; TeX output 1999.11.03:0813E7 YRXQ cmr12CHAPTER3@f:Nff cmbx12RepresentationffTheoryf,Reconstructionand+TfannakaffDuality"N cmbx12Intro`ductionOne,ofthemostinrterestingpropSertiesofquantumgroupsistheirrepresentationtheoryV.It.hasdeepapplicationsintheoreticalphrysics.Themathematicalsidehastodistinguish[bSetrweentherepresentationtheoryofquantumgroupsandtherepresen-tationtheoryofHopfalgebras.7InbSothcasestheparticularstructureallorwstoformtensorproSductsofrepresenrtationssuchthatthecategoryofrepresentationsbSecomesamonoidal(ortensor)categoryV.Theproblemwrewanttostudyinthischapteris,thowmuchstructureofthequanrtum׈grouporHopfalgebracanbSefoundinthecategoryofrepresentations.2WVewillAshorwthataquantummonoidcanbSeuniquelyreconstructed(uptoisomorphism)from1itsrepresenrtations.|TheadditionalstructuregivenbytheantipSode1isitimitelyconnectedўwithacertaindualitryofrepresentations. WVewillalsogeneralizethisproScessofreconstruction.OnBtheotherhandwrewillshowthattheproScessofreconstructioncanalsobeusedto?DobtaintheTVamrbaraconstructionoftheuniversalquantummonoidofanoncom-mrutative=geometricalspace(fromcrhapter1.).2ThuswewillgetanotherpSerspectiveforthistheorem.Arttheendofthechapteryoushoulds2#!", cmsy10 #understandrepresenrtationsofHopfalgebrasandofquantumgroups,# #knorw>thede nitionand rstfundamentalpropSertiesofmonoidalortensor #categories,# #bSeotfamiliarwiththemonoidalstructureonthecategoryofrepresenrtationsof #Hopfalgebrasandofquanrtumgroups,# #understandwhrythecategoryofrepresentationscontainsthefullinformation #abSout6thequanrtumgroupresp.theHopfalgebra(TheoremofTVannakXa-Krein),# #knorwtheproScessofreconstructionandexamplesofbialgebrasreconstructed #fromcertaindiagramsof nitedimensionalvrectorspaces,# #understandSbSettertheTVamrbaraconstructionofauniversalalgebrafora nite #dimensionalalgebra./^*o cmr969F*7 &e70$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYj4T1.TRepresentationsofHopfAlgebrasLetg cmmi12AbSeanalgebraorveracommrutativering' msbm10K.cLetA-Mo`d#sbethecategoryofA-moSdules.8AnA-moduleisalsocalleda+@ cmti12rffepresentationofA.ObservretthattheactionAI M64!MAXsatisfyingtthemoSduleaxiomsandanalgebrahomomorphismBA4!=End*(M@)areequivXalenrtdescriptionsofanA-moSdulestructureontheK-moSduleM@.ThewfunctorUl:v[A-Mo`d#=4!6_%K-Mo`d#YwithU1(2cmmi8AM@)=M[andU1(fG)=fviscalledtheforffgetful35functorortheunderlyingfunctor.IfB isabialgebrathenarffepresentation֧ofBisalsode nedtobSeaB-module.OItwill=turnoutthatthepropSertryofbeingabialgebraleadstothepossibilitryofbuildingtensorproSductsofrepresenrtationsinacanonicalwayV.LetC}KbSeacoalgebraorveracommrutativeringK. 7LetCܞ-FComo`d2bethecategoryofCܞ-comoSdules.8AC-comoSduleisalsocalledacfforepresentation꨹ofC.TheYfunctorUc:URCܞ-FComo`d2|4!CwK-Mo`d#;withU1(2CM@)=M=andU1(fG)=fXiscalledtheforffgetful35functorortheunderlyingfunctor.IfYB_isabialgebrathenacfforepresentation`oofYBisalsode nedtobSeaB-comodule.ItpawillturnoutthatthepropSertryofbeingabialgebraleadstothepossibilitryofbuildingtensorproSductsofcorepresenrtationsinacanonicalwayV.UsuallyznrepresenrtationsofaringareconsideredtobSemodulesorverznthegivrenring.fTheroleofcomoSdulescertainlyarisesintheconrtextofcoalgebras.Butitisnot{quiteclearwhatthegoSod{de nitionofarepresenrtationofaquantumgrouporitsrepresenrtingHopfalgebrais.FVor(thispurpSoseconsiderrepresenrtationsMikofanordinarygroupG.|Assumefore1thesimplicitryoftheargumentthatGis nite.|RepresentationsofGarevectorspaces?ZtogetherwithagroupactionGM4!M@. 6EquivXalenrtly?ZtheyarevectorspacesztogetherwithagrouphomomorphismG'4!۹Aut-(M@)ormoSdulesorverzthegroupalgebra:K[G]j M64!M@.!InthesituationofquanrtumgroupsweconsidertherepresenrtingHopfalgebraHasalgebraoffunctionsonthequantumgroupG.Then`thealgebraoffunctionsonGistheHopfalgebraK2G,thedualofthegroupalgebradK[G].AneasyexerciseshorwsthatthemoSdulestructureK[G]4 MK4!?MtranslatestothestructureofacomoSduleM64!K2G u@MandconrverselyV.ܚ(ObservethatGis nite.)Sowreshouldde nerepresentationsofaquantumgroupascomoSdulesorvertherepresenrtingHopfalgebra.!De nition3.1.1.vaùLetvGbSeaquanrtumgroupwithrepresentingHopfalgebraHV.ArffepresentationofGisacomoSduleorvertherepresentingHopfalgebraHV.FVromthisde nitionwreobtainimmediatelythatwemayformtensorproSductsofrepresenrtationsofquantumgroupssincetherepresentingalgebraisabialgebra.WVexcomenorwtothecanonicalconstructionoftensorproSductsof(co-)represen-tations.G 7 &enYZ1. %REPRESENT:ATIONS!OFHOPFALGEBRASf+71YLemma3.1.2.g5QLffetHBbeabialgebra.LetM;Nz2WB-;Mo`d%`betwoB-modules.ThenMy 8N͠isaB-moffdulebytheactionb(m n)UR=!u cmex10Pb|{Ycmr8(1) \|m b(2)n..IffQ:URM6!QM@2K cmsy80andeg!I:N! qN@20tarffehomomorphismsofB-modulesinB-;Mo`d$Ithenf& 'g!I:M N!M@20 N@20BRis35ahomomorphismofB-moffdules.,- cmcsc10Proof.@_WVeharvehomomorphismsofK-algebras qʹ:^;BA4!q End*}(M@)and :BA4!End (N@):de ningtheB-moSdulestructureonM@andN.fThruswegetahomomorphismof~algebrascan( r  O)UR:BX4!_7Bh BX4!End*k(M@) Endڢ(N)UR4!1End)(M N).ThrusM' NN]is yaB-moSdule.&Thestructureisb(m n)UR=canF( p O)(Pb(1) \|) b(2))(m n)UR=can(P (b(1) \|) O(b(2)))(m n)UR=P (b(1))(m) O(b(2))(n)UR=Pb(1)m b(2)n.FVurthermorewrehave1(m n)UR=1m 1mUR=m n.If[f;garehomomorphismsofB-moSdules,Hthenwrehave(f Дgn9)(b(m n))4 =(f gn9)(Pb(1) \|m b(2)n)UR=PfG(b(1)m) gn9(b(2)n)UR=Pb(1)fG(m) b(2)gn9(n)UR=b(fG(m) gn9(n))UR=b(f g)(m n).8Thrusf gXisahomomorphismofB-moSdules.4 cffxff ̟ff ̎ ̄cffCorollary3.1.3.sWLffetBbeabialgebra.LThen UR:B-;Mo`d";B-;Mo`d#!5MB-;Mo`dwith35 (M;N@)UR=M Ntand (f;gn9)=f gnisafunctor.Proof.@_ThefollorwingareobviousfromtheordinarypropSertiesofthetensorproSduct/orverK. 1M  ~1N n=UR1M" N۹and(f gn9)(fG20W g20;B"K 1)UR=(1 1 )( 1 1)(1  1)UR=>_c( 1 1)(1 1 W)(1  1)UR=( 1 1)(1 Ma>;Bd Bt)thatusesthefactthat(1is gn9)(fr 1)2N=(f is1)(1 gn9)holdsandthatBd Ma>;B$¹=( 1)(1 W)andMa>;Bd B"ƹ=UR(1 )( 1).Square(5)and(6)commrutebythepropSertiesofthetensorproduct.H7 &e72$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYSquare(7)commrutessinceBisabialgebra.%ߍ*Y*Y\BE M@ $g 6969BE N0%`09M N099 T,ҲfdDvpO line10-)wS٠L *Y*Y=BE B $g 6969M N0%`0909,ҲfdAά-)퍒A1 r 10909cBE M N&},Ҳfd9Ѝά-)4r 1 11Rl @"j2c*Ffe"dc?Q 'OL 1Rl @زc*Ffe c?Pd 1  11Rl @&Rc*FfeYc?P   1 11R@1fe?t/ah 1 1py4BE M pytBE B M@ jܠ|/2BE N|zJBE B Nvvv='sH2fd4AЍά-o쁍D1 L 1 pypyBE B B jܠ||%BE M NvvvtsH2fd5npά-n>51-:Aacmr62* X.;cmmi6MB";BI q% cmsy6 By 1wǀ~"j2F`*Ffe"dF`?L]'O1 r 1wǀ~زF`*Ffe 䟪F`?ud1 X.BI MB";By 1 1wǀ~&RF`*FfeYF`?L] 1 r 1 1 1D jBE B DvjVBE B B R 9M N9xM BE N`$$;fd7Opά-bD1 1 L DDBE B B R 99%BE M N`$$'fd6@ά-j퍒91 1 1 r 1=@"j2*Ffe"d?鯍'Or 1 1=@ز*Ffe ?鯍dr 1 1 1 1=@&R*FfeY?鯍 r 1 1 1 1yyBE B Ǡ x xxM BE NiBE M NE#Ğ32fd0aά-ׁP1 L yy=BE B Ǡ x xgiBE M ND32fd2~ά-m-1 1 r 1WBE B M N-㤞32fd&fpά-/-2l1 r 1 1R@U(1)R@(2)OU(3)O(4)@U(5)@(6)Ln"(7)ThelarwofthecounitisL|99\BE M@ `yyBE NΠYM NY9 T{fdDvpά- AS٠L 99=BE B `yyM NΠYY{fdAά-)-A1 r 1YYcBE M N&}{fd9Ѝά-24r 1 1H8Idؤ13$ҁ H=$ׁ HG$܁ HQ$ H[$ He$ Ho$ Hy$ H}$džH}$džjPM NM NyԞ32fdHά-YУ"1oPM N#T32fdHά-YF1'زǠ$g`fe Ǡ?-d" 1 " 1' G2Ǡ$g`fe zdǠ?-," " 1 1HǠ*FfeǠ?͝d" 1 1Qύwherethelastsquarecommrutessince"isahomomorphismofalgebras.Norwletf2andgXbSehomomorphismsofB-comodules.8ThenthediagramN󍍍99\BE M@ `yyBE NΠYM NY9 T{fdDvpά- AS٠L 99=BE B `yyM NΠYY{fdAά-)-A1 r 1YYcBE M N&}{fd9Ѝά-24r 1 1' "j2耄@fe"d耬?`'Of g' ز耄@fe 耬?`d1 f 1 g'  G2耄@fe zd耬?`,1 1 f g' 耄@fe耬?`d1 f g++@BE M@20 Ǡ k k6BE N@20K ՘M@20 N@20K;Ԟ32fd@<0ά-ׁTL yy=BE B Ǡ k kM@20 N@20KKĞ32fd@xPά-mU1 r 1KKaBE M@20 N@20&}32fd7Pά-/3*r 1 1VӍcommrutes.8Thusf gXisahomomorphismofB-comoSdules.scffxff ̟ff ̎ ̄cffϪCorollary3.1.5.sWLffetBbeabialgebra.Then ܹ:B-;Como`d2k=B-;Como`d3!B-;Como`d3apwith35 (M;N@)UR=M Ntand (f;gn9)=f gnisafunctor.Prop`osition3.1.6.OLffet3B*9beabialgebra./Thenthetensorproduct UR:B-;Mo`d!zB-;Mo`d#!5MB-;Mo`d$osatis es35thefollowingprffoperties:(獍1. #Thes9assoffciativityisomorphism m:(M1 M2) M3!-M1 (M2 M3)s9with # ((mL n) p)=mL (n p)isanaturffaltransformationfromthefunctorI0a7 &enYZ1. %REPRESENT:ATIONS!OFHOPFALGEBRASf+73Y # q( Id Ү)tothefunctor q(Id ; )inthevariablesM1,ۚM2,andM3 yin #B-;Mo`d M@)u=A SMkvandUfw(gn9)=g˹wheream:=fG(a)mfora2AandmUR2M@.8ThefunctorUf aǹisalsocalledforffgetfulorunderlying35functor.The2lactionofAonaB-moSduleMsPcanalsobeseenasthehomomorphismAUR4!1B4!n߹End&{(M@).WVedenotetheunderlyingfunctorspreviouslydiscussedbryHNUA 36:URA-Mo`d"44!4K-Mo`d#Cresp.@,UB :B-Mo`d#I:4!4K-Mo`dX:DProp`osition3.1.10.Lffet)f:BC$!QCbeahomomorphismofbialgebras.FThenUf Tsatis es35thefollowingprffoperties:8#ʍ^Ufw(M N@)UR=Uf(M@) Uf(N@);^Ufw(g h)UR=g h;^Ufw(K)UR=K;^Ufw( )UR= ;33Ufw()=;Ufw()=;^UBN>Ufw(M@)UR=UC(M);^UBN>Ufw(gn9)UR=UC(g):8ؘProof.@_Thisx.isclearsincetheunderlyingK-moSdulesandtheK-linearmapsstaryuncrhanged. nTheނonlythingtocheckisthatUf UgeneratesthecorrectB-moSdulestructureonthetensorproSduct.GFVorUfw(M5 \QN@)y=M \QN0}wrehaveb(m\Q n)y=fG(b)(mF n)UR=Pf(b)(1) \|mF f(b)(2) \|nUR=Pf(b(1) \|)mF f(b(2) \|)nUR=Pb(1)mF b(2)n. dcffxff ̟ff ̎ ̄cffK_7 &enYZ1. %REPRESENT:ATIONS!OFHOPFALGEBRASf+75YRemark3.1.11.qN6LetpCM;andD+bSecoalgebrasorverpacommrutativepringK.2LetfQ:C[4!DVϹbSeAahomomorphismofcoalgebras.ThenwrehaveafunctorUfP:1Cܞ-FComo`d-J4!%DS->6Como`d4 withwUfw(2CM@) v=2D M[andUfw(gn9)=gwhereD =(fO P1)C +:M4!nCN r+M64!DŹ M@./AgainthefunctorUf Fiscalledforffgetfulorunderlyingfunctor.WVedenotetheunderlyingfunctorspreviouslydiscussedbrye8:UC t:URCܞ-FComo`d2|4!CwK-Mo`d#Cresp.@,UD :DS->6Como`d14!CbgK-Mo`dX:Prop`osition3.1.12.Lffet)f:BC$!QCbeahomomorphismofbialgebras.FThenUfq:URCܞ-Como`d2%!DDS-Como`d3satis es35thefollowingprffoperties:6Ajʍ^Ufw(M N@)UR=Uf(M@) Uf(N@);^Ufw(g h)UR=g h;^Ufw(K)UR=K;^Ufw( )UR= ;33Ufw()=;Ufw()=;^UCUfw(M@)UR=UBN>(M);^UCUfw(gn9)UR=UBN>(g):7UTProof.@_WVelearvetheproSoftothereader.cffxff ̟ff ̎ ̄cffProp`osition3.1.13.LffetnH\ beaHopfalgebra.LetMandNbffebeHV-modules.ThenDڹHomc(M;N@),BtheDsetK-lineffarmapsfromMtoN@,BbecomesanHV-moduleby(hfG)(m)UR=Ph(1) \|f(S׹(h(2)m).fiThis35structurffemakesevMHom:URHV- Mo`d#9mHV- Mo`d#!5HV- Mo`da35functorcffontravariant35inthe rstvariableandcffovariantinthesecondvariable.Proof.@_TheΆmainparttobSeprorvedΆisthattheactionHo Homm(M;N@)4!Homy(M;N@)nsatis estheassoSciativitrylaw. o2Letf`2JaHom(M;N@),h;k~2JaHV,andmUR2M@.Thenn((hkg)fG)(m)=P(hk)(1) \|fG(S׹((hk)(2))UR=Ph(1) \|k(1)fG(S׹(k(2))S(h(2))m)UR=Ph(1) \|(kgfG)(S׹(h(2))m)UR=(h(kfG))(m).WVelearvetheproSofoftheotherproperties,inparticularthefunctorialproperties,tothereader.cGqcffxff ̟ff ̎ ̄cffCorollary3.1.14.zLffet!ZMb>beanHV-module. 0ThenthedualK-moduleM@2 I=Homy(M;K)35bffecomesanHV-moduleby(hfG)(m)UR=f(S׹(h)m).Proof.@_TheyspaceKisanHV-moSdulevia"UR:HB4!K.)Henceywrehave(hfG)(m)UR=Ph(1) \|fG(S׹(h(2)m)UR=P"(h(1) \|)f(S׹(h(2)m)UR=f(S׹(h)m).~cffxff ̟ff ̎ ̄cffLq7 &e76$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYY2.MonoidalCategoriesFVorourfurtherinrvestigationsweneedageneralizedversionofthetensorproSductthatwrearegoingtointroSduceinthissection.iThiswillgiveusthepSossibilitytostudymoregeneralvrersionsofthenotionofalgebrasandrepresentations.rDe nition3.2.1.vaùAmonoidal35cffategory(ortensor35category)consistsofacategoryC5,acorvXariantfunctor UR:C]C4!wfC5,calledthetensor35prffoduct,anobjectIF2URC5,calledtheunit,naturalisomorphisms ;ʍjVD (A;B;Cܞ)UR:(A B) C14!A (BE Cܞ);jVD(A)UR:I+ A4!1A;jVD(A)UR:A IF4!A;calledeassoffciativity,Bleftfunitandrightunit,sucrhthatthefollowingdiagramscommute:A9K((A B) Cܞ) Dx(A (BE Cܞ)) Dol:2fd3ά-̯qі (A;Bd;C) 1E[A ((BE Cܞ) DS)=̟:2fd3ά-̯ (A;Bd C;D| (A Bd;C;DInparticulartheJ8K-moSdulesformamonoidalcategoryV.WThisisourmostimportanrtexampleofamonoidalcategoryV.2.Let9BǹbSeabialgebraandB-Mo`d#-bethecategoryofleftB-modules.WVede nethestructureofaB-moSduleonthetensorproductM N6=URM K cN+bry~ mBE M Ns6 1X.M 1X.N:0!3B B M Ns61X.BX r 1X.N c!/B M B Nh6X.M X.NJ q!&MM Nasintheprevioussection.8SoB-Mo`d#ސisamonoidalcategorybry3.1.73. Let BibSeabialgebraandB-Como`d3bethecategoryofB-comodules. ThetensorproSductM N6=URM K cN+carriesthestructureofaB-comodulebry@ M Ns6X.M X.N (ځ!%BE M BE Ns61X.BX r 1X.N c!/B B M Ns6r 1X.M 1X.N:0!3B M N:asintheprevioussection.8SoB-Como`d2Visamonoidalcategorybry3.1.84.Let| GbSeamonoid.A|K-moSduletogetherwithafamilyofsubmodules(VgjgË2G)iscalledG-grffaded꨹ifV¹=URgI{2GFVg.LetiVtٹandWz/bSeG-gradedK-modules."A,homomorphismofK-modulesf2:V4!nWniscalledG-gradediffG(Vg)URWgforallgË2URG.-JTheG-gradedK-moSdulesandtheirhomomorphismsformthecategory(K-Mo`dX)2GofG-grffaded35K-modules.There7isamonoidalstructureon(K-Mo`dX)2G WgivrenbytheordinarytensorproSductV  *Wƹ.TheysubmoSdulesonthetensorproductV  *Waregivrenby(V  *Wƹ)g :=P h2G ZVh Wh1 g=URPh;k62G;hk=gAVh Wk#.c5.8Achain35cffomplexofK-moSdulesʃM6=UR(:::'0x"@q3ЍuK!'ʟM2'0 m@q2ЍVj!M1'0 m@q1ЍVj!M0)consistsZofafamilyofafamilyofK-moSdulesMie4andafamilyofhomomorphisms@n O:GMn4!kMn16with5@n1@n=0.This5crhaincomplexisindexedbythemonoidN0..Onemmaryalsoconsidermoregeneralchaincomplexesindexedbyanarbitrarycyclic monoid.: ChaincomplexesindexedbryN0<Y8N0 arecalleddoublecomplexes.SoAmruchmoregeneralchaincomplexesmaybSeconsidered.=WVerestrictourselvestocrhaincomplexesoverN0.LetfvMZandNbSecrhaincomplexes.KAfWhomomorphismofchaincffomplexesfp :(M!NconsistsԦofafamilyofhomomorphismsofK-moSdulesfn:URMn4!lNn |sucrhԦthatfnP@n+1=UR@n+1fn+1otforalln2N0.Thecrhaincomplexeswiththesehomomorphismsformthecategoryofchaincom-plexesK-Comp'*.Ife@M$andNarecrhaincomplexesthenweformanewchaincomplexMP lNwith`(M5 sQN@)n H:=JL*Pn U_Pi=0"Mi+ Nni;and@:J(M5 N@)n H4!W͹(M N@)n1,givren`byN7 &e78$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYY@(miq  Enni))I\:=(1)2id@M (mi) nni5+miq @(nni)).ThiszisoftencalledthetotalcomplexrassoSciatedwiththedoublecomplexofthetensorproductofMZVandN@.#ThenitiseasilycrheckedthatK-Comp+PisamonoidalcategorywiththistensorproSduct.aJProblem3.2.2.nR1.Prorve thatthecategory(K-Mo`dX)2G 6ofG-gradedK-moSdulesislequivXalenrttothecategoryKG-Como`d14ofKG-comoSdulesbythefollowingconstruc-tion.2IfVrisaG-gradedK-moSduletheVbSecomesaKG-comodulebrythemapȄ:URV4!nKGX Vp,#s2(vn9)UR:=g7 Xv6;forallvË2URVgandallg2URG.ConrverselyifV;Ȅ:V4!`KGX VisaKG-comoSdulethenVtogetherwiththesubmodulesVg*P:=URfvË2Vpjs2(vn9)=g VvgisaG-gradedK-moSdule.SinceKGisabialgebrathecategoryofKG-comoSdulesisamonoidalcategoryV.Shorw8thattheequivXalencede nedabSovebSetween(K-Mo`dX)2G WandKG-Como`d2pre-servresthetensorproSducts,hencethatitisamonoidalequivXalence.2.0LetBX=URKhx;yn9i=IÀwhereIisgeneratedbryx22;xy+xFyn9x.0ThenBmisabialgebrawith $thediagonal(yn9)UR=yRb )y,8(x)=x) 1+yRb x.The $counitis"(y)UR=1;"(x)=0.WVeinrtroSduced(thecooppositebialgebraof8)thisbialgebrainA.72.Shorw[AthatthecategoryK-Comp*ofchaincomplexesisequivXalenttothecategoryB-Como`d3qйof"B-comoSdulesbrythefollowingconstruction.OIfMisachaincomplexthende neaB-comoSduleonM6=URi2N֥MiwiththestructuremapȄ:M64!Bנ =yn92i1 Em20RAi1foralli2Nwreseethat@(midڹ)2Mi1AV.SowrehaveOlde ned@O:Mi k4!ωMi1AV.˙FVurthermorewreseefromthisequationthat@22g (midڹ)=0foralliUR2N.8Hencewrehaveobtainedachaincomplexfrom(M;s2).IfȪwreapply( 1)s2(m)=mȪthenwegetm=P-mi -withȪmi 2MihenceMչ=uLIi2N#Midڹ.This*togetherwiththeinrverse*constructionleadstotherequiredMequivXalence.)3.8AcoScrhaincomplexhastheform:~M6=UR(M0'0 m@q0ЍVj!M1'0 m@q1ЍVj!M2'0 m@q2ЍVj!:::.)O67 &e2. %MONOID9AL!CA:TEGORIESq79Ywith u@i+1AV@ii=0.HShorwthatthecategoryK-Co`comp8XofcoSchaincomplexesisequiv-alenrttoComo`d.J-25PBwhereBiscrhosenasinexample5.ҍLemma3.2.5.g5QLffetbCbeamonoidalcategory.W#Thenthefollowingdiagramscom-mute+0(I+ A) BoxI+ (A B)SF:2fd0ά-Í]< OvA Bꃀq(A) 1X.B97?`@C7?`@M7?`@MV@MVRꃀ~(A Bd)|V?`rV?`hV?`h0h0 08(A B) I:8A (BE I)=џ:2fd0ά-Í(u eA Bꃀ|(A Bd)?`@?`@?`@@RꃀJ1X.A ^ (Bd)H?`>?`4?`33 1Kand35wehave(I)UR=(I).ҍProof.@_First)wreobservethattheidentityfunctorIdCandthefunctorI/ լ- *areisomorphicSHbrythenaturalisomorphism.rThuswehaveIg fOf=gI gu=)gfOf=gn9.Inthefollorwingdiagramډvu&((I+ I) A) Bv`&(I+ (I A)) B}sH2fd)ά-om0 1vv7K&I+ ((I A) B) sH2fd)ά-pÍ sv ( 1) 1X8쟁 Qb8쟈Ql8쟏M^Qv8쟕Q|L0Q|L0ss(1 ) 1z̟ z̟z̟M^z̟l0l0+s1 ( 1)Oe̟ Ee̟;e̟M^1e̟+l0+l0+rӹ(I+ A) BI+ (A B):lA2fdG)ά-Í sK*Ǡ|z@feK\Ǡ?CcAE se*Ǡ|z@fee\Ǡ?jl1 &Π@feYܟΠ?Cc \ Π@feDܟΠ?iRX1rI+ (A B)I+ (A B):l:2fdG)ά-Y֯Z1u&(I+ I) (A B)7K&I+ (I (A B))}32fdά-Í ꃀv  (1 1)X8Ǡb8l8rLv8Ǣ|Lz|Lz3+G1 +l Q5l攴Q?l?^QIlQOe̟0QOe̟0sYallsubSdiagramscommruteexceptfortherighthandtrapSezoid.SinceallmorphismsareisomorphismstherighrthandtrapSezoidmustcommutealso.Hencethe rstdiagramoftheLemmacommrutes.Inasimilarwrayoneshorwsthecommutativityoftheseconddiagram.FVurthermorethefollorwingdiagramcommutesfj^pI+ (I I)b(I+ I) I\A2fdάÍ TI+ (I I)\A2fdά-Í Ԡ`I+ IWM{-1 F`@F`@F`@@R|GȚƖF`F`F` ԠRI+ IWM- 1F`@F`@F`@@R&1 $F`F`F` տFIuG?`@?`@?`@Ɩ@ƖRuG?`?`?` HerethelefthandtrianglecommrutesbythepreviouspropSertyV.UThecommuta-tivitryoftherighthanddiagramisgivenbytheaxiom.ThelowersquarecommutesPm7 &e80$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYsince%isanaturaltransformation.Inparticular(1 )=(1 ).Since%isanisomorphismandI+ -P ԰ =KId!YC*wregetUR=.Єcffxff ̟ff ̎ ̄cffProblem3.2.3.nRFVoruBmorphismsf<:I~4!M&andgc4:I~4!N&inamonoidalcategorydwrede ne(f9 1:Nz4!iM2 N@):=(f9 1IM)(I)21'andd(1 gϹ:M4!nM N@)UR:=(1 gn9)(I)21 \|.8ShorwthatthediagramN׍N㚠M Nd32fd'8ά- Ġf 1bYIbYDM4{fd6"Ѝά-i9fH낟Ǡ*FfeǠ?'6gHaǠ*Ffe4Ǡ?q-F1 grcommrutes.De nition3.2.6.vaùLet(C5; )and(DUV; )bSemonoidalcategories.8Afunctor?Fc:URC4!wfDtogetherwithanaturaltransformationqs(M;N@)UR:F1(M) F1(N)UR4!1F(M N)andamorphism2}0V:URID 4!F1(ICm)8discalledweffakly35monoidalifthefollorwingdiagramscommuteB^덍}(F1(M@) F(N@)) F(Pƹ)tF1(M N@) F(Pƹ)L:2fd2ά-'t4O 1BH(F1((M N@) Pƹ)̟:2fd'8ά-')9U4:Ǡ@feUglǠ?Ǡ@fe>괟Ǡ??;C4rEMandhasaunitË:URIF4!AsucrhthatthefollowingdiagramcommutesXፍY"I+ APUR԰n:=APUR԰n:=A IYgA A苼{fd,cά-iidh HZǠ*Ffe⌟Ǡ?`xI{ idH(XǠ*Ffe(̟Ǡ?`->LrPA A"A:~32fd{ά-kݹ.rHk}XidÎ ҁ H͎ ׁ H׎ ܁ H H H H H H džH džjR.7 &e82$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYLet#AandBE)bSealgebrasinC5.#^Amorphismofalgebrffas#fQ:URA4!1Bis#amorphisminCݹsucrhthat/[ikABu32fd6^ά- V{fY]/A AYkoBE Bo۟{fd@ά-inf fHm9Ǡ*FfemkǠ?\1rX.AHǠ*FfeHǠ?krX.BF*and[bY;BI'+X.A56ׁ 06 +6 &6 %n>%n> H'T*~X.BBoׁ AGo ALo AQo ARN>ARN>UAVjB(n32fd*ά- ;.f1commrute.TRemark3.2.8.j6ItisobrviousthatthecompSositionoftwomorphismsofalgebrasis0againamorphismofalgebras. Theidenrtity0alsoisamorphismofalgebras.ThruswreobtainthecategoryAlgo(C5)ofalgebrasinC.De nition3.2.9.vaùLetp@C#ubSeamonoidalcategoryV.ɨApcffoalgebrap@oracffomonoidinC?consistsofanobjectChntogetherwithacomrultiplicationg:A4!Aa AйthatiscoassoSciativreCw`CF CCF C C/32fdAJά-aid< YxCYnCF C5<{fdY?ά-ˍJNH纟Ǡ*FfeǠ?`SH V:Ǡ*Ffe lǠ?э; idy&ormoreprecisely>ٍd(CF Cܞ) C VCF (C Cܞ)+t32fdά-Í( aCF C32fd%9Ѝά-ׁ idԠmCԠCF CzD:2fdά-ζ .FŸǠ@fe.yǠ?񍍒id rbŸǠ@ferǠ??;wHt8鍹andhasacounit"UR:C14!I+sucrhthatthefollowingdiagramcommutesCw`YCY@>CF CKL{fd|z@ά-ˍHʟǠ*Ffe0Ǡ?`}i H! Ǡ*Ffe!<Ǡ?э&id/ xCF CKI+ CP1԰Jع=ܙCP1԰Jع=CF I:E32fd) ά-a} idHk}lA5rΟ>UH-Tf"X.DRnׁ Mn Hn Cn B.>B.> bYCbYVD)M{fd*(pά-i;NfS/7 &e2. %MONOID9AL!CA:TEGORIESq83Ycommrute.Remark3.2.10.qN6ItisobrviousthatthecompSositionoftwomorphismsofcoalge-brasisagainamorphismofcoalgebras.Theidenrtityalsoisamorphismofcoalgebras.ThrusweobtainthecategoryCoalg%z(C5)ofcoalgebrasinC.Remark3.2.11.qN6Observrethatthenotionsofbialgebra,4Hopfalgebra,andco-moSduleC$algebracannotbegeneralizedtoanarbitrarymonoidalcategorysincewreneed1mtoharve1manalgebrastructureonthetensorproSductoftrwo1malgebrasandthisrequires9&ustoinrterchange9&themiddletensorfactors.$ZTheseinrterchanges9&or ipsareknorwnFeunderthenamesymmetryV,]TquasisymmetryorbraidingandwillbSediscussedlateron.TĠ7 &e84$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYY3.DualObjectsArt!qtheendofthe rstsectioninCorollary3.1.15wesawthatthedualofanHV-moSduleScanbeconstructed.WVedidnotshorwthecorrespondingresultforcomodules.InkfactsucrhaconstructionforcomoSdulesneedssome nitenessconditions.Withthisrestriction'thenotionofadualobjectcanbSeinrtroducedinanarbitrarymonoidalcategoryV.CCDe nition3.3.1.vaùLet(C5; )bSeamonoidalcategoryM;2aWC;Tbeanobject.DAnobject,sM@2 72OCߨtogetherwithamorphismev:M@2\ tM34!IiscalledaleftodualforM+ifthereexistsamorphismdbN*:URIF4!M M@2 됹inCݹsucrhthatsʍx(M26dbr 1p ]!M M@2 M21 evp6 !@M@)UR=1Mm߹(M@22 V:1 dbp Ҳ( ! M@2 M M@22 V:ev" 1p Ua!!TM@2)UR=1M" D:Amonoidalcategoryiscalledleft35rigidifeacrhobjectM62URCݹhasaleftdual.Symmetricallywrede ne:canobject2oM62URCbtogetherwithamorphismev2:Mh3 'O2SM4!Iis,calledarightodualforMmqifthereexistsamorphismdb7:zI4!2VMi ׅMinCsucrhthat7ȍʍxU(M261 dbp h!9M 2jM M26evb 1p@ ]!M@)UR=1MmW(2M26dbr 1p ]!2"dM M 2jM21 evp6 !@2 M@)UR=1UTM D:NAmonoidalcategoryiscalledright35rigidifeacrhobjectM62URCݹhasaleftdual.ThemorphismsevkanddbarecalledtheevaluationrespSectivrelythedual35bffasis.CCRemark3.3.2.j6If(M@2;ev /)isaleftdualforMǹthenobrviously(M;ev)isarighrtdualforM@2 됹andconrverselyV.8OneusesthesamemorphismdbN*:URIF4!M M@2.Lemma3.3.3.g5QLffet(M@2;ev /)bealeftdualforM@.{Thenthereisanaturaliso-morphismuf\MorYC (- M;-33)PUR԰n:=Mor%5C*(-35;- M@ );i.35e.fithefunctor-  M6:URC!FCjisleftadjointtothefunctor- M@2 V::URC!FC5.Proof.@_WVe# givretheunitandthecounitofthepairofadjointfunctors.WWVede ne(A)%:=1A ۫ ǹdb1:A4!cA M> M@2 eandd (B)%:=1B L ǹev:B M@2 Mf4!B.Theseareobrviouslynaturaltransformations.8WVehavecommutativediagrams\ J(A MQlA M M@2 M5vl32fdY]ά-QF((A)=I51X.A ^ db | 1X.MSA M@)UR=1A M:|32fdZPά- F((A)=lÍ l1X.A ^ 1X.M evOanda\(BE M@2h*BE M@2 M M@26<32fdSά-QGv(Bd)=;F1X.BX 1"MG s0 dbM3BE M@2)UR=1Bd M"L32fdS0ά-"ZGv (Bd)=s鍒 1X.BX ev { 1"MGRthrustheLemmahasbSeenprovedbyCorollaryA.9.11.Icffxff ̟ff ̎ ̄cff#Theconrverseholdsaswell.GIf- m EMisleftadjointto- m EM@2 ϲthentheunitgivresamorphismdbw:= (I):I4!oPMw 6.M@2 |andthecounit givesamorphismev:=UR (I):M@2 M64!I+satisfyingtherequiredpropSerties.8ThruswehaveU7 &e*3. %DUAL!OBJECTSפ85YCorollary3.3.4.sWIf,- h KM6:URC!FCais,leftadjointto- KM@2 V::URC!FCathenM@2 isa35leftdualforM@.NCorollary3.3.5.sW(M@2;ev /)`isaleftdualforMDifandonlyiftherffeisanaturalisomorphismh ιMorH˟C(M@  -;-33)PUR԰n:=Mor%5C*(-35;M -);=i.ze.`άmZg1 evFTN@2 M M@2d32fd$>pά 1 f 1H`*Ffe64`? 1 gI{ 1 f 1O@^gN@2~ 퍍uevpO 1>_ZHgZRnZ\vZf}ZpZzZZZ'dZ'd~ Su1 db'd'd%'dƥv'd%l'dեb'd%X'd䥀N'd%D'd>>= %cffxff ̟ff ̎ ̄cffTProblem3.3.5.nR1.)In:thecategoryofN-gradedvrectorspacesdetermineallob-jectsM+thatharvealeftdual.2.mInthecategoryofcrhaincomplexesK-Comp,J*determineallobjectsM`fthatharvealeftdual.3.7In=thecategoryofcoScrhaincomplexesK-Co`comp7TdetermineallobjectsM~thatharvealeftdual.4.LetZ(M@2;ev /)bSealeftdualforM@.Shorwthatdb{:I`4!0M7] yM@2 ZisuniquelydeterminedbryM@,M2,andev.8(Uniquenessofthedualbasis.)5.Leth(M@2;ev /)bSealeftdualforM@.Shorwthatev+o:,JM@2  Mm.4!IZwisuniquelydeterminedbryM@,M2,anddb.W?7 &e*3. %DUAL!OBJECTSפ87YCorollary3.3.9.sWLffet<M;N|havetheleftduals(M@2;ev /M)and(N@2;ev /Ns)andletfQ:URM6!QNtbffe35amorphisminC5.fiThenthefollowingdiagramcommutesMPE0N N@2@N M@2:z̞32fd2ά-漍1 fǟ-:YIY\M M@27{fd& @ά-udb綟X.MHpǠ*FfeǠ?edbjX.NHjǠ*FfeǠ?`f 1NProof.@_ThefollorwingdiagramcommutesY:P$M$N N@2 M,Tfd@`ά-bdb̄ 1^ٍYN^ٍٚN N@2 N;ԟrfd!`ά-ٜdb̄ 1@9Ԍfem$Ԍ?'Åf@rԌfe⤟Ԍ?'$1 1 f@NrǠfe⤟Ǡ?ݍ$1 ev@1HHHqƕHqƕj܍This"implies(f ޹1M" D)db.MF=((1N " evgNS)(1N 1N"?v fG)(db 0N'r 1M ) 1M" D)db.MF=(1N b) Hev NX 1M" D)(1N H1N" } f 1M")(db 0N'r 1M 1M")db.Mp=UR(1N b) ev NX 1M")(1N b) 1N"X 4f| 1M" D)(1N N 1N" dbBM)db.N=(1N N (ev 1Ns 1M" D)(1N" f| 1M" D)(1N" db 0M ))db.N|Ĺ=UR(1N fG2)db.N'r.oDŽcffxff ̟ff ̎ ̄cff Corollary3.3.10.zLffetIM;N-havetheleftduals(M@2;ev /M)and(N@2;ev /Ns)andletfQ:URM6!QNtbffe35amorphisminC5.fiThenthefollowingdiagramcommutesN0phN@2 NrI:32fd$ߠά-MשTev #X.NYN@2 MY.M@2 MԄ{fdؐά-q<͝Lfǟ-:7 1H"Ǡ*Ffe TǠ?`91 fHOǠ*FfeԟǠ?-5Tev#X.MoProof.@_This:statemenrtfollowsimmediatelyfromthesymmetryofthede nitionofaleftdual.ccffxff ̟ff ̎ ̄cff Example3.3.11.uQLetk1M$2ٟR lMR ^ĹbSeanRJ-R-bimodule. |Thenk1Hom亟R$M(M:;R:)isǃanRJ-R-bimoSduleǃbry(rfGs)(x)=rfG(sx). qFVurthermoreǃwrehavethemorphismev:URHom۟R"n(M:;RJ:) R ;M64!Rde nedbryev(f Rm)UR=fG(m).(DualBasisLemma:)AtmoSduleM62URMR \iscalled nitelySgenerffatedandprojectiveifthereareelemenrtsm1;:::ʚ;mn2URM+undm21;:::ʚ;m2n2URHom۟R"n(M:;RJ:)sucrhthatcڍM8mUR2M6: knX ㇍Si=1midm i(m)UR=m:*ꍑActually;thisisaconsequenceofthedualbasislemma. ,>Butthisde nitionisequivXalenrttotheusualde nition.X07 &e88$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYLetM62URR HMR.+MasarighrtRJ-moSduleis nitelygeneratedandprojectivei Mhasaleftdual.8TheleftdualisisomorphictoHomd1R#WĹ(M:;RJ:).IfjMR is nitelygeneratedprojectivrethenweusedb:URRn4!{M R ;Hom!:R(͹(M:;RJ:)withdbN(1)=P*In U_Ii=1#mi`) R m2idڹ. BInfactwrehave(1O R ev)(db. R1)(m)=(1 R$ev 1)(Pmi R 9m2i Rm)ٝ=PHmidm2i(m)ٝ=m.>WVeharvefurthermore(ev / R1)(1E Rdb 0)(fG)(m)UR=(ev / R1)(P* n U_ i=1f\ R ~miy Rm2idڹ)(m)UR=Pf(midڹ)m2i(m)UR=f(Pmidm2i(m))=URfG(m)forallm2M+hence(ev / R1)(1 R ;dbk)(fG)=f.ConrverselyifMhasaleftdualM@2 thenev:M@2 7C R )Mw4!R"de nesahomo-morphismUR:M@2 V:4!Hom3>R:25(M:;RJ:)inR "MRbry(m2)(m)UR=ev(m2] R m). WVede neP* n U_ i=1miw{ m2i,:=URdbc(1)UR2MS M@2,then:m=(1 ev )(db. 1)(m)=(1 ev )(Pmiw{ $m2i% m)=P7jmid(m2i)(m)9sothatm1;:::ʚ;mn52M<and(m21);:::ʚ;(m2nP)2HomyRm(M:;RJ:)GlformadualbasisforM@,i.e. O-MPis nitelygeneratedandprojec-tivreasanRJ-moSdule.8ThusM@2 됹andHomd1R#WĹ(M:;RJ:)areisomorphicbythemap.AnalogouslyoHom0R#ù(:M;:RJ)oisarighrtdualforMi Mis nitelygeneratedandprojectivreasaleftRJ-moSdule.-׍Problem3.3.6.nRFindanexampleofanobjectMpinamonoidalcategoryCdthathasaleftdualbutnorighrtdual.De nition3.3.12.}!ùGivrenW objectsM;NinC5. ~ Anobject[M;N@]iscalledaleftS/innerHom#ofMdڹandNifthereisanaturalisomorphismMorYCDz(-j M;N@)Pj԰=Mor5C(-;[M;N@]),i.e.8ifitrepresenrtsthefunctorMor Cd(-P M;N@).IfthereisanisomorphismMorNCw(P xRM;N@)PW|԰pd=OMor):LC. (PS;[M;N@])naturalinthethreevXariableM;N;PnthenthecategoryCݹiscalledmonoidal35andleftcloseffd.If%thereisanisomorphismMor[ϟCɎ(M K#h (PS;HomyK!2i(M;N@))P԰N=Hom*:JK0,(P/ iM;N@)n|canbSere- #strictedtoanisomorphismtbHom{\NHu(PS;HomyK!2i(M;N@))PUR԰n:=Hom(yH0ȹ(PLn M;N@); #bSecause m))G=(f)(Ph(1) \|p> h(2)m)G=PwfG(h(1)p)(h(2)m)= #P,5(h(1) \|(fG(p)))(h(2)m)=PVh(1) \|(f(p)(S׹(h(2))h(3)m))=h(f(p)(m))=h((f) #(p m))p]andconrverselyp](h(fG(p)))(m)UR=Ph(1) \|(f(p)(S׹(h(2))m))UR=Ph(1) \|((f) #(pֱ S׹(h(2) \|)m))i=Pn(fG)(h(1)pֱ h(2)S׹(h(3))m)i=(fG)(hpֱ m)i=f(hp)(m). #ThrusHV-Mo`d$0isleftclosed.2#If M42PHV-Mo`d$Eisa nitedimensionalvrectorspacethenthedualvector #spaceM@2 N:=M+HomƴK$(M;K)againisanHV-moSdulebry(hfG)(m)M+:=f(S׹(h)m): #FVurthermoreM@2 됹isaleftdualforM+withthemorphismsfx^db:URK317!X ㇍ imi m i,2M M@ g #and}uev:URM@  M63f m7!fG(m)2KA #where=miչandm2iareadualbasisofthevrectorspaceM@. 2Clearlywehave #(1 ev [)(db. 1)UR=1M 5andE(ev / 1)(1 db )UR=1M"։sinceM>)isa nitedimensional #vrectorfspace.WVehavetoshowthatHdbandHev'areHV-moSdulehomomorphisms. #WVeharve&tʍ+#5(hdb.(1))(m)UR=(h(Pmi m2idڹ))(m)=(Ph(1) \|mi h(2)m2idڹ)(m)UR=+#5P7(h(1) \|midڹ)((h(2)m2i)(m))UR=P(h(1) \|mi)(m2i(S׹(h(2))m))UR=+#5P9h(1) \|S׹(h(2))mUR="(h)m="(h)(Pmi m2idڹ)(m)="(h)db.(1)(m)=+#5db81e("(h)1)(m)UR=dbc(h1)(m);)t #hencehdb.(1)UR=dbc(h1).8FVurthermorewrehavetʍ(hev /(f m)UR=hfG(m)=Ph(1) \|f(S׹(h(2))h(3)m)UR=P(h(1) \|f)(h(2)m)UR=(ꉟP74evC+e(h(1) \|f h(2)m)UR=ev(h(f m)):4. #Let^HL>bSeaHopfalgebra. KThenthecategoryofleftHV-comodules(seeExample #3.2.433.)isamonoidalcategoryV.LetM2џHV-Como`d3kbSea nitedimensional #vrectorspace. $LetmibSeabasisforMϕandletthecomultiplicationofthe #comoSdule9bes2(midڹ)UR=Phij z /8mjf .$Thenwrehave(hikl)UR=Phij z /8hjvk .$M@2 V::= #Hom8K?U(M;K)tbSecomesaleftHV-comodules2(m2jf )UR:=PS׹(hijJ){ m2idڹ.ZOnetvreri es #thatM@2 됹isaleftdualforM@.0Lemma3.3.14.mQLffet35M62URCjbeanobjectwithleftdual(M@2;ev /).fiThen1.fiM M@2 4is35analgebrffawithmultiplicationtL rUR:=1M . ev@ 1M".:M M@  M M@  V:!M M@ ZXO7 &e90$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYand35unit3ߍuUR:=db:IF!M M@ ;2.fiM@2 Mtis35acffoalgebra35withcffomultiplicationDKyйUR:=1M" db 1M B:M@  M6!QM@  M M@  Mand35cffounit5"UR:=ev@:M@  M6!QI:B5Proof.@_1.nThe#-assoSciativitryisgivenby(ro 1)riD=(1M _ oev 1M"X o1M 1M" D)(1M " ev4 1M")'=1M ev4 ev- 1M"k=(1M 1M"w 1M ev4 1M" D)(1M ev / 1M" D)UR=(1 r)r.The&axiomfortheleftunitisr(u 1)UR=(1M ev 1M")(db. 1M . 1M" D)UR=1M 1M" D.2.8isdualtothestatemenrtforalgebras.ʜDŽcffxff ̟ff ̎ ̄cffB5Lemma3.3.15.mQ1.PLffetAbeanalgebrainCandleftM62URCbffealeftrigidobjectwith)ileftdual(M@2;ev /).c%TherffeisabijectionbetweenthesetofmorphismsfQ:URA MU!M making(MaleftA-moffduleandthesetofalgebramorphismsVze*f !:URA!M| ;M@2.2. (LffetpCbeacoalgebrainCΥandleftM62URCbffealeftrigidobjectwithleftdual(M@2;ev /).TherffeɋisabijectionbetweenthesetofmorphismsfQ:URM6!QM C)makingM oaright*Cܞ-cffomodule35andthesetofcffoalgebra35morphismsV\e*f.:URM@2 M6!QC.B5Proof.@_1." By$Lemma3.3.14theobjectM_ M@2 isanalgebra.GivrenfQ:URA M4!Msucrh;thatMbSecomesanA-module. ByLemma3.3.3wreassociateVe*f1ܹ:=(f2 ꊹ1)(1 db):A4!A M+n M@2 4!M M@2.R]TheH|compatibilitryofVqe*fwiththemrultiplicationisgivenbythecommutativediagramvJA AvpAnLsH2fd[`ά-p rYsWA A M M@2Y A M M@2wfd6ά-92r 1 1$A M M@2$|M M@2ƹfdM@ά-`后f 1^ٍxA M M@2 M M@2^ٍ A M M@2rfd ά-l==md1 1 ev { 1&jM M@2 M M@2^1M M@232fdȒά-r b1 ev { 1sZ} Ǡ|z@feZ<Ǡ?+㎍F"a6cmex8erEsfJj% ㎍erfO򊟱R fe%R ? <1 f 1@Ԍfe%Ԍ?̈́Q<1 db | 1 1O-݊R fe.R ? f 1stS Ǡ|z@fet<Ǡ?+㎍z„ery8fsOэn1 1 dbk M`Hu M`H M`HUHUjsOb蜟M`@l蜟M`@v蜟M`@蜟M`@蜟M`@\O@\ORs8r1 ㎍erf@\\z\p\f\b蜟ob蜟o 胀1㎍i; ergEfld 1 1@bnf 1 1|rk ƕk ƕ@ͻɍ|1|U>|UY)X\16I@@I@JI@TI@^I@bMܟo@bMܟoR@b9f 1>|HH|HR|HZ|ƕHZ|ƕjTheunitaxiomisgivrenby;I}%IM M@2ȴRfd1ʀά-Ԡ db{AWA M M@2d32fd$ά-ׁ=1 dbHtM M@232fdά- x&f 1 Ǡ fe54Ǡ?+vu 5ŸǠ fehǠ?Mu 1 M1 1ٴHٴHٴHٴH +uH +uj[k@7 &e*3. %DUAL!OBJECTSפ91YConrverselyletg::NA4!)M @M@2 ޹bSeanalgebrahomomorphismandconsidervegԹ:=(1 evJ)(g" 1)nR:A M64!7M M@2 M64!M@.dThenXM:LF`9LF`4LF`/LF`*LF`&欟`&欟` s^MgI{ g 1C}`XMM`XW`XaM`Xk`XuM`X`XM`X`XM`X`XM`XXzPMigI{ 1CǠMGWǠaGkǠuGǠGǠGǠGۿۿ:ꬽD1 evۿ`X ?`X`X?`X'`X1?`X;`XE?`XO`XY?`Xc`Xm?`XoXoz0=h1 evoF`AtF`AyF`A~F`AF`AF`AF`AvL`AvL`Us^MlgI{ 1ZM`~ZM`tZM`sܟsܟ eand3 oA MM M@2 Mu32fd ά-o͍MgI{ 17M_$32fd,Ѝά-r 1 ev {MǠ@fe$Ǡ?ꬽm}Ju 1񍍒,db 1h?`Hh?`Hh?`Hh?`Hh?`Hh?`HydHydjid1٤ۿ`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`XXzycommrute.2.(M@2;ev /)isaleftdualforMĖinthecategoryC6ifandonlyif(M@2;db.)istherighrtTdualforMinthedualcategoryC52op R.wXSoifwedualizetheresultofpart1.wXweharvetocrhangesides,hence2.cffxff ̟ff ̎ ̄cff\7 &e92$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYY4.[FinitereconstructionTheEendomorphismringofavrectorspaceenjoysthefollowinguniversalpropSertyV.Itisavrectorspaceitselfandallowsahomomorphismq:End~(Vp) V^4!V..Itisunivrersaly5withrespSecttothispropertryV,i.e.ifZUӹisavectorspaceandfQ:URZ V4!`Visahomomorphism,+thenthereisauniquehomomorphismg3:ŬZJ4!End-B(Vp)sucrhthatD xEnd-7(Vp) V2V\Ğ32fd&ά-gۍ H`tf@ԟ>RYZF VŸǠ*FfeǠ?q-gI{ 1+commrutes.ThealgebrastructureofEndg(Vp)comesforfreefromthisunivrersalpropSertyV.IfHwrereplacethevectorspaceVbyadiagramofvectorspaces! :D54!Vec.wegetasimilarunivrersalobjectEnd(!n9).>AgaintheuniversalpropSertyinducesauniquealgebrastructureonEndg(!n9).Problem3.4.7.nR1.@녤>RYZF VӒǠ*FfeğǠ?q-"gI{ 1commrutes).8WVecallEandUR:E^ V4!`Vaveffctor35spaceactinguniversallyonVp.2.*LetfEعand(:E= &V4!V1bSefavrectorspaceactinguniversallyonVp.*Showthat"EuniquelyhasthestructureofanalgebrasucrhthatVnbSecomesaleftE-module.3. Let!Ƴ:XzD4!Vec6 7bSeadiagramofvrectorspaces.ShorwthatthereisaunivrersalkvectorspaceE andnaturaltransformation1b:E !4!!+(sucrhthatforeacrhU^vectorspaceZ1andeachnaturaltransformationfQ:URZVW y!Ë4!2j!×thereisauniquehomomorphismgË:URZ14!EsucrhthatLE^ !V!T32fd ά-gۍH`"f˷dׁ @շd @߷d @d @>@>RYVZF !KҟǠ*FfeǠ?q- gI{ 1׍commrutes).8WVecallEandUR:E^ !Ë4!2j!Xaveffctor35spaceactinguniversallyon!n9.]S7 &e[:4.pFINITE!RECONSTR9UCTION93Y4.J Let EO#and:E "!4!! EbSe avrectorspaceactinguniversallyon!n9.J Showthat_EAvuniquelyhasthestructureofanalgebrasucrhthat!bSecomesadiagramofleftE-moSdules.{Similar^considerationscanbSecarriedoutforcoactionsV4!`V( C:or!Ë4!2j! Cand#HacoalgebrastructureonCܞ.Thereisonerestriction,1phorwever.WVe#Hcanonlyuse nitedimensionalvrectorspacesVordiagramsof nitedimensionalvectorspaces.ThiswillbSedonefurtherdorwn.Aswrehaveseen,???WVeiwrantto ndauniversalnaturaltransformation:,|!4!!n coSend 3(!n9).FVorthispurpSosewreconsidertheisomorphisms>RMorh'CmD(!n9(X);!(X) M@)PUR԰n:=Mor%5C*(!n9(X) j !n9(X);M@)thatmaregivrenbyf7!Y(ev / 1)(1ߛ fG)mandasinverseg7!Y(1ߛ gn9)(db. 1). 0WVe rstGdevreloptechniquestodescribSethepropertiesofanaturaltransformation :!4!! MθaspropSertiesoftheassociatedfamilygn9(X)k:!(X)2 !(X)k4!M@."eWVewillseethatgË:UR!n92{K M!4!2jMwillbSeacffone.)ThenwrewillshowthatisauniversalnaturalytransformationifandonlyifitsassoSciatedconeisunivrersal.OIntheliteraturethisiscalledacoSend!".Throughout thissectionassumethefollorwing.FLetD_ubSeanarbitrarydiagramscrheme. eLetCW=bSeacocompletemonoidalcategorysucrhthatthetensorproductpreservrescolimitsinbSotharguments.(~LetC0ybSethefullsubcategoryofthoseobjectsinaC_thatharveaaleftdual.$Let!Ë:URD4!CbSeadiagraminCsucrhthat!n9(X)UR2C0leforallXF2URDUV,Li.e./!>.isgivrenbyafunctor!0V:URD4!C0./WVecallsuchadiagrama nitediagrffam;yinC5.+RFinallyforanobjectM2Clet!O M:D464!,CbSethefunctorwith(! M@)(X)UR=!n9(X) M@..čRemark3.4.1.j6ConsidercthefollorwingcategoryxSeDtq.FVoreachmorphismf:z:{X*4!YutherePisanobjectVe*fׂ2wGeD.TheobjectcorrespSondingtotheidenrtityP1X :X4!X{isdenotedbrywDeXI2xuebDp.'FVoreachmorphismfGa:bX4! YhinD:NtherearetwomorphismsQf1Ï:V,e*f4!w$5!e!6X09ǹandf2:V,e*f4!w"Te!6Y.inx6eD .kFVurthermoretherearetheidenrtities1fq:V~e*URf K4!V!U|e*,*f*)yinx`eD .9Since therearenomorphismswithweXasdomainotherthan(1X)i~:weX74!w&le#nhX2*andU1f:Ve*f4!V!e*f*swre onlyhavetode nethefollowingcompSositions(1X)i fj揹:=fjf .Then8xv3eD bSecomesacategoryandwrehave1'eX r۹=UR(1X)1V=(1X)2.]ߍWVede neadiagram!n92 !Ë:x˅eURDA4!!Cݹasfollorws.8IffQ:URXF4!YisgiventheniO}(!n9  !n9)(V)Re*f)UR:=!(Yp) j !(X)>andXAʍ!n9(f1)UR:=!(fG)2j !(1X);4-!n9(f2)UR:=!(1YP)2j !(fG):^(7 &e94$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYThemwcolimitof!n921 !۰consistsofanobjectcoSend" (!n9)32C togethermwwithafamilyofmorphisms(XJg;X)UR:!n9(X)2j !(X)UR4!1coSend2w(!)sucrhthatthediagrams\Gxf2MorXD;?!n9(Zi q(fG)) j !(Qu(f))%a'؍X2Ob D^!n9(X) j !(X)@cc+ܟfd[ά- pFcFc+ܟfd[ά-׍-q&&pcoSend|r(!n9)@Kƌ2fd Pά-A{Proof.@_This{isjustareformrulationofRemarkA.10.11,&sincethecolimitmayalsobSebuiltfromthecommrutativesquaresgivrenaborve.ObservrethatfortheconstructionofthecolimitnotallobjectsofthediagramharvetobSeusedbutonlythoseoftheform!n9(X)2j !(X).{cffxff ̟ff ̎ ̄cff*cTheorem3.4.3.p(Tannak@a-Krein)Lffet!Ë:URD!fgC0VCFbea nitediagram.1BThenthereexistsanobjectcoSend!G9(!n9)UR2Cand#{anaturffaltransformationȄ:UR!Ë!J! coSend ;0(!n9)suchthatforeachobjectM62URCand:feffachnaturaltransformation'<:!!N ! mM{Jthere:fexistsauniquemorphism~e' ȹ:URcoSend!(!n9)UR!Mtsuch35thatthediagrffamRH@!HW! coSend ^"(!n9)ht{fd 0ά--0H'Z'Xׁ @X @X @X @ԟt>@ԟt>RHSj! M椢Ǡ*FfeԟǠ?ҍT1 e'0cffommutes.*cProof.@_LetjcoSend#n(!n9)p2C)togetherjwithmorphisms(V)Re*f)p:!(Yp)2pR N!(X)4!3coSendz(!n9)YbSethecolimitofthediagram!2o' @!;0:xC*eD04!&C5.Sowregetcommutative_q7 &e[:4.pFINITE!RECONSTR9UCTION95Ydiagramsr䍍!n9(X)2j !(X)3!I{(f)-:UT 1lΠlΠlΠlΠlΠÇlΠė쟯Fė쟯F*(X&;X)ylF`HylF`HylF`H ylF`HylF`H!ylF`H"H"j!n9(Yp)2j !(Y)ꃀ1 !I{(f)l?`Hl?`Hl?`Hl?`Hl?`HÇl?`HėHėjꃀ(Yx;Y)ylǠylǠylǠ ylǠylǠ!ylǠ"?"?*\n!n9(Yp)2j !(X)&coSendDc(!n9)@ :2fdx`ά-̰獒o@(㎍erf)FforeacrhfQ:URXF4!YinC5.FVor@X2xCwrede neamorphisms2(X):!n9(X)4!P}!n9(X) coSend!F(!)@bry(1 (XJg;X))(db. 1)UR:!n9(X)4!1!n9(X)_ !n9(X)2c !n9(X)UR4!1!(X)_ coSendu(!n9).ThenxwregetasinCorollary3.3.5(XJg;X)UR=(1 ev@)(1 s2(X)).WVeshorwthat]ڹisanaturaltransformation.8ForeacrhfQ:URXF4!YthesquareW9s!n9(Yp) !(Yp)2h!n9(Yp) !(X)2:\32fd/逍ά-W`J(1 !I{(f)-:HFIH !n9(X) !(X)2ܟ{fdQِά-HDdbǺX.XH*Ǡ*Ffe \Ǡ?e~dbX.YHǠ*Ffe9Ǡ?`!I{(f) 1$commrutesbyCorollary3.3.9.8Thusthefollowingdiagramcommutes|1N!n9(X)!n9(X) !(X)2j !(X)PܟA2fd=ά-偍ddboyL 1,!n9(X) coSend ^"(!) !A2fdά-1 (X&;X)L!n9(Yp) !(Yp)2j !(X)!n9(Yp) !(X)2j !(X)al:2fd ά-έ''ЀJJ1 !I{(f)-:UT 12+q!n9(Yp)R!n9(Yp) !(Yp)2j !(Yp)P32fd@<0ά-ׁe3 dbp 1-`!n9(Yp) coSend ^"(!): L32fdά-1 (Yx;Y)? zǠM@fe?SǠ?){!I{(f)XzǠM@feY)Ǡ?],!I{(f) 1񍍑i\dbt 1K< QU!_C)beadiagraminC5.AssumethattherffeisauniversalobjectcoSend"m(!n9)andnaturaltransformationȄ:UR!Ë!J! coSend ^"(!n9).ThenptherffeisexactlyonecoalgebrastructureoncoSend!(!n9)suchthatthediagramsKl䍍fe! coSend ^"(!n9)! coSend ^"(!n9) coSend(!n9)32fdά-aL 1H2!H|! coSend ^"(!n9)-{fdSά--0HǠ*FfeԟǠ?k}#H bǠ*Ffe Ǡ?o1 xandCH@!HW! coSend ^"(!n9)ht{fd 0ά--0H}id]!Xׁ @X @X @X @ԟt>@ԟt>RHؒ! I椢Ǡ*FfeԟǠ?͝T1 Qcffommute.Proof.@_BecauseloftheunivrersalpropSertyofcoSend"(!n9)therearestructuremor-phisms}[):coSend"(!n9)4!߹coSend4Y(!)e coSend (!)}and[):coSend"(!)4!I._This}impliesthecoalgebrapropSertrysimilartotheproofofCorollary3.3.8.l#Մcffxff ̟ff ̎ ̄cffObservre,thatbythisconstructionallobjectsandallmorphismsofthediagram!۹::D4!'C0 CQcare.comoSdulesormorphismsofcomodulesorver.thecoalgebracoSendz(!n9). EInDZfactC~Z:=coSend#U6(!)istheunivrersalcoalgebraoverwhichthegivendiagrambSecomesadiagramofcomodules.Corollary3.5.2.sWLffetS(DUV;!n9)beadiagramC:withobjectsinC0.ThenallobjectsofthediagrffamarecomodulesoverthecoalgebraC|+:=coSend"S(!n9)andallmorphismsarffemorphismsofcomodules.KIfD7OisanothercoalgebraandallobjectsofthediagramarffeDS-comodulesby'(X):!n9(X)!!n9(X)/ D:Qandallmorphismsofthediagrffamarffe3morphismsofDS-comodulesthenthereexistsauniquemorphismofcoalgebras~e' ȹ:URcoSend!(!n9)UR!Dsuch35thatthediagrffamKl䍍H@!HW! coSend ^"(!n9)ht{fd 0ά--0H'Z'Xׁ @X @X @X @ԟt>@ԟt>RH֘[! D椢Ǡ*FfeԟǠ?ҍT1 e'cffommutes.Proof.@_The͈morphisms'(X)UR:!n9(X)4!1!n9(X)o* D!de ne͈anaturaltransforma-tionsinceallmorphismsofthediagramaremorphismsofcomoSdules..Sotheexistencec7 @z5.pTHE!CO9ALGEBRAcoAend99Yandtheuniquenessofamorphismfe':URcoSend!(!n9)UR4!1Dgeisclear.ETheonlythingtoshorwisthatthisisamorphismofcoalgebras.sThisfollorwsfromtheuniversalpropSertyofC1=URcoSend!(!n9)andthediagram_2R!_2M! CzD[˲fdCgά-YR퍒&`[ujĠm@jĠw@jğ@X@XR[uf1 \Ġm@\Ġw@\ğ@X@XR[2Πdfe2dΠ?㩍}?1[2Πdfe$dΠ?[A n1 e'+T! C+Np! CF CmIJfd+s ά-8@iL 12Ǡdfe+dǠ?@ҍn1 e'2ǠdfedǠ?e21 e' e'ԠR!Ԡ! DzD:2fdBά-`@W''uG 'jğ?`@jğ?`@jğ?`@@Rԏg,1 \ğ?`@\ğ?`@\ğ?`@@R! D~! D6 D32fd*(pά-o͍շ' 1(ywheretherighrtsideofthecubSecommutesbytheuniversalpropSertyV.-eSimilarlywegetthati7e'ƹpreservresthecounitsincethefollowingdiagramcommutesjY+!+Ł9! C$IJfd - ά-K퍒Y!Y ! D${fdά-i'Hn`*FfeD`? 1H%R`*FfeX`?@ҍ?1 e'HܒǠ*FfeğǠ?8ID1H]1 "ݐ䟜@䟦@䟰@䟺@d@dRH͝1 "ݐׁ @ @ @ @d>@d>R⍒0#! K HHHHHHHHFHFj`Y 1H"0#! K8Iɔ1ҁ Hׁ H܁ H H H H H HdžHdžje %cffxff ̟ff ̎ ̄cffd-7 &e100$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYI.6..ThebialgebracoSendLet!R:uDo4!YCsand!n920 :DUV20 4!'NCsbSediagramsinC5. }WVecallthediagram(DUV;!n9)w (D20#;!n920!n9(X+ Yp) BbL32fdЍά-'(X Y)뭍andG*$⍒R0K⍒K K,{fdY?ά-‚.⍍hh=HɪǠ*FfeܟǠ?H 8*Ǡ*Ffe k\Ǡ?{`!n9(I)y!n9(I) B̞32fdKd ά-]'(I)commrute.WVedenotethesetofmonoidalnaturaltransformationsbryNat+Qx Gɹ(!n9;! B).Problem3.6.8.nRShorwthatNat+Qx Gɹ(!n9;! B)isafunctorinB.Theorem3.6.5.pLffet(DUV;!n9)beareconstructive,RmonoidaldiagraminVec.2`ThencoSendz(!n9)GisabialgebrffaandȄ:UR!Ë!J!' )coSendh(!)isamonoidalnaturffaltransforma-tion.If2LBRisabialgebrffaand@ι:-!!0 ! gBisamonoidalnaturffaltransformation,thenjtherffeisauniquehomomorphismofbialgebrasf:coend(!n9)!ߍBsuchjthatthef7 &e102$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYdiagrffamE̴H@!HW! coSend ^"(!n9)ht{fd 0ά--0Hk}A@Xׁ @X @X @X @ԟt>@ԟt>RH! B椢Ǡ*FfeԟǠ?`T1 fs2cffommutes.Proof.@_ThemrultiplicationofcoSend!"(!n9)arisesfromthefollowingdiagramLNEH>x!n9(X) !(Yp)Hxe !n9(X) !(Yp) coSend ^"(!) coSend ^"(!)Y8t{fdά- A_L !n9(X+ Yp)v!n9(X+ Yp) coSend ^"(!)P32fdK(ά-mtn0H3S2Ǡ*Ffe3dǠ?H׺Ǡ*FfeǠ?HP԰չ=!n9(X) !(Yp) coSend ^"(!)`&ğׁ @0ğ @:ğ @Dğ @FD>@FD>R㕍FVoritheconstructionoftheunitwreconsiderthediagramD0u=-q(fIg;fid ʤg)togetherwith#!0D:D04!Vec/k, !0(I)=K,the#monoidalunitobjectinthemonoidalcategoryof.diagramsinVeciL.Then(K4!K K)=(!0 4!k!0 coSend j(!0)).istheunivrersalmap.8ThefollorwingdiagramtheninducedtheunitforcoSend!"(!n9)K]ō⍑j 8K⍒:K Kv4{fdAά-‚.⍍cxcx=c4h!n9(I)!n9(I) coSend ^"(!)}Ԟ32fd!wЍά-HnǠ*FfenǠ?HtǠ*FfȩǠ?HPN԰36=K coSend ^"(!n9)`t+ʬQttQtQt+QtrQ tQf4Qf4s8鍹ByusingtheunivrersalpropSertyonechecksthelawsforbialgebras.TheuWabSorvediagramsshowinparticularthatthenaturaltransformationhQ:!4!n! coSend ^"(!n9)ismonoidal.Scffxff ̟ff ̎ ̄cffg,֠7 &e_F7.pTHE!QUANTUMMONOIDOFAQUANTUMSP:A9CENQh103YT7.eThequantummonoidofaquantumspaceProblem3.7.9.nRIf!uAisa nitedimensionalalgebraandو:fVA4!9M@(A)~C AtheunivrersalcoSoperationoftheTVambarabialgebraonAfromtheleftthenW:iA4!fAtK M@(A)(withthesamemrultiplicationonM(A))isaunivrersalcoSoperationofM@(A)onAfromtherighrt.Thecomultiplicationde nedbythiscoSoperationisWUR:M@(A)4!1M(A)nW M(A)./Thrus wehavetodistinguishbSetweentheleftandtherighrtTVambarabialgebraonAandwehaveMrb(A)UR=Ml!ȹ(A)2cop .RNorwconsiderthespSecialmonoidaldiagramschemeD:=DUV[X;m;u].ǔTVomakethings0simplerwreassumethatVecisstrictmonoidal. ThecategoryDUV[X;m;u]0hastheobjectsX !::: !Xn=}mX2 nforalln2N(andIn:=X2 0 M)andthemorphismsm*:XѴ 1X̭4!dX,Lu:I̭4!X*͹and9Jallmorphismsformallyconstructedfromm;u;idbrytakingtensorproSductsandcompositionofmorphisms.LetAbSeanalgebrawithmrultiplicationmA 4ѹ:VAL A4!gA뙹andunituA 4ѹ:K4!gA.Then^!A 36:URD4!CNde nedbry!n9(X)=A,:!(X2 n 6K)=A2 n Dȹ,:!(m)=mA yBand^!(u)=uAisϲastrictmonoidalfunctor./IfAis nitedimensionalthenthediagramis nite.WVegetRTheorem3.7.1.pLffeth.Abea nitedimensionalalgebra.UThenthealgebraM@(A)cffoactingFuniversallyfrffomtherightonA(therightTambarabialgebra)M@(A)andcoSendz(!A)35arffeisomorphicasbialgebras.Proof.@_WVeSharvestudiedtheTVambarabialgebraforleftcoactionfQ:URA4!1M@(A)' A butherewreneedtheanalogueforuniversalrightcoactionfQ:URA4!1A& M@(A) (seeProblem3.9).LetrBs׍l@ >RH7A ((BE Cܞ) M@)|:2fd+7ά-̯ h i>(A;Bd C;M")?Ǡ@fe@&Ǡ?ꃀD؜ i>(A Bd;C;M")qEjǠ@feqxǠ?ꃀ;m"1 i>(Bd;C;M") F(A B) (CF M@)>7A (BE (CF M@))v<32fd@ά-} i>(A;Bd;C M")MKqVr(A I) MArA (I+ M@)L:2fdC ά-̯ƴ i>(A;I;M")A Mꃀ(A) 1l Ql攴Ql?^QlQz̟0Qz̟0sꃀp 1 I{(M")L L攴L?^L#0#0+͍AC5-categoryiscalledstrictifthemorphisms O;Xaretheidenrtities.Leth(DUV; )and(D20#; )bSeC5-categories.AhfunctorF]9:,(D~4!3D20ntogetherwithanaturaltransformation(A;M@)UR:A F1(M)UR4!1F(A M)iscalledaweffakSC5-functorifthefollorwingdiagramscommuteDT(A B) F1(M@)9F1((A B) M@){:2fd{@ά-'֪K*Ǡ@feK\Ǡ?ꃀB e*Ǡ@fee\Ǡ?ꃀjlF(( i>)A (BE F1(M@))/lA F1(BE M@){32fd2$ά- Z1 9F1(A (BE M@))L32fd2$ά- (Lˍ2I+ F1(M@)2F1(I+ M@)n:2fd"€ά-'֪F1(M@)uG}?`@?`@?`@Ɩ@ƖRꃀF((I{)?`?`?` QIf,ʬinaddition,߹isanisomorphismthenwrecallFaC5-functor.RThefunctoriscalledastrict35C5-functorifҩistheidenrtitymorphism.kZ7 &ev8.pRECONSTR9UCTION!AND1 cmsy9Cf-CA:TEGORIESeX107YAonaturaltransformation'q:FL4!F120 %bSetrween(weak)C5-functorsiscalledaC5-trffansformation꨹ifHA F120J(M@)F120J(A M@)>32fdά-h2Qq-:0HNA F1(M@)H@F1(A M@){fdά-iةH:Ǡ*FfelǠ?`1X.A ^ '(M")H :Ǡ*Ffe lǠ?`y'(A M")commrutes.ٍExample3.8.2.oQLetpCybSeacoalgebraandC:=URVec.xThenthecategoryComo`d,p-0Cof~righrtCܞ-comoSdulesisaC5-categorysinceN62URComo`d-R-1C}andV2URC=Vec0zimpliesthatVG N+isacomoSduleswiththecomodulestructureofN@.TheQunderlyingfunctor!Ë:URComo`d-R-1C14!Vec.ʹisastrictC5-functorsincewrehaveV% !n9(N@)3=!(V% N@).ASimilarlyms!q Mtֹ:3Como`d.-2~C4!^Vec1,isaC5-functorsinceVG (!n9(N@) M)PUR԰n:=!n9(VG N) M.Lemma3.8.3.g5QLffetCz)beacoalgebra. jLet!b:KComo`d0TK-4C!+Vec6betheun-derlyingKBfunctor.Lffet'[:!%!~!R zM&beKBanaturaltransformation.Then'isaC5-trffansformation35withC=URVec.Proof.@_Itcsucestoshorw1V ~) 'a'(N@)UR='(V N@)cforanarbitrarycomoSduleN.WVeshorwthatthediagramNzv{VG NVG N M32fd2Bά-W`1X.Vn '(N")Y{VG NYVG N M{fd2Bά-`'(V N")HjǠ*Ffe'Ǡ?8I51HǠ*FfeܟǠ?8I \1m⍹commrutes.ZuLet(vidڹ)bSeabasisofVp.FVoranarbitraryvrectorspaceWletpi::h`VN GW4!nWxbSetheprojectionsde nedbrypidڹ(t)UR=pi(P jvjv lwjf )=wi;ϹwherePjvj lwjNo0N M:,32fdT0ά-W`'(N")Y[VG NYVG N M/{fd>ά-`'(V N")HJǠ*FfeH|Ǡ?'"-p8:iH ʟǠ*Ffe Ǡ?i|p8:i,r Mlky7 &e108$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYcommrute. oExpressedinformulasthismeans'(N@)pidڹ(t)F=pi'(Vy  N@)(t)foralltUR2VG N@.8Hencewrehave0ƍʍ2e(1V p '(N@))(t)UR=(1V '(N@))(Pvi pidڹ(t))UR=Pvi '(N@)pidڹ(t)p=URPidvi pid'(VG N@)(t)UR='(V N@)(t)0ƍSowrehave1V p '(N@)UR='(VG N@)asclaimed.լcffxff ̟ff ̎ ̄cff[WVe)prorvethefollowingTheoremonlyforthecategoryC$=qjVec=ofvectorspaces.The`Theoremholdsingeneralandsarysthatinanarbitrarysymmetricmonoidalcategory]CthecoalgebraC:orepresenrtsthefunctorC5-Natބ(!n9;!gI M@)PW԰2?=Mor&C,+(C5;M)ofnaturalC5-transformations.*Theorem3.8.4.p޹(Reconstructionofcoalgebras)(LffetCbeacoalgebra.GLet!:Como`d*`-.5C1!Vec/[bffe35theunderlyingfunctor.fiThenCP1԰Jع=ܙcoSend.(!n9).Proof.@_LetM,iinVecandlet'Vʹ:!4!5Z!w >MbSeanaturaltransformation.;vWVede ne(thehomomorphismAe':URC14!M bryAe'=UR(Y 1)'(Cܞ)(usingthefactthatCƹisacomoSdule.LetNbSeaCܞ-comodule.*ThenNisasubcomoduleofN U-CmbryȄ:URN64!N U-CsincethediagramFN CN CF C<32fd(eά-a^L 1YaNY~N Cܟ{fd@Z@ά--(HǠ*Ffe,Ǡ?k}HǠ*Ffe,Ǡ?o}1 Çcommrutes.8Thusthefollorwingdiagramcommutes+`ΉN+E`N CnIJfdWά-K퍒ՐYS)N MYN CF M|D{fd<@ά- AgRL 1He"`*Ffef#T`? M'(N")H^`*Ffeۑԟ`? ^'(N" C)=1X.N2 '(C)H>N M8IU1ԟ Xԟҁ Xԟ Xԟׁ Xԟ Xԟ܁ Xԟ Xԟ Xԟ Xԟ Xԟ Xԟ Xԟ X ԟ Xԟ Xԟ X&ԟdžX&ԟdžzH͝t1  1ҁ Hׁ H܁ H  H H H( H2 H6džH6džje21 e'4@4@4@4@ 4@4@4@)4@34@=4@>P@>PR0ƍInparticularwrehaveshownthatthediagramJzY!Yݲ! C{fd - ά--(H'[' ׁ @ @ @ @Q>@Q>RH! MV2Ǡ*FfedǠ?ҍ;1 e'commrutes.mz7 &ev8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESeX109YTVo-shorwtheuniquenessofe'йletgr:G9C#4!MPbSeanotherhomomorphismwith(1 gn9)+=n'. mFVorQc2C.Mwrehavegn9(c)n=g( 1)(c)n=( 1)(1 g)(c)n=( 1)'(Cܞ)(c)UR=e' ȹ(c).ThecoalgebrastructurefromCorollary3.5.1istheoriginalcoalgebrastructureof&TCܞ.ThiscanbSeseenasfollorws.Thecomrultiplication.:!)4!!A ICisanaturaltransformationn hence(w\ *1C)J:5!Q4!!rc C CJisn alsoanaturaltransformation.As^inCorollary3.5.1thisinducedauniquehomomorphism1:Cb4!C 4Cso^thatthediagramQUkfe! coSend ^"(!n9)! coSend ^"(!n9) coSend(!n9)32fdά-aL 1H2!H|! coSend ^"(!n9)-{fdSά--0HǠ*FfeԟǠ?k}#H bǠ*Ffe Ǡ?o1  ݍcommrutes..JInasimilarwaythenaturalisomorphism!P԰=.! tKinducesauniquehomomorphismUR:C14!KsothatthediagramV;ЍH@!HW! coSend ^"(!n9)ht{fd 0ά--0H}id]!Xׁ @X @X @X @ԟt>@ԟt>RHؒ! I椢Ǡ*FfeԟǠ?͝T1 ocommrutes. BecausewoftheuniquenessthesemustbSethestructurehomomorphismsofCܞ.6/cffxff ̟ff ̎ ̄cff"WVeneedamoregeneralvrersionofthisTheoreminthenextchapter.'SoletC'bSeaUncoalgebra."Let!Ë:URComo`d-R-1C14!Vec.0bSetheunderlyingfunctorandȄ:UR!4!2j! yCtheunivrersalnaturaltransformationforCP1԰Jع=ܙcoSend.(!n9).WVeusethepSermrutationmaponthetensorproductthatgivresthenaturalisomorphismEo:URN1j T1 N2 T2 ::: NnR TnP԰ =KN1 N2 ::: NnR T1 T2:::5J TnwhicrhGisuniquelydeterminedbythecoherencetheoremsandisconstructedbysuitableapplicationsofthe ipo:URN TP԰=TLn N@.Letǀ!n92n :Como`d0-4˟CPComo`d.OP-29CP:::@PComo`d.OP-C]4!Vec6CbSeǀthefunctor!n92n(N1;N2;:::ʚ;NnP)=!n9(N1) !(N2) :::px !(NnP).MFVor%notationalconrvenience%weabbreviatefN@g2n :=.N1' gN2 ::: NnP,F>similarlyfCܞg2n =.CD C ::: CWandffGg2n:=URf1j f2 ::: fnP.8Sowregeto:URfN Tg2nP԰ =KfN@g2nR fTg2nP:n$7 &e110$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYLemma3.8.5.g5QLffetO':!n92n !!n92nE MbeOanaturaltransformation.Then'isaC5-trffansformation35inthesensethatthediagramsDVifVpg2nR fN@g2n``fVpg2nR fN@g2n Mq\32fdU}ά-W`fVg-:n7 '(Nq1*;:::;Nn)_fVG N@g2n4fVG N@g2nR M:2fdgά-̯'(Vq1* Nq1;:::;Vn7 Nn)yǠ@feyܟǠ?@َ>RHm!n92n1 MBǠ*FfetǠ?ҍx1 e'4TVoshorwtheuniquenessof?)e'3letgr:u9Cܞ2n '4!!M~bSeanotherhomomorphismwith(1!I{nL gn9)s22(n)&=n'.WVe&Qharvegֹ=ngn9("2n ) 1Cn =Ϲ)W2n =g("2n ) 1Cn =Ϲ)s22(n) (C5;:::ʚ;Cܞ)=Z΍("2nR 1M )(1Cnw gn9)s22(n) (C5;:::ʚ;Cܞ)UR=("2n 1M )'(C5;:::ʚ;Cܞ)=e' ȹ.acffxff ̟ff ̎ ̄cffKԍNorwweprovethe nitedimensionalcaseofreconstructionofcoalgebras.ڍProp`osition3.8.7.O(Reconstruction)֔LffetC2beacoalgebra.PLetComo`d/603-8)C2betheOcffategoryof nitedimensionalCܞ-comodulesand!0:QComo`d/"Q03U-8C!ǭVec2betheunderlying35functor.fiThenwehaveCP1԰Jع=ܙcoSend.(!n9).Proof.@_Let5MvbSeinVecandlet':!C14!1!K ݷMbSe5anaturaltransformation.WVebde nethehomomorphisme':_C4!Masfollorws. 5Letc2Cܞ. 5LetNbSea nite)MdimensionalCܞ-subScomodule)MofCconrtainingc.Thenwede negn9(c):=(jN 1)'(N@)(c).If?N20\isanother nitedimensionalsubScomoduleofCݹwithc2N@20\andwithN6URN@20ŹthenthefollorwingcommutesB7ԠNԠhN Ml:2fdzά-̯'(N")Ǡ@fe׼Ǡ?[ʟǠ@feǠ?KN@20K.LN@20 M,32fd ά-*'(N"-:0)胀@?`Hά-c퍒  1Thrusλthede nitionofMJe' 1(c)isindepSendentofthechoiceofN@.FVurthermoreMJe'Z:هN4!}[M`ܹisobrviouslyalinearmap.FVoranytwoelementsc;c2022cCthereisa nitedimensionalusubScomoduleN6URCwithc;c20#2NYe.g.thesumofthe nitedimensionalsubScomodulesconrtainingcandc20separatelyV.8Thusi7e'ƹcanbSeextendedtoallofCܞ.p7 &e112$3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYYTheBrrestoftheproSofisessenrtiallythesameastheproofofthe rstreconstructiontheorem.|cffxff ̟ff ̎ ̄cffFTherepresenrtationsallowtoreconstructfurtherstructureofthecoalgebra.EWVeprorve|areconstructiontheoremabSoutbialgebras. RecallthatthecategoryofB-comoSdulesorverabialgebraB͹isamonoidalcategoryV,:furthermorethattheunderlyingfunctor)! :^Como`d-^-1B:d4!OVec/D"isamonoidalfunctor.bFVromthisinformationwrecanreconstructthefullbialgebrastructureofB.8WVeharvepTheorem3.8.8.pLffetBbeacoalgebra.LetComo`d/g-3Bbeamonoidalcategorysuch thattheunderlyingfunctor!Ë:URComo`d-R-1BX!Vec.uisamonoidalfunctor.XThentherffelisauniquebialgebrastructureonBthatinducesthegivenmonoidalstructureon35thecfforepresentations.Proof.@_Firstwwreprovetheuniquenessofthemultiplicationr\$:B5 z/B*4!lBandKcoftheunith0:K4! {B.[TheKcnaturaltransformationm):!4!{!Z BibSecomesamonoidaljnaturaltransformationwithrUR:B@ BX4!_7BandjË:K4!1BWVejshorwthatr꨹andXareuniquelydeterminedbry!ands2.Letr20<:B 9BV 4!*BWӹandn920u:BV 4!*KbSemorphismsthatmakre/amonoidalnaturaltransformation.8ThediagramsVH@!n9(X) !(Yp)Hq!n9(X) !(Yp) BE B{fdoˀά-`lL(X) (Y)HbzǠ*FfecǠ?'YH63Ǡ*Ffe6g,Ǡ?R; r-:0I0&!n9(X+ Yp)w>!n9(X+ Yp) BbL32fdЍά-=L(X Y)tandM덍⍒K⍒JRK K̟{fdY?ά-‚.⍍PP=HSJǠ*Ffe|Ǡ?HʟǠ*FfeǠ? |1 n920jj!n9(K)ᅢ!n9(K) BL32fdH8pά-L(K)hAcommrute.8InparticularthefollowingdiagramscommuteVHAFd!n9(B) !(B)Ht!n9(B) !(B) BE B@|{fdpά-`L(Bd) (B)HbzǠ*FfecǠ?'YH63Ǡ*Ffe6g,Ǡ?R; r-:0I!n9(BE B)!n9(BE B) B32fdpά-)"L(Bd B)q=7 &ev8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESeX113YandS@⍒K⍒JRK K̟{fdY?ά-‚.⍍PP=HSJǠ*Ffe|Ǡ?HʟǠ*FfeǠ? |1 n920jj!n9(K)ᅢ!n9(K) BL32fdH8pά-L(K)"GHence uwregetP b(1) j#  c(1) b(2) \|c(2)ι=URPb(1) c(1) r209(b(2) c(2) \|) uand1 1UR=1 n920l"Π@feTΠ?ۏ1 "Ǡ@feTǠ?ɞ⍍=sa"ΠM@feaTΠ?ɻfnԍandh50"v2K0"2KA2fd`ά-Y֯Z1ԩ" IK BZ\L(K)NF`HNF`HNF`HNF`HNF`HNF`H^H^j81 @Π@Π@Π@Π@Π@ΠPFPF*ǍF`@F`@F`@F`@F`@F`@F`@Ɩ`@Ɩ`Rҿ㍒<ǠǠǠǠ ǠǠǠ$G$G HB؜*Ǡ@fe\Ǡ?ɞ⍍݁݁=nԍHenceXandrarecoalgebrahomomorphisms.s㺠7 &ev8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESeX115YTVo}shorwtheassoSciativityofrweidentifyalongthemaps c:O(MO N@) PP԰ =Ml (N Pƹ)^andfurthermoresimplifytherelevXanrtdiagramby xingthatIrepresentsasuitablepSermrutationofthetensorfactors.8Thenthefollowingcommute+BE B B+)BE B B B B BNIJfdV0ά-: U9jI{(L(Bd) (B) (B)++fBE B B/LIJfd3pά-q퍒7   1YBE B BY)BE B B B B B\y\yNfdV0ά-kX\I{(L(Bd B) (B))h9h9N҄fdV0ά-&X\I{(L(Bd) (B B))YYr8BE B/L{fd?ά-)-=    1BE B B\rBE B B BN32fdnά-p7L(Bd B B)~QBl32fdc0ά-m7   1H.U`*Ffe.`? 3;<1H.UǠ*Ffe.Ǡ?8I3;<1H泪`*Ffeܟ`? ɐ1 (r 1)Hj`*Ffe򜟽`? 1 (1 r)H\ꟽ`*Ffe`?>/hr 1Hh`*Ffeܟ`?>/N\1 rH9Ǡ*FfelǠ?o<1 rHʟǠ*FfeǠ?`|reTheuppSerrorwistheidentityhencewegettheassoSciativelaw.FVor39theproSofthatrhasthepropertiesofaunitwremustexplicitlyconsiderthecoherence morphismsandByreasonsofsymmetrywrewillonlyshowonehalfofoftheunitaxiom.]ThisaxiomfollorwsfromthecommutativityofthefollowingdiagramYv#Bv1BE B psH2fdoЍά-n!L(Bd)v"v"kyBE B KV1sH2fdـά-o0ōW-:1v"v"VBE BsH2fdά-o67vvB{ sH2fd/pά-omv 1"#BE K"__BE B K B'ПA2fd4 ά--L(Bd) (K)"".BE K B B0A2fd@ά-mA1 r 1""J~mBE B B*(A2fd`ά-7͍+ 1 1tBE BސA2fdά-m 1 1ԩ"#BE Kԩ"BE K B'П:2fd_ά-̯stL(Bd K)#BVBE B p32fd8ά-VAL(Bd)B{ 32fd/pά-mv 1sޟՠ@feՠ?"}-:1s#^ՠ@feVՠ? 1 1 L(K)sgBՠ@feguPՠ?^MH|p1 1 sՠ@feՠ?^M1 ޟΠ@feΠ? =ޟΠ@feΠ?ۏw1 1 rgBǠM@feguPǠ?XP1 rǠM@feǠ?»rޟǠ@feǠ?uG =s)vL(Bd) 1/ՠ9ՠCՠMՠWՠaՠb'Mb'M*PM:!0 1?`H?`H(?`H2?`H<?`HF?`HG%pHG%pje %cffxff ̟ff ̎ ̄cff;7 /1 cmsy9-%n eufm10,- cmcsc10+@ cmti12*o cmr9(ppmsbm8' msbm10$ msam10"a6cmex8!u cmex10 q% cmsy6K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8N cmbx12Nff cmbx12XQ cmr12O line10