Milnor-Moore categories and monadic decomposition

Contents lists available atScienceDirect

Journal

of

Algebra

www.elsevier.com/locate/jalgebra

Milnor–Moore

categories

and

monadic

decomposition

Alessandro Ardizzonia,∗_{, Claudia Menini}b

a _{UniversityofTurin,DepartmentofMathematics“G.Peano”,viaCarloAlberto}

10,I-10123Torino,Italy

b

Universityof Ferrara,DepartmentofMathematics,ViaMachiavelli35,Ferrara, I-44121,Italy

a r t i c l e i n f o a bs t r a c t

Article history:

Received29January2015 Availableonline3December2015 CommunicatedbyNicolás Andruskiewitsch MSC: primary18C15 secondary17B75 Keywords: Monads Milnor–Moorecategory GeneralizedLiealgebras

InthispapermonoidalHom-Liealgebras,Liecoloralgebras, LiesuperalgebrasandothertypeofgeneralizedLiealgebras are recovered by meansof an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg–Moore categories. To this aim we introduce the notionofMilnor–Moorecategoryasamonoidalcategoryfor which a Milnor–Moore type Theorem holds. We also show how to liftthe propertyof being a Milnor–Moore category wheneverasuitablemonoidalfunctorisgivenandweapply thistechniquetoprovideexamples.

© 2015ElsevierInc.All rights reserved.

Contents

Introduction . . . 489

✩ _{ThispaperwaswrittenwhilebothauthorsweremembersofGNSAGA.Thefirstauthorwaspartially} supported by the research grant “Progetti di Eccellenza 2011/2012” from the “Fondazione Cassa di RisparmiodiPadovaeRovigo”.

* Correspondingauthor.

E-mailaddresses:alessandro.ardizzoni@unito.it(A. Ardizzoni),men@unife.it(C. Menini).

URLs:http://sites.google.com/site/aleardizzonihome(A. Ardizzoni),

http://sites.google.com/a/unife.it/claudia-menini(C. Menini).

http://dx.doi.org/10.1016/j.jalgebra.2015.09.031

1. Preliminaries . . . 493

2. Commutationdata . . . 497

3. Braidedobjectsandadjunctions . . . 502

4. Braidedcategories . . . 509

5. Liealgebras . . . 513

6. Adjunctionsforenvelopingfunctors . . . 521

7. Stationarymonadicdecomposition . . . 526

8. LiftingthestructureofMM-category . . . 536

9. ExamplesofMM-categories . . . 541

9.1. Quasi-bialgebras . . . 541

9.2. Dualquasi-bialgebras . . . 545

Acknowledgments . . . 550

Appendix A. (Co)equalizersand(co)monadicity . . . 550

Appendix B. Braided(co)equalizers . . . 554

References . . . 562

Introduction

ThecelebratedMilnor–MooreTheorem[32,Theorem5.18]establishes,in characteris-ticzero,anequivalencebetweenthecategoryofprimitivelygeneratedbraidedbialgebras andthe category ofLie algebras.Thefunctors givingtheequivalence arethe universal enveloping algebra functor U, associating the universal enveloping algebra U (L) to a Lie algebra L, and the primitive functor P which gives the primitive part P (B) of a given bialgebraB. The factthat thecounit UP → Id oftheadjunction involvedis an isomorphismjustencodesthefactthatthebialgebras consideredareprimitively gener-ated.Ontheotherhandthecrucial pointintheproof ofthetheorem isthatthemaps

ηL : L → P (U (L)) giving the unit of the adjunction (U, P) are isomorphisms. Now observethatthetensoralgebraT (V ),definedforanyvectorspaceV ,yieldsafunctorT

fromthecategoryofvectorspacestothecategoryofbialgebraswhichisaleftadjointof thefunctorP obtainedfromP forgettingtheLiealgebrastructure.Inthiscasetheunit

V → P (T (V )) failstobe anisomorphismingeneral.NotealsothatU (L) isaquotient ofT (L).Thuswecouldsaythat(U, P) isarefinementoftheadjunction(T, P ) obtained byrestrictingthecodomainofP andchangingtheleftadjointinordertoobtainanew adjunctionwith invertibleunit.

ConsideringthewidercontextofamonoidalcategoryM,bialgebrasandLiealgebras aresubstitutedbytheirsymmetricbraided analogue(forinstance abraidedsymmetric bialgebra inM isan objectequipped with an algebrastructure,acoalgebra structure and asymmetric Yang–Baxter operatorsatisfyingthe expectedcompatibilityaxioms); thesamehappensfortheprimitivefunctorandtheenvelopingfunctor.Suchacategory

M iscalledMilnor–Moore(MM)exactlywhentheunitofthisadjunctionisan isomor-phism. In this case, the category of symmetric braided Lie algebras can be described bytheso-calledmonadicdecompositionoftheprimitivefunctor(seebelow).Whenthe category M isalsosymmetric, then wecanconsider bialgebrasand Lie algebras inM as inthe caseofvector spaces.Inthis casewe provethatif M isaMM-category then the unit η : Id → PU of the corresponding adjunction (U, P) is an isomorphism too

and thatthe category of Lie algebras canbe recovered from the starting category M by means of the same iterated procedure. For this reason, the remaining part of our investigationfocusesongivingexamplesofMM-categories.Thefirstexampleisthe cat-egory M of vector spaces in characteristic zero. Then, under mild conditions, we find that amonoidalcategory M endowed withaconservative and exactmonoidalfunctor

M→ M preservingdenumerablecoproductsisstillMM.Asaconsequencewecanprove thatmonoidalHom-Liealgebras,Liecoloralgebras,Liesuperalgebrasandothertypeof generalizedLiealgebras arerecoveredbymeansofthesameiteratedconstructionbased onEilenberg–Moorecategories.

Inordertoexplainourresultsmoreprecisely,weneednowtoenterintothetechnical details of our setting. Let (L :B → A, R : A → B) be an adjunction with unit η and

counit .Then RL isamonad onB (withmultiplication RL andunitη) andonecan consider theEilenberg–Moorecategory RLB associatedto thismonadand theso-called

comparison functor K : A→RLB which is definedby KX := (RX, RX) and Kf :=

Rf .Thisgivesthediagram

A R A K IdA B L RLB RLU

where theundashedpartcommutes.InthecasewhenK itselfhasaleftadjointΛ,one canrepeatthisconstructionstartingfromthenewadjunction(Λ,K).Goingonthisway onepossiblyobtainsadiagramoftheform

A R0 A R1 IdA A R2 IdA . . . IdA B0 L0 B1 L1 U0,1 B2 L2 U1,2 . . . U2,3

whereitismoreconvenienttorelabel(L,R) and(Λ,K) as(L0,R0) and(L1,R1) respec-tively. If there isa minimalN ∈ N such thatLN isfull and faithful,then R is saidto

have monadic decomposition of monadic length N .This is equivalent to requiring that theforgetfulfunctorUN,N +1 isacategoryisomorphismandnoUn,n+1hasthisproperty

for 0≤ n≤ N − 1 (see e.g.[4,Remark 2.4]). In[4, Theorem 3.4], we investigated the particular case BialgM P BialgM P1 Id_BialgM BialgM P2 Id_BialgM M T M1 U0,1 T1 M2 U1,2 T2

where M denotesthe category of vectorspaces overafixed base fieldk,BialgM is the categoryofk-bialgebras,T isthetensorbialgebrafunctor(thebarrednotationservesto

distinguishthisfunctorfrom thetensoralgebrafunctor T : M→ AlgM whichgoesinto k-algebras) and P is theprimitive functor which assigns to eachk-bialgebra its space ofprimitive elements.WeprovedthatthisP hasamonadicdecompositionof monadic lengthatmost2.Moreover,whenchar (k) = 0,foreveryV2= ((V, μ) , μ1)∈ M2onecan define[x, y] := μ(xy− yx) foreveryx,y∈ V .Then(V, [−, −]) isanordinaryLiealgebra and T2V2 = T V /(xy− yx − [x, y] | x, y ∈ V ) is the corresponding universalenveloping algebra. This suggests a connection between the category M2 and the category LieM ofLiek-algebras. Itisthen naturalto expecttheexistenceof acategory equivalenceΛ suchthatthefollowingdiagram

Bialg_M P Bialg_M P1 Id_BialgM Bialg_M P2 Id_BialgM Id_BialgM Bialg_M P Id_BialgM M T M1 T1 U0,1 M2 T2 U1,2 Λ LieM U HLie

commutesinitsundashedpart,whereHLiedenotestheforgetfulfunctor,U theuniversal envelopingbialgebrafunctorandP thecorresponding primitivefunctor.

A first investigation showed that, in order to solve the problem above, it is more naturaltoworkwithbraidedk-vectorspacesinsteadofordinaryk-vectorspacesandto replacethecategoriesM,Bialg_MandLieMwiththeirbraidedanaloguesBrM,BrBialgM and BrLieM consisting of braided vector spaces, braided bialgebras and braided Lie algebras respectively.We werefurtherled to substitute M with anarbitrary monoidal categoryM.Wepointoutthat,inordertoproduceabraidedanalogueoftheuniversal envelopingalgebrawhich furthercarriesa braidedbialgebra structure,the assumption thattheunderlying Yang–Baxter operatorissymmetric is alsoneeded. Thus letBrs_M, BrBialgs_M and BrLies_M be the analogue of Br_M, BrBialg_M and BrLie_M consisting of objects with symmetric Yang–Baxter operator. Let Ts

Br : BrsM → BrBialgsM be the symmetricbraidedtensorbialgebrafunctorandletP_Brs beitsrightadjoint,theprimitive functor.WelookforaconditionforPs

Brtohavemonadicdecompositionofmonadiclength at mosttwo. Onthe other handthe functor Ps

Br induces afunctor PBrs : BrBialg

s

M →

BrLies_M which turns outto havea left adjointUs

Br, theuniversalenveloping bialgebra functor.

In view of the celebrated Milnor–Moore Theorem, see Remark 7.5, we say that a category M isaMilnor–Moorecategory (MM-categoryforshort) whenevertheunitof the adjunction (Us

Br,PBrs ) is a functorial isomorphism (plus other conditions required for the existence of the functors involved). One of the main results in the paper is

decomposition of monadic length at most two. Moreover, inthis case, we can identify thecategory (Brs_M)₂ withBrLies_M throughanequivalenceΛBr: (Brs_M)2→ BrLie

s M. BrBialgs_M Ps Br BrBialgs_M (PBrs)1 IdBrBialgs_M BrBialgs_M (PBrs)2 IdBrBialgs_M IdBrBialgs_M BrBialgs_M Ps Br IdBrBialgs_M Brs_M Ts Br (Brs_M)1 (Ts Br)1 U0,1 (Brs_M)2 (Ts Br)2 U1,2 ΛBr BrLies_M Us Br HBrLies

Hence MM-categories,besides having aninterest intheirown, give us anenvironment where thefunctorP_Brs hasabehavior completelyanalogoustotheclassicalvectorspace situationwe investigatedin[4, Theorem3.4].Inthe caseofasymmetric MM-category

M the connection with Milnor–MooreTheorem becomesmoreevident. Infact,inthis case, we canapply Theorem 7.2 to obtain that the unitof the adjunction U, P is a functorialisomorphism. Bialg_M P Bialg_M P1 Id_BialgM Bialg_M P2 Id_BialgM Id_BialgM Bialg_M P Id_BialgM M T M1 T1 U0,1 M2 T2 U1,2 Λ Lie_M U HLie

The next step is to provide meaningful examples of MM-categories. A first result in this direction is Theorem 8.1, based on a result by Kharchenko, which states that the categoryM of vectorspacesover afieldof characteristic0 isaMM-category.Note that the Lie algebras involved are notordinary ones but theydepend on a symmetric Yang–Baxter operator.

Muchofthematerialdevelopedinthepaper(seee.g.Proposition 3.7,Theorem 8.3and the constructionofthe adjunctionsusedtherein)is devotedto theproofof ourcentral resultnamelyTheorem 8.4whichallowsustoliftthepropertyofbeingaMM-category whenever asuitablemonoidal functorisgiven. Amain tool inthis proofis theconcept of commutation datum which we introduce and investigate in Section 2. We use this

Theorem 8.4 inthecaseoftheforgetfulfunctorF :M→ M whereM isasubcategory ofM.Thegoalistoprovide,inthisway,meaningfulexamplesofMM-categoriesM and, inthe casewhenM is symmetric, torecognize thecorresponding typeofLie algebras. A firstexampleofMM-categoryobtainedinthiswayisthecategoryofYetter–Drinfeld

modules, over aHopfalgebra over afieldof characteristic zero, which isconsidered in

Example 9.1.Subsection 9.1 (resp.9.2)dealswith thecase whenM isthe category of modules(resp.comodules)overaquasi-bialgebra(resp.overadualquasi-bialgebra).We provethattheforgetful functor satisfiesthe assumptions ofTheorem 8.4 ifand onlyif thequasi-bialgebra(resp.thedualquasi-bialgebra)isadeformationofausualbialgebra, see Lemma 9.4(resp. Lemma 9.13). Asparticular casesof this situationweprovethat thecategory H (M) of [16,Proposition 1.1]is anMM-category, seeRemark 9.10.Note thatan object inLie_M, for M= H (M), is nothingbut amonoidal Hom-Lie algebra. In Remark 9.17, we recover (H, R)-Lie algebras in the sense of [13, Definition 4.1] by consideringthecategoryofcomodulesoveraco-triangularbialgebra(H, R) regardedas aco-triangulardual quasi-bialgebrawithtrivialreassociator. Inparticular, letG be an abelian group endowed with an anti-symmetric bicharacter χ : G× G → k\ {0} and extendχ by linearityto ak-linear map R :k[G]⊗ k[G]→ k,where k[G] denotes the group algebra. Then (k [G] , R) is a co-triangular bialgebra and, as a consequence, we recover(G, χ)-Liecoloralgebras inthesenseof[33,Example10.5.14],inExample 9.18, andinparticularLiesuperalgebrasinExample 9.19.

The appendices contain general results regarding the existence of (co)equalizers in the category of (co)algebras, bialgebras and their braided analogue over a monoidal category. These results are applied to obtain Proposition B.11, which is used in the proofofTheorem 7.1.

1. Preliminaries

Inthissection,weshallfixsomebasicnotationandterminology.

Notation1.1.Throughoutthispaperk willdenoteafield.Allvectorspaceswillbedefined overk.Theunadornedtensorproduct⊗ willdenotethetensorproductoverk ifnotstated otherwise.

1.2. Monoidal Categories. Recall that (see [26, Chap. XI]) a monoidal category is a category M endowedwithanobject1∈ M (calledunit),afunctor⊗:M× M→ M (calledtensorproduct),andfunctorialisomorphismsaX,Y,Z : (X⊗Y )⊗Z → X ⊗(Y ⊗Z),

lX : 1⊗X → X,rX : X⊗1→ X,foreveryX,Y ,Z inM.Thefunctorialmorphisma is

calledtheassociativityconstraint andsatisfiesthePentagonAxiom,thatistheequality

(U⊗ aV,W,X)◦ aU,V⊗W,X◦ (aU,V,W ⊗ X) = aU,V,W⊗X◦ aU⊗V,W,X

holds true, for every U , V , W , X in M. The morphisms l and r are called the unit constraints andtheyobeytheTriangle Axiom,thatis(V ⊗ lW)◦ aV,1,W = rV ⊗ W ,for

everyV , W inM.

Amonoidalfunctor (alsocalled strongmonoidalintheliterature)

between two monoidal categories consists of a functor F : M → M, an isomorphism

φ2(U,V ) : F (U )⊗ F (V ) → F (U ⊗ V ), natural in U,V ∈ M, and an isomorphism

φ0: 1 → F (1) suchthatthediagram

(F (U )⊗F (V ))⊗F (W ) aF (U ),F (V ),F (W ) φ2(U,V )⊗F (W ) F (U⊗ V ) ⊗F (W ) φ2(U⊗V,W ) F ((U⊗ V ) ⊗ W ) F (aU,V,W) F (U )⊗(F (V )⊗F (W ))F (U )⊗ _φ 2(V,W ) F (U )⊗F (V ⊗ W ) φ2(U,V⊗W ) F (U⊗ (V ⊗ W ))

is commutative,and thefollowing conditionsare satisfied:

F (lU)◦ φ2(1, U )◦ (φ0⊗F (U )) = lF (U ), F (rU)◦ φ2(U, 1)◦ (F (U)⊗φ0) = rF (U ).

Themonoidalfunctor iscalledstrict iftheisomorphismsφ0,φ2 areidentitiesofM. Thenotionsofalgebra,moduleoveranalgebra,coalgebraandcomoduleovera coal-gebracanbeintroducedinthegeneralsetting ofmonoidalcategories.

As it is noticedin [28, p.420], thePentagon Axiom solves theconsistency problem that appearsbecausethere aretwo ways to go from ((U⊗ V )⊗ W )⊗ X toU ⊗ (V ⊗

(W ⊗ X)). The coherence theorem,due to S. Mac Lane [31, Chapter VII, Section 2], solves the similar problem for the tensor product of an arbitrary number of objects inM.Accordinglywiththistheorem,wecanalwaysomitallbracketsandsimplywrite

X1⊗ · · · ⊗ Xn foranyobjectobtainedfromX1,. . . ,Xn byusing⊗ andbrackets.Alsoas

aconsequenceofthecoherence theorem,themorphismsa,l,r takecareofthemselves, sotheycanbeomittedinanycomputationinvolvingmorphismsinM.Thus,forsakeof simplicity,fromnowonwewillomittheassociativityandunit constraintsunlessneeded to clarifythecontext.

Let V bean objectinamonoidal category (M, ⊗, 1).Define iteratively V⊗n for all

n∈ N bysettingV⊗0:= 1 forn= 0 andV⊗n:= V⊗(n−1)⊗ V forn> 0.

Remark1.3. LetM beamonoidal category.Denote byAlg_M thecategory ofalgebras inM and theirmorphisms. AssumethatM hasdenumerable coproductsand thatthe tensor products (i.e.M⊗ (−) : M→ M and (−) ⊗ M : M→ M,for everyobjectM

inM)preservesuchcoproducts.By[31, Theorem2,page172], theforgetfulfunctor Ω : Alg_M→ M

hasaleftadjointT :M→ Alg_M.ByconstructionΩT V =⊕n∈NV⊗n foreveryV ∈ M.

Foreveryn∈ N,wewill denoteby

thecanonicalinjection.Givenamorphismf : V → W inM,wehavethatT f isuniquely determinedbythefollowingequality

ΩT f◦ αnV = αnW ◦ f⊗n, for every n∈ N. (1)

ThemultiplicationmΩT V andtheunituΩT V areuniquelydeterminedby

mΩT V ◦ (αmV ⊗ αnV ) = αm+nV, for every m, n∈ N, (2)

uΩT V = α0V. (3)

Notethat(2)shouldbeintegratedwiththeproperunitconstrainswhenm orn iszero. Theunitη andthecounit oftheadjunction(T, Ω) are uniquelydetermined,forall

V ∈ M and(A, mA, uA)∈ Alg_M bythefollowingequalities

ηV := α1V and Ω (A, mA, uA)◦ αnA := mnA−1 for every n∈ N (4)

wheremn_A−1: A⊗n→ A istheiteratedmultiplicationofA definedbym−1_A := uA,m0A:=

IdAand, forn≥ 2,mn−1A = mA(mn−2A ⊗ A).

Definition1.4.Recallthatamonad onacategoryA isatripleQ:= (Q, m, u),whereQ:

A→ A isafunctor,m: QQ→ Q andu:A→ Q arefunctorialmorphismssatisfyingthe associativityandtheunitalityconditionsm◦mQ= m◦Qm andm◦Qu= IdQ = m◦uQ.

Analgebra over amonad Q onA (orsimplyaQ-algebra)isapair(X, μ) where X∈ A

andμ: QX→ X isamorphisminA suchthatμ◦ Qμ= μ◦ mX andμ◦ uX = IdX.A

morphismbetween twoQ-algebras (X, μ) and (X, μ) is amorphismf : X → X inA suchthatμ◦ Qf = f ◦ μ. Wewill denote by_QA the category of Q-algebrasand their morphisms. This is the so-called Eilenberg–Moore category of the monad Q (which is sometimesalso denoted byAQ inthe literature).When themultiplication and unit of themonadareclearfromthecontext,wewilljustwriteQ insteadofQ.

Amonad Q onA gives risetoan adjunction(F, U ) := (_QF,_QU ) where U :_QA → A

istheforgetfulfunctorandF :A→_QA isthefreefunctor. Explicitly:

U (X, μ) := X, U f := f and F X := (QX, mX) , F f := Qf.

Note that U F = Q. The unit of the adjunction (F, U ) is given by the unit u : A →

U F = Q of the monad Q. The counit λ : F U → QA of this adjunction is uniquely determined by the equality U (λ (X, μ)) = μ for every (X, μ) ∈ _QA. It is well-known thattheforgetfulfunctorU :_QA→ A isfaithfulandreflectsisomorphisms(seee.g.[12, Proposition4.1.4]).

Let (L :B → A, R : A → B) be an adjunction with unit η and counit . Then (RL, RL, η) is a monad on B and we can consider the so-called comparison functor

K : A → RLB of the adjunction (L, R) which is defined by KX := (RX, RX) and

Kf := Rf .NotethatRLU◦ K = R.

Definition 1.5. An adjunction (L : B → A,R : A → B) is called monadic (tripleable

in Beck’sterminology [10,Definition 3,page 8])whenever thecomparison functor K : A→RLB isanequivalence ofcategories. A functorR is called monadic ifithasaleft

adjointL suchthattheadjunction(L,R) ismonadic,see[10,Definition3’,page8].Ina similar wayonedefines comonadic adjunctionsand functorsusing theEilenberg–Moore category LR_{A ofcoalgebrasoverthecomonadinduced by(L, R).}

Thenotionofanidempotentmonad istightlyconnectedwith themonadiclengthof afunctor.

Definition 1.6.(See[8,page231].)A monad(Q, m, u) iscalled idempotent wheneverm

is an isomorphism. Anadjunction (L, R) iscalled idempotent whenever the associated monad isidempotent.

The interestedreader canfindresultson idempotentmonads in[8,34]. Herewe just note that(L, R) isidempotent ifand onlyifηR isafunctorialisomorphism.

Definition 1.7. (See[4, Definition 2.7], [5, Definition 2.1] and [34, Definitions 2.10and 2.14].)FixaN ∈ N. WesaythatafunctorR has amonadic decompositionof monadic length N whenever thereexistsasequence(Rn)_n≤N offunctors Rn suchthat

1) R0= R;

2) for0≤ n≤ N,thefunctorRn hasaleftadjointfunctorLn;

3) for 0≤ n ≤ N − 1, the functor Rn+1 is the comparison functor induced by the

adjunction(Ln, Rn) withrespect toitsassociated monad;

4) LN isfull andfaithfulwhile Ln isnotfullandfaithfulfor0≤ n≤ N − 1.

Compare withtheconstructionperformedin[29,1.5.5,page49].

Note that for functor R : A → B having a monadic decomposition of monadic length N ,wehaveadiagram

A R0 A R1 IdA A R2 IdA · · · · IdA A RN IdA B0 L0 B1 L1 U0,1 B2 L2 U1,2 · · · · U2,3 BN LN UN−1,N (5) where B0=B and,for1≤ n≤ N,

• Bn isthecategoryof(Rn−1Ln−1)-algebrasRn−1Ln−1Bn−1;

Wewill denote by ηn : IdBn → RnLn and n : LnRn → IdA the unitand counit of the adjunction(Ln, Rn) respectivelyfor 0≤ n ≤ N. Note thatone canintroduce the

forgetfulfunctorUm,n:Bn→ Bm forallm≤ n with0≤ m,n≤ N.

Proposition1.8.(See[4, Proposition 2.9].)Let(L :B → A, R : A → B) beanidempotent adjunction.ThenR :A→ B hasamonadicdecompositionofmonadiclengthatmost1. Wereferto [4, Remarks 2.8and 2.10]for furthercommentsonmonadic decomposi-tions.

Definition1.9. Wesaythatafunctor R iscomparable whenever thereexists asequence (Rn)_n∈N offunctorsRn suchthatR0= R and,forn∈ N,

1)thefunctorRn hasaleftadjointfunctorLn;

2) the functor Rn+1 is the comparison functor induced by the adjunction (Ln, Rn)

withrespectto itsassociatedmonad.

Inthis casewe have adiagram as (5) butnot necessarily stationary. Hencewe can considertheforgetfulfunctorsUm,n:Bn → Bmforallm≤ n withm,n∈ N.

Remark1.10. Fixa N ∈ N. A functor R havinga monadicdecomposition of monadic lengthN iscomparable, see[4,Remark2.10].

BytheproofofBeck’sTheorem[10,ProofofTheorem1]onegetsthefollowingresult. Lemma 1.11. Let A be a category such that, for any (reflexive) pair (f, g) [15, 3.6, page 98]wheref ,g : X→ Y aremorphismsinA,onecan chooseaspecificcoequalizer. Thenthe comparisonfunctor K :A→RLB of an adjunction (L, R) is aright adjoint.

Thusany rightadjointR :A→ B iscomparable.

LetF :A→ B beafunctor.WedenotebyIm(F ),theimageofF ,thefullsubcategory ofB whoseobjectsarethose oftheformF A forsomeA∈ A.

Lemma 1.12. Let F : C → B be a full and faithful functor which is also injective on objects.

1) Let G : A → B be a functor such that Im(G) ⊆ Im(F ). Then there is a unique functorG : A→ C suchthat F G = G.

2)LetG,G:A→ B befunctorsasin1).Foranynaturaltransformationγ : G→ G thereisaunique naturaltransformation γ : G→ G suchthat Fγ = γ.

2. Commutationdata

Lemma2.2.Let(L, R) and(L, R) beadjunctionsthatfitintothefollowingcommutative diagram of functors A F R A R B G B (6)

Then thereisauniquenatural transformationζ : LG−→ F L such that

Rζ◦ ηG = Gη (7) holds,namely ζ :=  LGL−→ LGη GRL = LRF L−→ F LF L  . (8)

Moreover wehavethat

F = F ◦ ζR. (9)

Definition 2.3.Wewillsaythat(F, G) : (L, R)→ (L, R) isacommutationdatum if

1) (L, R) and(L, R) areadjunctionsthatfitintothecommutativediagram(6). 2) Thenaturaltransformation ζ : LG−→ F L of Lemma 2.2is afunctorial isomor-phism.

Themap ζ willbecalledthecanonicaltransformation ofthedatum.

Proposition 2.4. Let (F, G) : (L, R)→ (L, R) and(F, G) : (L, R)→ (L, R) be a commutation data.Then(FF, GG) : (L, R)→ (L, R) isa commutationdatum.

InthefollowingresultwewilladoptthenotationsofDefinition 1.7forL1,R1,B1and theiranaloguewithprimes.

Proposition 2.5. Let(F, G) : (L, R) → (L, R) be a commutation datumof functors as in(6).AssumealsothatF preservescoequalizersofreflexivepairsofmorphismsinA and that thecomparisonfunctorsR₁ andR1 haveleft adjointsL1 andL1 respectively.Then

G liftstoafunctorG1:B1→ B1 suchthatG1(B, μ) := (GB, Gμ◦ RζB),G1(f ) = Gf

and thefollowingdiagramscommute.

B1 G1 U B 1 U B G B A F R1 A R1 B1 G1 B 1

Furthermorethefunctor G1 isconservative(resp.faithful)whenever G is.

IfG is faithfulthenG1 isfull (resp. injectiveon objects)whenever G is.

Proof. Denote by ζ the canonical map of the datum (F, G) : (L, R) → (L, R). Set

λ:= Rζ : (RL) G → RF L= G(RL). ByLemma 2.2, ζ fulfills (9). By(7), we have

λ◦ ηG= Gη and

GRL◦ λRL ◦ RLλ = GRL◦ RζRL◦ RLRζ = R[F L◦ ζRL ◦ LRζ]

(9)

= R[F L◦ LRζ] = R[ζ◦ LG] = λ◦ RLG

Hencewe canapply[23, Lemma1]to the case”K” = RL,”H” = RL and ”T ” = G. Thus we get a functor G1 : B1 → B1 such that UG1 = GU . Explicitly G1(B, μ) := (GB, Gμ◦ RζB),G1(f ) = Gf .Wehave

G1R1A = G1(RA, RA) = (GRA, GRA◦ RζRA) = (RF A, R[F A◦ ζRA])(9)= (RF A, RF A) = R₁F A

and G1R1f = GRf = RF f = R1F f so that G1R1 = R1F . By the proof of [10,

Theorem 1], if we set π := L1◦ LUη1, we get the following coequalizerof areflexive pairofmorphismsinA.

LRLB

Lμ LB

LB = LU (B, μ) π(B,μ) L1(B, μ)

SinceF preservescoequalizersofreflexivepairsofmorphismsinA,wegetthebottom forkinthediagrambelowisacoequalizer.

LRLGB LGμ◦R_ζB LGB ζRLB◦L_R_ζB LGB ζB F π(B,μ)◦ζB F L1(B, μ) Id_{F L1(B,μ)} F LRLB F Lμ F LB F LB F π(B,μ) F L1(B, μ) (10) Wecompute F Lμ◦ (ζRLB ◦ LRζB) = ζB◦ LGμ◦ LRζB = ζB◦ L(Gμ◦ RζB) , F LB◦ (ζRLB ◦ LRζB)(9)= F LB◦ LRζB = ζB◦ LGB

so thatdiagram (10)serially commutes.Since, inthis diagram,thevertical arrows are isomorphisms, the upper fork is a coequalizer too. In a similar way, if we set π :=

LRLB Lμ LB LB π _B_,μ L₁(B, μ)

For(B, μ) := G1(B, μ) wegetthecoequalizer

LRLGB LGμ◦R_ζB LGB LGB π _G 1(B,μ) L₁G1(B, μ)

Bytheforegoing,F π (B, μ)◦ ζB coequalizesthepair(L(Gμ◦ RζB) , LGB).Bythe universalpropertyofcoequalizers,thereisauniquemorphismζ1(B, μ) : L1G1(B, μ)−→

F L1(B, μ) suchthatζ1(B, μ)◦ πG1(B, μ) = F π (B, μ)◦ ζB.Bytheuniquenessof the coequalizers,ζ1(B, μ) isanisomorphism.

Letus checkthatζ1(B, μ) isnatural.Letf : (B, μ)→ (B, μ) inB1.Then

F L1f◦ ζ1(B, μ)◦ πG1(B, μ) = F L1f◦ F π (B, μ) ◦ ζB = F π (B, μ)◦ F LUf ◦ ζB = F π (B, μ)◦ ζB◦ LGU f = ζ1(B, μ)◦ πG1(B, μ)◦ LUG1f

= ζ1(B, μ)◦ L1G1f◦ πG1(B, μ)

sothatF L1f◦ζ1(B, μ) = ζ1(B, μ)◦L1G1f andhencewegetafunctorialisomorphism

ζ1: L1G1−→ F L1.Wehave

1◦ πR1= 1◦ L1R1◦ LUη1R1= ◦ LR1◦ LUη1R1= ◦ LU [R11◦ η1R1] = ,

Rπ◦ ηU = RL1◦ RLUη1◦ ηU = RL1◦ ηUR1L1◦ Uη1= RL1◦ ηRL1◦ Uη1= U η1 so that, we obtain that 1◦ πR1 =  and Rπ◦ ηU = Uη1 and similar equations for (L, R).Wecompute

U(R₁ζ1◦ η1G1) = Rζ1◦ RπG1◦ ηUG1 def. ζ1

= RF π◦ RζU◦ ηGU

(7)

= RF π◦ GηU = G [Rπ ◦ ηU] = GUη1= UG1η1

so thatR₁ζ1◦ η1G1 = G1η1. Let us check that G1 is conservative whenever G is.Let

f : (B, μ)→ (B, μ) inB1 besuchthatG1f isanisomorphism.ThenUG1f = GU f is anisomorphism.SinceG andU areconservative(see[12,Proposition4.1.4,page189]), we getthatf isanisomorphism.

If G isfaithful,from UG1= GU andthe factthatU is faithful,wededucethatG1 is faithful.

Assume G is faithful and full. Let f ∈ B₁ (G1(B, μ) , G1(B, μ)). Then Uf ∈

G (μ◦ RLh) ◦ RζB = Gμ◦ GRLh ◦ RζB = Gμ◦ RF Lh◦ RζB

= Gμ◦ RζB◦ RLGh = Gμ◦ RζB◦ RLUf

= Uf ◦ Gμ ◦ RζB = Gh◦ Gμ ◦ RζB = G (h◦ μ) ◦ RζB.

Sinceζ isanisomorphismandG isfaithful,wegetthatμ◦ RLh= h◦ μ so thatthere is aunique morphism k ∈ B1((B, μ) , (B, μ)) such that U k = h.Hence Uf = Gh =

GU k = UG1k and hencef = G1k.ThusG1 isfaithfulandfull.

Assume G is faithful and injective on objects. If G1(B, μ) = G1(B, μ) i.e. (GB, Gμ◦ RζB) = (GB, Gμ◦ RζB) thenGB = GB andGμ◦ RζB = Gμ◦ RζB. InviewoftheassumptionsonG andsinceζ isanisomorphism,weget(B, μ) = (B, μ) sothatG1isfaithfulandinjectiveonobjects. 2

Lemma 2.6. Let (L, R) and (L, R) be adjunctions of functors as in (6). Assume that RζR isa functorial isomorphism where ζ : LG−→ F L is the natural transformation ofLemma 2.2.Assume alsothat G isconservative.

1)LetA∈ A besuchthatηRF A isanisomorphism.ThenηRA isanisomorphism. 2)If theadjunction(L, R) isidempotent then(L, R) isidempotent.

Proof. 1) Since ηRF A = ηGRA is an isomorphism and RζR is an isomorphism, we get thatRζRA◦ ηGRA is an isomorphism. By (7) this means that GηRA is an isomorphism.SinceG isconservative,weconclude.

2)(L, R) isidempotentifandonlyifηR isafunctorialisomorphismandsimilarlyfor (L, R). Thus (L, R) isidempotent ifand only if ηR is afunctorial isomorphism. If thelatter condition holdsthen ηRF isa functorialisomorphism and, by1),so isηR

andhence(L, R) isidempotent. 2

Lemma2.7.Let(F, G) : (L, R)→ (L, R) beacommutationdatum.If G isconservative andη isanisomorphismso isη.

Proof. By(7),wehaveRζ◦ ηG= Gη. 2

Corollary2.8. Let(F, G) : (L, R)→ (L, R) beacommutationdatum. Assumealso that F preservescoequalizersofreflexivepairsofmorphismsinA andthatG isconservative. Assumethat both R andR arecomparable. LetN∈ N.

1)LetA∈ A besuchthatη_NR_N F A isanisomorphism.ThenηNRNA is an

isomor-phism.

2)If (L_N, R_N) isidempotent sois(LN, RN).

Proof. ApplyProposition 2.5andLemma 2.6. 2

Lemma 2.9. Let(L, R) bean adjunction andlet F and G befull and faithfulfunctors which are also injective on objects and have domain and codomain as in the following diagrams. Assume that Im(LG) ⊆ Im(F ) and that Im(RF ) ⊆ Im(G). Set L := LG andR := RF withnotation asinLemma 1.12sothatL andR aretheuniquefunctors which makethefollowingdiagrams commute

A F A B L G B L A F R A R B G B

Then (L, R) is an adjunction with unit η : Id_B → RL and counit : LR→ Id_A which satisfy

Gη = ηG and F  = F (11)

where η and  are the corresponding unit and counit of (L, R). Moreover (F, G) : (L, R)→ (L, R) is acommutationdatum andthecanonicaltransformation ζ : LG→ F L is IdLG.

Proof. Apply Lemma 1.12onceobservedthatRL= RLG,LR = LRF ,G = Id _Band 

F = Id_A. Thendefineη := ηG and:= F . 2

3. Braidedobjectsandadjunctions

Definition 3.1. Let (M, ⊗, 1) be a monoidal category (as usual we omit the brackets althoughwearenotassumingtheconstraintsaretrivial).

1) LetV beanobjectinM.Amorphismc= cV : V ⊗ V → V ⊗ V iscalledaYang–

Baxter operator (see [26, Definition XIII.3.1]) ifit satisfies the quantum Yang–Baxter equation

(c⊗ V ) (V ⊗ c) (c ⊗ V ) = (V ⊗ c) (c ⊗ V ) (V ⊗ c) (12) on V ⊗ V ⊗ V . We further assume that c is invertible. The pair (V, c) will be called a braided object in M. A morphism of braided objects (V,cV) and (W,cW) in M is a

morphismf : V → W suchthatcW(f⊗ f)= (f⊗ f)cV.ThisdefinesthecategoryBrM

of braidedobjectsandtheirmorphisms.

2) [9]Aquadruple(A,m,u,c) is calledabraidedalgebra if

• (A,m,u) isanalgebrainM; • (A,c) isabraidedobjectinM;

c(m⊗ A) = (A ⊗ m)(c ⊗ A)(A ⊗ c), (13)

c(A⊗ m) = (m ⊗ A) (A ⊗ c) (c ⊗ A), (14)

c(u⊗ A)l−1_A = (A⊗ u) r_A−1, c(A⊗ u)r−1_A = (u⊗ A) l−1_A . (15) A morphism of braided algebras is, by definition, amorphism of algebras which, in addition,isamorphismofbraidedobjects.ThisdefinesthecategoryBrAlg_Mofbraided algebrasand theirmorphisms.

3) Dually one introduces the category BrCoalg_M of braided coalgebras and their morphisms.

4)[39,Definition5.1] Asextuple(B,m,u,Δ,ε,c) is acalled abraided bialgebra if

• (B,m,u,c) isabraidedalgebra; • (B,Δ,ε,c) isabraidedcoalgebra; • thefollowingrelationshold:

Δm = (m⊗ m)(B ⊗ c ⊗ B)(Δ ⊗ Δ), Δu = (u⊗ u)Δ1, (16)

εm = m1(ε⊗ ε) , εu = Id1. (17)

Amorphism ofbraided bialgebrasis bothamorphism ofbraided algebras and coal-gebras.Thisdefinesthecategory BrBialg_M ofbraidedbialgebras.

Recall thata Yang–Baxter operatorc is called symmetric or a symmetry whenever

c2_{= Id.DenotebyBr}s

M,BrAlgsM,BrCoalgsMandBrBialgsMthefullsubcategoriesofthe

respectivecategoriesaboveconsistingofobjectswithsymmetricYang–Baxteroperator. Denoteby Is Br: Br s M→ BrM, IsBrAlg : BrAlg s M→ BrAlgM, Is

BrCoalg: BrCoalgsM→ BrCoalgM, IsBrBialg: BrBialgsM→ BrBialgM

theobviousinclusionfunctors.Note thattheyarefull,faithful,injectiveonobjectsand conservative.

Remark3.2. LetM be amonoidal category. Let A be oneof the following categories Br_M,BrAlg_M,BrCoalg_M andBrBialg_M, letAs bethecorrespondingfullsubcategory ofobjectswithsymmetricYang–BaxteroperatoranddenotebyIs

A:As→ A theobvious

inclusionfunctor. LetD_A:A → M betheforgetfulfunctor.

1)LetX ∈ A, Ys∈ Asandletα : X→ I_AsYsbeamorphisminA suchthatα :=DAα

is a monomorphism. Set X := D_AX and Y := D_AIs

AYs. Since α is braided we have

(α⊗ α) c2

X = c2Y (α⊗ α) = α ⊗ α wherecX andcY aretheYang–BaxteroperatorsofX

andY respectively.Assumethatα⊗α isamonomorphism.Thenweobtainc2

X = IdX⊗X

reflects monomorphisms,wehaveprovedthatAsis closedinA forthose subobjectsin

A whicharepreservedbyD_Aandby(−)⊗2◦ D_Awhere(−)⊗2:M→ M: V → V ⊗ V . 2) DuallyAs_{is closedinA forthose quotientsinA whichare preservedbyD}

Aand

by(−)⊗2◦ D_A.

3.3.LetM andMbemonoidalcategories.Following[6,Proposition2.5],everymonoidal functor (F, φ0, φ2) : M → M induces ina naturalway suitable functors BrF , AlgF , BrAlgF andBrBialgF suchthatthefollowingdiagramscommute

Br_M BrF H Br_M H M F M Alg_M AlgF Ω Alg_M Ω M F M BrAlg_MBrAlgF HAlg BrAlg_M HAlg Alg_M AlgF Alg_M BrAlg_MBrAlgF ΩBr BrAlg_M ΩBr Br_M BrF Br_M BrBialg_MBrBialgF 0Br BrBialg_M 0 Br BrAlg_MBrAlgFBrAlg_M where theverticalarrowsdenote theobvious forgetfulfunctors.Moreover (1) ThefunctorsH,Ω,HAlg, ΩBr,0Br areconservative.

(2) BrF , AlgF ,BrAlgF andBrBialgF areequivalences (resp.isomorphismsor conser-vative)wheneverF is.

(3) F preserves symmetricobjects (thisfollowsby definitionoftheYang–Baxter oper-atorinducedbyF ).ThuswecandefineBrsF ,BrAlgsF andBrBialgsF suchthat

Brs_MBr s_F Is Br Brs_M Is Br BrM BrF BrM BrAlgs_MBrAlg s_F Is BrAlg BrAlgs_M Is BrAlg BrAlg_MBrAlgFBrAlg_M

BrBialgs_MBrBialg s_F Is BrBialg BrBialgs_M Is BrBialg BrBialg_MBrBialgFBrBialg_M

(18) Next aim is to recall some meaningful adjunctions that will be investigated in the paper.

3.4.Let M beamonoidal category. AssumethatM has denumerablecoproducts and that thetensor products preserve suchcoproducts. Inview of[6, Proposition 3.1], the functor ΩBr hasaleftadjointTBr andthefollowing diagramscommute.

BrAlg_M HAlg Alg_M

BrM TBr H M T BrAlg_M HAlg ΩBr Alg_M Ω BrM H M (19)

TheunitηBr andthecounitBrareuniquelydetermined bythefollowing equations

HηBr= ηH, HAlgBr= HAlg, (20)

whereη and denotetheunitandcounitoftheadjunction(T, Ω) ofRemark 1.3.Using

Lemma 2.9,oneshows thatthe adjunction(TBr, ΩBr) inducesanadjunction(TBrs , ΩsBr) suchthatthefollowingdiagrams commute.

BrAlgs_M I s BrAlg BrAlg_M Brs_M Ts Br Is Br Br_M TBr BrAlgs_M I s BrAlg Ωs Br BrAlg_M ΩBr Brs_M I s Br Br_M (21)

Thelemma canbeappliedbythe followingargument. It isclearthatIm(ΩBrIsBrAlg)⊆ Im(Is

Br).Let(M, c)∈ Br

s

M andset (A, mA, uA, cA) := TBrIsBr(M, c).

Using[6,(42)],wehavecA(αmM⊗ αnM ) = (αnV ⊗ αmM ) cm,nA sothat

c2_A(αmM⊗ αnM ) = cA(αnV ⊗ αmM ) cm,nA = (αmM⊗ αnM ) cn,mA c m,n A

and cn,m_A cm,n_A = Id_M⊗(m+n).The latteris provedby inductionont = m+ n ∈ N using

[6,Proposition2.7]. Thusc2

A(αmM⊗ αnM ) = (αmM⊗ αnM ) foreverym,n∈ N andhencec2A= IdA⊗A.

Therefore(A, mA, uA, cA)∈ BrAlgs_M and TBrIsBr(M, c) =IsBrAlg(A, mA, uA, cA).Hence

Im(TBrIsBr)⊆ Im(IsBrAlg).Thus,byLemma 2.9wehavethedesiredadjunctionwithunit

ηs

Br : IdBrs

M → Ω

s

BrTBrs and counitsBr : TBrs ΩsBr → IdBrAlgs

M whichare uniquelydefined by

Is

BrAlgsBr= BrIsBrAlg and IsBrηsBr= ηBrIsBr. (22) FurthermoreIs_BrAlg,Is_Br: (T_Brs , Ωs_Br)→ (TBr, ΩBr) isacommutationdatumwith canon-icaltransformation givenbytheidentity.

Definition3.5.LetM beapreadditive monoidalcategorywithequalizers.Assumethat the tensorproducts areadditive. Let C:= (C, ΔC, εC, uC) be a coalgebra(C, ΔC, εC)

endowed with acoalgebra morphism uC : 1 → C. Inthis setting we always implicitly

assumethatwecanchooseaspecificequalizer

P (C) ξC C

ΔC

(C⊗uC)r−1C +(uC⊗C)l−1C

C⊗ C (23)

Wewillusethesamesymbolwhen C comesouttobe enrichedwith anextrastructure suchuswhenC willdenote abialgebraor abraidedbialgebra.

Wenow investigatesomepropertiesof TBr.

3.6. Let M be a preadditive monoidal category with equalizers and denumerable co-products. Assume that the tensor products are additive and preserve equalizers and denumerable coproducts.By3.4, theforgetfulfunctor ΩBr: BrAlg_M→ BrM hasaleft adjoint TBr : BrM → BrAlg_M. In view of [6, Lemma 3.4], TBr induces afunctor TBr suchthat

BrBialg_M 0Br BrAlg_M

Br_M

TBr TBr

(24)

Explicitly, forall (V, c)∈ Br_M,wecanwrite TBr(V, c) intheform (A,mA,uA,ΔA,εA,

cA) whereΔA: A→ A⊗ A andεA: A→ 1 areuniquealgebramorphismssuchthat

ΔA◦ α1V = δVl + δVr, (25)

εA◦ α1V = 0, (26)

where δl

V := (uA⊗ α1V )◦ l−1_V andδVr := (α1V ⊗ uA)◦ r−1_V .Moreover

εA◦ αnV = δn,0Id1, for every n∈ N. (27)

In viewof [6, Theorem 3.5], the functor TBr has aright adjoint PBr : BrBialg_M → Br_M, which is constructed in [6, Lemma 3.3]. The unit ηBr and the counit Br are uniquelydetermined bythefollowing equalities

ξTBr◦ ηBr= ηBr, (28)

Br0Br◦ TBrξ =0BrBr, (29) where (V, c)∈ Br_M,B∈ BrBialg_M whileηBr and Br denotetheunitandcounitof the adjunction(TBr, ΩBr) respectively.Moreoverξ : PBr→ ΩBr0Br isanatural transforma-tioninducedbythecanonicalmorphismin(23).

Note that from 3.4 it is clear that Im(TBrIsBr) ⊆ Im(IsBrBialg). Let B ∈ BrBialg

s M

and set (P, cP) := PBrIsBrBialgB. Since thetensor products preserveequalizers,we have that ξB⊗ ξB is a monomorphism so thatwe canapply 1) in Remark 3.2 to get that (P, cP) ∈ Brs_M. Thus Im(PBrIBrBialgs ) ⊆ Im(IsBr). Hence, by Lemma 2.9 we have an adjunctionTs_Br, P_Brs suchthatthediagrams

BrBialgs_M I s BrBialg BrBialg_M Brs_M Ts Br Is Br Br_M TBr BrBialgs_M I s BrBialg Ps Br BrBialg_M PBr Brs_M I s Br Br_M (30)

commuteand theunitηs_Br: IdBrs

M → P

s

BrTsBr andthecounitBrs : TsBrPBrs → IdBrBialgs M areuniquelydefinedby

Is

BrBialgsBr= BrIsBrBialg and IsBrηsBr= ηBrIsBr. (31) MoreoverIs BrBialg,IsBr  :Ts Br, PBrs  →TBr, PBr 

isacommutationdatumwith canon-icaltransformation given by the identity. Note thatthe functor 0Br induces a functor 0s

Brsuchthatthefollowingdiagramscommute.

BrBialgs_M Is BrBialg 0s Br BrAlgs_M Is BrAlg BrBialg_M 0Br BrAlg_M Brs_M TBrs TsBr BrBialgs_M 0s Br BrAlgs_M (32)

Furthermore, byLemma 1.12, thenaturaltransformation ξ : PBr → ΩBr0Br induces a naturaltransformation ξ := ξIs

BrBialg: PBrs → ΩsBr0sBr suchthatIsBrξ = ξIsBrBialg. Proposition 3.7. Let (F, φ0, φ2) : M → M be a monoidal functor between monoidal

categories. Assume that M and M have denumerable coproducts and that F and the tensor productspreserve suchcoproducts.Then both

(AlgF, F ) : (T, Ω)→ (T, Ω) and (BrAlgF, BrF ) : (TBr, ΩBr)→ (TBr , ΩBr)

arecommutationdata.

Proof. First we deal with (AlgF, F ) : (T, Ω) → (T, Ω). By 3.3, we have that Ω ◦ AlgF = F ◦ Ω. By Remark 1.3, we have that Ω and Ω have left adjoints T and T

respectively.Thestructuremorphismsφ0,φ2 induce,foreveryn∈ N, theisomorphism  φnV : (F V )⊗n→ F (V⊗n) givenby  φ0V : = φ0, φ1V := IdF V, φ2V := φ2(V, V ) , and, for n > 2  φnV : = φ2 V⊗(n−1), V◦φn−1⊗ F V  .

Using the naturality of φ2 and (2) it is straightforward to check, by induction on

n∈ N,that

mn_{(AlgF )T V}−1 ◦ (F α1V )⊗n= F αnV ◦ φnV. (33)

Letζ bethemapofLemma 2.2i.e.ζ = (AlgF ) T◦ TF η.Wecompute ΩζV ◦ αnF V = Ω(AlgF ) T V ◦ ΩTF ηV ◦ αnF V (4)

(4) = mn_{(AlgF )T V}−1 ◦ (F α1V )⊗n (33) = F αnV ◦ φnV = (∇t∈NF αtV )◦ jnV ◦ φnV = (∇t∈NF αtV )◦ ⊕t∈NφtV  ◦ αnF V

wherejnV : F (V⊗n)→ ⊕t∈NF (V⊗t) denotesthecanonicalmorphism.Sincethis

equal-ityholdsforanarbitraryn∈ N,weobtainΩζV = (∇n∈NF αnV )◦

⊕n∈NφnV

.Nowφn

is anisomorphismbyconstructionand ∇_n∈NF αnV :⊕n∈NF (V⊗n)→ F (⊕n∈NV⊗n) is

anisomorphismasF preservesdenumerablecoproducts.HenceΩζV isanisomorphism. This clearly impliesζV is anisomorphismand hence(AlgF, F ) : (T, Ω)→ (T, Ω) is a commutation datum.

Now, let us consider (BrAlgF, BrF ) : (TBr, ΩBr) → (TBr , ΩBr). By 3.4, the functor ΩBr : BrAlg_M → BrM has a left adjoint TBr : BrM → BrAlg_M and the (co)unit of the adjunctionobeys(20).Moreover HAlgTBr= T H.By3.3, wehaveH(BrF ) = F H, Ω(AlgF ) = F Ω, H_Alg (BrAlgF ) = (AlgF ) HAlg and ΩBr(BrAlgF ) = (BrF ) ΩBr. In viewofLemma 2.2thediagrams

BrAlg_M BrAlgF ΩBr BrAlg_M ΩBr BrM BrF BrM Alg_M AlgF Ω Alg_M Ω M F M (34)

inducethemapsζBr: TBr (BrF )→ (BrAlgF ) TBrand ζ : TF → (AlgF ) T definedby

ζBr= Br(BrAlgF ) TBr◦ TBr (BrF ) ηBr and ζ = (AlgF ) T◦ TF η. (35) Oneeasilychecks that

H_Alg ζBr= ζH. (36)

Bythefirstpartoftheproof,ζ isafunctorialisomorphismsothatwegetthatH_Alg ζBr is afunctorialisomorphismtoo. Since H_Alg triviallyreflects isomorphisms,weget that

ζBr isafunctorialisomorphism. 2

Proposition 3.8. LetM and M be preadditive monoidalcategorieswith equalizers. As-sume thatthetensorfunctorsareadditiveandpreserveequalizersinbothcategories.For any monoidal functor (F, φ0, φ2) : M → M which preserves equalizers, the following

diagram commutes BrBialg_M BrBialgF PBr BrBialg_M PBr BrM BrF BrM (37)

whereBrBialgF and BrF arethefunctors of3.3.Moreoverwehave

ξ(BrBialgF ) = (BrF ) ξ. (38)

Assume also that the categories M and M have denumerable coproducts and that F andthetensor productspreservesuchcoproducts. Then(BrBialgF, BrF ) :TBr, PBr



→



T_Br, P_Br  isacommutationdatum.

Proof. The first part is [6, Proposition 3.6]. Let us prove the last assertion. As-sume thatthe monoidal category M has denumerablecoproducts and thatthe tensor products preserve such coproducts. By 3.6, we have that PBr and PBr have left ad-joints TBr and TBr respectively. By3.3, we have 0Br(BrBialgF ) = (BrAlgF )0Br and Ω_Br(BrAlgF ) = (BrF ) ΩBr.By(24),wehave0BrTBr= TBr.Thecommutativediagrams

(37)and(34)-leftinducethenaturaltransformationsζBr: TBr(BrF )→ (BrBialgF ) TBr andζBr: TBr (BrF )→ (BrAlgF ) TBr ofLemma 2.2i.e.

ζBr= Br(BrBialgF ) TBr◦ TBr(BrF ) ηBr and

ζBr= Br(BrAlgF ) TBr◦ TBr (BrF ) ηBr.

Using(29), (38)and(28),oneeasily checks that0_BrζBr= ζBr. ByProposition 3.7, we knowthatζBrisafunctorialisomorphism.Since0Bristriviallyconservative,wededuce thatζBr isafunctorialisomorphismtoo. 2

4. Braidedcategories

4.1.Abraidedmonoidalcategory (M,⊗,1,c) isamonoidalcategory(M,⊗,1) equipped withabraiding c,thatisanisomorphismcU,V : U⊗ V → V ⊗ U,naturalinU,V ∈ M,

satisfying,forallU,V,W ∈ M,

cU,V⊗W = (V ⊗ cU,W)◦ (cU,V ⊗ W ) and cU⊗V,W = (cU,W ⊗ V ) ◦ (U ⊗ cV,W).

Abraidedmonoidalcategoryiscalledsymmetric ifwefurtherhavecV,U◦ cU,V = IdU⊗V

foreveryU,V ∈ M.

A (symmetric) braided monoidal functor is amonoidal functor F : M → M such that F (cU,V)◦ φ2(U,V ) = φ2(V,U )◦ c_{F (U ),F (V )}. More details on these topics can be foundin[26, ChapterXIII].

Remark4.2.Givenabraidedmonoidalcategory(M,⊗,1,c) thecategoryAlg_Mbecomes monoidalwhere,foreveryA,B ∈ Alg_M themultiplicationand unitof A⊗ B are given by

mA⊗B : = (mA⊗ mB)◦ (A ⊗ cB,A⊗ B) : (A ⊗ B) ⊗ (A ⊗ B) → A ⊗ B,

MoreovertheforgetfulfunctorAlg_M→ M isastrictmonoidalfunctor,cf.[25,page60]. Definition 4.3. A bialgebra in a braided monoidal category (M,⊗,1,c) is a coalge-bra (B,Δ,ε) in the monoidal category Alg_M. Equivalently a bialgebra is a quintuple (A, m, u, Δ, ε) where(A, m, u) isanalgebrainM and(A, Δ, ε) isacoalgebrainM such that Δ and ε aremorphisms of algebras where A⊗ A is an algebraas inthe previous remark.DenotebyBialg_MthecategoryofbialgebrasinM andtheirmorphisms,defined intheexpectedway.

4.4. Let M be abraided monoidal category. In view of [6, Proposition4.4], there are obvious functorsJ ,JAlg andJBialgsuchthatthediagrams

Bialg_M JBialg 0

BrBialg_M 0Br

Alg_M JAlg BrAlg_M

Alg_M JAlg Ω BrAlg_M ΩBr M J Br_M (39)

commute. Infact thefunctors J ,JAlg and JBialg add theevaluation of the braidingof

M on theobjectonwhichtheyact.Moreovertheyarefull,faithful,injectiveonobjects and conservative.

Assume thatM has denumerable coproducts and thatthe tensor functors preserve suchcoproducts.Then,by[6,Proposition4.5],thefollowingdiagram

Alg_M JAlg BrAlg_M

M

T J

Br_M

TBr (40)

is commutative. WhenM is symmetric thefunctors J , JAlg and JBialg factor through functors Js_,Js

Alg andJBialgs i.e.thefollowingdiagramscommute(applyLemma 1.12).

M J Js Brs_M Is Br Br_M Alg_M JAlg JAlgs BrAlgs_M Is BrAlg BrAlg_M Bialg_M JBialg Js Bialg BrBialgs_M Is BrBialg BrBialg_M (41)

Note thattheyarefull, faithful,injectiveonobjectsandconservativeandthefollowing diagram commutes.

Bialg_M 0 Js Bialg BrBialgs_M 0s Br Alg_M Js Alg BrBialgs_M (42)

4.5. Let M bea preadditive braided monoidal category with equalizers. Assumethat thetensorproductsareadditiveandpreserveequalizers.Define thefunctor

P := H◦ PBr◦ JBialg: Bialg_M→ M

For any B := (B, mB, uB, ΔB, εB) ∈ Bialg_M one easily gets that P (B) =

P (B, ΔB, εB, uB), see [6, 4.6]. The canonical inclusion ξP (B, ΔB, εB, uB) :

P (B, ΔB, εB, uB)→ B willbedenotedbyξB.Thuswehavetheequalizer

P (B) ξB B

ΔB

(B⊗uB)r−1B +(uB⊗B)lB−1

B⊗ B

By[6,Proposition4.7],wehaveacommutativediagram

Bialg_M JBialg P BrBialg_M PBr M J Br_M (43)

wherethehorizontalarrowsarethefunctorsof4.4.Furthermore

ξJBialg= J ξ. (44)

AssumefurtherthatM hasdenumerablecoproducts andthatthetensorproducts pre-servesuchcoproducts.ByRemark 1.3, theforgetfulfunctor Ω: Alg_M → M has aleft adjointT :M→ Alg_M.Note that

Is BrAlgTBrsJs (21) = TBrIsBrJs (41) = TBrJ (40) = JAlgT (41) = Is_BrAlgJ_Algs T

and hence, since Is

BrAlg is both injective on morphisms and objects, we get that the followingdiagram commutes

Alg_M JAlgs BrAlgs_M M T Js Brs_M Ts Br (45)

T :M → Bialg_M

suchthatthefollowingdiagramscommute.

Bialg_M JBialg BrBialg_M

M T J Br_M TBr (46) Bialg_M 0 Alg_M M T T (47)

By[6,Theorem4.9],thefunctorT isaleftadjointofthefunctorP : Bialg_M→ M.The unitη andcounit oftheadjunctionareuniquelydeterminedbythefollowingequalities

ξT◦ η = η, 0 ◦ T ξ = 0, (48) where η and denotetheunitandcounitoftheadjunction(T, Ω) respectively.Wehave that Is BrBialgTsBrJs (30) = TBrIsBrJs (41) = TBrJ (46) = JBialgT (41) = Is_BrBialgJ_Bialgs T and that Is BrPBrs JBialgs (30) = PBrIsBrBialgJBialgs (41) = PBrJBialg (43) = J P(41)= IsBrJsP so thatthefollowing diagramcommutes.

BialgM JBialgs BrBialgs_M M T Js Brs_M TsBr BialgM P JBialgs BrBialgs_M Ps Br M Js Brs_M (49)

Proposition 4.6. Let M be a preadditive braided monoidal category with equalizers. Assume that the tensor products are additive and preserve equalizers. Assume further that M hasdenumerable coproducts andthat thetensor productspreserve such coprod-ucts. Then the morphism ζ : TBrJ −→ JBialgT of Lemma 2.2 is IdTBrJ. In particular (JBialg, J ) :



T , P→TBr, PBr 

isacommutationdatum.

Proof. Consider thecommutativediagram (43). ByLemma 2.2,then thereis aunique naturaltransformation ζ : TBrJ −→ JBialgT suchthatPBrζ◦ ηBrJ = J η. By[6,

Proposition 4.7. Let M be a preadditive symmetric monoidal category with equalizers. Assumethatthetensorproductsareadditiveandpreserveequalizers.Assumefurtherthat M has denumerable coproducts and that thetensor products preserve such coproducts. Then the morphism ζs _{: T}s

BrJs −→ JBialgs T of Lemma 2.2 is IdTs BrJs. In particular  Js Bialg, Js  :T , P→Ts Br, PBrs  isacommutation datum.

Proof. Considerthecommutativediagram (49).ByLemma 2.2,then there isaunique naturaltransformation ζs_{: T}s

BrJs−→ JBialgs T suchthatPBrsζs◦ ηsBrJs= Jsη.Now Is BrηsBrJs ( 31) = ηBrIsBrJs (41) = ηBrJ (∗) = J η(41)= Is_BrJsη

wherein(∗) weused[6,Equality(75)].Thusηs

BrJs= Jsη.Byuniquenessofζs,wehave

ζs _{= Id} Ts

BrJs (note thatweare using thatthedomain and codomain ofζ

s _coincideby (49)). 2

4.8. Let M and M be braided monoidal categories. Following [6, Proposition 4.10], everybraidedmonoidalfunctor(F, φ0, φ2) :M→ Minducesinanaturalwayafunctor BialgF andthefollowingdiagramscommute.

M F J M J Br_M BrF Br_M Bialg_M BialgF JBialg Bialg_M JBialg BrBialg_MBrBialgFBrBialg_M

Bialg_M BialgF 0

Bialg_M

0 Alg_M AlgF Alg_M Moreover

1) BialgF is an equivalence (resp.category isomorphismor conservative) wheneverF

is.

2) IfF preservesequalizers,thefollowing diagramcommutes.

Bialg_M BialgF P Bialg_M P M F M 5. Liealgebras

The following definition extends the classical notion of Lie algebra to a monoidal category which is not necessarily braided. We expected this notion to be well-known, but we could not find any reference. We point out that in the following definition we shouldmoreproperlyspeakof“rightbraided Lie”algebraascondition(51)and itsleft analogue(56)seemnotto beequivalentingeneral,seeLemma 5.3.

Definition5.1.1)GivenanabelianmonoidalcategoryM abraidedLiealgebra inM

con-sistsofatern(M, c, [−] : M ⊗ M → M) where(M, c)∈ Br_Mandthefollowingequalities hold true: [−] = − [−] ◦ c (skew-symmetry); (50) [−] ◦ (M ⊗ [−]) ◦Id(M⊗M)⊗M + (M⊗ c) (c ⊗ M) + (c ⊗ M) (M ⊗ c) = 0 (Jacobi condition); (51) c◦ (M ⊗ [−]) = ([−] ⊗ M) ◦ (M ⊗ c) ◦ (c ⊗ M) ; (52) c◦ ([−] ⊗ M) = (M ⊗ [−]) ◦ (c ⊗ M) ◦ (M ⊗ c) . (53) Of courseoneshouldtakecareoftheassociativityconstraints,butaswedidbefore,we continuetoomitthem.AmorphismofbraidedLiealgebras(M, c, [−]) andM, c, [−]

in M is a morphism f : (M, c) → (M, c) of braided objects such that f ◦ [−] =

[−]◦ (f ⊗ f).Thisdefinesthecategory BrLie_M ofbraidedLiealgebrasinM andtheir morphisms. Denoteby

HBrLie: BrLieM→ BrM: (M, c, [−]) → (M, c)

theobviousfunctorforgettingthebracketandactingastheidentityonmorphisms.Note thatHBrLie isfaithfulandconservative.

Denote by BrLies_M the full subcategory BrLie_M consisting of braided Lie algebras with symmetricYang–Baxteroperator.Denoteby

Is

BrLie: BrLie

s

M→ BrLieM

the inclusion functor. It is clear that, by Lemma 1.12, the functor HBrLie induces a functor H_BrLies suchthatthediagram

BrLies_M Is BrLie Hs BrLie Brs_M Is Br BrLie_M HBrLie Br_M (54)

commutes.SinceHBrLieandbothverticalarrowsarefaithfulandconservative,thesame is trueforHs

BrLie.

2) Let M be anabelian braided monoidal category. A Lie algebra in M consistsof a pair(M, [−] : M ⊗ M → M) such that(M, cM,M, [−]) ∈ BrLieM, where cM,M is the

braiding c of M evaluated onM . A morphism of Lie algebras (M, [−]) andM, [−]

inM is amorphism f : M→ M inM suchthatf ◦ [−] = [−]◦ (f ⊗ f). Thisdefines thecategory Lie_M ofLiealgebras inM and theirmorphisms.Notethatthereisafull, faithful,injectiveonobjectsandconservativefunctor

JLie: LieM→ BrLieM: (M, [−]) → (M, cM,M, [−])

whichactsastheidentityonmorphisms.Thisnotionalreadyappearedin[30,c)page 82], whereaLiealgebrainM iscalled anM-Lie algebra.Denote by

HLie: LieM→ M : (M, [−]) → M

theobviousfunctorforgettingthebracketandactingastheidentityonmorphisms.Note thatHBrLieJLie= J HLie.

3) Let M be anabelian symmetric monoidal category. Given (M, [−]) ∈ Lie_M it is clearthat(M, cM,M, [−]) ∈ BrLies_MsothatJLiefactorsthroughafunctorJLies suchthat thefollowingdiagramscommute.

Lie_M JLies JLie BrLie_M BrLies_M I s BrLie BrLies_M H s BrLie Brs_M Lies_M Js Lie HLie M Js (55)

Remark 5.2. We point out that BrLies_M = YBLieAlg(M) with the notations of [21, Definition2.5](notethat(52)follows from(53)asweareinthesymmetriccase). Lemma 5.3. Let M be an abelian monoidal category. Consider a tern(M,c,[−] : M ⊗

M → M) where (M, c) ∈ Br_M. If c2 _{= Id and (50) holds, then we have that (51) is}

equivalent to

[−] ◦ ([−] ⊗ M) ◦Id(V⊗V )⊗V + (M⊗ c) (c ⊗ M) + (c ⊗ M) (M ⊗ c)

= 0. (56)

Proof. Thisproof isessentiallythesameas[20, Lemma2.9]. 2

Remark5.4. Inview of Lemma 5.3, inthe particular case when M is the category of vector spaces and (M, c) ∈ Br_M, conditions (50) and (56) encode the notion of Lie algebrainthesenseofGurevich’s [19].

Definition5.5.LetM apreadditivemonoidalcategorywithequalizersanddenumerable coproducts.Let(M, c)∈ Br_M.Forα2M asinofRemark 1.3, weset

θ(M,c):= α2M◦ (IdM⊗M− c) : M ⊗ M → ΩT M. (57)

WhenM isbraidedanditsbraidingonM iscM,MwewillsimplywriteθM forθ(M,cM,M). Definition5.6.LetM beamonoidalcategory. Let(A, mA, uA) beanalgebrainM and

letf : X → A beamorphisminM.Weset