; TeX output 1999.09.17:13317 YRXQ cmr12CHAPTER8&Nff cmbx12Tfos3olbox"N cmbx121.ijCategoriesDe nition8.1.1.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:Examples8.1.2.th1.8ThecategoryofsetsSetӋ.2.The categoriesofRJ-moSdulesR-Mo`dn:,Ikg-vrectorspacesk-Vec[^ork-Mo`dn:,IgroupsGr ,aabSelian>groupsAb{,monoidsMonL,commrutative>monoidscMon$L,ringsRi|, eldsFldr,topSologicalspacesTopn5.Since^moSdulesarehighlyimportanrtforallwhatfollows,{werecallthede nitionandsomebasicpropSerties.De nitionandRemark8.1.3.ßLetR޹bSearing(alwraysassociativewithunit).ADleftRJ-moffduleR 'M`xisan(additivrelywritten)abSeliangroupMtogetherwithanopSerationRM63UR(rr;m)7!rm2M+sucrhthat1. #(rSs)mUR=r(sm),2. #(r6+s)mUR=rSm+sm,3. #rS(m+m209)UR=rm+rm209,4. #1mUR=mforallrr;sUR2RJ,m;m20#2M@.EacrhabSeliangroupisa( msbm10Z-moduleinauniquewrayV.],o cmr91*7 &e28. %TOOLBO9XYAhomomorphism*ofleftRJ-moffdulesfga:bR M`F4!5R N8isagrouphomomorphismsucrhthatfG(rSm)UR=rfG(m).Analogouslywrede nerightRJ-moSdulesMR ;andtheirhomomorphisms.WVeudenotebryHomUR"I(:M;:N@)thesetofhomomorphismsofleftRJ-moSdulesR MYandRN@.PSimilarlyHomlR#_(M:;N:)denotesthesetofhomomorphismsofrighrtRJ-moSdulesMR ;andNR.8BothsetsareabSeliangroupsbry(f+gn9)(m)UR:=fG(m)+gn9(m).fFVorarbitrarycategorieswreadoptmanyofthecustomarynotations.ʴNotation8.1.4.p>6fQ2URMorOC(A;B)willbSewrittenasf:URA4!1BorArf ㎍4!B.8Aꔹiscalledthedomain,BtherffangeoffG.TheDcffompositionoftrwomorphismsfu:;vA4!yBJandg:B|4!+Ciswrittenasgn9fQ:URA4!1CForasgf:URA4!1Cܞ.fDe nitionandRemark8.1.5.ßAmorphismf鴹:A4!\BHiscalledanisomor-phismaTifthereexistsamorphismg:PBV4!3AinCsucrhthatfGg=P1B andgn9fgO=1A.TheYmorphismgisuniquelydeterminedbryfsincegn920Ĺ=URgn920:8(M;T1)4!'(M;T2)isbijectivreandcontinuous.%Theinversemap,hhowever,isnotcontinuous,hhencef׹isnoisomorphism(homeomorphism).ManrywellknownconceptscanbSede nedforarbitrarycategories.WVearegoingtoapplysomeofthem.8Herearetrwoexamples.De nition8.1.7.vaù1. ]?Amorphismrf0:@1A4!B#d> H'Egڤׁ @ڤ @ڤ @ڤ @!$>@!$>RA!hAB432fdpά!7pX.;cmmi6ApB32fd@ά-!7pX.BHoǠ*FfeآDǠ?`T(fh;gI{)ecommrutes.AnobjectE:2h#C?Riscalleda nalǝobjeffctifforevreryobjectT 2h#C?RthereexistsauniquemorphismeUR:T4!eE(i.e.8MornݟCܜ(T;E)consistsofexactlyoneelemenrt).AcategoryCwhicrhhasaproSductforanytwoobjectsAandBandwhichhasa nalobjectiscalledacategorywith niteproSducts.I8Remark8.1.9.j6IfSEtheproSduct(AuqB;pA;pBN>)SEoftrwoSEobjectsAandBKinCzexiststhenitisuniqueuptoisomorphism.Ifthe nalobjectEinCݹexiststhenitisuniqueuptoisomorphism.Problem8.1.1.nRLetmCbSeacategorywith niteproducts.6wGivreade nitionofaproSduct*ofafamilyA1;:::ʚ;An nz(nUR0).,Shorw*thatproductsofsucrhfamiliesexistinC5.De nitionandRemark8.1.10._LetnCabSeacategoryV.1ThenC52opDwiththefol-lorwing}DdataOb'C52op 뤹:=URObC5,%Mor"Cmrop&>(A;B)UR:=MorOC(B;A),%and}Df8op VgË:=g9r9fCde nesanewcategoryV,thedual35cffategory꨹toC5.Remark8.1.11.qN6Anry#~notionexpressedincategoricalterms(withobjects,1mor-phisms,?andtheircompSosition)hasadualnotion,i.e. }thegivrennotioninthedualcategoryV.MonomorphismsHfGinthedualcategoryC52op fareepimorphismsintheoriginalcat-egoryCandconrverselyV.H9A nalobjectsIIinthedualcategoryC52op isaninitial7objeffctintheoriginalcategoryC5.De nition8.1.12.}!ùThecffoproductoftrwoobjectsinthecategoryCչisde nedtobSeaproductoftheobjectsinthedualcategoryC52op R.Remark8.1.13.qN6EquivXalenrt"totheprecedingde nitionisthefollowingde ni-tion.GivrenA;BX2URC5.!AnobjectAqB?inCW̹togetherwithmorphismsjA 36:A4!1AqBandsjB :URBX4!_7AqBX4!Byissa(categorical)coproSductofAandBifforevreryobjectT 2kTC˹andevrerypairofmorphismsfS:A4!5T\andgٍ:BZ4!;T\thereexistsauniquemorphism[f;gn9]UR:AqBX4!_7TnsucrhthatthediagramLƅH_T`f#dׁ @#d @#d @#d @i>@i>RH'Eg!$ׁ !$ !$ !$ ڤ>ڤ> bYAbYLAqB4{fdά-ejX.AbYbYpBzԟ{fdPYάeښjX.BHoǠ*FfeآDǠ?`T[߱8fh;gI{]7 &e48. %TOOLBO9XYcommrutes.ThecategoryCissaidtoharve nite2cffoproductsifC52op0isacategorywith niteproSducts.8Inparticularcoproductsareuniqueuptoisomorphism./;7  -%n eufm10,o cmr9+@ cmti12( msbm10 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12|{Ycmr8N cmbx12Nff cmbx12XQ cmr12O line100a