; TeX output 1999.09.29:1823'7 YRXQ cmr12CHAPTER2Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr939(*7 &e401:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYeFN cmbx122.{MonoidsandGroupsinaCategoryBeforewreuseHopfalgebrastodescribSequantumgroupsandsomeofthebSetterknorwn#groups,Ȃsuchasanealgebraicgroupsandformalgroups,ȂweintroSducetheconcept/gofageneralgroup(andofamonoid)inanarbitrarycategoryV.UsuallythisconceptJisde nedwithrespSecttoacategoricalproductinthegivrencategoryV.ButinʊsomecategoriesthereareingeneralnoproSducts..+Still,onecande netheconceptofagroupinavrerysimplefashion.&HWVewillstartwiththatde nitionandthenshowthatitcoincideswiththeusualnotionofagroupinacategoryincasethatcategoryhas niteproSducts.De nition2.2.1.vaùLet5 !", cmsy10CbSeanarbitrarycategoryV. Letg cmmi12G2CbSe5anobject. WVeuse)thenotationG(X)t:=Mor~!K cmsy8C =(XJg;G))forallXf2tC5,yG(fG):=Mor~C(f;G))forallmorphismsfQ:URXF4!Y8finC5,andfG(X):=MorOC(XJg;f)forallmorphismsfQ:G4!1G209.G!͹togetherwithanaturaltransformationmf:G(-)~G(-)f4!gG(-)!iscalleda,@ cmti12grffoup(monoid)inthecategoryC5,\ifEfthesetsG(X)togetherwiththemrultiplicationm(X)UR:G(X)G(X)UR4!1G(X)aregroups(monoids)forallXF2URC5.Let"(G;m)and(G209;m20)"bSegroupsinC5.XOA morphismf':g(G4!G20[in"CWiscalledahomomorphism35ofgrffoups꨹inC5,ifthediagramsMHyHQG(X)G(X)H&*G(X)ޮ|{fd$ 0O line10-`ƪ2cmmi8m|{Ycmr8(X)G209(X)G20(X)G209(X)32fdά-"&m-:"q% cmsy60(X)H Ǡ*Ffe<Ǡ?`f(X)f(X)H Ǡ*Ffe<Ǡ?`ef(X)8鍹commruteforallXF2URC5.Let?(G;m)and(G209;m20)?bSemonoidsinC5.A9morphismf:G4!OG20xin?Ctiscalledahomomorphism35ofmonoidsinC5,ifthediagramsMHyHQG(X)G(X)H&*G(X)ޮ|{fd$ 0ά-`ƪm(X)G209(X)G20(X)G209(X)32fdά-"&m-:0(X)H Ǡ*Ffe<Ǡ?`f(X)f(X)H Ǡ*Ffe<Ǡ?`ef(X)8鍹andApHfg]CW~uׁ̟ ̟ ̟ ̟ ی>ی> Hnu-:07,ׁ A7, A7, A7, AZl>AZl>U2JG(X)nG209(X)̶32fd_`ά- f(X)ŶcommruteforallXF2URC5.Problem2.2.1.nR1)IfasetZf6togetherwithamrultiplicationmUR:ZfZ14!Zisamonoid,]then9theunitelemenrteUR2ZIis9uniquelydetermined.Ifitisagroupthenalso)7 &eph2. %MONOIDS!ANDGR9OUPSINACA:TEGORYc41YtheVinrversei :Z?4!mZ3isuniquelydetermined.}Unitelemenrtandinversesofgroupsarepreservredbymapsthatarecompatiblewiththemultiplication.2):FindanexampleofmonoidsYandZHandamapf%:݁Yy4!pZwithfG(y1y2)݁=fG(y1)f(y2)forally1;y2V2URYp,butfG(eYP)6=eZ8.3)bIf(G;m)isagroupinCbandiX :-G(X)4!aG(X)bistheinrverse,thenbiisanatural6itransformation.#TheYVonedaLemmaprorvidesamorphismS:CG4!G6isuchthatiX r۹=URMorOC(XJg;S׹)UR=S(X)forallXF2URC5.Prop`osition2.2.2.OLffetC6beacategorywith nite(categorical)products. WA2nobjeffctoGinCcarriesthestructuremq:G(-35)(G(-)q!HG(-)oofagrffoupinCifandonly=iftherffearemorphismsmgw:GGgw!5[G,?uugw:E!rG,and=SN:G!5[Gsuchthatthe35diagrffamsU)YCGGGYjGGl{fdQά-)-m1O\GGbGt 32fd*Fά-ÍmH`\*Ǡ*Ffe`\Ǡ?͝I{1mHѪǠ*FfeܟǠ?]C\mY|E^GPUR԰n:=GPUR԰n:=GEYLoGG>cl{fd ά-)-;1u"GGXgfG̞32fd_ά-Í!mHǠ*FfeǠ?͝ABu1H\֊Ǡ*Ffe] Ǡ?]Ca,Ǡ?1mi?-H:Ǡ*FfelǠ?(mi?-ecommrutesifandonlyifMorѨC?g(-;m(m R1))UR=MorOC(-;m)(Mor5C(-;m) R1)UR=m̹-j(m̹-s1)_R=m̹-j(1m̹-)_R=MorOC(-;m)(1MorK Cʹ(-;m))_R=MorOC(-;m(1m))ifandonlyif*g7 &e421:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYm(m1)UR=m(1m)ifandonlyifthediagramMY#GGGYmLGG۠ܟ{fdQά-)-ۙm1GGdGϦ|32fd*Fά-ÍZmH^Ǡ*Ffe̟Ǡ?͝~ 1mHǠ*FfeLǠ?]Cm^ߍcommrutes.8InasimilarwayoneshowstheequivXalenceoftheotherdiagram(s).cffxff ̟ff ̎ ̄cffProblem2.2.2.nRLetCbSeacategorywith niteproducts.HShorwthatamorphismfQ:URG4!1G20inCݹisahomomorphismofgroupsifandonlyifJzYGGYIGI|{fd@ά-:mK̔G20xG20KG2032fdNά-.&m-:0HǠ*Ffe4̟Ǡ?`ffHڟǠ*Ffe Ǡ?`focommrutes.De nition2.2.3.vaùA.Ocffogroup.`(comonoid)GinCᕹisagroup(monoid)inC52op R,?Ni.e.anaGobjectGUR2ObC=ObC52op togetheraGwithanaturaltransformationm(X)UR:G(X)G(X)UR4!1G(X)whereG(X)UR=MorOCmr;cmmi6op&(XJg;G)=MorOC(G;X),sucrhthat(G(X);m(X))isagroup(monoid)foreacrhXF2URC5.Remark2.2.4.j6Let C7@bSeacategorywith nite(categorical)coproducts.  Anobject1GinCfcarriesthestructuremӹ:G(-)ɨG(-)4!_3G(-)1ofacogroupinCfifandonly%*iftherearemorphisms:G4!]G~qG,3":G4!I,3and%*Sk:G4!G%*sucrhthatthediagramsL#7GqGq9GqGqG[d32fdά- [q1bYBGbY}8GqGO\{fd*ά-ˍa\HZǠ*FfeǠ?` 1qHGXڟǠ*FfeG Ǡ?`;0ٙۄ8GqGٙI+qGPUR԰n:=GPUR԰n:=GqI32fdqά- "q1bYQ6GbYVGqG ܟ{fd_Zά-ˍ 0Hg Ǡ*FfegA,Ǡ?`k1q"HZǠ*FfeǠ?`+H8Ib1H+ʬQHtQ HQH+Q HrQ*HQ-ܟQ-ܟsNa bY4GbYտFI{fd K0ά-bYbY&Gbܟ{fd K0ά-"gGqGYGqG32fd6@ά- бw1qSkбwSr}q1H*Ǡ*Ffe\Ǡ?`H*Ǡ*Ffe\ׁ 6` zrlcommruteo^whererisdualtothemorphismde nedinA.2.ThemultiplicationsarerelatedbryX r۹=URMorOC(;X)UR=(X).Let"CoWbSeacategorywith nitecoproductsandletGandG20[becogroupsinC5.ThenElahomomorphismofgroupsf7й:G20 4!hGisamorphismf:G4!/G20inElCsucrh+"7 &eph2. %MONOIDS!ANDGR9OUPSINACA:TEGORYc43YthatthediagramE̴YVGYGG{fd@ά-ˍɆnK:G20KG20xG20532fdNά-.y*-:0H6Ǡ*FfeiǠ?`lffHzǠ*FfeǠ?` fcommrutes.8Ananalogousresultholdsforcomonoids.Remark2.2.5.j6Obrviously/similarobservXationsandstatementscanbSemadeforotheralgebraicstructuresinacategoryC5.CSoonecaninrtroSducevectorspacesandcorvectorO spaces,&monoidsandcomonoids,ringsandcoringsinacategoryC5. fThestructures