturaltransformation m׹:URKC t !FYois(^calledalimitoraprfpojectivelimit(^ofthediagramFwiththeprfpojection ^ifforeacwhobjectCܞ20)2URC`andforeachnaturaltransformation':URKC0 &?d!FthereisaluniquemorphismfQ:URCܞ20) !n0C sucwhthatN33KC33F̙32fd& ά-c-LR'2ׄ|@2|@2|@2|@uƟ @uƟ Rۍ7KC0,*CfeB,?Ki?";cmmi6f/7 O10. %LIMITSANDCOLIMITS,PRڄfd~ά-Lކ m8:iLX'ч8iׄ|@|@|@|@ @ RLfKJCܞ20:!P,*CfeT,?TfڍcommwuteslforallobjectsDiOFinDUV.The@spSecialcolimitsthatcanbeformedowver@thediagramsasinExample8.10.4arelcalledcfpoproduct,2initialobjefpct,lresp.8coequalizer.#;Example8.10.6.uIn/Vecthe/object0isaninitialobject.3In, msbm10K-lAlgʋtheobjectK#isaninitialobject.InGeom'theone-elemenwtfunctorAW7!fg#isanalobject.InDK-lAlg߬theobjectfa#2AjfG(a)=gn9(a)gDĺistheequalizerofthetwwoDalgebrahomo-morphismsf:ZAs!ΥBúandg1:As!ΥB.BInKAlg9thecartesian(setofpairs)andthecategoricallproSductscoincide.Remark8.10.7.qHBA1colimit9ofadiagramCnisalimitofthecorrespSonding(dual)diagraminthedualcategoryC52op R.gThwustheoremsabSoutlimitsinarbitrarycategoriesautomaticallyɿalsoproSduce(dual)theoremsaboutcolimits.ևHowwever,observeɿthattheoremsTabSoutlimitsinaparticularcategory(forexamplethecategoryofvwectorspaces)translateonlyinwtotheoremsabSoutcolimitsinthedualcategorye,awhichmostoftenlisnottoSouseful.Prop_osition8.10.8.RLimitsandcfpolimitsofdiagramsareuniqueuptoisomor-phism.wЍ] dccsc12Proof.@,YLetPFN:=D $F!څCIbSeadiagramandletC5;handx~C;C~bSelimitsofF1.TThen$thereareuniquemorphismsfҲ:x7~CZr!$yC|ºandg:CgQ!x~ C%with Kf ҹ=&~C\andE~Kg*P=UR m׺.3Thisۆimplies K1X.CM=UR mչid8yKX.C&=UR )=~URKg*P=URKfwKg=URKfg andۆanalogouslyE~K1Kɍ~FCM=~UR mKgI{f .8Becauseloftheuniquenessthisimplies1C t=URfGgXand1;U ~C=URgn9fG.Remark8.10.9.qHBNowwithatwehavetheuniquenessofthelimitresp.9colimit(upto,~isomorphism)wwecanintroSduceauniednotation.ThelimitofadiagramF߹:D!nCwilllbSedenotedbwylimi c<((F1),thecolimitbylimi<&!<((F1).Theorem8.10.10.~If ChasarbitrfparyproductsandequalizersthenChasarbitrarylimits,2i.e.fC%iscfpomplete.Proof.@,YLetDqbSeadiagramscwhemeandFA:Df. ~!Cߺadiagram. HFirstweform*theproSductsQDcoSdomain!(range)ofthemorphismfQ:URDS20w! DS20ǟ20inDwEsointhiscaseCodom(fG)UR=DS20ǟ20p.WeeldeneforeacwhmorphismfQ:URDS20w! DS20ǟ20 ZltwomorphismsasfollowsL4pfq:=UR mןF((DDS20ǟ20 hencepg=qn9g. FThwusTgcanbSeuniquelyfactorizedthroughtheequalizer :Ker(p;qn9)!QH@Dh20t"= h=gn9.Because{oftheuniquenessofthefactorizationofgthrough  }wwegethL=h209.Thus(Ker(p;qn9);r)listhelimitofF1.Remark8.10.11.xֺThelproSofoftheprecedingTheoremgivwesanexplicitcon-structionlofthelimitofF}asanequalizer%uKer2(p;qn9)uNşY'؍D