Hopf monads on monoidal categories

HOPF MONADS ON MONO IDAL CA TEGORIES ALAIN BRUGUI ` ERES, STEVE LA CK, AND ALEXIS VIRELIZI ER Abstract. W e deﬁne Hopf monads on an arbitrary monoidal category , ex- tending the d eﬁnition give n in [BV07] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal mon ad) whose fusion operators are inv ertible. This deﬁnition can b e formulated i n terms of H opf adjunc- tions, whic h are comonoidal adjunctions with an inv ertibil it y condition. On a monoidal cat egory wi th int ernal Homs, a Hopf monad is a bimonad admitting a lef t and a right an tip o de. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (whic h are linear Hopf monads on a c ategory of bimo dules admitting a right adjoint ). W e sho w that any ﬁnite tensor category is the category of ﬁnite-dimensional mo dules ov er a Hopf algebroid. Any Hopf algebra in the cen ter of a monoidal category C gives ri s e to a Hopf monad on C . The Hopf monads so obtained are exactly the augmen ted Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P , then P is a cross pro duct of T b y a Hopf al gebra of the cen ter of the category of T - modules (generalizing the Radford-Ma j id b osonization of Hopf algebras). W e sho w that the comonoidal comonad of a Hopf adjunction is canonically represen ted by a cocommutat ive cen tral coalgebra. As a corollary , we obtain an extension of Sweedler’s Hopf mo dule decomp osition theorem to Hopf monads (in f act to the weak er notion of pre-Hopf monad). Contents Int ro duction 1 1. Preliminarie s a nd nota tions 3 2. Hopf monads 5 3. Hopf monads on closed mono idal categories 11 4. Cross pro ducts a nd r elated constructions 19 5. Hopf monads asso ciated with Hopf alg ebras and b osonization 24 6. Induced coa lgebras a nd Hopf mo dules 33 7. Hopf alg ebroids a nd ﬁnite ab elian tensor categ ories 41 References 45 Introduction Hopf monads o n auto nomous categ ories (that is, mono idal catego ries with du- als) were introduced in [BV0 7] as a to ol for unders ta nding and compa ring quantum inv ar ian ts of 3 manifolds, namely the Reshetikhin-T ur aev inv a riant a s socia ted with a mo dular categor y a nd the T urae v-Viro inv ariant asso ciated with a spher ical cat- egory (as revisited by Ba rrett-W estbury). In this pap er w e extend the no tion of Hopf monad to any mono idal categ ory . Hopf mo nads g eneralize classica l Hopf algebr as, a s well as Hopf alg ebras in a braided category . Hopf alg ebras ar e bia lgebras with an extra co ndition: the ex istence of an Date : October 23, 2018. 2000 Mathematics Subje ct Classiﬁc ation. 18C20,16T05,18D10,18D15. 1 2 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER inv ertible antipo de. Similar ly , one exp ects Ho pf mo nads to b e bimonads satisfying some extra condition. The concept of bimonad (also called opmonoidal monad) was intro duced by Mo erdijk in [Mo e02] 1 . Recall that if T is a mo nad on a ca tegory C , then one deﬁnes a category C T of T - mo dules in C (often c a lled T - algebras ). A bimonad on a mono ida l category C is a mona d on C such that C T is mono idal and the for getful functor U T : C T → C is strict mono idal. This means that T is a como noidal mo nad: it comes with a coa sso ciativ e natura l tr ansformation T 2 ( X, Y ) : T ( X ⊗ Y ) → T X ⊗ T Y and a counit T 0 : T 1 → 1 . F or example, a bialgebra A in a braided catego ry B g ives rise to bimonads A ⊗ ? and ? ⊗ A o n B . More ge ne r ally , bialgebr oids in the sense of T akeuchi are a lso exa mples of bimo na ds. More gener ally still, an y c omonoidal adjunction deﬁnes a bimonad, so that bimona ds e x ist in many settings . The ‘extra condition’ a bimona d s hould satisfy in o rder to deser v e the title of Hopf monad is no t o b vious, as there is no straightforward gener alization of the notion o f antipo de to the mono idal setting. When C is autono mous, acc ording to T anna k a theory , one ex pects that a bimona d T be Hopf if and only if C T is autonomous. This turns out to b e equiv alent to the existence of a left antipo de and a right antipo de, which ar e natural transformations s l X : T ( ∨ T ( X )) → ∨ X and s r X : T ( T ( X ) ∨ ) → X ∨ . That was pre cisely the deﬁnition o f a Hopf monad given in [BV07]. While it is satisfactor y for applications to quantum topolo gy , as the categorie s in volv ed are autonomous, this deﬁnition has s ome drawbacks for other applications: for instance, it do esn’t encompa ss inﬁnite-dimensional Hopf alg e- bras since the category of vector s paces o f ar bitrary dimension is not autonomous. Therefore one is prompted to ask several questions: • What ar e Hopf monads on ar bitrary monoidal ca teg ories? • What are Hopf mo na ds on c losed monoidal catego ries (with internal Homs)? • Is it p ossible to characterize Ho pf monads obtained from Hopf algebras? • Can one extend clas s ical results of the theory of Hopf alg ebras to Hopf monads on monoidal catego r ies? • When do es a bia lgebroid deﬁne a Hopf monad? The aim of this pap er is to ans wer these questions. In Section 2, we deﬁne Hopf monads on a n arbitrar y mono idal ca tegory . O ur deﬁnition is inspired by the fact that a bialgebr a A is a Hopf a lgebra if a nd only if its fusion morphisms H l , H r : A ⊗ A → A ⊗ A , deﬁned by H l ( x ⊗ y ) = x (1) ⊗ x (2) y and H r ( x ⊗ y ) = x (2) y ⊗ x (1) , ar e inv ertible. If T is a bimonad, we in tro duce the fusion op er ators H l and H r , which are natura l tr ansformations H l X,Y = ( T X ⊗ µ Y ) T 2 ( X, T Y ) : T ( X ⊗ T Y ) → T X ⊗ T Y , H r X,Y = ( µ X ⊗ T Y ) T 2 ( T X , Y ) : T ( T X ⊗ Y ) → T X ⊗ T Y , and decree that T is a Hopf monad if H l and H r are invertible. W e also in tro duce the related notion of Hopf adjunction . The monad of a Hopf adjunction is a Hopf monad, a nd a bimona d is a Ho pf monad if and only if its a djunction is a Hopf adjunction. It turns out that cer ta in classica l results on Hopf alg ebras extend nat- urally to Hopf monads (or more gener ally to pre-Hopf monads), such as Maschk e ’s semisimplicity cr iterion and Sw eedler’s theorem on the s tructure of Hopf mo dules (see Section 6). In Sectio n 3, we study Hopf mona ds on closed monoidal categ ories. Hopf monads on such ca tegories can b e c haracter iz ed, as in the a utonomous case, in categor ical terms and also in terms of antipo des. More prec is ely , let T be a bimonad o n a 1 Bimonads were in tro duced in [M oe02] under the name ‘Hopf monads’, whic h we prefer to reserve for bimonads with antipo des b y analogy wi th Hopf algebras. HOPF M ONADS ON MONOIDAL CA TEGORIES 3 closed monoidal category C , that is, a monoidal categor y with in ternal Homs. W e show that T is a Hopf mona d if and only if its categ ory of mo dules C T is closed a nd the forg etful functor U T preserves internal Homs. Also T is a Hopf mo nad if and only if it admits a left antip o de and right antip o de , that is , natural transforma tions in tw o v aria bles: s l X,Y : T [ T X , Y ] l → [ X , T Y ] l and s r X,Y : T [ T X , Y ] r → [ X , T Y ] r where [ − , − ] l and [ − , − ] r denote the left and r igh t internal Homs, each of them satisfying t wo axioms a s exp ected. The pro of of these re sults relies on a class iﬁcation of adjunction liftings. In the s pecial case where C is a utonomous, we s ho w tha t the deﬁnition of a Hopf mo na d given in this pap er sp ecializes to the one g iv en in [BV07]. In the sp ecial case where C is ∗ -autonomo us (and so mono idal closed), a Hopf monad in the sens e of [DS04] is a Hopf mona d in our sense (but the conv erse is not true). In Section 5, we study the rela tions b etw een Ho pf algebras and Hopf monads . Given a lax central bialgebra o f a monoidal catego ry C , that is, a bialg ebra A in the lax cen ter Z lax ( C ) of C , with lax 2 half-braiding σ : A ⊗ ? → ? ⊗ A , the endofunctor A ⊗ ? of C is a bimonad, deno ted by A ⊗ σ ? on C . This bimonad is augmented, that is, endow ed with a bimo nad morphis m A ⊗ σ ? → 1 C . It is a Hopf monad if a nd only if A is a Hopf alg ebra in the center Z ( C ) o f C . The main r esult of the section is tha t this constr uc tio n deﬁnes an equiv alence of categor ies b et ween cen tral Hopf a lgebras of C (that is, Hopf algebras in the center Z ( C )) and augmented Hopf monads o n C . More generally , given a Hopf monad T on C and a central Hopf alg e bra ( A , σ ) of the category of T - mo dules, we construct a Hopf monad A ⋊ σ T on C of which T is a retra ct. Conv er sely , under suitable exactness conditions (inv o lving reﬂexive co equalizers), a n y Hopf monad P of which T is a r etract is of the form A ⋊ σ T . The pro of of this r e sult is based on tw o gener al constructions in volving Hopf monads : the cro ss pro duct and the cro ss quotient, which a re studied in Section 4. In Sectio n 6, we show that the co monoidal comona d o f a pre-Hopf adjunction is canonically repres en ted by a coc omm utativ e central coalg ebra. Combining this with a descent r esult for monads, w e obtain a generalization of Sweedler’s Hopf mo dule decomp osition theor em to Hopf monads (in fact to pre- Ho pf monads). W e study the close relations hips b et ween Hopf adjunctions, Hopf monads, and co comm utative central co algebras. Finally , in Section 7, we study bialgebro ids which, accor ding to Szlach´ anyi [Szl03], are linear bimo nads on catego r ies of bimo dules admitting a right a djoin t. A bial- gebroid corr esponds with a Hopf mo na d if a nd only if it is a Ho pf algebroid in the sense of Schauen burg [Sch00]. W e als o use Hopf monads to pr o ve tha t any ﬁnite tensor categ ory is naturally equiv a len t (as a tensor categ ory) to the categ o ry o f ﬁnite-dimensional mo dules o ver some ﬁnite dimensional Hopf algebroid. 1. Preliminaries a nd not a tions Unless otherw is e sp eciﬁed, categories ar e small, and mo noidal ca tegories are strict. W e denote by Ca t the categ ory of small categ ories (whic h is not small). If C is a catego ry , we deno te by Ob( C ) the set of ob jects of C and by Hom C ( X, Y ) the set of morphisms in C fro m an ob ject X to an ob ject Y . The identit y functor of C is denoted by 1 C . If C is a catego ry a nd c an ob ject of C , the c ate gory of obje cts of C over c is the category C /c whose ob jects are pair s ( a, φ ), with a ∈ Ob( C ) and φ ∈ Hom C ( a, c ). Morphisms from ( a, φ ) to ( b, ψ ) in C /c ar e mor phism f : a → b in C satisfying the condition ψ f = φ . T he y are called morphi sms over c . 2 Here lax means that σ need not be inv ertible. 4 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER Similarly the c ate gory of obje ct s of C un der c is the c ategory c \C whose ob jects are pair s ( a, φ ), with a ∈ Ob( C ) a nd φ ∈ Hom C ( c, a ). A pair of parallel morphisms X f / / g / / Y is r eﬂexive (resp. c or eﬂexive ) if f and g have a common section (resp. a common retraction), that is, if there ex ists a morphism h : Y → X such that f h = g h = id Y (resp. hf = hg = id X ). A r eﬂexive c o e qualizer is a co equalizer of a re ﬂexiv e pair . Similarly a c or eﬂexive e qualizer is an equa liz er of a coreﬂex iv e pa ir. 1.1. Monoi dal categories and functors. Given an ob ject X of a monoidal cat- egory C , we denote by X ⊗ ? the endofunctor of C deﬁned on o b jects by Y 7→ X ⊗ Y and on morphisms by f 7→ X ⊗ f = id X ⊗ f . Similarly one deﬁnes the endo functor ? ⊗ X of C . Let ( C , ⊗ , 1 ) and ( D , ⊗ , 1 ) b e tw o mo noidal categories . A monoidal functor from C to D is a triple ( F, F 2 , F 0 ), where F : C → D is a functor, F 2 : F ⊗ F → F ⊗ is a natural tra nsformation, and F 0 : 1 → F ( 1 ) is a morphism in D , such that: F 2 ( X, Y ⊗ Z )(id F ( X ) ⊗ F 2 ( Y , Z )) = F 2 ( X ⊗ Y , Z )( F 2 ( X, Y ) ⊗ id F ( Z ) ); F 2 ( X, 1 )(id F ( X ) ⊗ F 0 ) = id F ( X ) = F 2 ( 1 , X )( F 0 ⊗ id F ( X ) ); for all o b jects X , Y , Z of C . A mono idal functor ( F , F 2 , F 0 ) is said to be st r ong (resp. strict ) if F 2 and F 0 are isomorphisms (resp. iden tities). A natura l transformation ϕ : F → G betw een mo noidal functors is monoidal if it satisﬁes : ϕ X ⊗ Y F 2 ( X, Y ) = G 2 ( X, Y )( ϕ X ⊗ ϕ Y ) and G 0 = ϕ 1 F 0 . W e de no te by MonCat the categ ory of sma ll monoidal categ ories, morphisms being strong mono idal functors. 1.2. Comonoi dal functors. Le t ( C , ⊗ , 1 ) a nd ( D , ⊗ , 1 ) b e t wo monoida l c ate- gories. A c omonoidal fun ctor (also called opmonoi dal functor ) fr o m C to D is a triple ( F , F 2 , F 0 ), where F : C → D is a functor, F 2 : F ⊗ → F ⊗ F is a na tural transformatio n, and F 0 : F ( 1 ) → 1 is a morphism in D , s uc h that:  id F ( X ) ⊗ F 2 ( Y , Z )  F 2 ( X, Y ⊗ Z ) =  F 2 ( X, Y ) ⊗ id F ( Z )  F 2 ( X ⊗ Y , Z ); (id F ( X ) ⊗ F 0 ) F 2 ( X, 1 ) = id F ( X ) = ( F 0 ⊗ id F ( X ) ) F 2 ( 1 , X ); for all o b jects X , Y , Z of C . A comono idal functor ( F, F 2 , F 0 ) is said to b e str ong (resp. strict ) if F 2 and F 0 are isomorphisms (resp. identities). In that case, ( F, F − 1 2 , F − 1 0 ) is a strong (resp. strict) mono idal functor. A natural transformatio n ϕ : F → G betw een mo noidal functors is c omonoidal if it satisﬁes: G 2 ( X, Y ) ϕ X ⊗ Y = ( ϕ X ⊗ ϕ Y ) F 2 ( X, Y ) a nd G 0 ϕ 1 = F 0 . Note that the no tions o f co monoidal functor and comonoida l natura l transfor - mation are dua l to the no tions of monoidal functor and monoidal na tural transfor - mation. HOPF M ONADS ON MONOIDAL CA TEGORIES 5 2. Hopf mona ds In this section, we deﬁne Hopf monads on an arbitrar y monoidal ca tegory: they are the bimo na ds whose fusion op erators ar e inv er tible. W e also introduce the related no tion of Hopf adjunction: the monad of a Ho pf adjunction is a Hopf monad, a nd a bimona d is a Ho pf monad if and only if its a djunction is a Hopf adjunction. 2.1. Monads. Let C b e a categor y . Recall that the catego ry End( C ) of endofunc- tors of C is strict mono idal with comp osition for monoidal pr o duct a nd identit y functor 1 C for unit ob ject. A monad on C is a n a lgebra in End( C ), tha t is, a triple ( T , µ, η ), wher e T : C → C is a functor, µ : T 2 → T and η : 1 C → T are natural transformatio ns, such that: µ X T ( µ X ) = µ X µ T X and µ X η T X = id T X = µ X T ( η X ) for any ob ject X of C . Monads on C form a categor y Mon( C ), a morphism from a mo nad ( T , µ, η ) to a monad ( T ′ , µ ′ , η ′ ) b eing a na tural transfor mation f : T → T ′ such that f η = η ′ and f µ = µ ′ T ( f ) f T . The identit y functor 1 C is a mona d (with the identit y for pr o duct and unit) and it is an initial ob ject in Mon( C ). 2.2. Mo dules ov er a monad. Let ( T , µ, η ) b e a mo na d on a catego ry C . An action of T on an ob ject M of C is a morphism r : T ( M ) → M in C such that: rT ( r ) = r µ M and r η M = id M . The pair ( M , r ) is then called a T - mo dule in C , or just a T -mo dule 3 . Given tw o T -mo dules ( M , r ) and ( N , s ) in C , a morphism of T - mo dules from ( M , r ) to ( N , r ) is a morphism f ∈ Hom C ( M , N ) whic h is T - line ar , that is, such that f r = sT ( f ). This gives ris e to the category of T -mo dules (in C ), with co mpo- sition inherited from C . W e denote this categor y b y C T (the notation T - C is used in [BV07]) . The for getful fun ct or U T : C T → C o f T is deﬁned by U T ( M , r ) = M for any T -mo dule ( M , r ) and U T ( f ) = f for any T -linear morphism f . It has a left adjoint F T : C → C T , ca lled the fr e e mo dule functor , deﬁned by F T ( X ) = ( T X , µ X ) for any ob ject X of C and F T ( f ) = T f for a ny morphism f of C . 2.3. Monads, adjunctions, and monadicity. Let ( F : C → D , U : D → C ) be an adjunction, with unit η : 1 C → U F and counit ε : F U → 1 D . Then T = U F is a monad with pro duct µ = U ( ε F ) and unit η . There exists a unique functor K : D → C T such that U T K = U and K F = F T . This functor K , called the c omp arison functor of the a djunction ( F, U ), is deﬁned b y K ( d ) = ( U d, U ε d ). An adjunction ( F , U ) is monadic if its co mparison functor K is an equiv alence of categorie s. F o r exa mple, if T is a monad on C , the adjunction ( F T , U T ) has mona d T and compar ison functor K = 1 C T , a nd so is mo nadic. A functor U is monadic if it a dmits a left adjoint F and the a djunction ( F, U ) is monadic. If such is the c a se, the mona d T = U F of the adjunction ( F , U ) is called the monad of U . It is well-deﬁned up to unique isomorphism o f monads (as the left adjoint F is unique up to unique natura l iso morphism). Theorem 2. 1 (Beck) . An adjunction ( F : C → D , U : D → C ) is monadic if and only if the functor U satisﬁes the fol lowing c onditions: (a) The functor U is c onservative, t hat is, U r eﬂe cts isomorphisms; (b) Any r eﬂexive p air of morphisms in D whose image by U has a split c o e qual- izer has a c o e qualizer, which is pr eserve d by U . 3 Pa irs ( M , r ) ar e usually called T -algebras in the literature (see [ Mac98]). 6 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER Mor e over, if ( F, U ) is monadic, the c omp arison funct or K is an isomorphism if and only if the functor U satisﬁes the tr ansp ort of structu r e c ondition: (c) F or any isomorphism f : U ( d ) → c in C , wher e c ∈ Ob( C ) and d ∈ Ob( D ) , ther e exist a unique e c ∈ Ob( D ) and a unique isomorphism e f : d → e c in D such that U ( e f ) = f . 2.4. Bimonads. A bimonad on a monoidal categor y C is a monad ( T , µ, η ) on C such that the functor T : C → C is comonoidal a nd the na tural transfor mations µ : T 2 → T and η : 1 C → T are comono idal. In other words, T is endowed with a natural tra nsformation T 2 : T ⊗ → T ⊗ T and a morphism T 0 : T ( 1 ) → 1 in C such that:  T X ⊗ T 2 ( Y , Z )  T 2 ( X, Y ⊗ Z ) =  T 2 ( X, Y ) ⊗ T Z  T 2 ( X ⊗ Y , Z ) , ( T X ⊗ T 0 ) T 2 ( X, 1 ) = id T X = ( T 0 ⊗ T X ) T 2 ( 1 , X ) , T 2 ( X, Y ) µ X ⊗ Y = ( µ X ⊗ µ Y ) T 2 ( T X , T Y ) T ( T 2 ( X, Y )) , T 0 µ 1 = T 0 T ( T 0 ) , T 2 ( X, Y ) η X ⊗ Y = η X ⊗ η Y , T 0 η 1 = id 1 . Remark 2.2 . A bimonad T on a monoidal catego ry C = ( C , ⊗ , 1 ) may b e view ed as a bimonad T cop on the monoidal categor y C ⊗ op = ( C , ⊗ op , 1 ), with comonoidal structure T cop 2 ( X, Y ) = T 2 ( Y , X ) and T cop 0 = T 0 . The bimonad T cop is ca lled the c o opp osite of the bimo nad T . W e hav e: ( C ⊗ op ) T cop = ( C T ) ⊗ op . Remark 2 .3. The dual notion of a bimonad is that of a bicomo nad, that is, a monoidal como na d. An endofunctor T of a monoida l category C = ( C , ⊗ , 1 ) is a bicomo nad if and only if the opp osite endo functor T op is a bimonad on C op = ( C op , ⊗ , 1 ). Bimonads on C for m a category BiMon( C ), mo rphisms of bimonads being como- noidal morphis ms of mo nads. The ident ity functor 1 C is a bimonad on C , which is an initial ob ject of BiMon( C ). 2.5. Bimonads and comonoi dal adjunctions. A c omonoidal adjunction is an adjunction ( F : C → D , U : D → C ), where C and D ar e monoidal ca tegories, F and U are comono idal functors, and the a djunction unit η : 1 C → U F and counit ε : F U → 1 D are co monoidal na tural transformatio ns. If ( F, U ) is a comonoidal adjunction, then U is in fact a strong comonoidal functor, which w e ma y view as a strong monoidal functor. Con versely , if a strong monoidal functor U : D → C admits a left adjoint F , then F is co monoidal, with comonoidal structur e given by: F 2 ( X, Y ) = ε F X ⊗ F Y F U 2 ( F X , F Y ) F ( η X ⊗ η Y ) and F 0 = ε 1 F ( U 0 ) , and ( F, U ) is a co monoidal adjunction (viewing U as a str ong comonoida l functor), see [McC02]. A c o monoidal a djunction is an instance of a do ctrinal a djunction in the sense of [Kel74]. The mona d T = U F of a co monoidal adjunction ( U, F ) is a bimonad, and the compariso n functor K : D → C T is stro ng mono idal and sa tisﬁes U T K = U as monoidal functors a nd K F = F T as comonoidal functors (see fo r instance [BV07, Theorem 2.6 ]). The co monad ˆ T = F U of a comonoida l a djunction ( U, F ) is a c omonoidal c omonad , that is, a comona d whose underly ing endofunctor is endow ed w ith a comonoidal structur e so that its copro duct and co unit are comonoida l. Example 2.4. The adjunction ( F U , U T ) of a bimona d T is a co monoidal adjunction (beca use U T is str ong mono idal). HOPF M ONADS ON MONOIDAL CA TEGORIES 7 Remark 2 .5. Comonoidal adjunctions ar e s omewhat mislea dingly called monoidal adjunctions in [BV07]. 2.6. F usion o p erators. Let T b e a bimo nad on a monoida l categor y C . The left fusion op er ator of T is the natural transformation H l : T (1 C ⊗ T ) → T ⊗ T deﬁned by: H l X,Y = ( T X ⊗ µ Y ) T 2 ( X, T Y ) : T ( X ⊗ T Y ) → T X ⊗ T Y . The right fusion op er ator of T is the natural transformatio n H r : T ( T ⊗ 1 C ) → T ⊗ T deﬁned by: H r X,Y = ( µ X ⊗ T Y ) T 2 ( T X , Y ) : T ( T X ⊗ Y ) → T X ⊗ T Y . F rom the axioms of a bimonad, we easily deduce: Prop osition 2. 6. The left fusion op er ator H l of a bimonad T satisﬁes: H l X,Y T ( X ⊗ µ Y ) = ( T X ⊗ µ Y ) H l X,T Y , H l X,Y T ( X ⊗ η Y ) = T 2 ( X, Y ) , H l X,Y η X ⊗ T Y = η X ⊗ T Y , ( T 2 ( X, Y ) ⊗ T Z ) H l X ⊗ Y , Z = ( T X ⊗ H l Y ,Z ) T 2 ( X, Y ⊗ T Z ) , ( T 0 ⊗ T X ) H l 1 ,X = µ X , ( T X ⊗ T 0 ) H l X, 1 = T ( X ⊗ T 0 ) , and the left pentagon equation: ( T X ⊗ H l Y ,Z ) H l X,Y ⊗ T Z = ( H l X,Y ⊗ T Z ) H l X ⊗ T Y ,Z T ( X ⊗ H l Y ,Z ) . Similarly the right fusion op er ator H r of T satisﬁes: H r X,Y T ( µ X ⊗ Y ) = ( µ X ⊗ T Y ) H r T X,Y , H r X,Y T ( η X ⊗ Y ) = T 2 ( X, Y ) , H r X,Y η T X ⊗ Y = T X ⊗ η Y , ( T X ⊗ T 2 ( Y , Z )) H r X,Y ⊗ Z = ( H r X,Y ⊗ T Z ) T 2 ( T X ⊗ Y , Z ) , ( T X ⊗ T 0 ) H r X, 1 = µ X , ( T 0 ⊗ T X ) H r 1 ,X = T ( T 0 ⊗ X ) , and the right pentagon equa tion: ( H r X,Y ⊗ T Z ) H r T X ⊗ Y ,Z = ( T X ⊗ H r Y ,Z ) H r X,T Y ⊗ Z T ( H l X,Y ⊗ Z ) . Remark 2.7. A bimonad can b e r eco vered from its left (or right) fusio n op erator. More precisely , let T b e a n endofunctor of a monoidal categor y C endow ed with a natural trans fo rmation H X,Y : T ( X ⊗ T Y ) → T X ⊗ T Y sa tisfying the left pe n ta gon equation: ( T X ⊗ H Y ,Z ) H X,Y ⊗ T Z = ( H X,Y ⊗ T Z ) H X ⊗ T Y ,Z T ( X ⊗ H Y ,Z ) , and with a morphism T 0 : T 1 → 1 and a natural tra ns formation η X : X → T X satisfying: H X,Y η X ⊗ T Y = η X ⊗ T Y , T 0 η 1 = id 1 , ( T X ⊗ T 0 ) H X, 1 = T ( X ⊗ T 0 ) , ( T 0 ⊗ T X ) H 1 ,X T ( η X ) = id T X . Then T admits a unique bimonad structure ( T , µ, η , T 2 , T 0 ) having left fusion o per- ator H . The pr o duct µ and comonoida l str uctural mor phism T 2 are given b y: µ X = ( T 0 ⊗ T X ) H 1 ,X and T 2 ( X, Y ) = H X,Y T ( X ⊗ η Y ) . 8 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER 2.7. Hopf monads and pre-Hopf monads. Let C b e a monoidal catego ry . A left (resp. a right ) Hopf monad on C is a bimonad on C who s e left fusion op erator H l (resp. right fusion o perato r H r ) is a n isomorphism. A Hopf monad on C is a bimonad o n C such that b oth left and r igh t fusio n op erators ar e isomorphisms. Hopf mona ds on C form a full sub category HopfMon( C ) of the categor y BiMo n( C ) o f bimonads. The identit y functor 1 C is a Hopf monad on C , which is an initial ob ject of HopfMon( C ). It is conv enien t to consider a weak er notion: a left (resp. right ) pr e-Hopf monad on C is a bimona d o n C such that, for a n y ob ject X of C , the morphism H l 1 ,X (resp. H r X, 1 ) is invertible. A pr e-Hopf monad is a bimonad which is a left and a right pre-Hopf monad. Clearly any Hopf monad is a pr e - Hopf mona d, but the c on verse is false: Example 2.8. W e provide an ex a mple of a pre-Hopf mo nad on a monoidal (even autonomous) categor y which is no t a Hopf mo nad. Let Z - vect k be the autonomous category of ﬁnite dimensiona l Z - graded vector space s o n a ﬁeld k , and let N - vect k be its full sub c a tegory of gra ded vector spaces with supp ort in N . The inclusion functor ι : N - vect k → Z - vect k has a left a djoin t π , which s ends a Z - graded vector space to its non-nega tiv e part. The adjunction ( π , ι ) is monoidal. The bimonad T = ιπ o n Z - vect k of this adjunction (see Section 2.5 ) is a pre-Ho pf monad but not a Hopf mo nad. Remark 2.9. Certain genera l results on Hopf alg ebras extend naturally to pre- Hopf monads, suc h as Sweedler’s theorem o n the structure o f Hopf mo dules (see Sec- tion 6). Also, Maschk e’s s emisimplicit y theorem for Hopf mona ds o n autonomous categorie s given in [BV07, Theor em 6 .5] holds word for word fo r pre-Hopf mo nads in arbitra ry monoidal catego ries. Indee d the pro of given in [B V07], which relied on the pr o perties of a certa in na tural transfor mation Γ X : X ⊗ T 1 → T 2 X , extends in a straig h tforward wa y , o bserving that Γ X = H r − 1 X, 1 ( η X ⊗ T 1 ). Example 2.10. Given a Hopf algebra A in a braided categor y , we depic t its pro duct m , unit u , copro duct ∆, c o unit ε , and in vertible antipo de S as follows: m = P S f r a g r e p la c e m e n t s A A A , u = P S f r a g r e p la c e m e n t s A , ∆ = P S f r a g r e p la c e m e n t s A A A , ε = P S f r a g r e p la c e m e n t s A , S = P S f r a g r e p la c e m e n t s A A , S − 1 = P S f r a g r e p la c e m e n t s A A . Let B b e a braided categor y with braiding τ , and A a bialg ebra in B . As shown in [BV07], the endofunctor A ⊗ ? of B is a bimona d on B , with s tr ucture maps: µ X = P S f r a g r e p la c e m e n t s A A A X X Y , η X = P S f r a g r e p la c e m e n t s A X X Y , ( A ⊗ ?) 2 ( X, Y ) = P S f r a g r e p la c e m e n t s A A A X X Y Y , ( A ⊗ ?) 0 = P S f r a g r e p la c e m e n t s A X Y . Its fusion oper ators are: H l X,Y = P S f r a g r e p la c e m e n t s A A A A X X Y Y and H r X,Y = P S f r a g r e p la c e m e n t s A A A A X X Y Y . If A is a Hopf algebr a with invertible antipo de S , then A ⊗ ? is a Hopf mona d, the inv ers es of the fusion o p erator s b eing: H l − 1 X,Y = P S f r a g r e p la c e m e n t s A A A A X X Y Y and H r − 1 X,Y = P S f r a g r e p la c e m e n t s A A A A X X Y Y . HOPF M ONADS ON MONOIDAL CA TEGORIES 9 Similarly , if A is a Hopf algebra in B with inv ertible antipo de, then ? ⊗ A is a Hopf monad o n B . Th us Hopf monads generalize Hopf a lgebras in braided c a tegories. In particular, a Hopf algebra ov er a commu tative ring k deﬁnes a Hopf monad on the categ ory o f k -mo dules. See Section 5 for a detailed discus sion of Hopf monads asso ciated with Hopf algebras. Remark 2.11. Let T b e a bimonad on a mo no idal catego ry C . Then T is a rig h t (pre- )Hopf monad if and only if its opp osite bimonad T cop on C ⊗ op (see Remark 2.2) is a left (pre- )Hopf monad. 2.8. Hopf monads and Hopf adjunctions. In view of the relation betw een bi- monads a nd comono idal adjunctions recalled in Section 2.5, it is natural to lo ok for a characterization of Hopf mona ds in ter ms of a djunctions. This lea ds to the notion of a Hopf adjunction. Let ( F : C → D , U : D → C ) b e a comonoida l a djunction b et w een monoidal categorie s (see Section 2.5). The left H opf op er ator and the right Hopf op er ator of ( F, U ) are the na tural transformatio ns H l : F (1 C ⊗ U ) → F ⊗ 1 D and H r : F ( U ⊗ 1 C ) → 1 D ⊗ F deﬁned by: H l c,d = ( F c ⊗ ε d ) F 2 ( c, U d ) : F ( c ⊗ U d ) → F c ⊗ d, H r d,c = ( ε d ⊗ F c ) F 2 ( U d, c ) : F ( U d ⊗ c ) → d ⊗ F c, for c ∈ Ob( C ) a nd d ∈ O b( D ). Remark 2. 12. Ho pf adjunctions w ere initially introduced by L awvere in the c o n- text of ca rtesian catego ries under the name of F rob enius adjunctions [La w70]. Remark 2.13. Let T = U F b e the bimonad of the como noidal a djunction ( F , U ). The fusion op erator s H l and H r of T are related to the Hopf op erators H l and H r of ( F, U ) as follows: H l X,Y = U 2 ( F X , F Y ) U ( H l X,F Y ) and H r X,Y = U 2 ( F X , F Y ) U ( H r F X,Y ) for all X , Y ∈ O b( C ). A left (r esp. right) Hopf adjunction is a comono idal adjunction ( F , U ) such that H l (resp. H r ) is invertible. A Hopf adjunction is a comonoida l adjunction such that bo th H l and H r are inv er tible. A left (r esp. right) pr e-H opf adjunction is a comono idal adjunction ( F, U ) such that H l 1 , − (resp. H r − , 1 ) is inv ertible. A pr e-Hopf adjunction is a c omonoidal adjunc- tion such that b oth H l 1 , − and H r − , 1 are inv ertible. F rom Remark 2.13, we easily deduce: Prop osition 2. 14. (a) The monad of a left (r esp. right) Hopf adjunction is a left (r esp. right) H opf monad. In p articular the monad of a Hopf adjunction is a Hopf monad. (b) The monad of a left (r esp. right) pr e-Hopf adjunction is a left (r esp. right) pr e-Hopf monad. In p articular the monad of pr e-Hopf adjunction is a pr e- Hopf monad. On the other hand, a bimonad is a Hopf monad if and o nly if its ass o ciated comonoidal adjunction is a Hopf adjunction: Theorem 2 .15. L et T b e a bimonad on a monoidal c ate gory C . (a) T is a left (r esp. right) Hopf monad if and only if the c omonoidal adjunction ( F T , U T ) is a left (r esp. right) Hopf adjunction. In p articular T is a Hopf monad if and only if ( F T , U T ) is a Hopf adjunction. 10 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER (b) T is a left (r esp. right) pr e-Hopf monad if and only if t he c omonoidal ad- junction ( F T , U T ) is a left (re sp. right) pr e-Hopf adjunction. In p articular T is a pr e-Hopf monad if and only if ( F T , U T ) is a pr e-Hopf adjunction. W e prov e Theorem 2 .1 5 in Section 2.9. Hopf adjunctions are stable under compo sition: Prop osition 2.16. The c omp osite of two left (r esp. right) Hopf adjunctions is a left (r esp. right) Hopf adjunction. In p articular the c omp osite of two Hopf adjunctions is a Hopf adjunction. Prop osition 2 .16 is a dir ect consequence of the following lemma: Lemma 2.17. L et ( F : C → D , U : D → C ) and ( G : D → E , V : E → D ) b e two c omonoidal adjunctions. Denote by H l (r esp. H ′ l , r esp. H ′′ l ) and H r (r esp. H ′ r , r esp. H ′′ r ) the left and right Hopf op er ators of ( F , U ) (r esp. ( G, V ) , r esp. ( GF, U V ) ). Then H ′′ l c,e = H ′ l F c,e G ( H l c,V e ) and H ′′ r e,c = H ′ r e,F c G ( H r V e,c ) for al l c ∈ Ob( C ) and e ∈ Ob( E ) . 2.9. Pro of of Theorem 2 . 15 . The ‘if ’ par t of each a ssertion res ults immediately from P ropo sition 2 .14, since T is the bimonad o f its comonoidal a djunction. The ‘only if ’ part, les s straightforw ard, results from the following lemma: Lemma 2. 1 8. L et T b e a bimonad on a monoidal c ate gory C . Denote by H l and H r its fusion op er ators and H l , H r the Hopf op er ators of t he adjunction ( F T , U T ) of T . L et X b e an obje ct C . Then H l X, − is invertible if and only if H l X, − is invertible, and in that c ase their inverses ar e r elate d by: H l − 1 X,Y = H l − 1 X,F T Y and H l − 1 X, ( M ,r ) = T (id X ⊗ r ) H l − 1 X,M (id T X ⊗ η M ) . Similarly H r − ,X is invertible if and only if H r − ,X is invertible, and in that c ase: H r − 1 Y ,X = H r − 1 F T Y ,X and H r − 1 ( M ,r ) , X = T ( r ⊗ id X ) H r − 1 M ,X ( η M ⊗ id T X ) . Pr o of. By Remark 2.13, the for getful functor U T being strict monoidal, we have H l X,Y = H l X,F T ( Y ) and H r X,Y = H r F T ( X ) ,Y . Hence the ‘if ’ parts a nd the expressions given for in verses o f fusion op erators . Let us prov e the ‘only if ’ part of the left-ha nded ca se (the right-handed case ca n be done similarly ). Assume H l X, − is inv er tible. Set A = T ( X ⊗ ?), B = T X ⊗ ?, and α = U T H l X, − : AU T → B U T . W e hav e α F T = H l X, − and so α F T is inv e r tible. Therefore α is invertible by Lemma 2.19 b elow. Thus H l is inv ertible ( U T being conserv a tiv e) and H l − 1 X, ( M ,r ) = T (id X ⊗ r ) H l − 1 X,M (id T X ⊗ η M ) for a n y T - mo dule ( M , r ).  Lemma 2 .19. L et α : AU T → B U T b e a natur al t r ansformation, wher e T i s a monad on a c ate gory C and A, B : C → D ar e two functors. If α F T is invertible, so is α , and α − 1 ( M ,r ) = A ( r ) α − 1 F T M B ( η M ) for any T - mo dule. Pr o of. Let ( M , r ) b e a T - mo dule. The for k T 2 M µ M , , T r 2 2 T M r / / M HOPF M ONADS ON MONOIDAL CA TEGORIES 11 in C is split by T 2 M T M η T M o o M η M o o . As a result, in the diagram: AT 2 M α F T T M   Aµ M / / AT r / / AT M α F T M   Ar / / AM α ( M,r )   B T 2 M B µ M / / B T r / / B T M B r / / B M the tw o rows ar e split co equalizers and the ﬁrs t tw o columns are inv ertible by assumption. Therefore the third co lumn is als o inv ertible. Since r : F T M → ( M , r ) is T - linear, we obtain: α − 1 ( M ,r ) = α − 1 ( M ,r ) B ( rη M ) = A ( r ) α − 1 F T M B ( η M ).  3. Hopf mona ds on closed mo noid al ca tegories In this section we deﬁne binar y left a nd right antipo des for a bimonad T on a closed mono ida l categ ory C and show that T is a Hopf monad if and o nly if T admits binary left and r igh t antipo des, or eq uiv alen tly , if the category of T - mo dules is closed mo noidal and the forgetful functor U T preserves internal Homs. When C is autonomous, Hopf mo nads as deﬁned in the present pap er coincide w ith Hopf monads deﬁned in [BV07] in ter ms of unary ant ip odes. The general results on Hopf mona ds on clo sed mono ida l categ o ries ar e stated in Section 3 .3 and the a utonomous case is studied in Section 3.4. The rest of the section is devoted to the pro ofs whic h ar e bas ed on a classiﬁcatio n of adjunction liftings (see Section 3.5). 3.1. Close d m onoidal categories. See [EK66] for a gener al reference . Let C b e a monoida l c ategory . Let X , Y be tw o ob jects of C . A left internal Hom fr om X to Y is an ob ject [ X, Y ] l endow ed with a mor phism ev X Y : [ X , Y ] l ⊗ X → Y such that, for each ob ject Z of C , the mapping  Hom C ( Z, [ X , Y ] l ) → Hom C ( Z ⊗ X , Y ) f 7→ ev X Y ( f ⊗ id X ) is a bijection. If a left in ternal Ho m from X to Y exists, it is unique up to unique isomorphism. A monoidal categor y C is left close d if left internal Homs exist in C . This is equiv alent to saying that, for every o b ject X of C , the endofunctor ? ⊗ X a dmits a right adjo int [ X , ?] l , with adjunction unit and counit: ev X Y : [ X , Y ] l ⊗ X → Y and co ev X Y : Y → [ X , Y ⊗ X ] l , called resp ectively the left evaluation and the left c o evaluation . Let C b e a left closed mono idal category . The left internal Homs of C give r is e to a functor: [ − , − ] l : C op × C → C where C op is the c a tegory o pposite to C . Moreover, from the ass ociativity and unitarity of the monoida l pro duct o f C , w e deduce isomorphisms [ X ⊗ Y , Z ] l ≃ [ X , [ Y , Z ] l ] l and [ 1 , X ] l ≃ X which we will a bstain from writing down in fo r m ulae. The co mposition c X,Y ,Z : [ Y , Z ] l ⊗ [ X , Y ] l → [ X , Z ] l of internal Homs is the natural tr a nsformation deﬁned by: ev X Z ( c X,Y ,Z ⊗ X ) = ev Y Z ([ Y , Z ] l ⊗ ev X Y ) . 12 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER Remark 3.1. If X is an ob ject of a mono idal categ ory C admitting a left dua l ( ∨ X , ev X , co ev X ) then, for ev ery o b ject Y of C , [ X , Y ] l = Y ⊗ ∨ X is a left internal Hom fro m X to Y , with ev aluation morphism ev X Y = Y ⊗ ev X . T her efore any left autonomous catego ry is left clos ed mo noidal. Remark 3.2. A left closed monoidal category C is left autono mous if and only if c X, 1 ,X : [ 1 , X ] l ⊗ [ X, 1 ] l → [ X , X ] l is an iso morphism for a ll ob ject X of C . In that case, ∨ X = [ X , 1 ] l is a le ft dua l of X , with ev aluatio n ev X = ev X 1 and co ev aluation co ev X = (ev X 1 ⊗ id ∨ X ) c − 1 X, 1 ,X co ev X 1 . One deﬁnes similarly right internal Homs and right close d monoidal ca tegories. A mo noidal category is r ig h t closed if and o nly if, fo r every ob ject X of C , the endofunctor X ⊗ ? ha s a right adjoint [ X , ?] r , with adjunction unit and counit: e ev X Y : X ⊗ [ X , Y ] r → Y and g co ev X Y : Y → [ X , X ⊗ Y ] l , called resp ectiv ely the right evaluation and the right c o evaluation . The right inter- nal Homs of a monoidal left right clo sed category C g iv e rise to a functor: [ − , − ] r : C op × C → C . Remark 3 .3. A right in ternal Hom in a monoidal categ ory C is a left internal Hom in C ⊗ op , and C is right closed if and only if C ⊗ op is left clo sed. A close d monoidal c ate gory is a monoidal categ ory which is b oth left and rig h t closed. 3.2. F unctors prese rving in ternal Hom s. Let X , Y b e ob jects of a monoida l category D which have a left internal Hom [ X , Y ] l , with ev a luation mor phism ev X Y : [ X , Y ] l ⊗ X → Y . A mono idal functor U : D → C is said to pr eserve the left internal Hom fr om X to Y if U [ X, Y ] l , endow e d with the ev alua tion U (ev X Y ) U 2 ([ X, Y ] l , X ) : U [ X , Y ] l ⊗ U X → U Y , is a left internal Hom from U X to U Y . A monoidal functor U : D → C b et ween left closed mono ida l categ o ries is left close d if it pres e rv es all left in ternal Homs. Let U : D → C b e a monoidal functor b etw een left closed mo no idal ca tegories. The natural tr ansformation U (ev X Y ) U 2 ([ X, Y ] l , X ) : U [ X , Y ] l ⊗ U X → U Y induces by universal prop ert y of internal Homs a natural tr ansformation: U l X,Y : U [ X , Y ] l → [ U X , U Y ] l . The mono idal functor U is left close d if and only if U l is an iso morphism. Similarly one deﬁnes monoidal functors pr eserving right internal H oms and right close d mo noidal functors. Lemma 3.4. L et U : D → C b e a st r ong monoidal funct or b etwe en left close d monoidal c ate gories. If U is c onservative, left close d, and C is left autonomous, then D is left autonomous. Pr o of. According to Remark 3.2, it is enough to show that, for any ob ject X of D , the comp osition morphism c X, 1 ,X : [ 1 , X ] l ⊗ [ X, 1 ] l → [ X , X ] l is an iso morphism. Since U is strong mono idal, U 2 and U 0 are isomorphisms. Consider the following HOPF M ONADS ON MONOIDAL CA TEGORIES 13 commutativ e diagram: U ([ 1 , X ] l ⊗ [ X , 1 ] l ) U ( c X, 1 ,X ) / / U [ X, X ] l U l X,X / / [ U X, U X ] l U [ 1 , X ] l ⊗ U [ X , 1 ] l U 2 ([ 1 ,X ] l , [ X, 1 ] l ) O O U l 1 ,X ⊗ U l X, 1 T T T T T * * T T T T T [ 1 , U X ] l ⊗ [ U X , 1 ] l c U X, 1 ,U X O O [ U 1 , U X ] l ⊗ [ U X , U 1 ] l [ U 0 ,U X ] l ⊗ [ U X,U − 1 0 ] l j j j j j 4 4 j j j j j Since U l is an is omorphism ( U being left clo sed) a nd c U X, 1 ,U X is inv ertible (by Remark 3.2), we obtain that U ( c X, 1 ,X ) is inv ertible. Now U is conser v ative. Hence c X, 1 ,X is a n isomorphism.  Prop osition 3.5. L et ( F : C → D , U : D → C ) b e a c omonoidal adj unction b etwe en monoidal left (r esp. right) close d c ate gories. Then ( F, U ) is a left (r esp. right) Hopf adjunction if and only if U is left (r esp. right) close d. Pr o of. W e pr o ve the left-handed version (the right one can b e done similarly ). Let ( F : C → D , U : D → C ) b e a comonoida l adjunction b et w een left closed mo no idal categorie s. F or a ny c ∈ Ob( C ) and d, e ∈ Ob( D ), set h e c,d :  Hom D ( F ( c ) ⊗ d, e ) → Hom D ( F ( c ⊗ U d ) , e ) α 7→ α H l c,d where H l is the left Hopf op erator of ( F, U ). Note that h e c,d is natural in c, d, e and one veriﬁes easily that it is the comp osition: Hom D ( F c ⊗ d, e ) ∼ − → Hom D ( F c, [ d, e ] l ) ∼ − → Hom C ( c, U [ d, e ] l ) u c,d,e / / Hom C ( c, [ U d, U e ] l ) ∼ − → Hom C ( c ⊗ U d, U e ) ∼ − → Hom D ( F ( c ⊗ U d ) , e ) , where u d,e c = Hom C ( c, U l d,e ) and a ll other maps are adjunction bijections . Assume that U is left close d. Let c ∈ Ob( C ) a nd d ∈ Ob( D ). Since U l d, − is a n isomorphism, u d, − c is an isomor phism, and so is h − c,d . Therefore H l c,d is inv er tible by the Y oneda lemma. Hence ( F, U ) is a left Hopf adjunction. Conv er sely , suppo se that ( F , U ) is a left Hopf adjunction. Let d, e ∈ Ob( D ). Since H l − ,d is an iso morphism, h e − ,d is an isomorphism, and so is u d,e − . Therefore U l d,e is inv ertible by the Y o neda lemma. Hence U is left closed.  3.3. Hopf monads and an tip o des i n the closed monoidal setti ng. Let T b e a bimonad o n a monoida l ca tegory C . If C is left closed, a binary left antip o de for T , or simply left antip o de for T , is a natural tra nsformation s l = { s l X,Y : T [ T X , Y ] l → [ X , T Y ] l } X,Y ∈ Ob( C ) satisfying the following tw o axioms: T  ev X Y ([ η X , Y ] l ⊗ X )  = ev T X T Y ( s l T X,Y T [ µ X , Y ] l ⊗ T X ) T 2 ([ T X , Y ] l , X ) , (1a) [ X , T Y ⊗ η X ] l co ev X T Y = [ X , ( T Y ⊗ µ X ) T 2 ( Y , T X )] l s l X,Y ⊗ T X T (co ev T X Y ) , (1b) for all o b jects X , Y of C . Similarly if C is right closed, a binary right antip o de for T , or simply right antip o de for T , is a natural transformation s r = { s r X,Y : T [ T X , Y ] r → [ X , T Y ] r } X,Y ∈ Ob( C ) 14 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER satisfying: T  e ev X Y ( X ⊗ [ η X , Y ] r )  = e ev T X T Y ( T X ⊗ s r T X,Y T [ µ X , Y ] r ) T 2 ( X, [ T X , Y ] r ) , (2a) [ X , η X ⊗ T Y ] r g coev X T Y = [ X , ( µ X ⊗ T Y ) T 2 ( T X , Y )] r s r X,T X ⊗ Y T ( g co ev T X Y ) , (2b) for all o b jects X , Y of C . With this deﬁnition of (binary ) antipo des, we hav e: Theorem 3. 6. L et T b e a bimonad on a left (r esp. right) close d monoidal c ate- gory C . The fol lowing assertions ar e e qu ival ent: (i) The bimonad T is a left (r esp. right) Hopf monad on C ; (ii) The monoidal c ate gory C T is left (r esp. right) close d and t he for getful func- tor U T is left (r esp. right) close d; (iii) The bimonad T admits a left (r esp. right) binary antip o de. This theorem is prov ed in Section 3 .6. Remark 3.7. If the equiv alent conditions of Theorem 3.6 are satisﬁed, in ternal Homs in C T may be constructed in terms of the an tipo des of T a s follows. If T is a left Hopf monad a nd C is left closed monoida l, then a left internal Hom for any t wo T - mo dules ( M , r ) and ( N , t ) is g iven by: [( M , r ) , ( N , t )] l =  [ M , N ] l , [ M , t ] l s l M ,N T [ r, N ] l  , ev ( M ,r ) ( N ,t ) = ev M N , and co ev ( M ,r ) ( N ,t ) = co ev M N . Similarly , if T is a right Hopf monad and C is rig h t closed monoidal, then a right int ernal Hom for any tw o T - mo dules ( M , r ) a nd ( N , t ) is given by: [( M , r ) , ( N , t )] r =  [ M , N ] r , [ M , t ] r s r M ,N T [ r, N ] r  , e ev ( M ,r ) ( N ,t ) = e ev M N , and g co ev ( M ,r ) ( N ,t ) = g co ev M N . In a ddition to characterizing Hopf monads on clo sed monoida l catego ries, the left and right antipo des, when they exist, a re unique a nd w ell-b ehav ed with resp ect to the bimo na d structure: Prop osition 3. 8. L et T b e a bimonad on a monoidal c ate gory C . (a) If C is left (r esp. right) close d and T admits a left (r esp. right) antip o de, then this antip o de is u nique. (b) Assu me C is left close d and T is a left Hopf monad. Then t he left antip o de s l for T satisﬁ es: s l X,Y µ [ T X,Y ] l = [ X , µ Y ] l s l X,T Y T ( s l T X,Y ) T 2 [ µ X , Y ] l , s l X,Y η [ T X,Y ] l = [ η X , η Y ] l , s l X ⊗ Y , Z T [ T 2 ( X, Y ) , Z ] l = [ X , s l Y ,Z ] l s l X, [ T ( Y ) ,Z ] l , s l T ( 1 ) ,X [ T 0 , X ] l = id T X , for al l obje cts X , Y , Z of C . (c) Assume C is right close d and T is a right Hopf monad. Then the right antip o de s r for T satisﬁ es: s r X,Y µ [ T X,Y ] r = [ X , µ Y ] r s r X,T Y T ( s r T X,Y ) T 2 [ µ X , Y ] r , s r X,Y η [ T X,Y ] r = [ η X , η Y ] r , s r X ⊗ Y , Z T [ T 2 ( X, Y ) , Z ] r = [ X , s r Y ,Z ] r s r X, [ T ( Y ) ,Z ] r , s r T ( 1 ) ,X [ T 0 , X ] r = id T X , for al l obje cts X , Y , Z of C . HOPF M ONADS ON MONOIDAL CA TEGORIES 15 The prop osition is pr o ved in Sectio n 3.6. Lastly , the antipo des and the in verses of the fusion op erators of a Hopf monad can b e expres sed in terms o f one another, as follows: Prop osition 3. 9. If T is a left Hopf monad on a left close d monoidal c ate gory C , then the inverse of the left fusion op er ator H l and the left antip o de s l ar e r elate d as fol lows: H l − 1 X,Y = T ( X ⊗ µ Y )ev T Y T ( X ⊗ T 2 Y ) ( s l T Y ,X ⊗ T 2 Y T (co ev T 2 Y X ) ⊗ id T Y ) , s l X,Y = [ X , T ev T X Y ] l [ η X , H l − 1 X, [ T X,Y ] l ] l co ev T X T [ T X,Y ] l . Similarly if T is a right Hopf monad on a right close d monoidal c ate gory C , then the inverse of the right fusion op er ator H r and the right ant ip o de s l ar e r elate d as fol lows: H r − 1 X,Y = T ( µ X ⊗ Y ) e ev T X T ( T 2 X ⊗ Y ) (id T X ⊗ s r T X,T 2 X ⊗ Y T ( g co ev T 2 X Y ) , s r X,Y = [ X , T e ev T X Y ] r [ η X , H r − 1 X, [ T X,Y ] r ] r g co e v T X T [ T X,Y ] r . The prop osition is pr o ved in Sectio n 3.6. 3.4. Hopf m onads on autonomous categories. The notion of Ho pf monad int ro duced in this pap er is a genera lization of the notion of Hopf monad on an autonomous catego ry intro duced in [BV07]. If T is a bimonad o n a left auto nomous ca tegory C , a unary left antip o de fo r T , or simply left antip o de for T , is a natural transformation s l = { s l X : T ( ∨ T X ) → ∨ X } X ∈ Ob( C ) satisfying: T 0 T (ev X ) T ( ∨ η X ⊗ X ) = ev T X  s l T X T ( ∨ µ X ) ⊗ T X  T 2 ( ∨ T X , X ); ( η X ⊗ ∨ X )co ev X T 0 = ( µ X ⊗ s l X ) T 2 ( T X , ∨ T X ) T (co ev T X ); for every ob ject X of C . Similarly if T is a bimonad on a r igh t autono mo us categ ory C , a unary right antip o de for T , or simply left antip o de for T , is a natural tra nsformation s r = { s r X : T (( T X ) ∨ ) → X ∨ } X ∈ Ob( C ) satisfying: T 0 T ( e ev X ) T ( X ⊗ η ∨ X ) = e ev T X  T X ⊗ s r T X T ( µ ∨ X )  T 2 ( X, ( T X ) ∨ ); ( X ∨ ⊗ η X ) g coev X T 0 = ( s r X ⊗ µ X ) T 2 (( T X ) ∨ , T X ) T ( g co ev T X ); In [BV07], a left (res p. right) Hopf monad T on a left (resp. right) auto nomous category C is deﬁned as a bimonad on C which admits a left (resp. right) unary antipo de or, e quiv alent ly by [BV07, Theo rem 3.8], whose categor y o f mo dules C T is left (resp. r igh t) autonomo us. This de ﬁnitio n, which makes sense only in the autonomous setting, agrees with that given in Section 2.7: Theorem 3. 10. L et C b e a lef t (r esp. right) aut onomous c ate gory and T b e a bimonad on C . Then the fol lowing assertions ar e e quivalent: (i) The bimonad T has a left (r esp. right) u nary ant ip o de; (ii) The bimonad T has a left (r esp. right) binary antip o de; (iii) The bimonad T is a left (re sp. right) Hopf monad. The theorem is prov ed in Section 3.6 16 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER Remark 3.11. The bina ry left a n tipo de s l X,Y and unary left a n tipo de s l X of a left Hopf monad T o n a left autonomous category are r elated as follows: s l X,Y = ( T X ⊗ s l Y ) T 2 ( X, ∨ T Y ) a nd s l X = ( T 0 ⊗ ∨ X ) s l X, 1 . Similarly the binary right antipo de s r X,Y and unary r ig h t antipo de s l X of a r igh t Hopf monad T o n a right autono mo us categ o ry a re related as follows: s r X,Y = ( s r Y ⊗ T X ) T 2 ( ∨ T Y , X ) and s r X = ( ∨ X ⊗ T 0 ) s r X, 1 . 3.5. Lifting adjunctions. In this sec tio n, ( T , µ, η ) is a monad o n a catego r y C and ( T ′ , µ ′ , η ′ ) is a monad on a categor y C ′ . A lift of a functor G : C → C ′ along ( T , T ′ ) is a functor e G : C T → C ′ T ′ such that U T ′ e G = GU T . It is a well-kno wn fact that such lifts e G ar e in bijective corres pon- dence with natural transforma tions ζ : T ′ G → GT sa tisfying: ζ µ ′ G = G ( µ ) ζ T T ′ ( ζ ) and ζ η ′ G = G ( η ) . Such a natural trans fo rmation ζ is called a lifting datum for G along ( T , T ′ ). The lift e G ζ corres p onding with a lifting datum ζ is deﬁned b y e G ζ ( M , r ) = ( G ( M ) , G ( r ) ζ M ) . Conv er sely , the lifting datum asso ciated with a lift e G is ζ = U T ′ ( ε ′ e GF T ) T ′ G ( η ) , where ε ′ denotes the counit of the adjunction ( F T ′ , U T ′ ). Consider tw o functors G, G ′ : C → C ′ , a lifting datum ζ for G , and a lifting datum ζ ′ for G ′ . Then a natural tra ns formation α : G → G ′ lifts to a natura l transformatio n e α : e G ζ → f G ′ ζ ′ (in the sense that U T ′ ( e α ) = α U T ) if and only if it satisﬁes ζ ′ T ′ ( α ) = α T ζ . Example 3.12. Let T b e a bimonad on a monoidal c a tegory C and ( M , r ) b e a T - mo dule. Then the endofunctors ? ⊗ M a nd M ⊗ ? of C lift to endofunctor s ? ⊗ ( M , r ) and ( M , r ) ⊗ ? of C T . The lifting da ta co rresp onding with these lifts a re the Hopf oper ators H l − , ( M ,r ) and H r ( M ,r ) , − of the comonoidal adjunction ( F T , U T ). Now let ( G : C → C ′ , V : C ′ → C ) b e an adjunction, with unit h : 1 C → V G and counit e : GV → 1 C ′ . A lift of t he adjunction ( G , V ) alo ng ( T , T ′ ) is an adjunction ( e G, e V ), where e G : C T → C ′ T ′ is a lift of G a long ( T , T ′ ), e V : C ′ T ′ → C T is a lift o f V along ( T ′ , T ), and the unit e h and counit e e of ( e G, e V ) a re lifts of h and e resp ectiv ely . Lifts of the adjunction ( G, V ) are in bijective cor r espo ndence with pair s ( ζ , ξ ), where ζ : T ′ G → GT and ξ : T V → V T ′ are natur al transformations sa tisfying the following axio ms : ζ µ ′ G = G ( µ ) ζ T T ′ ( ζ ) , (3a) ζ η ′ G = G ( η ) , (3b) ξ µ V = V ( µ ′ ) ξ T ′ T ( ξ ) , (3c) ξ η V = V ( η ′ ) , (3d) T ′ ( e ) = e T ′ G ( ξ ) ζ V , (3e) h T = V ( ζ ) ξ G T ( h ) . (3f ) Such a pair ( ζ , ξ ) is called a lifting datum for the adjunction ( G, V ) along ( T , T ′ ). HOPF M ONADS ON MONOIDAL CA TEGORIES 17 By adjunction, w e hav e a bijection Φ :  Hom ( T V , V T ′ ) → Hom ( GT , T ′ G ) ξ 7→ Φ( ξ ) = e T ′ G G ( ξ G ) GT ( h ) whose inv erse is given by Φ − 1 ( α ) = V T ′ ( e ) V ( α V ) h T V . Theorem 3.13. L et ζ : T ′ G → GT ′ b e a lifting datum for G along ( T , T ′ ) . Then the fol lowing assertions ar e e quivalent: (i) Ther e exists a natur al t r ansformation ξ : T V → V T ′ such that ( ζ , ξ ) is a lifting datum for the adjunction ( G, V ) along ( T , T ′ ) . (ii) ζ is invertible. If such is the c ase, ξ is unique and ξ = Φ − 1 ( ζ − 1 ) . The theo rem, whic h may be interpreted in terms of do ctrinal adjunctions, results immediately fro m the following lemma: Lemma 3. 14. L et ζ : T ′ G → GT and ξ : T V → V T ′ b e natura l tr ansformations. (a) Axiom (3 e) is e quivalent to Φ( ξ ) ζ = id T ′ G , and (3f) to ζ Φ( ξ ) = id GT . (b) If (3e) and (3f) hold, t hen (3a) is e quivalent to (3c) , and (3b) to (3d) . Pr o of. The adjunction bijection Hom ( T ′ GV , T ′ ) ∼ → H om ( T ′ G, T ′ G ), deﬁned by β 7→ β G T ′ G ( h ), sends T ′ ( e ) to id T ′ G , a nd e T ′ G ( ξ ) ζ V to e T ′ G G ( ξ G ) ζ V G T ′ G ( h ) = Φ( ξ ) ζ . Similarly the adjunction bijection H om ( T , V GT ) ∼ → Hom ( GT , GT ) sends h T to id GT and V ( ζ ) ξ G T ( h ) to ζ Φ( ξ ). Hence Part (a). Now assume that Axioms (3e) and (3 f) hold. In other words, ζ is inv ertible and ζ − 1 = Φ( ξ ). Then Axiom (3 a) and Axiom (3b) can b e re-w r itten as Φ( ξ ) G ( µ ) = µ ′ G T ′ (Φ( ξ ))Φ( ξ ) T and Φ( ξ ) G ( η ) = η ′ G , which transla te resp ectively to Axiom (3c) and Axio m (3d) via the adjunction bijections H om ( GT 2 , T ′ G ) ∼ → H om ( T 2 V , V T ′ ) and Hom ( G, T ′ G ) ∼ → Hom ( V , V T ′ ). Hence Part (b).  3.6. Bimonads and l ifting adjunctions. Here , by applying the results of Sec- tion 3.5 to bimonads in closed monoidal catego ries, we prov e Theorems 3.6 and 3.1 0 and Pr opos itions 3 .8 and 3.9. W e dea l with the left closed cas e , fro m which the right clo sed case results using the co opp osite bimonad (see Remark s 2.11 and 3.3). Let C b e a monoidal categ o ry and T b e a bimonad on C . Note that T × 1 C T is a bimo na d on C × C T . The monoida l tensor pro duct ⊗ : C T × C T → C T of C T is a lift of the functor ⊗ (1 C × U T ) : C × C T → C alo ng ( T × 1 C T , T ): C T × C T U T × 1 C T   ⊗ / / C T U T   C × C T ⊗ (1 C × U T ) / / C The co r resp o nding lifting datum ζ : T (1 C ⊗ U T ) → T ⊗ U T is given b y: ζ ( M ,r ) X = ( X ⊗ r ) T 2 ( X, M ) : T ( X ⊗ M ) → T ( X ) ⊗ M . Note that ζ = U T ( H l ), where H l denotes the left Hopf op erator o f the comonoidal adjunction ( F T , U T ), and s o , U T being conserv a tiv e, ζ is inv ertible if a nd only if T is a left Hopf mona d. Assume now that C is left clo s ed, that is, we have an adjunction (? ⊗ X , [ X , ?] l ) for ea ch X ∈ Ob( C ). In particular (? ⊗ M ) ( M ,r ) ∈ C T is a family of e ndo functors of C admitting r ig h t adjoints indexed by C T . Lemma 3. 15. The fol lowing assertions ar e e quivalent: (i) The c ate gory C T is left close d monoidal and U T is left close d; 18 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER (ii) F or e ach T - mo dule ( M , r ) , the adjunction (? ⊗ M , [ M , ?] l ) lifts to an ad- junction (? ⊗ ( M , r ) , e V ( M ,r ) ) . Pr o of. Let us prove that (i) implies (ii). Recall that since U T is left clos ed, w e have a natural isomorphism U l T : U T [ , ] l → [ U T , U T ] l , s ee Section 3.2. Thus, by transp ort of structure, we may cho ose left internal Homs in C T so that U T [( M , r ) , ( N , r )] l is equal to [ M , N ] l , U l T being the identit y . Then the adjunction (? ⊗ ( M , r ) , [( M , r ) , ?] l ) is a lift of the adjunction (? ⊗ M , [ M , ?] l ). Conv er sely (ii) implies (i) since the existence of an adjunction (? ⊗ ( M , r ) , e V ( M ,r ) ) lifting the a djunction (? ⊗ M , [ M , ?] l ) means ﬁr s tly tha t C T is left close d mono ida l, with [( M , r ) , ?] l = e V ( M ,r ) , and sec ondly that U l T is the identit y (and so U T is left closed).  Let us pr o ve Theorem 3 .6, Pr opos itions 3.8 and 3.9, a nd Theorem 3.10. Pr o of of The or em 3.6. Acco r ding to Theo rem 3.13, given a T - mo dule ( M , r ), the adjunction (? ⊗ M , [ M , ?] l ) lifts to a n a djunction (? ⊗ ( M , r ) , e V ( M ,r ) ) if and o nly if the lifting datum ζ ( M ,r ) is in vertible. Ther efore, b y Lemma 3.15, C T is left closed monoidal and U T is left clo sed if and only if ζ is inv ertible, and so if and only if T is a left Hopf mona d. Hence the equiv alence o f assertions (i) and (ii). Assume (i) holds, so tha t ζ is inv ertible. By Theo rem 3.13, for any T - mo dule ( M , r ), there e x ists a unique natural transfor mation ξ ( M ,r ) : T [ M , ?] l → [ M , T ] l such tha t ( ζ ( M ,r ) , ξ ( M ,r ) ) is a lifting datum for the a djunction (? ⊗ M , [ M , ?] l ) a long ( T , T ), which is given by ξ ( M ,r ) X = [ M , T (ev M X )] l [ M , ζ ( M ,r ) − 1 [ M ,X ] l ] l co ev M T [ M , X ] l . Note that ξ is natural in ( M , r ). Axioms (3e) and (3f) fo r this lifting datum are: T (ev M X ) = ev M T X ( ξ ( M ,r ) X ⊗ M ) ζ ( M ,r ) [ M ,X ] l , (4a) co ev M T X = [ M , ζ ( M ,r ) X ] l ξ ( M ,r ) X ⊗ M T (co ev M X ) . (4b) They translate to Axioms (1a) and (1b) of a left antipo de under the adjunction bijection: Ψ :  Hom ( T [ U T , 1 C ] l , [ U T , T ] l ) → Hom ( T [ T , 1 C ] l , [1 C , T ] l ) ξ 7→ s l = { s l X,Y = [ η X , T Y ] l ξ F T X Y } X,Y ∈ Ob( C ) Hence as sertion (iii). Conv er sely a ssume (iii) holds. Denote by s l the le ft antipo de o f T . Set ξ = Ψ − 1 ( s l ), that is, ξ ( M ,r ) Y = s l M ,Y T [ r, Y ] l . Under Ψ − 1 , Axioms (1a) and (1b) for s l translate to (4 a) and (4b). In particula r, for any T - mo dule ( M , r ), ξ ( M ,r ) satis- ﬁes (3e) a nd (3f). F urther more, Axioms (3 a) and (3b) ho ld for ζ ( M ,r ) as it is a lifting datum for ? ⊗ M . Th us, by Lemma 3.14, Axioms (3 c) a nd (3d) ho ld for ξ ( M ,r ) , that is: ξ ( M ,r ) X µ [ M ,X ] l = [ M , µ X ] l ξ ( M ,r ) T X T ( ξ ( M ,r ) X ) , (5a) ξ ( M ,r ) X η [ M ,X ] l = [ M , η X ] l . (5b) Therefore ( ζ ( M ,r ) , ξ ( M ,r ) ) is a lifting datum fo r the adjunction (? ⊗ M , [ M , ?] l ) along ( T , T ). Hence (ii) by Lemma 3.15. This concludes the pro of of Theorem 3 .6.  HOPF M ONADS ON MONOIDAL CA TEGORIES 19 Pr o of of Pr op osition 3.8. Part (a) results fr om the fact that if a natur al transfor- mation ξ satisfying (4a) and (4b) exists, it is uniq ue b y Theo rem 3.1 3. Let us prov e Part (b). Ass ume that T admits a left an tip o de s l . When tra nslated in ter ms of s l , Axioms (4a) and (4b) yield the compa tibility o f s l with µ and η . Given tw o T - mo dules ( M , r ) and ( N , t ), the T - action o f the left internal Hom [( M , r ) , ( N , t )] l obtained by lifting [ M , N ] l is [ M , t ] l s l M ,N T [ r, N ] l . Given a third T - mo dule ( P , p ), the T - linearity of the canonical isomor phism [( M , r ) ⊗ ( N , t ) , ( P , p )] l ≃ [( M , r ) , [( N , t ) , ( P , p )] l ] l , translated in terms of s l , yields the compatibility of s l to T 2 . Simila rly the T - linearity of the ca nonical isomorphis m [( 1 , T 0 ) , ( M , r )] l ≃ ( M , r ) yields the compatibilit y o f s l to T 0 . Hence Prop osition 3.8.  Pr o of of Pr op osition 3.9. Deno te b y s l the left antipo de o f T and set ξ = Ψ − 1 ( s l ). Recall that ξ ( M ,r ) Y = s l M ,Y T [ r, Y ] l and s l X,Y = [ η X , T Y ] l ξ F T X Y . By Theore m 3.13, ξ ( M ,r ) X = [ M , T (ev M X )] l [ M , ζ ( M ,r ) − 1 [ M ,X ] l ] l co ev M T [ M , X ] l , ζ ( M ,r ) − 1 X = ev M X ⊗ M ( ξ ( M ,r ) X ⊗ M ⊗ M ) T (co ev M X ) , where ζ ( M ,r ) X = H l X, ( M ,r ) . By Lemma 2 .1 8, we hav e: H l − 1 X,Y = ζ F T Y X − 1 and ζ ( M ,r ) − 1 X = T (id X ⊗ r ) H l − 1 X,M (id T X ⊗ η M ) . Hence the expression of s l in terms of H l − 1 , and c on versely .  Pr o of of The or em 3.10. W e pr o ve the left handed version. Asse r tions (ii) and (iii) are equiv alent by Theo rem 3.6. Assertion (iii) is equiv alent to C T and U T being left closed, and so to C T being left autonomous (using L emma 3.4 and the fact that a strong monoidal functor preserves left duals ). Hence (ii) is equiv a len t to (i) by [BV07, Theor em 3 .8].  4. Cr oss pr oducts and rela ted con str uctio ns In this section w e study the cr oss pro duct of Ho pf monads (previously intro duced in [BV09] for Hopf monads on autonomous catego ries). In par ticular we introduce the inv erse op eration, called the cross quo tien t. 4.1. F unctoriality o f categories o f mo dules . Let C b e a categ o ry . If T is a monad on C , then ( C T , U T ) is a ca tegory ov er C , that is, an ob ject of Ca t / C . An y morphism f : T → P o f monads on C induces a functor f ∗ :  C P → C T ( M , r ) 7→ ( M , rf M ) ov er C , that is, U T f ∗ = U P . Moreov er , any functor F : C P → C T ov er C is o f this form. This construction deﬁnes a fully faithful functor  Mon( C ) op → Cat / C T 7→ ( C T , U T ) If f : T → P is a morphism of bimonads o n a monoida l ca tegory C , then f ∗ : C P → C T is a str ic t monoidal functor over C , and any s trong mo noidal functor F : C P → C T ov er C (that is, such that U T F = U P as monoida l functor s) is o f this form (see [BV07, Lemma 2 .9]). Hence a fully faithful functor BiMon( C ) op → MonCat / C . 20 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER 4.2. Exactness prop erties of monads. A Hopf mo nad T on a monoidal cate- gory C admits a right adjoint if C is a utonomous (see [BV07, Co r ollary 3.12]), but not in genera l. In many cases, the ex istence o f a right adjoint can b e replaced by the weaker condition o f preserv a tion of reﬂexive coeq ualizers (deﬁned in Section 1). Lemma 4 .1 ([Lin69 ]) . L et C b e a c ate gory and T b e a monad on C pr eserving r eﬂexive c o e qualizers. Then: (a) A r eﬂexive p air of morphisms of C T whose image by U T has a c o e qualizer, has a c o e qualizer, and this c o e qualizer is pr eserve d by U T ; (b) If r eﬂexive c o e qualizers exist in C , they exist also in C T and U T pr eserves them. Lemma 4.2. L et C b e a monoidal c ate gory admitting r eﬂexive c o e qualizers, which ar e pr eserve d by m onoid al pr o duct on the left (r esp. right). I f T is a bimonad on C pr eserving r eﬂexive c o e qualizers, then C T has r eﬂex ive c o e qualizers which ar e pr eserve d by monoidal pr o duct on t he left (re sp. right). Pr o of. Let us prov e the r igh t-handed v ersion. Accor ding to Lemma 4.1, C T has reﬂexive co equalizers and U T preserves them. Let h b e a coe qualizer of a reﬂexive pair ( f , g ) in C T , and d b e an o b ject of C T . Denoting k b e a co equalizer o f the reﬂexive pair ( f ⊗ d, g ⊗ d ), the morphism h ⊗ d factoriz es uniquely as φk . Both U T ( h ⊗ d ) a nd U T k ar e co equalizers o f ( U T f , U T g ) (b ecause U T and U T ⊗ U T d pre- serve reﬂexive co equalizer s) s o U T φ is an is omorphism. Hence φ is an isomorphism, since U T is conser v ativ e. Thus h ⊗ d is a co equalizer o f ( f ⊗ d, g ⊗ d ).  4.3. Cross pro ducts. Let T b e a mo nad on a categ ory C . If Q is a monad on the category C T of T - mo dules, the monad of the comp osite adjunction  C T  Q U Q ' ' F Q g g C T U T ' ' F T g g C is called the cr oss pr o duct of T by Q and denoted b y Q ⋊ T (see [BV09, Section 3.7]). As an endofunctor of C , Q ⋊ T = U T QF T . The pro duct p and unit e of Q ⋊ T are: p = q F T Q ( ε QF T ) and e = v F T η , where q and v are the pro duct and the unit of Q , a nd η a nd ε ar e the unit and counit o f the adjunction ( F T , U T ). Note that the comp osition of tw o mona dic functors is not monadic in gener al. How ever: Prop osition 4.3 ([BW85]) . If T is a monad on a c ate gory C and Q is a monad on C T which pr eserves r eﬂexive c o e qualizers, then the for getful functor U T U Q is monadic with monad Q ⋊ T . Mor e over the c omp arison functor K : ( C T ) Q → C Q ⋊ T is an isomorphism of c ate gories. If T is a bimona d on a mono idal ca tegory C and Q is a bimonad on C T , then Q ⋊ T = U T QF T is a bimonad on C (since a comp osition of como noidal adjunctions is a comonoidal adjunction), with comonoidal structur e g iv en by: ( Q ⋊ T ) 2 ( X, Y ) = Q 2  F T ( X ) , F T ( Y )  Q  ( F T ) 2 ( X, Y )  , ( Q ⋊ T ) 0 = Q 0 Q  ( F T ) 0  . In that case the compa rison functor K : ( C T ) Q → C Q ⋊ T is strict monoidal. The cr oss pro duct is functorial in Q : the assignment Q 7→ Q ⋊ T deﬁnes a functor ? ⋊ T : BiMon( C T ) → BiMon( C ). HOPF M ONADS ON MONOIDAL CA TEGORIES 21 F rom Prop osition 2 .16 a nd P ropo sition 2.1 4, we deduce: Prop osition 4. 4. The cr oss pr o du ct of two left (r esp. right) Hopf monads is a left (r esp. right) H opf m onad . In p articular, the cr oss pr o duct of two Hopf monads is a Hopf monad. Example 4.5. Let H b e a bialg ebra over a ﬁeld k and A b e a H -module a lgebra, that is, an algebra in the monoida l categor y H Mo d of left H - mo dules. In this situation, we may form the cross pro duct A ⋊ H , which is a k -alg ebra (se e [Ma j9 5 ]). Recall that H ⊗ ? is a monad on V ect k and A ⊗ ? is a mona d o n H Mo d. Then: ( A ⊗ ?) ⋊ ( H ⊗ ?) = ( A ⋊ H ) ⊗ ? as monads. Moreover, if H is a quasitriangula r Hopf a lgebra and A is a H -module Hopf algebra , that is, a Ho pf a lgebra in the bra ided category H Mo d, then A ⋊ H is a Hopf algebr a ov er k , and ( A ⊗ ?) ⋊ ( H ⊗ ?) = ( A ⋊ H ) ⊗ ? as Hopf monads. Example 4.6. Let T b e a Hopf mo nad o n an autono mous category C . Assume T is c ent r alizable , that is , for all ob ject X o f C , the co end Z T ( X ) = Z Y ∈C ∨ T ( Y ) ⊗ X ⊗ Y exists (see [BV09]). In that case, the assig nment X 7→ Z T ( X ) is a Hopf monad on C , called the cen tralizer of T and denoted b y Z T . The centralizer Z T of T lifts canonically to a Hopf mo nad f Z T on C T , which is the cen tralizer of 1 C T . Then, by [BV09, Theo rem 6.5], D T = f Z T ⋊ T is a quasitria ngular Hopf monad, c a lled the double of T , and satis ﬁe s Z ( C T ) ∼ = C D T as bra ided categorie s, where Z denotes the categoric al center. Distributive laws, intro duced by Beck [Bec69] to enco de comp osition and lifting of monads, have b een ada pted to Hopf mona ds on autono mous ca tegories in [BV09 ]. The next corollar y deals with the ca se of Hopf monads on ar bitrary monoida l ca t- egories. Corollary 4 .7. L et P and T b e Hopf monads on a monoidal c ate gory C and Ω : T P → P T b e a c omonoida l distributive law of T over P . (a) If P is a Hopf monad, then the lift ˜ P Ω of P is a Hopf monad on C T . (b) If P and T ar e Hopf monads, then t he c omp osition P ◦ Ω T is a Hopf monad on C and ˜ P Ω ⋊ T = P ◦ Ω T as Hopf monads. Pr o of. Recall from [BV09, Theorem 4.7] that Ω deﬁnes a bimona d ˜ P Ω on C T , which is a lift of the bimo nad P , and a bimonad P ◦ Ω T on C , with underlying functor P T . Moreover P ◦ Ω T = ˜ P Ω ⋊ T as bimo na ds on C . The forg etful functor U T maps the fusion op erators of ˜ P Ω to those of P . Therefor e if P is a Hopf mona d, s o is ˜ P Ω (as U T is conserv a tive). If b oth P a nd T ar e Hopf mo nads, then ˜ P Ω ⋊ T is a Hopf monad by P rop osition 4 .4, and so is P ◦ Ω T .  Lemma 4.8. L et T b e a bimonad on a monoidal c ate gory C and Q b e a bimonad on C T . Assume that the monoidal pr o ducts of C T and ( C T ) Q pr eserve r eﬂexive c o e qualizers in the left (r esp. right) variable. If the adjunction ( F Q F T , U T U Q ) is a left (r esp. right) Hopf adjunction and T is a left (r esp. right) Hopf monad, then Q is a left (r esp. right) Hopf monad. Pr o of. Let us prove the left handed v ersion. Denote by H l , H ′ l , a nd H ′′ l the left Hopf op erators of the adjunctions ( F T , U T ), ( F Q , U Q ), and ( F Q F T , U T U Q ) resp ectiv ely . Assume that ( F Q F T , U T U Q ) is a left Hopf adjunction, that is H ′′ l is inv ertible. Assume a lso that T is a left Hopf monad. B y Theore m 2.15, H l is in vertible and it is enough to show that H ′ l is also in vertible. Let e be an arbitrar y ob ject of ( C T ) Q . 22 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER The na tural transfor mation H ′ l − ,e : F Q (? ⊗ U Q e ) → F Q ⊗ e is inv ertible on the imag e of F T , since H ′ l F T ,e = H ′′ l − ,e F Q ( H l − ,U Q ( e ) ) − 1 by Lemma 2.17. No w let ( M , r ) b e a T - mo dule. The co equalize r F T T ( M ) µ M / / F T ( r ) / / F T ( M ) r / / ( M , r ) is reﬂex ive b ecause F T ( r ) F T ( η M ) = id F T ( M ) = µ M F T ( η M ). This reﬂexive co equal- izer is pres erv ed by the functors F Q (? ⊗ U Q ( e )) and F Q ⊗ e , beca use F Q is a left adjoint and ? ⊗ U Q ( e ) and ? ⊗ e pr eserve reﬂexive co equalizers (by h yp o thesis). Hence H ′ l ( M ,r ) , e is inv e r tible.  4.4. Cross quotients. Let f : T → P b e a mo rphism of mona ds on a category C . W e say that f is cr oss qu otient able if the functor f ∗ : C P → C T is monadic. In that case, the mona d of f ∗ (on C T ) is ca lled the cr oss quotient of f and is denoted b y P ÷ | f T or simply P ÷ | T . No te that the comparison functor C P K / / f ∗   ; ; ; ; ; ( C T ) P ÷ | T U P ÷ | T { { v v v v v C T is then an isomor phism of categories (b y the last ass ertion of Theorem 2.1). Lemma 4. 9 ([Lin69]) . L et f : T → P b e a morphism of monads on a c ate gory C . The fol lowing assertions ar e e quivalent: (i) The morphism f is cr oss quotientable; (ii) The functor f ∗ admits a left adjoint; (iii) F or any T - mo dule ( M , r ) , the r eﬂexive p air F P T M P ( r ) / / p M P ( f M ) / / F P M admits a c o e qualizer F P M → G ( M , r ) in C P , wher e p is the pr o duct of P . If these c onditions hold, a left adjoi nt f ! of f ∗ is given by f ! ( M , r ) = G ( M , r ) . Pr o of. It r esults from Bec k’s theor e m (see Theor em 2 .1) that if U and V a r e com- po sable functor s s uc h that b oth U V and U are monadic, then V is monadic if a nd only if it admits a left adjoint. Thus (i) is equiv alent to (ii). Now let ( M , r ) b e a T - mo dule and ( N , ρ ) b e a P - mo dule. The pair of Asser- tion (iii) is reﬂexive (since F P ( η M ) is a co mmon retraction). Via the adjunction bijection Hom C P ( F P M , ( N , ρ )) ≃ Hom C ( M , U P ( N , ρ )) = Hom C ( M , N ) , morphisms F P M → ( N , ρ ) which co equalize that pair co rresp ond with T - linear morphisms ( M , r ) → f ∗ ( N , ρ ). Ther e fore (ii) is equiv a len t to (iii). W e co nc lude using the last assertio n of Theorem 2.1.  Remark 4 .10. A mor phism f : T → P of monads o n C is cr oss quotientable whenever C admits co equalizers of reﬂexive pairs a nd P preserve them. A cross quotien t of bimonads is a bimonad: let f : T → P b e a cross quotientable morphism o f bimonads on a mo noidal category C . Since f ∗ is stro ng monoidal, P ÷ | f T is a bimo nad on C T and the co mparison functor K : C P → ( C T ) P ÷ | f T is an isomorphism of monoidal categ o ries. The cross quo tien t is inv erse to the cross pro duct in the following sense: Prop osition 4. 11. L et T b e a (bi)monad on a (monoidal) c ate gory C . HOPF M ONADS ON MONOIDAL CA TEGORIES 23 (a) If T → P is a cr oss quotientable morphism of (bi)monads on C , then ( P ÷ | T ) ⋊ T ≃ P as (bi)monads. (b) L et Q b e a (bi)monad on C T such that U T U Q is monadic. Then the u nit of Q deﬁnes a cr oss quotientable morphism of (bi)monads T → Q ⋊ T and ( Q ⋊ T ) ÷ | T ≃ Q as (bi)monads. Pr o of. Let us prove Part (a). Since C P ≃ ( C T ) P ÷ | T , the functor U P ÷ | T U T is monadic. Hence an isomo rphism C P ≃ C ( P ÷ | T ) ⋊ T of (monoidal) ca tegories ov er C , which induces an isomorphis m ( P ÷ | T ) ⋊ T ≃ P of (bi)monads on C . Let us prov e Part (b). Set f = u ⋊ T : T → Q ⋊ T , where u is the unit of Q . W e hav e a commutativ e diag ram o f (monoidal) functors: ( C T ) Q K / / U Q   C Q ⋊ T f ∗ z z v v v v v v v v v U Q ⋊ T   C T U T / / C where K is the comparison functor of the adjunction ( F Q F T , U T U Q ). Since K is a equiv ale nce, the functor f ∗ is monadic, with (bi)monad ( Q ⋊ T ) ÷ | T . Hence K induces a n iso morphism o f (bi)monads ( Q ⋊ T ) ÷ | T ≃ Q .  Remark 4.1 2 . Let T b e a bimo nad on a monoidal category C . Let BiMon( C T ) m be the full sub category o f BiMon( C T ) who se ob jects ar e monads Q on C T such that U T U Q is monadic. Let T \ BiMon( C ) q be the full s ubcatego ry of T \ B iMon ( C ) whose ob jects are quotientable morphisms o f bimonads from T . Then the functor  BiMon( C T ) → T \ BiMon( C ) Q 7→ ( Q, T → Q ⋊ T ) induces an equiv alence of categor ies BiMon( C T ) m ≃ T \ BiMon( C ) q , with quasi- inv ers e given by ( T → P ) 7→ ( P ÷ | T ). Under suitable exactness a ssumptions, if P and T are Ho pf monads, so is P ÷ | T : Prop osition 4.1 3. L et C b e a monoidal c ate gory admitting r eﬂex ive c o e qualizers, and whose monoida l pr o duct pr eserves r eﬂexive c o e qualizers in t he left (r esp. right) variable. L et T and P b e two left (r esp. right) Hopf monads on C which pr eserve r eﬂexive c o e qu alize rs. Then any morphism of bimonads T → P is cr oss quotientable and P ÷ | T is a left (r esp. right) H opf m onad . Pr o of. Let us prov e the left-handed version. The mo rphism f is cr oss q uotien table by Remar k 4.10, and so P ≃ ( P ÷ | f T ) ⋊ T a s bimonads. The monoida l pro ducts of C T and C P preserve reﬂexive co equalizers in the left v ariable by Lemma 4 .2. Applying Lemma 4.8 to the bimonads T and P ÷ | T , we get that P ÷ | T is a left Hopf monad.  Example 4.14 . Let f : L → H b e a morphis m of Hopf a lgebras over a ﬁeld k , s o that H b ecomes a L - bimo dule b y setting ℓ · h · ℓ ′ = f ( ℓ ) hf ( ℓ ′ ). The morphis m f induces a morphism of Hopf monads on V ect k : f ⊗ k ? : L ⊗ k ? → H ⊗ k ? 24 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER which is cr oss quotientable, and ( H ⊗ ?) ÷ | ( L ⊗ ?) is a k - linear Hopf monad on the monoidal categ ory L Mo d given by N 7→ H ⊗ L N . This construction deﬁnes an equiv alence o f categor ies L \ HopfAlg k → HopfMon k ( L Mo d) , where L \ HopfAlg k is the categ o ry o f Hopf k -a lgebras under L and HopfMon k ( L Mo d) is the ca tegory of k -line a r Hopf monads on L Mo d. 5. Hopf monads associa ted with Hopf algebras a nd bosoniz a tion Examples o f Hopf monads may be o bta ined from Hopf algebra s. F or insta nce, any Hopf alg ebra A in a br aided ca tegory B gives rise to Hopf mona ds A ⊗ ? a nd ? ⊗ A on B , see E xample 2.10. More gene r ally , any Hopf algebra ( A, σ ) in the center Z ( C ) of a mono ida l C gives rise to a Hopf monad A ⊗ σ ? on C (see Sec tion 5 .3, or [BV0 9 ] for the autono mous cas e). Hopf monads of this form ar e c a lled representable. The main result of this section asserts that a Hopf mona d on a mo noidal catego ry is representable if and o nly if it is augmented, that is, endow e d with a Hopf mona d morphism from itself to the identit y (see Theorem 5 .17) . More g enerally , given a Hopf mo nad T o n C a nd a Hopf algebra ( A , σ ) in the center Z ( C T ) of the categor y of T - mo dules, we construct a Ho pf monad A ⋊ σ T on C of which T is a retra ct. Co n versely , under suitable exactness conditions (inv o lving reﬂexive co equalizer s), any Hopf mona d P o f whic h T is a r e tr act is of the form A ⋊ σ T for so me Hopf algebra ( A , σ ) in Z ( C T ). 5.1. Lax braidings, lax half-braidings and lax cen ter. A lax br aiding of a monoidal ca teg ory C is a natura l tr ansformation τ = { τ X,Y : X ⊗ Y → Y ⊗ X } X,Y ∈ Ob( C ) satisfying: τ X,Y ⊗ Z = (id Y ⊗ τ X,Z )( τ X,Y ⊗ id Z ) , τ X ⊗ Y , Z = ( τ X,Z ⊗ id Y )(id X ⊗ τ Y ,Z ) , τ X, 1 = id X = τ 1 ,X . A lax br aide d c ate gory is a monoidal ca tegory endowed with a la x braiding. A br aiding is a n in vertible lax br aiding, and a br aide d c ate gory is a mo noidal catego ry endow ed with a braiding . Let C be a monoidal category and M an ob ject of C . A lax half-br aiding for M is a natural transforma tio n σ : M ⊗ 1 C → 1 C ⊗ M such that σ Y ⊗ Z = (id Y ⊗ σ Z )( σ Y ⊗ id Z ) and σ 1 = id M . The pair ( M , σ ) is then ca lled a lax half-br aiding of C . The lax c enter of C (see [Sch00, DPS07]) is the la x braided ca tegory Z lax ( C ) deﬁned as follows. Ob jects of Z lax ( C ) are left half-br aidings of C . A mor phism in Z lax ( C ) from ( M , σ ) to ( M ′ , σ ′ ) is a mor phism f : M → M ′ in C suc h that: (id 1 C ⊗ f ) σ = σ ′ ( f ⊗ id 1 C ). The monoidal pr oduct a nd lax bra iding τ a re: ( M , σ ) ⊗ ( N , γ ) =  M ⊗ N , ( σ ⊗ id N )(id M ⊗ γ )  and τ ( M ,σ ) , ( N , γ ) = σ N . A half br aiding is a lax half braiding ( M , σ ) such that σ is inv ertible. The c enter of C is the full monoidal sub category Z ( C ) ⊂ Z lax ( C ) whose ob jects a re half braidings of C . It is a bra ide d category , with braiding induced b y τ . Note that if C is a utonomous, lax half braidings a re in fact half br aiding, so that the lax cen ter Z lax ( C ) co incides with the center Z ( C ). HOPF M ONADS ON MONOIDAL CA TEGORIES 25 5.2. Hopf alg ebras in lax braided categorie s . Let B b e a lax braided ca tegory , with lax braiding τ . A bialgebr a in B is an ob ject A of B endow ed with a n alg ebra structure ( m, u ) and a coa lgebra str uc tur e (∆ , ε ) in B s atisfying: ∆ m = ( m ⊗ m )(id A ⊗ τ A,A ⊗ id A )(∆ ⊗ ∆) , ∆ u = u ⊗ u, εm = ε ⊗ ε, εu = id 1 . Bialgebra s in B , together with morphisms of bialgebr as (deﬁned in the o b vious wa y), form a catego ry BiAlg ( B ). Let A be a bia lg ebra in B . An antip o de o f A is a morphism S : A → A in B such that: m ( S ⊗ id A )∆ = uε = m (id A ⊗ S )∆ . If it exists, an an tipo de for A is unique, it s atisﬁes: S m = mτ A,A ( S ⊗ S ) , S u = u, ∆ S = ( S ⊗ S ) τ A,A ∆ , εS = ε , and we hav e: τ A,A = ( m ⊗ m )( S ⊗ ∆ m ⊗ S )(∆ ⊗ ∆). If τ A,A is in vertible, an op antip o de of A is a morphism S ′ : A → A in B s uch that: mτ − 1 A,A ( S ′ ⊗ id A )∆ = uε = mτ − 1 A,A (id A ⊗ S ′ )∆ . If it exists, an opant ip ode for A is unique. If τ A,A is invertible, the bialgebra A admits an a n tipo de and an opantipo de if a nd only if it a dmits an invertible a n tipo de, or eq uiv alen tly , an in vertible opantipo de. When such is the case, the opant ip ode is the inv erse of the antipo de. Let A b e a bialgebra o n a lax braided category B , with la x half-bra iding τ . The fusion op er ator of A is the morphism H = ( A ⊗ m )(∆ ⊗ A ) = P S f r a g r e p la c e m e n t s A A A A : A ⊗ A → A ⊗ A. The opfusion op er ator of A is the morphism H ′ = ( m ⊗ A )( A ⊗ τ A,A )(∆ ⊗ A ) = P S f r a g r e p la c e m e n t s A A A A τ A,A : A ⊗ A → A ⊗ A. Lemma 5. 1. (a) The bialgebr a A admits an antip o de S if and only if its fusion op er ator H is invertible. If su ch is t he c ase, S = ( ε ⊗ A ) H − 1 ( A ⊗ u ) , H − 1 = ( A ⊗ m )( A ⊗ S ⊗ A )(∆ ⊗ A ) . (b) If τ A,A is invertible, the bialg ebr a A admits an op antip o de S ′ if and only if its opfusion op er ator H ′ is invertible. If su ch is t he c ase, S ′ = ( ε ⊗ A ) H ′ − 1 ( u ⊗ A ) , H ′ − 1 = τ − 1 A,A ( m ⊗ A )( S ′ ⊗ A ⊗ A )( τ − 1 A,A ⊗ A )( A ⊗ ∆) . A Hopf algebr a in a lax braide d categ ory B , with lax bra iding τ , is a bialgebra A in B admitting an inv ertible antipo de and such that τ A,A is inv ertible. Hopf algebras in B form a full sub categor y o f BiAlg( B ) denoted b y HopfAlg ( B ). Remark 5 .2. If B is a bra ide d catego ry , the mirro r B of B is the braided ca t- egory obtained when the br aiding τ of B is replaced by its mirror τ (deﬁned by τ X,Y = τ − 1 Y ,X ). If ( A, m, u, ∆ , ε ) is a bialgebra in a braided category B , one deﬁnes a bialgebra A op in B by setting A op = ( A, m op , u, ∆ , ε ), with m op = mτ − 1 A,A . W e hav e ( A op ) op = A . An opantipo de for A is an an tip o de for A op . The bimonads 26 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER A ⊗ ? and ? ⊗ A op are iso morphic via τ A, − . See [BV09, Sectio n 1.11 and E x ample 2.3] for details. 5.3. Hopf m onads represented b y cen tral Ho pf algebras. Let C b e a monoidal category . A (lax) c entr al algebr a (resp. c o algebr a , r esp. bialgebr a , resp. H opf alge- br a ) of C is a n algebra (r e sp. coalge bra, r e sp. bialgebra , resp. Hopf alg ebra) in the (lax) center of C . An y lax central coalgebr a ( A, σ ) o f C gives ris e to a como noidal endofunctor of C , denoted by A ⊗ σ ?, deﬁned b y A ⊗ ? as a functor and endow ed with the comono idal structure: ( A ⊗ σ ?) 2 ( X, Y ) = ( A ⊗ σ X )(∆ ⊗ X ) ⊗ Y = P S f r a g r e p la c e m e n t s A A A X X Y Y σ X , ( A ⊗ σ ?) 0 = ε = P S f r a g r e p la c e m e n t s A X Y σ X A , where ∆ and ε deno te the copro duct and co unit o f ( A, σ ). F or a n y lax cent ral bialgebra ( A, σ ) o f C , the comonoidal endo functor A ⊗ σ ? is a bimonad on C with monad structure given b y: µ X = m ⊗ X = P S f r a g r e p la c e m e n t s A A A X X and η X = u ⊗ X = P S f r a g r e p la c e m e n t s A X X , where m and u are the pro duct and unit of A . Denote b y A Mo d σ the mo no idal category C A ⊗ σ ? , that is, the category of left A - mo dule (in C ) with monoidal pr oduct ( M , r ) ⊗ ( N , s ) = ( M ⊗ N , ω ), where ω = P S f r a g r e p la c e m e n t s A M M r s N N σ M , and monoidal unit ( 1 , ε ). The bimonads o f the form A ⊗ σ ? can be c haracter ized as follows: Lemma 5.3. L et A b e an obje ct of C and c onsider the endofunctor T = A ⊗ ? of C . L et ∆ : A → A ⊗ A and ε : A → 1 b e morphisms in C and σ : A ⊗ ? → ? ⊗ A b e a natur al tr ansformation such that σ 1 = id A . Set T 2 ( X, Y ) = ( A ⊗ σ X ⊗ Y )(∆ ⊗ X ⊗ Y ) and T 0 = ε. Then the fol lowing c onditions ar e e qu iva lent: (i) ( T , T 2 , T 0 ) is a c omonoidal endofunctor of C ; (ii) σ is a lax half br aiding for A and ( A, σ ) is a c o algebr a in Z lax ( C ) with c opr o duct ∆ and c ounit ε . Assume these e quivalent c onditions hold. Then T = A ⊗ σ ? as c omonoidal funct ors. F u rthermor e, let m : A ⊗ A → A and u : 1 → A b e morphisms in C and set: µ = m ⊗ ? : T 2 → T and η = u ⊗ ? : 1 C → T . Then the fol lowing c onditions ar e e qu iva lent: (iii) T is a bimonad with pr o duct µ , unit η , and c omonoida l structure ( T 2 , T 0 ) ; (iv) ( A, σ ) is a lax c entra l bialgebr a of C with pr o duct m , unit u , c opr o duct ∆ , and c ounit ε . If these e quivalent c onditions hold, T = A ⊗ σ ? as bimonads. Pr o of. The veriﬁcation, lengthy but stra igh tforward, is left to the r eader.  HOPF M ONADS ON MONOIDAL CA TEGORIES 27 Let ( A, σ ) b e a lax cent ral bialgebra o f C , that is, a bia lgebra in Z lax ( C ). The left and right fusion o pera to rs o f the monad A ⊗ σ ? ar e: H l X,Y = ( A ⊗ X ⊗ m )( A ⊗ σ X ⊗ A )(∆ ⊗ X ⊗ A ) ⊗ Y = P S f r a g r e p la c e m e n t s A A A A X X Y Y σ X , H r X,Y = ( m ⊗ X ⊗ A )( A ⊗ σ A ⊗ X )(∆ ⊗ A ⊗ X ) ⊗ Y = P S f r a g r e p la c e m e n t s A A A A X X Y Y σ A ⊗ X . Prop osition 5.4. L et ( A, σ ) b e a lax c entr al bialgebr a in C , and let A ⊗ σ ? b e the c orr esp onding bimonad on C . Then: (a) The fol lowing c onditions ar e e quivalent: (i) A ⊗ σ ? is a left Hopf monad; (ii) A ⊗ σ ? is a left pr e-Hopf monad; (iii) A admits an antip o de; (b) The fol lowing c onditions ar e e quivalent: (i’) A ⊗ σ ? is a right Hopf monad; (ii’) A ⊗ σ ? is a right pr e-H opf m onad; (iii’) σ is invertible and A admits an op antip o de. In p articular, t he bimonad A ⊗ σ ? is a Hopf monad if and only if A ⊗ σ ? is a pr e-Hopf monad, if and only if ( A, σ ) is a c entr al Hopf algebr a of C , that is, a Hopf algebr a in the c enter Z ( C ) . Remark 5.5 . Let ( A, σ ) b e a central Hopf algebr a of C a nd A Mo d σ be the mono idal category o f left A - mo dules (with mono idal pro duct induced by σ ). Then the full sub c ategory A mo d σ ⊂ A Mo d σ of le ft A - mo dules ( M , r ) whose underling ob ject M has a left and a right dual is autonomous. Remark 5.6. If B is a br a ided category , then its braiding τ deﬁnes a fully faithful braided functor  B → Z ( B ) X 7→ ( X , τ X, − ) which is a mo noidal s ection of the forgetful functor Z ( B ) → B . In particular if A is a bialgebra in B , then ( A, τ A, − ) is a cen tral bialg ebra of B and w e hav e A ⊗ ? = A ⊗ τ A, − ? as bimonads on B , w her e A ⊗ ? is the bimonad construc ted in Example 2.10. Also, if A is a bialg e bra in B , then A op is a bialg ebra in the mirror B of B (see Remark 5 .2), ( A op , τ A, − ) is a cen tral bialg ebra of B , where τ is the mirror braiding of τ , and ? ⊗ A ≃ A op ⊗ τ A, − ? as bimonads on B . Mor e o ver A is a Hopf alg ebra in B if and only if ( A, τ A, − ) is a central Hopf alge br a of B , if and only if A ⊗ ? is a Hopf mona d on B , if and only if ? ⊗ A is a Hopf mo nad on B . Pr o of of Pr op osition 5.4. Le t H l be the left fusion op erator of T = A ⊗ σ ? and H be the fusion op erator of A . W e hav e: H l X,Y = H l X, 1 ⊗ Y and H l 1 , 1 = H . Thus the bimona d T is a left Hopf mo na d if and only if H l − , 1 is an isomor phism, and T is a left pre-Ho pf monad if and only if H is an isomorphism. Hence (ii) is equiv alent to (iii) since, by Lemma 5.1, H is inv ertible if and o nly if A admits an antipo de. Assuming (iii) and denoting S the antipo de of A , o ne veriﬁes easily that 28 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER ( A ⊗ X ⊗ m )( A ⊗ σ X ⊗ A )(( A ⊗ S )∆ ⊗ X ⊗ A ) is inv er se to H l X, 1 . Therefor e (iii) implies (i). Hence Part (a) of the pr opos ition, since (i) implies (ii) is tr ivial. Let us prov e Part (b). Denote by H r the r igh t fusion op erator of T a nd H ′ the opfusion op erator o f A . Since H r X,Y = H r X, 1 ⊗ Y , the bimonad T is a r igh t Hopf monad if and only if it is a right pre-Hopf monad. Hence (i’) is equiv a le nt to (ii’). Mo reov er , we hav e: H r X, 1 = ( A ⊗ σ X )( H ′ ⊗ X ). If (iii’) holds, then σ and H ′ are inv er tible b y Lemma 5.1, and so H r − , 1 is an iso morphism. Hence (iii’) implies (ii’). Co n versely , if H r − , 1 is an iso morphism, then in particular H ′ = H r ( 1 , 1 ) is inv ertible, and A ⊗ σ is invertible. Since 1 is a retract of A , this implies that σ is inv ertible. Hence (ii’) implies (iii’). This completes the pro of of Part (b). In pa r ticular T is a Ho pf monad if and only if σ is inv e r tible and ( A, σ ) admits an antipo de and an opant ip ode, in other words, ( A, σ ) is a Hopf alg ebra in Z ( C ).  5.4. Characterization of representable Ho pf monads. L e t C b e a monoida l category . A bimonad T on C is augmente d if it is endow ed with an augmentation , that is, a bimonad mo r phism e : T → 1 C . Augment ed bimona ds on C form a c a tegory B iMon( C ) / 1 C , whose o b jects ar e augmented bimonads on C , and morphisms betw een tw o augmented bimo na ds ( T , e ) and ( T ′ , e ′ ) ar e morphisms of bimonads f : T → T ′ such that e ′ f = e . If ( A, σ ) is a lax c e ntral bia lgebra of C , the bimona d A ⊗ σ ? (se e Section 5.3) is augmented, with a ugmen tation e = ε ⊗ ? : A ⊗ σ ? → 1 C , where ε is the counit of ( A, σ ). Hence a functor BiAlg ( Z lax ( C )) → BiMon( C ) / 1 C which, acco rding to Prop osition 5.4, induces b y r estriction a functor R :  HopfAlg( Z ( C )) → Ho pfMo n( C ) / 1 C ( A, σ ) 7→ ( A ⊗ σ ? , ε ⊗ ?) where HopfMon( C ) / 1 C denotes the category of aug men ted Hopf monads on C . Theorem 5 .7. The functor R is an e quivalenc e of c ate gories. In o ther words, Hopf monads repr e sen table by central Hopf algebr as are no thing but augmented Hopf monads . Theorem 5.7 is prov ed in Section 5.6. Remark 5.8. Hopf monads a re not repr esen table in g eneral. A c o un terexample is given in [BV09] in terms of centralizers. Let T b e a centralizable Hopf monad on an autono mous categor y C (se e E x ample 4.6). In genera l the centralizer Z T of T is not r e presen table by a Hopf alg e br a. F o r example, let C = G - vect b e the categor y of ﬁnite-dimensiona l G - graded vector spaces over a ﬁeld k for so me ﬁnite group G . The iden tit y 1 C of C is centralizable and its cen tralizer Z 1 C is representable if and only if G is a belian (see [BV09, Remark 9.2]). Hopf mona ds on a braided ca teg ory B whic h ar e representable by Ho pf alg ebras in B can also b e characterized as follows: Corollary 5.9. L et T b e a Hopf monad on a br aide d c ate gory B . Then T is isomorphi c to the Hopf monad A ⊗ ? for some Hopf algebr a A in B if and only if it is endowe d with an augmentation e : T → 1 C c omp atible with the br aiding τ of B in the fol lowing sense: ( e X ⊗ T 1 ) T 2 ( X, 1 ) = ( e X ⊗ T 1 ) τ T 1 ,T X T 2 ( 1 , X ) for al l obje ct X of B . The coro lla ry is prov ed in Section 5.6. Remark 5.10. Let T be a cen tralizable Hopf monad on a br aided autonomous B (see Remark 5.8). Then the centralizer Z T of T is repres e ntable by a Hopf algebra HOPF M ONADS ON MONOIDAL CA TEGORIES 29 C T = R Y ∈B ∨ T ( Y ) ⊗ Y in B (see [B V09, Theo r em 8 .4]). This repres en tabilit y result may b e re c o vered fr o m Co rollary 5.9, obser ving that e X = Z Y ∈C (ev Y ⊗ X )( ∨ η Y ⊗ τ − 1 Y ,X ) : Z T ( X ) → X deﬁnes a n aug men tation o f Z T which is compatible with the braiding τ of B . 5.5. Bosoni zat ion. Let C b e a mono ida l categor y . Given a Hopf monad ( T , µ, η ) on C a nd a central Hopf a lgebra ( A , σ ) of C T (that is, a Hopf algebra in the center Z ( C T ) of C T ), set: A ⋊ σ T = ( A ⊗ σ ?) ⋊ T . As a cr o ss pro duct of Hopf monads , A ⋊ σ T is a Hopf monad on C (see Prop osi- tion 4.4). Set A = ( A, a ), where A = U T ( A ) and a is the T - action on A . As a n endofunctor of C , A ⋊ σ T = A ⊗ T . The pro duct p and unit v o f A ⋊ σ T are: p X = P S f r a g r e p la c e m e n t s A A T X T ( A ⊗ T X ) a µ X T 2 ( A, T X ) and v X = P S f r a g r e p la c e m e n t s A T X T ( A ⊗ T X ) a µ X T 2 ( A , T X ) A X T X η X . The co mo noidal str ucture of A ⋊ σ T is given by: ( A ⋊ σ T ) 2 ( X, Y ) = P S f r a g r e p la c e m e n t s A A A T X T Y T ( X ⊗ Y ) T 2 ( X, Y ) σ F T ( X ) and ( A ⋊ σ T ) 0 = P S f r a g r e p la c e m e n t s A T X T Y T ( X ⊗ Y ) T 2 ( X , Y ) σ F T ( X ) A T ( 1 ) T 0 . Denoting by u and ε the unit and co unit o f ( A , σ ), the mor phisms ι = u ⊗ T : T → A ⋊ σ T and π = ε ⊗ T : A ⋊ σ T → T are Ho pf monads morphisms such that π ι = id T . Hence T is a retract o f A ⋊ σ T in the categ ory Ho pfMon( C ) of Hopf monads o n C . Example 5.11. Let T b e a centralizable Hopf monad on a autonomo us categor y C and D T be the double of T (see E x ample 4.6). If T is quasitriangula r (see [BV07]), then C T is braided and T is a re tr act of D T . In that case, the braided catego ry C T admits a co end C , which is a Hopf algebr a , and D T = C ⋊ τ C, − T where τ is the braiding o f C T . Conv er sely , under exactness assumptions, a Ho pf monad whic h admits T as a retract is of the for m A ⋊ σ T for some central Hopf alg ebra ( A , σ ) of C T . This results from the fact that augmented Hopf monads ar e repres e n ta ble, using the notio n of cross quotient studied in Sectio n 4.4: Corollary 5 .12. L et P and T b e Hopf monads on a monoidal c ate gory C such that T i s a r et ra ct of P . Assume that r eﬂexive c o e qualizers exist in C and ar e pr eserve d by P and the monoidal pr o duct of C . Then ther e exists a c entra l Hopf algebr a ( A , σ ) of C T and an isomorphism of Hopf monads P ≃ A ⋊ σ T such that we have a c ommutative diagr am of Hopf monads: P ≃ / / ' ' O O O O O O O O O O O A ⋊ σ T        T Z Z 4 4 4 4 4 o o o 7 7 o o o o o o = / / T 30 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER Pr o of. Denote by f : T → P and g : P → T the morphisms of Hopf monads making T a retr a ct of P , tha t is, g f = id T . B y ass umption P preserves reﬂexive co equalizers and so, sinc e T is a r etract of P , the Hopf monad T preser ves reﬂexive co equalizers to o. By Le mma 4 .2, reﬂexive co equalize r s exis t in C T and C P and ar e pr eserved b y the mo no idal pro duct. By Pr o positio n 4.1 3, the crosse d quotient P ÷ | T (relative to f : T → P ) exists and is a Hopf mona d on C T . On the other hand, by functoriality of the cross quotient (see Remark 4.12), g : P → T induces a morphism of bimonads g ÷ | T : P ÷ | T → T ÷ | T ∼ = 1 C T . In other w ords the Hopf monad P ÷ | T is augmented. By Theor em 5.7, there exists a Hopf algebra ( A , σ ) in Z ( C T ) such that P ÷ | T = A ⊗ σ ?. By P ropo sition 4.11, P = ( P ÷ T ) ⋊ T = ( A ⊗ σ ?) ⋊ T = A ⊗ σ T as Hopf monads. The co mmutativit y of the diagram is straightforw ard.  Remark 5.13. Let H b e a Hopf alge bra o ver a ﬁeld k , and A a Hopf alg ebra in the braided ca teg ory of Y etter-Drinfeld mo dules H H Y D . In that situation, Radford constructed a Hopf algebra A # H , known as Ra dfor d’s bipr o duct , or R adfor d-Maji d b osonization . Radford [Rad85] (see also [Ma j94]) show e d that if K is a Ho pf alg ebra on a ﬁeld k a nd p is a pro jection of K , that is, an idempo ten t endomorphism of the Hopf a lg ebra K , then K may b e describ ed as a bipro duct as follows. Deno te by H the image of p , which is a Hopf subalg ebra of K . Then there exists a Hopf alge br a A in H H Y D such that K = A # H . Corollar y 5.12 generalizes Radford’s theorem. Indeed, in the situation of the theor e m, the Hopf mona d H ⊗ ? is a retra ct of the Hopf monad K ⊗ ? o n V ect k . Hence, by Cor ollary 5.12, there exists a Hopf algebra ( A , σ ) in Z ( H Mo d ) such that K ⊗ = A ⋊ σ ( H ⊗ ?). Identifying the center of H Mo d with the categor y of Y e tter -Drinfeld mo dules, we view ( A , σ ) as a Hopf algebra A in H H Y D . Then K ⊗ ? = A ⋊ σ ( H ⊗ ?) = A # H ⊗ ? as Hopf mo nad, and so K = A # H . 5.6. Regul ar augmenta tions. In this section w e prove Theorem 5.7 and Cor ol- lary 5 .9 us ing the notion of regula r a ugmen tation. Let T b e a comonoida l endofunctor of a monoida l catego ry C and e : T → 1 C be a comonoidal natural tra nsformation. Deﬁne natural transfo r mations u e : T → T 1 ⊗ ? and v e : T → ? ⊗ T 1 by: u e X = ( T 1 ⊗ e X ) T 2 ( 1 , X ) a nd v e X = ( e X ⊗ T 1 ) T 2 ( X, 1 ) . W e say that e is left r e gular if u e is inv e r tible. Lemma 5. 14. Assu m e e is left r e gular and set σ = v e ( u e ) − 1 : T 1 ⊗ ? → ? ⊗ T 1 . Then the natu r al tr ansformation σ is a lax half br aiding in C and ( T 1 , σ ) is a lax c ent r al c o algebr a of C with c opr o duct T 2 ( 1 , 1 ) and c ounit T 0 . F urthermor e the natur al tr ansformation u e : T → T 1 ⊗ σ ? is a c omonoida l isomorphi sm. Pr o of. By tra nspor t of str ucture, the endo functor P = T 1 ⊗ ? of C a dmits a unique comonoidal str ucture such that the natura l isomorphism u e : T → P is co monoidal, that is, P 0 u e 1 = T 0 and P 2 ( X, Y ) u e X ⊗ Y = ( u e X ⊗ u e Y ) T 2 ( X, Y ) . HOPF M ONADS ON MONOIDAL CA TEGORIES 31 W e have e 1 = T 0 (since e is c o monoidal) and so u e 1 = id T 1 and v e 1 = id T 1 . Hence P 0 = P 0 u e 1 = T 0 and σ 1 = v e 1 ( u e 1 ) − 1 = id T 1 . Moreov er, P 2 ( X, Y ) u e X ⊗ Y = ( u e X ⊗ u e Y ) T 2 ( X, Y ) = ( T 1 ⊗ e X ⊗ T 1 ⊗ e Y ) T 4 ( 1 , X, 1 , Y ) = ( T 1 ⊗ v e X ⊗ Y )( T 1 ⊗ T X ⊗ e Y ) T 3 ( 1 , X, Y ) = ( T 1 ⊗ v e X u e − 1 X ⊗ Y )( T 1 ⊗ u e X ⊗ e Y ) T 3 ( 1 , X, Y ) = ( T 1 ⊗ σ X ⊗ Y )( T 1 ⊗ T 1 ⊗ e X ⊗ e Y ) T 4 ( 1 , 1 , X, Y ) = ( T 1 ⊗ σ X ⊗ Y )( T 2 ( 1 , 1 ) ⊗ ( e X ⊗ e Y ) T 2 ( X, Y )) T 2 ( 1 , X ⊗ Y ) = ( T 1 ⊗ σ X ⊗ Y )( T 2 ( 1 , 1 ) ⊗ X ⊗ Y ) u e X ⊗ Y (since e is comono idal). Therefore P 2 ( X, Y ) = ( T 1 ⊗ σ X ⊗ Y )( T 2 ( 1 , 1 ) ⊗ X ⊗ Y ) b e cause u e is inv e r tible. W e conclude using Lemma 5.3.  Recall tha t an augmentation of a bimonad T on C is a morphism o f bimonads from T to 1 C . It is called left r e gular if it is le ft r egular as a comono idal natural transformatio n. Lemma 5.15. L et ( T , µ, η ) b e an augmen t e d bimonad on C . A s sume its augmenta- tion e : T → 1 C is left r e gular. Then σ = v e ( u e ) − 1 is a lax half br aiding for T 1 and ( T 1 , σ ) is a lax c entr al bialgebr a of C , with pr o duct m = µ 1 ( u e T 1 ) − 1 , un it u = η 1 , c opr o duct T 2 ( 1 , 1 ) , and c ounit T 0 . Mor e over u e : T → T 1 ⊗ σ ? is an isomorphism of bimonads. Pr o of. By tra nspor t of str ucture, the endo functor P = T 1 ⊗ ? of C a dmits a unique bimonad structure such that the natural tr ansformation u e : T → P is an isomor - phism of bimo nads. I n view of Lemmas 5.3 and 5.14, it is enough to verify that the pro duct µ ′ and unit η ′ of P are g iv en by µ ′ = µ 1 ( u e T 1 ) − 1 ⊗ ? and η ′ = η 1 ⊗ ?. Since u e is a mor phism o f monads, we hav e: η ′ X = u e X η X = ( T 1 ⊗ e X ) T 2 ( 1 , X ) η X = η 1 ⊗ e X η X = η 1 ⊗ X . Also, setting m = µ 1 ( u e T 1 ) − 1 , we hav e: µ ′ X u e T 1 ⊗ X T ( u e X ) = u e X µ X = ( T 1 ⊗ e X ) T 2 ( 1 , X ) µ X = ( µ 1 ⊗ e X µ X ) T 2 2 ( 1 , X ) = ( mu e T 1 ⊗ e X T ( e X )) T 2 ( T 1 , T X ) T ( T 2 ( 1 , X )) = ( m ⊗ X )  ( T 1 ⊗ e T 1 ) T 2 ( 1 , T 1 ) ⊗ e X  T 2 ( T 1 , X ) T ( u e X ) = ( m ⊗ X ) ( T 1 ⊗ e T 1 ⊗ X ) T 2 ( 1 , T 1 ⊗ X ) T ( u e X ) = ( m ⊗ X ) u e T 1 ⊗ X T ( u e X ) , and so µ ′ X = m ⊗ X since u e is inv e r tible.  Lemma 5.16. L et T b e an augmente d left pr e-Hopf monad on C . Then its aug- mentation e : T → 1 C is left r e gular and ( u e ) − 1 = T ( e ) H l − 1 1 , − ( T 1 ⊗ η ) . Pr o of. Let X b e a n o b ject of C a nd set θ e X = T ( e X ) H l − 1 1 ,X ( T 1 ⊗ η X ). W e hav e: u e X θ e X = ( T 1 ⊗ e X ) T 2 ( 1 , X ) T ( e X ) H l − 1 1 ,X ( T 1 ⊗ η X ) = ( T 1 ⊗ e X T ( e X )) T 2 ( 1 , T X ) H l − 1 1 ,X ( T 1 ⊗ η X ) = ( T 1 ⊗ e X µ X ) T 2 ( 1 , T X ) H l − 1 1 ,X ( T 1 ⊗ η X ) = ( T 1 ⊗ e X ) H l 1 ,X H l − 1 1 ,X ( T 1 ⊗ η X ) = ( T 1 ⊗ e X η X ) = id T 1 ⊗ X 32 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER and θ e X u e X = T ( e X ) H l − 1 1 ,X ( T 1 ⊗ η X e X ) T 2 ( 1 , X ) = T ( e X ) H l − 1 1 ,X ( T 1 ⊗ T ( e X ) η T X ) T 2 ( 1 , X ) = T ( e X ) T 2 ( e X ) H l − 1 1 ,T X ( T 1 ⊗ η T X ) T 2 ( 1 , X ) = T ( e X ) T ( µ X ) H l − 1 1 ,T X ( T 1 ⊗ η T X ) T 2 ( 1 , X ) = T ( e X ) H l − 1 1 ,X ( T 1 ⊗ µ X η T X ) T 2 ( 1 , X ) by Prop osition 2.6 = T ( e X ) H l − 1 1 ,X ( T 1 ⊗ µ X T ( η X )) T 2 ( 1 , X ) = T ( e X ) H l − 1 1 ,X H l 1 ,X T ( η X ) = T ( e X η X ) = id T X . Hence u e is inv e r tible with inv ers e θ e .  Theorem 5 .17. L et C b e a monoidal c ate gory. The fu n ctor R lax :  BiAlg( Z lax ( C )) → BiMon( C ) / 1 C ( A, σ ) 7→ ( A ⊗ σ ? , ε ⊗ ?) induc es an e quivalenc e of c ate gories fr om BiAlg ( Z lax ( C )) t o the ful l sub c ate gory of BiMon( C ) / 1 C of augmente d bimonads ( T , e ) such that e is left r e gular. Pr o of. If ( A, σ ) is a bia lgebra in Z lax ( C ), then e = ε ⊗ ? : A ⊗ σ ? → 1 C is a left regular bimonad mor phism (since u e = id A ⊗ ? ). Therefor e R lax takes v alues in the full sub category A ⊂ BiMon( C ) / 1 C of augmented bimonads ( T , e ) such that e is left regular . Con versely , le t ( T , e ) b e an ob ject of A . By Lemma 5.15, T 1 is endow ed with a half-braiding σ and ( T 1 , σ ) is a bia lgebra in Z ( C ). This construction is functorial, that is, gives rise to a functor I : A → BiAlg( Z lax ( C )) deﬁned on ob jects by I ( T , e ) = ( T 1 , σ = v e u e − 1 ) and on mo rphisms by I ( f ) = f 1 . Moreov er I is quasi-inv erse to R lax . Indeed, fo r ( T , e ) in A , u e is an isomorphis m from ( T , e ) to R lax I ( T , e ) a nd, for ( A, σ ) in BiAlg ( Z lax ( C )), w e have IR lax ( A, σ ) = ( A, σ ). Hence the Theor em.  Corollary 5.18. L et C b e a monoidal c ate gory. The funct or R lax induc es e quiva- lenc es of c ate gories b etwe en: (a) L ax c entr al left Hopf algebr as of C and augmente d left Hopf monads on C . (b) Centr al Hopf algebr as of C and augmente d Hopf monads on C . Mor e over an augmente d left (r esp. right) pr e-Hopf monad on C is in fact a left (r esp. right) Hopf monad. Pr o of. Let ( T , e ) b e a n a ugmen ted bimonad such that T is a left pre-Ho pf mo nad. Then e is left regular by Lemma 5.16. B y Theorem 5.17, T is of the form A ⊗ σ ? for some bialge br a ( A, σ ) in Z lax ( C ). By Prop osition 5.4, A admits an a n tip o de, and T is in fact a left Hopf mo nad. Hence the ﬁr st equiv a lence of catego ries. Mor e o ver, by Pr opo s ition 5.4, T is a Ho pf monad if and only if ( A, σ ) is a Hopf algebra in Z ( C ). Hence the second equiv ale nc e of categ o ries.  Pr o of of The or em 5.7. The theorem is just Assertio n (b) of Cor ollary 5.1 8 .  Pr o of of Cor ol lary 5.9. By Theo r em 5.7, the augmentation e : T → 1 B deﬁnes a Hopf algebra ( A = T 1 , σ ) in Z ( B ) such that T ≃ A ⊗ σ ?. In view of Remark 5.6, the ques tion is whether σ = τ A, − . Recall σ = v e ( u e ) − 1 . Therefore σ X = τ A,X if and only if v e X = τ A,X u e X , that is, ( e X ⊗ T 1 ) T 2 ( X, 1 ) = ( e X ⊗ T 1 ) τ T 1 ,T X T 2 ( 1 , X ).  HOPF M ONADS ON MONOIDAL CA TEGORIES 33 6. Induced coalgeb ras a nd Hopf modules A co comm utative coalge br a o f the center of a monoida l catego ry D g iv es ris e to a co monoidal co mo nad on D and, under cer tain exac tness assumptions, to a Hopf adjunction. On the other hand, we show that the comono idal comona d of a pre-Hopf adjunction ( F : C → D , U : D → C ) is represented b y its induc e d c o algebr a , which is a co commu tative co algebra o f the categorica l center of D . As an application, we o btain a structure theor e m for Hopf mo dules ov er pre- Hopf monads on monoidal ca tegories. It genera lizes Sw eedler’s Theo rem on the structure o f Hopf mo dules ov er a Hopf alge br a, and is an enhanced version of [BV07, Theor em 4 .6] which concer ns Hopf monads on a utonomous catego r ies. 6.1. F rom co commuta tiv e cen tral coalgebras to Hopf adjunctions. Let D be a monoidal categ ory and ( C, ∆ , ε ) b e a co algebra in D . Denote by C Como d the category of left C - como dules in D . The forgetful functor V : C Como d → D has a right adjo int, the cofr ee co module functor R :  D → C Como d X 7→ ( C ⊗ X , ∆ ⊗ X ) . The c omonad (on D ) of the adjunction ( V , R ) is ˆ T = ( C ⊗ ? , ∆ ⊗ ? , ε ⊗ ?). This adjunction is comonadic, since ˆ T - c o modules are just left C - como dules. On the o ther hand, the monad T = RV (on C Como d) o f the adjunction ( V , R ) is deﬁned by T ( M , δ ) = ( C ⊗ M , ∆ ⊗ M ) for any C - co module ( M , δ ), with pro duct µ ( M ,δ ) = C ⊗ ε ⊗ M and unit η ( M ,δ ) = δ . In general the adjunction ( V , R ) is not monadic. Remark 6.1. The adjunction ( V , R ) is monadic if, for instance, D admits reﬂexive co equalizers and C ⊗ ? is conser v ative a nd preserves re ﬂe x iv e co equalizers. Now let ( C, σ ) b e a la x central coalgebra of D , tha t is, a co algebra in Z lax ( C ). Then the endofunctor C ⊗ ? of D has b oth a comonad str ucture (b ecause C is a coalgebr a in D ) and a comono idal structure denoted by C ⊗ σ ? (see Le mma 5.3). A lax central coalg ebra ( C, σ ) is c o c ommu tative if its copro duct ∆ satisﬁes σ C ∆ = ∆. W e have: Lemma 6.2. L et ( C, σ ) b e a lax c entr al c o algebr a of D . Then C ⊗ σ ? is a c omonoidal c omonad if and only if ( C , σ ) is c o c ommu tative. Pr o of. One c hecks that the co product ∆ ⊗ ? of the comonad C ⊗ ? is comonoidal if and only if σ C ∆ = ∆, and tha t its co unit ε ⊗ ? is always comono ida l.  W e say that a co comm utative lax central coalgebr a ( C, σ ) of D is c otensor able if for each pair ( M , δ ), ( N , δ ′ ) of left C - como dules, the cor eﬂexiv e pair M ⊗ N σ M δ ⊗ N / / M ⊗ δ ′ / / M ⊗ C ⊗ N admits an eq ualizer, de no ted by M ⊗ σ C N → M ⊗ N , and the endofunctor C ⊗ ? preserves these equalizers. Let ( C, σ ) be a cotens o rable co commut ative lax central c o algebra of D . Given t wo left C - como dules ( M , δ ) and ( N , δ ′ ), there exists a unique left coa ction δ ′′ of C on M ⊗ σ C N such that the mor phism M ⊗ σ C N → M ⊗ N is a como dule morphism ( M ⊗ σ C N , δ ′′ ) → ( M ⊗ N , δ ⊗ N ). The assignment ( M , δ ) × ( N , δ ′ ) 7→ ( M ⊗ σ C N , δ ′′ ) deﬁnes a functor: ⊗ σ C : C Como d × C Como d → C Como d . Then the categor y C Como d of left C - como dules (in C ) is monoidal, with mono idal pro duct ⊗ σ C and unit ob ject ( C, ∆). W e denote this mono idal categor y by C Como d σ . 34 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER The c ofree co module functor R : D → C Como d σ is strong monoidal, so that the comonadic adjunction ( V , R ) is comono idal, with comonoidal comonad C ⊗ σ ?. Lemma 6.3. L et D b e a monoidal c ate gory admitting c or eﬂexive e qualizers which ar e pr eserve d by the monoidal pr o duct. L et ( C, σ ) b e a c o c ommutative c entr al c o alge- br a of D . Then ( C, σ ) is c otensor able and the monoidal c ate gory C Como d σ admits c or eﬂex ive e qualizers which ar e pr eserve d by the monoidal pr o duct ⊗ σ C and t he for- getful functor V : C Como d σ → D . Pr o of. By sta nda rd diagram chase left to the reader.  Theorem 6.4. L et D b e a monoidal c ate gory and ( C , σ ) b e a c otensor able c o c om- mutative lax c entr al c o algebr a of D . Then the c omonoidal adjunction ( V : C Como d σ → D , R : D → C Como d σ ) is a left Hopf adjunction, and its induc e d lax c ent r al c o algebr a is ( C, σ ) . Mor e over, if σ is invertible, ( V , R ) is a Hopf adjunction. Pr o of. Let d b e a n ob ject of D and ( M , δ ) b e a left C - como dule. Then the mo r- phism σ M δ ⊗ d : M ⊗ d → M ⊗ C ⊗ d is an equa lizer of the pair M ⊗ C ⊗ d σ M δ ⊗ C ⊗ d / / M ⊗ ∆ ⊗ d / / M ⊗ C ⊗ C ⊗ d . Hence an iso morphism M ⊗ σ C R ( d ) ∼ − → M ⊗ d which is the left fusion op erator of the comonoidal a djunction ( V , R ). Similarly if σ is inv ertible, the morphism ( σ − 1 d ⊗ M )( d ⊗ δ ) : d ⊗ M → C ⊗ d ⊗ M is a n eq ualizer of the pair C ⊗ d ⊗ M ( σ C ⊗ d ⊗ M )(∆ ⊗ d ⊗ M ) / / C ⊗ d ⊗ δ / / C ⊗ d ⊗ C ⊗ M . Hence a n isomor phism R ( d ) ⊗ σ C M ∼ − → d ⊗ M which is the r igh t fusion op erator of the como noidal adjunction ( V , R ).  6.2. Induced coalgebra and com onad o f a comonoi dal adjunction. Let C , D b e mo noidal ca tegories and ( F : C → D , U : D → C ) b e a co monoidal adjunction, with adjunction unit η : 1 C → U F and co unit ε : F U → 1 D . Being co monoidal, F sends the triv ial coalgebr a 1 in C to a coalg e br a ˆ C = F ( 1 ) in D , with copro duct ∆ = F 2 ( 1 , 1 ) and counit ǫ = F 0 , called the induc e d c o algebr a of the c omonoidal adjunction . The endofunctor ˆ T = F U of D is a co monoidal c o monad, with co product F ( η U ) : ˆ T → ˆ T 2 and co unit ε (see Section 2.5). In this situa tion we have three co monads o n the categor y D , na mely : • ? ⊗ ˆ C (with copro duct ? ⊗ ∆ and counit ? ⊗ ǫ ); • ˆ C ⊗ ? (with copro duct ∆ ⊗ ? and counit ǫ ⊗ ?); • the (como no idal) comona d ˆ T = F U of the adjunction ( F , U ). How ar e they related? Lemma 6. 5. The Hopf op er ators H l and H r deﬁne morphisms of c omonads: H l 1 , − : ˆ T → ˆ C ⊗ ? and H r − , 1 : ˆ T → ? ⊗ ˆ C . HOPF M ONADS ON MONOIDAL CA TEGORIES 35 Pr o of. The commut ativity o f the following diag rams: F U d F 2 ( 1 ,U d )   F η U d / / F 2 ( 1 ,U d ) ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q F U F U d F 2 ( 1 ,U F U d )   F 1 ⊗ F U F U d F 1 ⊗ ε F U d   F 1 ⊗ F U d F 1 ⊗ F 2 ( 1 ,U d )   F 1 ⊗ F U d F 1 ⊗ ε d   F 2 ( 1 , 1 ) ⊗ F U d / / F 1 ⊗ F 1 ⊗ F U d F 1 ⊗ F 1 ⊗ ε d   F 1 ⊗ d F 2 ( 1 , 1 ) ⊗ d / / F 1 ⊗ F 1 ⊗ d F U d ε d $ $ J J J J J J J J J J F 2 ( 1 ,U d )   F 1 ⊗ F U d F 1 ⊗ ε d   d F 1 ⊗ d F 0 ⊗ d : : u u u u u u u u u u which results fro m the fact that the adjunction ( F , U ) is co mo noidal, means that H l 1 , − is a morphism of como nads. The proo f for H r − , 1 is similar.  6.3. F rom Hopf adjunctions to co comm utativ e cen tral coalgebras. In the case of a left pre-Hopf adjunction, the induced coa lgebra is endow ed with a ca nonical lax half braiding, making it a co comm utative lax central coalgebra which represents the induced comonoidal comonad: Theorem 6.6. L et ( F : C → D , U : D → C ) b e a left pr e-Hopf adjunction, with induc e d c o algebr a ˆ C . Then: (a) The natu r al tr ansformation ˆ σ = H r − , 1 H l − 1 1 , − : ˆ C ⊗ ? → ? ⊗ ˆ C is a lax half- br aiding of D such that, for any obje ct c of C , the diagr am: F c F 2 ( 1 ,c ) | | x x x x x F 2 ( c, 1 ) " " F F F F F ˆ C ⊗ F c ˆ σ F c / / F c ⊗ ˆ C is c ommutative. (b) ( ˆ C , ˆ σ ) is a c o c ommutative lax c entr al c o algebr a of D and H l 1 , − : ˆ T → ˆ C ⊗ ˆ σ ? is an isomorphism of c omonoidal c omonads. Pr o of. Let ( F, U ) b e a left pre- Hopf adjunction, so that H l 1 , − is inv ertible. Since U is strong monoida l, we identify ˆ C = F ( 1 ) a nd ˆ T ( 1 ) = F U ( 1 ) as coalgebra s in D . W e apply Lemma 5 .14 to the como noidal endofunctor ˆ T of D and the comonoidal morphism ε : F U → 1 C . The natura l tr ansformations u ε : F U → F U ( 1 ) ⊗ ? and v ε : F U → ? ⊗ F U ( 1 ) of the lemma a r e nothing but H l 1 , − and H r − , 1 resp ectiv ely . Therefore u e = H l 1 , − being inv ertible, we conclude that ˆ σ = v e ( u e ) − 1 is a lax braiding on D and ( ˆ C , ˆ σ ) is a coa lgebra in Z ( D ) s uc h that u e is a comonoida l isomorphism. Now, for any ob ject c of C , we hav e H l 1 ,F c F ( η c ) = F 2 ( 1 , c ) and H r F c, 1 F ( η c ) = F 2 ( c, 1 ) , from which the equa lit y ˆ σ F c F 2 ( 1 , c ) = F 2 ( c, 1 ) follo ws directly . Hence Part (a). Applying this equality to d = 1 gives the co commutativit y o f the coa lgebra ( ˆ C , ˆ σ ), so that ˆ C ⊗ ˆ σ ? is a comonoidal co monad by Lemma 6.2. Thus H l 1 , − = u e is an isomor phism o f co mo noidal co monads, hence Part (b).  As a consequence, the comonoidal comonad of a pre-Hopf adjunction is canoni- cally r epresent ed by a co commutativ e central coalgebra of the upp er catego ry . Mo re precisely: Corollary 6.7. L et ( F : C → D , U : D → C ) b e a pr e-Hopf adjunction, with induc e d c o algebr a ˆ C . Then ˆ σ = H r − , 1 H l − 1 1 , − is a half-br aiding for ˆ C . Mor e over ( ˆ C , ˆ σ ) is a 36 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER c o c ommutative c entr al c o algebr a in D c al le d the induced c e n tr al coa lgebra of the pre-Hopf adjunction ( F, U ). Pr o of. Since ( F, U ) is a pr e -Hopf a djunction, the pre-Hopf op erators H r − , 1 and H l 1 , − are inv er tible. Thus ˆ σ is invertible and the corolla ry follows then dir e c tly from Theore m 6.6.  Example 6. 8. Let C be a monoida l catego ry a nd ( A, σ ) b e a Ho pf algebra in Z ( C ), with pro duct m , copr oduct ∆, a nd co unit ε . Co nsider the Ho pf monad T = A ⊗ σ ? on C (see Pr opos itio n 5 .4). Recall A Mo d σ denotes the mono idal categ ory C A ⊗ σ ? of left A - mo dules (in C ), with mono idal pro duct induced b y σ (see Section 5 .3). The induced coa lgebra ˆ C o f A ⊗ σ ? is the le ft A - mo dule ˆ C = ( A, m ), with copr oduct ∆ and counit ε . Its asso ciated half-braiding is given by ˆ σ ( M ,r ) = P S f r a g r e p la c e m e n t s A A M M σ M r σ A for any left A - mo dule ( M , r ). The n ( ˆ C , ˆ σ ) is a co comm utative coalgebr a in the center Z ( A Mo d σ ) of A Mo d σ . Prop osition 6 .9. L et ( F : C → D , U : D → C ) b e a c omonadic pr e-Hopf adjunction, with induc e d c ent r al c o algebr a ( ˆ C , ˆ σ ) . Assu me t hat for al l X , Y obje cts of C , t he morphism F 2 ( X, Y ) : F ( X ⊗ Y ) → F ( X ) ⊗ F ( Y ) is an e qualizer of the c or eﬂexive p air F X ⊗ F Y F 2 ( X, 1 ) ⊗ F Y / / F X ⊗ F 2 ( 1 ,Y ) / / F X ⊗ F 1 ⊗ F Y , and these e qu alize rs ar e pr eserve d by the endofunctor F ( 1 ) ⊗ ? . Then the c o c ommu- tative c entr al c o algebr a ( ˆ C , ˆ σ ) is c oten sor able and the c omp arison functor K : C → ˆ C Como d ˆ σ is a str ong monoidal e quivalenc e. In p articular ( F , U ) is a Hopf adjunction. Pr o of. The cotensorability assumption means that for each pair ( M , δ ), ( N , δ ′ ) of left ˆ C - como dules, the co reﬂexive pair M ⊗ N σ M δ ⊗ N / / M ⊗ δ ′ / / M ⊗ C ⊗ N admits an eq ualizer, and the endofunctor ˆ C ⊗ ? pre serves these equalizer s . Now recall that the comparison functor K is deﬁned by K ( X ) = ( F X , F 2 ( 1 , X )) for X in C . If X is an ob ject of C then by Theorem 6.6, Part (a), we hav e ˆ σ F X F 2 ( 1 , X ) = F 2 ( X, 1 ). Since K is an equiv a lence, w e conclude that ( ˆ C , ˆ σ ) is co tensorable. Mo re- ov er , we have K ( X ⊗ Y ) = K ( X ) ⊗ ˆ σ ˆ C K ( Y ) so that K is a strong monoidal equiv- alence. B y Theorem 6.4, ( F, U ) is a Hopf adjunction.  HOPF M ONADS ON MONOIDAL CA TEGORIES 37 6.4. Descent . Let ( T , µ, η ) be a mona d on a catego ry C . Its adjunction ( F T , U T ) has unit η : 1 C → U T F T = T and has counit denoted by ε : F T U T → 1 C T . Let ˆ T b e the co monad of the adjunction ( F T , U T ), that is, ˆ T = F T U T on C T , with copro duct δ = F T ( η U T ) a nd counit ε . Denote by H ( T ) the categor y ( C T ) ˆ T of ˆ T - c o modules in the categor y of T - mo dules in C . Ob jects of H ( T ) are triples ( B , r , ρ ), where B is an o b ject of C , r : T B → B , and ρ : B → T B ar e mo rphisms in C , s uch that ( B , r ) is a T - mo dule, that is, rT ( r ) = rµ B and r η B = id B , and ( B , ρ ) is a ˆ T - c o module who se coaction is T - linea r, that is, T ( ρ ) ρ = δ B ρ, rρ = id B , and ρr = µ B T ( ρ ) . Morphisms in H ( T ) fro m ( M , r, ρ ) to ( N , s,  ) are mo rphisms f : M → N in C which are mor phisms of T - mo dules a nd ˆ T - como dules: f r = sT ( f ) and T ( f ) ρ = f . The co mpa rison functor of the co monad ˆ T is the functor χ :  C → H ( T ) X 7→ ( T X , µ X , T η X ) . The ques tio n whether χ is a n equiv alence is a desce n t problem. The c oinvariant p art of an o b ject B = ( B , r , ρ ) of H ( T ) is the equa lizer of the coreﬂexive pa ir B η B / / ρ / / T B . If the co in v ariant part of B exists, it is denoted by i B : B T → B . W e say that T admits c oinvariant p arts if any ob ject o f H ( T ) admits a coinv ariant part. W e say that T pr eserves c oinvariant p arts if, for any o b ject B of H ( T ) which admits a co in v ariant pa rt i B : B T → B , the morphism T ( i B ) is an equalizer of the pair ( T η B , T ρ ). The following characteriza tion of monads T for which χ is an equiv alence is a reformulation o f [FM71, Theor em 1]. Theorem 6. 10. L et T b e a monad on a c ate gory C . The fol lowing assertions ar e e quivalent: (i) The functor χ : C → H ( T ) is an e quivalenc e of c ate gories; (ii) T is c onservative, admits c oinvariant p arts, and pr eserves c oinvaria nt p arts. If such is the c ase, t he fun ctor ‘c oinvariant p art’ B 7→ B T is quasi-inverse to χ . 6.5. Hopf mo dul es fo r pre-Hopf monads. Let T be a bimonad on a mo noidal category C . The induc e d c o algebr a of T , denoted by ˆ C , is the induced coalg ebra of the comonoidal a djunction ( F T , U T ). Explicitly ˆ C = ( T ( 1 ) , µ 1 ), with copro duct T 2 ( 1 , 1 ) and counit T 0 . Note that U T ( ˆ C ) = T ( 1 ) is a coalgebra in C . A left Hopf T -mo dule (as deﬁned in [BV07 ]) is a left ˆ C -como dule in C T , that is, a triple ( M , r, ρ ) such that ( M , r ) is a T - mo dule, ( M , ρ ) is a left T ( 1 )- como dule, and ρr = ( µ 1 ⊗ r ) T 2 ( T 1 , M ) T ( ρ ) . A morphism of Hopf T -mo dules betw e e n tw o left Hopf T -mo dules ( M , r, ρ ) and ( N , s,  ) is a morphism of ˆ C -como dules in C T , that is, a morphism f : M → N in C such that f r = sT ( f ) and (id T ( 1 ) ⊗ f ) ρ = f . W e denote by H l ( T ) the ca tegory of left Hopf T - mo dules. 38 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER The c oinvariant p art of a left Hopf mo dule M = ( M , r, ρ ) is the equalizer of the coreﬂexive pa ir M η 1 ⊗ M / / ρ / / T ( 1 ) ⊗ M . If it exists, it is denoted by M T . W e say that T preser v es co in v ariant parts of left Hopf mo dules if, whenever a left Hopf mo dule M = ( M , r, ρ ) admits a coinv a riant part i T : M T → M , then T ( i T ) is a n equalizer of ( T ( η 1 ⊗ M ) , T ρ ). Theorem 6. 11. L et T b e a left pr e-Hopf monad on a monoidal c ate gory C . The fol lowing assertions ar e e quivalent: (i) The functor h l :  C → H l ( T ) X 7→  T X , µ X , T 2 ( 1 , X )  is an e quivalenc e of c ate gories; (ii) T is c onservative, left Hopf T - mo dules admit c oinvariant p arts, and T pr e- serves them. If these hold, the functor ‘c oinvariant p art’ M 7→ M T is quasi-inverse to h l . Remark 6.12. Similar ly , we deﬁne the catego r y H r ( T ) of right Hopf T -mo dules . Since H r ( T ) = H l ( T cop ), Theor em 6.1 1 holds a lso fo r r igh t pre-Hopf monads and right Hopf modules (see Remar k 2 .11). Example 6. 13. Let ( A, σ ) b e a central Ho pf algebra in a monoidal categ ory C , tha t is, a Hopf alg ebra in the center Z ( C ) of C . Co nsider the left Hopf mo nad T = A ⊗ σ ? on C , see Pro position 5.4. A left Hopf mo dule over A is left Hopf T - mo dule, that is, a triple ( M , r : A ⊗ M → M , ρ : M → A ⊗ M ) such that ( M , r ) is a left A - mo dule, ( M , ρ ) is a left A -como dule, and ρr = ( m ⊗ r )(id A ⊗ σ A ⊗ id M )(∆ ⊗ ρ ), where m is the pro duct of A and ∆ is co pr oduct o f A . Assume now that C splits idempo ten ts. Then the morphism r ( S ⊗ id M ) ρ (where S denotes the antipo de of A ) is an idempo ten t of A ⊗ M and its image is the coinv ar ian t part of M . O ne veriﬁes that T is conser v ative and pr eserves coinv ariant parts. Applying Theor em 6.11, we obtain the fundamental theorem o f Hopf mo dules fo r cent ral Ho pf algebra s . In view of Remark 5 .6, we recover the fundamental decomp osition theorem of Hopf mo dules for Hopf algebr as in a braided categ ory (see [BK L T00]) which, for the category of vector spaces o ver a ﬁeld, is just Sw eedler’s classical theor em. F o r a detailed treatment o f the ca se of Ho pf a lg ebras over a ﬁeld, we r efer to [BN10, Examples 6.2 and 6.3]. Pr o of of The or em 6.11. Let ˆ T b e the comona d of the adjunction ( F T , U T ) and ˆ C be the induced coalgebra of T . Since T is a left pre-Hopf monad, H l 1 , − : ˆ T → ˆ C ⊗ ? is an isomorphism of comonads by Lemma 6.5. It induces an iso morphism of categor ies κ l T : H ( T ) = ( C T ) ˆ T → ( C T ) ˆ C ⊗ ? = H l ( T ) such that κ l T χ = h l . W e conclude using Theor em 6 .10.  6.6. Summary. In this section we summarize the re lationships betw een Hopf mo n- ads, Hopf adjunctions, and co commutativ e central co algebras. W e ha ve constructed s ev eral corr e spondences b et ween these ob jects: • A Ho pf adjunction ( F : C → D , U : D → C ) gives rise to a Hopf monad m ( F, U ) = U F on C by Prop osition 2.14, and to a co commutativ e central coalgebr a c ( F, U ) = ( ˆ C , ˆ σ ) in D by Co r ollary 6 .7; HOPF M ONADS ON MONOIDAL CA TEGORIES 39 • A Hopf monad T o n a monoida l ca tegory C deﬁnes a Hopf adjunction a ( T ) = ( F T : C → C T , U T : C T → C ) by Theor e m 2.1 5 ; • A c otensor able co commutativ e cen tral coalg ebra ( C, σ ) on a monoida l cat- egory D yields a Hopf adjunction o ( C, σ ) = ( U : C Como d σ → D , R : D → C Como d σ ) by Theor e m 6.4. Hence the following triang le: Hopf adjunctions m w w c   Hopf monads ca 1 1 a 7 7 co comm utative central co algebras mo r r [ \ ] ^ _ ` a b c d o \ \ M I D ? 9 W e ha ve: • ma ( T ) = T ; • am ( F , U ) ≃ ( F, U ) if and only if the adjunction ( F , U ) is monadic; • co ( C, σ ) = ( C, σ ); • assuming c ( F , U ) is cotenso r able, we have oc ( F , U ) ≃ ( F , U ) if the comonoidal adjunction ( F, U ) satisﬁes the conditions o f Pr opo s ition 6.9. With suitable exactness assumption, we hav e in fact equiv alences: Theorem 6 .14. The fol lowing data ar e e quivalent via the assignments a and c : (A) A Hopf monad T on a monoidal c ate gory C s u ch t hat: • C admits r eﬂexive c o e qualizers and c or eﬂexive e qualizers, and its monoidal pr o duct pr eserves c or eﬂexive e qualizers; • T is c onservative and pr eserves r eﬂexive c o e qualizers and c or eﬂexive e qual- izers; (B) A Hopf adjunction ( F : C → D , U : D → C ) such that: • C and D admit r eﬂexive c o e qualizers and c or eﬂexive e qualizers, and their monoidal pr o ducts pr eserve c or eﬂexive e qualizers; • F and U ar e c onservative, U pr eserves r eﬂexive c o e qualizers and F pr e- serves c or eﬂex ive e qualizers. (C) A c o c ommu tative c en tr al c o algebr a ( C , σ ) in a m onoidal c ate gory D su ch that: • D admits r eﬂex ive c o e qualizers and c or eﬂex ive e qu alizers, and its monoidal pr o duct pr eserves c or eﬂexive e qualizers (in p articular the c entra l c o algebr a ( C, σ ) is c otensor able); • the endofunctor C ⊗ ? of D is c onservative and pr eserves r eﬂexive c o e qual- izers. Mor e over, a Hopf adjunction satisfying t he c onditions of ( B) is a monadic and c omonadic Hopf adjunction. Pr o of. Firstly , we show the equiv ale nce o f (A) a nd (B). Let T b e a Hopf monad o n a monoidal categ ory C satisfying the conditions of (A). Then U T , b eing the for getful functor of a monad, preserves and creates limits and in particular equalizers. As a r esult, the mo no idal catego ry C T admits co reﬂexive equalizers a nd U T preserves them. F rom this one deduces that, since the monoidal pro duct of C prese rv es coreﬂexive equalizers, s o do es that of C T . Moreover, since T preserves reﬂexive 40 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER co equalizers, U T creates and pr eserves them. Co nsequen tly: C T admits reﬂexive co equalizers, and F T preserves reﬂexive co equalizers. The for getful functor U T is conser v ative, and since by as sumption T = U T F T is conserv a tiv e, so is F T . Thu s a ( T ) = ( F T , U T ) is a Hopf adjunction satisfying the conditions o f (B), and we hav e ma ( T ) = T . Conv ersely , let ( F, U ) b e a Hopf adjunction satisfying the conditions of (B). By adjunction F pr eserves colimits and U preser v es limits. The Hopf monad T = m ( F , U ) = U F is cons erv ative a nd pre s erv es r eﬂexiv e coeq ualizers and coreﬂexive equalizers , s o that it satisﬁes the conditions of (A). Moreov er by Beck’s monadicity theorem, the adjunction ( F , U ) is mona dic, so a m ( F, U ) ≃ ( F, U ), hence the equiv alence o f (A) and (B). Secondly , we show the eq uiv alence o f (B) and (C). Let ( C, σ ) b e a co commut ative central comonad in a mono idal category D satisfying the conditions of (C). Then ( C, σ ) is cotensor a ble, and the adjunction o ( C, σ ) = ( V : C Como d σ → D , R : D → C Como d σ ) is a Hopf adjunction. It is co monadic, with comonoida l comonad ˆ T = C ⊗ σ ?. It is also monadic, see Remark 6.1. Moreov e r the cotensor pro duct ⊗ σ C preserves core- ﬂexive equa lizer by Lemma 6 .3. Th us the adjunction ( V , R ) satisﬁes the conditions of (B). W e have co ( C, σ ) = ( C, σ ). Let us prov e co n v ersely that if ( F, U ) is a Hopf adjunction satisfying the con- ditions of (B), then its induced central co algebra ( ˆ C , ˆ σ ) = c ( F , U ) satisﬁes the conditions of (C) a nd o c ( F, U ) ≃ ( F, U ) as Hopf a djunctions. W e will need the following lemmas. Lemma 6.15. L et C b e a c ate gory admitting c or eﬂexive e qualizers and let T b e a c onservative monad on C pr eserving c or eﬂexive e qualizers. Then for e ach obje ct X of C , η X is an e qualizer of the p air ( T ( η X ) , η T X ) . Pr o of. Let X b e a n ob ject of C . Observe tha t T ( η X ) is a n equaliz er of the cor eﬂexiv e pair ( T 2 ( η X ) , T ( η T X )). Since T is co ns erv ative a nd C admits cor eﬂexiv e equalizers preserved by T , η X is an equalizer of the coreﬂexive pa ir ( T ( η X ) , η T X ).  Lemma 6.16. L et C b e a monoidal c ate gory whose monoidal pr o duct pr eserves c or eﬂex ive e qualizers in the left variable. L et T b e a left Hopf monad on C which pr eserves c or eﬂexive e qualizers. Assume furthermor e that for e ach obje ct X of C , η X is an e qualizer of the p air ( T ( η X ) , η T X ) . Then for al l obje cts X , Y of C , T 2 ( X, Y ) : T ( X ⊗ Y ) → T X ⊗ T Y is an e qualizer of t he c or eﬂexive p air T ( X ) ⊗ T ( Y ) T 2 ( X, 1 ) ⊗ T ( Y ) / / T ( X ) ⊗ T 2 ( 1 ,Y ) / / T ( X ) ⊗ T ( 1 ) ⊗ T ( Y ) . Pr o of. The following diagr am: T ( X ⊗ Y ) T ( X ⊗ η Y ) / / =   T ( X ⊗ T Y ) T ( X ⊗ η T Y ) / / T ( X ⊗ T ( η Y )) / / H l X,Y   T ( X ⊗ T 2 Y ) ( T X ⊗ H l 1 ,Y ) H l X,T Y   T ( X ⊗ Y ) T 2 ( X,Y ) / / T X ⊗ T Y T 2 ( X, 1 ) ⊗ T Y / / T X ⊗ T 2 ( 1 ,Y ) / / T X ⊗ T 1 ⊗ T Y , is commutativ e (in the sense that the left square and the tw o right squar es obtained by taking r espectively the top and b ottom a rrow of each pair , ar e comm utative); this results ea sily fr om Prop osition 2.6. The top row is an equalizer b ecause the endofunctor T ( X ⊗ ?) preserves co reﬂexive equa liz ers. Since H l is inv er tible, we conclude that the b ottom row is exact, hence the lemma.  HOPF M ONADS ON MONOIDAL CA TEGORIES 41 Now let T = U F b e the Hopf monad o f ( F , U ). Then T satisﬁe s the hypo theses of L e mma s 6.15 and 6.16, so that for all ob jects X , Y of C , T 2 ( X, Y ) : T ( X ⊗ Y ) → T ( X ) ⊗ T ( Y ) is an equalizer of the pair T ( X ) ⊗ T ( Y ) T 2 ( X, 1 ) ⊗ T ( Y ) / / T ( X ) ⊗ T 2 ( 1 ,Y ) / / T ( X ) ⊗ T ( 1 ) ⊗ T ( Y ) . Moreov er, the adjunction ( F, U ) is monadic. In particular the functor U cre a tes and preser v es e q ualizers; th us F 2 ( X ⊗ Y ) is an equalizer of the pair F ( X ) ⊗ F ( Y ) F 2 ( X, 1 ) ⊗ F ( Y ) / / F ( X ) ⊗ F 2 ( 1 ,Y ) / / F ( X ) ⊗ F ( 1 ) ⊗ F ( Y ) . W e may ther efore apply Pro position 6.9 to the adjunction ( F, U ), and we conclude that the co mpa rison functor C → ˆ C Como d ˆ σ is a str o ng mo noidal equiv alence and c ( F, U ) satisﬁes the c onditions of (C), hence oc ( F, U ) ≃ ( F , U ) a s Hopf adjunctions.  7. Hopf algebroids and finite abelian tensor ca tegories In this section, we s tudy bia lgebroids which, acc ording to Szlach´ anyi [Szl03], ar e linear bimonads on c ategories o f bimo dules admitting a right adjoint. A bialgebroid corres p onds with a Hopf monad if and o nly if it is a Hopf alg ebroid in the sense of Schauen burg [Sch00]. W e also use Hopf monads to prove that any ﬁnite tensor category is naturally equiv a len t (as a tenso r catego ry) to the catego r y of ﬁnite- dimensional mo dules over some ﬁnite dimensional Ho pf algebroid. 7.1. Bialgebroi ds and bimonads. Let k b e a commutative ring a nd R b e a k -algebra . Denote by R Mo d R the category of R - bimo dules. It is a mono idal cate- gory , with monoidal pr oduct ⊗ R and unit o b ject R . W e identif y R Mo d R with the category R e Mo d o f left R e - mo dules, wher e R e = R ⊗ k R op . Hence a monoidal pro d- uct ⊠ on R e Mo d (corr esponding to ⊗ R on R Mo d R ), with unit R (whose R e - ac tion is ( r ⊗ r ′ ) · x = r xr ′ ). If f : B → A is k -algebra mor phis m, we de no te by f A the left B - mo dule A with left a ction b · a = f ( b ) a , a nd b y A f the rig h t B - mo dule A with right a ction a · b = af ( b ). A left bialgebr oid with b ase R (a ls o called T akeuc hi × R -bialgebra ) co nsists of data ( A, s, t, ∆ , ε ) where: • A is a k -algebra ; • s : R → A a nd t : R op → A a re k -algebr a mo rphisms who s e images in A commute. Hence a k -a lgebra morphism e :  R e → A r ⊗ r ′ 7→ s ( r ) t ( r ′ ) , which gives rise to a R e - bimo dule e A e . • ( e A, ∆ , ε ) is a co algebra in the monoida l ca tegory ( R e Mo d , ⊠ , R ). In this situation the T akeuc hi pro duct A × R A ⊂ e A ⊠ e A , deﬁned b y A × R A = { P a i ⊗ b i ∈ e A ⊠ e A | ∀ r ∈ R, P a i t ( r ) ⊗ b i = P a i ⊗ b i s ( r ) } is a k -a lgebra, with pro duct deﬁned by ( a ⊗ b )( a ′ ⊗ b ′ ) = aa ′ ⊗ b b ′ , and one requires : • ∆( A ) ⊂ A × R A ; • ∆ : A → A × R A is a k -algebr a mor phism; • ε ( a s ( ε ( a ′ ))) = ε ( aa ′ ) = ε ( a t ( ε ( a ′ ))); • ε (1 A ) = 1 R . 42 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER The no tion of left bialgebro id ha s a nice interpretation in terms of bimonads. A bialgebroid A with base R gives r ise to an endofunctor of R e Mo d ≃ R Mo d R : T A :  R e Mo d → R e Mo d M 7→ T A ( M ) = e A e ⊗ R e M The ax ioms of a left bialgebro id ar e such that T A is a k -linear bimonad admitting a right adjoint. These pro perties c haracterize left bialg ebroids: Theorem 7.1 ([Szl03]) . L et k b e a ring and R a k - algebr a. Via the c orr esp ondenc e A 7→ T A , the fol lowing data ar e e quivalent: (A) A left bialgebr oid A with b ase R ; (B) A k - line ar bimonad T on the m onoid al c ate gory R Mo d R ≃ R e Mo d admitting a right adjoi nt. 7.2. Hopf al g ebroids. W e deﬁne a left , resp. right , (pr e- ) H opf algebr oid to b e a bialgebroid A whos e asso ciated bimonad T A is a left, r esp. right, (pre- )Hopf monad. A (pr e- ) H opf algebr oid is a left and right (pre- )Hopf algebro id. Let A be a bia lgebroid and T A be its a ssoc iated bimo nad o n R Mo d R ≃ R e Mo d. The fusion oper ators H l and H r of T A are: H l M ,N :  e A e ⊗ R e  M ⊠ ( e A e ⊗ R e N )  → ( e A e ⊗ R e M ) ⊠ ( e A e ⊗ R e N ) a ⊗ m ⊗ b ⊗ n 7→ a (1) ⊗ m ⊗ a (2) b ⊗ n and H r M ,N :  e A e ⊗ R e  ( e A e ⊗ R e M ) ⊠ N  → ( e A e ⊗ R e M ) ⊠ ( e A e ⊗ R e N ) a ⊗ b ⊗ m ⊗ n 7→ a (1) b ⊗ m ⊗ a (2) ⊗ n . Using the fa ct that R e and R are resp ectiv ely a pro jective ge nerator and the unit ob ject o f R e Mo d, we obtain the following characterizatio n of Hopf bia lgebroids a nd pre-Hopf alg e broids. Prop osition 7. 2. L et A b e a bialgebr oid with b ase R . Then: (a) The bialgebr oid A is a left Hopf algebr oid if and only if t he R e - line ar map H l R e ,R e :  e A t ⊗ R op t A → e A ⊠ e A a ⊗ b 7→ a (1) ⊗ a (2) b is bije ctive. (b) The bialgebr oid A is a right Hopf algebr oid if and only if t he R e - line ar map H r R e ,R e :  e A s ⊗ R s A → e A ⊠ e A a ⊗ b 7→ a (1) b ⊗ a (2) is bije ctive. (c) The bialgebr oid A is a left pr e-Hopf algebr oid if and only if the R e - line ar map H l R,R e :  e A e ⊗ R e e A → e ¯ A ⊠ e A a ⊗ a ′ 7→ a (1) ⊗ a (2) a ′ is bije ctive, wher e e ¯ A = e A e ⊗ R e R = A/ { as ( r ) = at ( r ) | a ∈ A, r ∈ R } . (d) The bialgeb r oid A is a right pr e-Hopf algebr oid if and only if the R e - line ar map and H r R e ,R :  e A e ⊗ R e e A → e A ⊠ e ¯ A a ⊗ a ′ 7→ a (1) a ′ ⊗ a (2) is bije ctive. Remark 7 .3. The notion o f × R -Hopf algebra in tro duced by Schauen bur g in [Sch00] coincides with our notion o f left Hopf algebroid. HOPF M ONADS ON MONOIDAL CA TEGORIES 43 Remark 7.4. The catego ry R e Mo d is monoida l clo sed with internal Homs: [ M , N ] l = Hom R op ( R e M , R e N ) and [ M , N ] r = Hom R ( R e M , R e N ) . By Theor em 3 .6, a left bia lg ebroid A with ba s e R is a Hopf a lgebroid if a nd only if it admits a left a n tip o de s l M ,N : e A e ⊠ Hom R op ( e A e ⊗ R e M , N ) → Hom R op ( M , e A e ⊗ R e N ) and a right antipo de s r M ,N : e A e ⊠ Hom R ( e A e ⊗ R e M , N ) → Hom R ( M , e A e ⊗ R e N ) . Remark 7.5. Let A b e a pre- Hopf algebroid with base R . Since R e Mo d is ab elian, the bimonad T A admits coinv ariant parts. If A e is a faithfully ﬂat rig h t R e - mo dule, then T A is cons e r v ative a nd pr eserves coinv ariant pa rts. Thus, the Hopf mo dule decomp osition theo rem (see Theor e m 6.11) applies to (pre- )Hopf a lgebroids which are faithfully ﬂat on the right over the base r ing. 7.3. Existence of ﬁbre functors for ﬁnite tensor categories. A tensor c ate- gory over k is an autono mous category endow ed with a structure o f k -linear ab elian category such that the monoidal pro duct ⊗ is bilinear and End( 1 ) = k . W e say that a k -linear ab elian categor y A is ﬁnite if it is k -linear ly equiv a len t to the categ ory R mo d of ﬁnite-dimensional left mo dules over some ﬁnite-dimensional k -algebra R . Note tha t if A is a ﬁnite, then so is A op , since the functor  ( R mo d ) op → R op mo d N 7→ Hom( N , k ) is a k -linear equiv alence. Theorem 7. 6. L et C b e a ﬁnite tensor c ate gory over a ﬁeld k . Then C is e quivalent, as a tensor c ate gory, to the c ate gory of mo dules over a ﬁn it e- dimensio nal left Hopf algebr oid over k . W e ﬁrst state a nd prove an ana logue of this theorem in terms of Hopf monads in a so mewhat more general setting. Let C b e a monoidal ca tegory . Recall that the categ ory En d ( C ) o f e ndofunctors of C is strict monoidal with c o mposition for monoidal pro duct and 1 C for monoida l unit. The functor Ω :  C → End ( C ) X 7→ X ⊗ ? is strong monoidal. Theorem 7. 7. L et E b e a ful l monoidal sub c ate gory of E nd ( C ) c ontaining Ω( C ) . Denote by ω : C → E the c or estriction of Ω to E . Then (a) If ω has a left adjoint F , the adjunction ( F , ω ) is monadic, i ts monad T = ω F is a bimonad on E , and t he c omp arison functor C → E T is a monoidal e quivalenc e. (b) If C is right autonomous, t hen ω has a left adjoi nt if and only if the c o end F ( e ) = Z X ∈C e ( X ) ⊗ X ∨ exists for al l e ∈ Ob( E ) . In that c ase, the assignment e 7→ F ( e ) deﬁnes a functor which is a left adjoint of ω , and t he bimonad T = ω F is a right Hopf monad. (c) If C is autonomous and ω has a left adjoi nt F , then the bimonad T = ω F is a Hopf monad. 44 A. BR UGUI ` ERES, S. LACK, AND A. VIRELIZIER Pr o of. Assume ω has a left adjoint F . Then the adjunction ( F , ω ) is a comonoida l adjunction, so that the comparison functor K : C → E T is stro ng monoidal. Besides, ω has a left qua si-in verse e 7→ e ( 1 ), a nd so satisﬁes conditions (a) and (b) of Beck’s monadicity Theore m 2.1, so that the a djunction ( F , ω ) is monadic and K is a monoidal equiv alence. Hence Part (a). Assume C is right autonomous. F or e ∈ O b( E ) a nd X ∈ Ob( C ), we hav e a natura l bijection b etw een na tural transfor mations e → X ⊗ ? and dinatural transfor mations { e ( Y ) ⊗ Y ∨ → X } Y ∈ Ob( C ) . Therefore ω has a right adjoint if and only if the co ends F ( e ) exist for any ob ject e o f E . Ass ume that such is the case. Then the assig nmen t e 7→ F ( e ) gives a le ft adjoint o f ω . F or X ∈ Ob( C ) and e ∈ Ob( E ), we hav e: F ( ω ( X ) ◦ e ) = Z Y ∈C X ⊗ e ( Y ) ⊗ Y ∨ ≃ X ⊗ Z Y ∈C e ( Y ) ⊗ Y ∨ = X ⊗ F e bec ause X ⊗ ? has a r igh t a djoin t X ∨ ⊗ ? and so pres erv es colimits. One checks that this iso mo rphism is the r igh t Hopf op erator H r e,X : F ( ω ( X ) ◦ e ) → X ⊗ F e of the adjunction ( F , ω ). Thus T is a right Hopf monad by Theorem 2.15. Hence Part (b). Finally assume that C is also left autono mous. Let X ∈ O b( C ) and e ∈ Ob( E ). Since the functor ∨ X ⊗ ? is left a djoin t to X ⊗ ? a nd the functor ? ⊗ X pr eserves colimits (b ecause it has a r igh t adjoint ? ⊗ ∨ X ), we have: F ( e ◦ ω ( X )) = Z Y ∈C e ( X ⊗ Y ) ⊗ Y ∨ ≃ Z Y ∈C e ( Y ) ⊗ ( ∨ X ⊗ Y ) ∨ ≃ Z Y ∈C e ( Y ) ⊗ Y ∨ ⊗ X ≃ F e ⊗ X. One chec k s that the comp osition of these isomorphisms is the left Hopf op erator H l X,e : F ( e ◦ ω ( X )) → F e ⊗ X o f the adjunction ( F , ω ). Therefore T is also a left Hopf monad b y Theorem 2 .15. Hence Part (c).  Pr o of of The or em 7.6. W e apply The o rem 7 .7 to a ﬁnite tensor categor y C over a ﬁeld k . If A is a k - linear ab elian categor y , we denote b y End r a k ( A ) the full monoida l sub c ategory of End ( A ) of k - linear endofunctor s whic h a dmit a right adjoint. Set E = End r a k ( C ). F o r X ∈ Ob( C ), the endofunctor X ⊗ ? is k - linea r a nd has a r igh t adjoint, namely X ∨ ⊗ ?, so we hav e Ω( C ) ⊂ E . Denoting by ω : C → E the corestrictio n o f Ω to E , we hav e a commutativ e triangle of stro ng mo noidal k - linear functors: C Ω / / ω " " E E E E E End ( C ) E inclusion 8 8 q q q q q q By as sumption, there exists a ﬁnite dimensional k - algebr a R and a k - linear equiv alence Υ : C → R mo d , with qua s i-in verse Υ ∗ of Υ, hence a k - linea r strong monoidal equiv alence:  E = E nd r a k ( C ) → End r a k ( R mo d ) E 7→ Υ ◦ E ◦ Υ ∗ Comp osing this with the well-known strong monoida l k - linea r equiv alence  End r a k ( R mo d ) → R mo d R e 7→ e ( R R ) R we obtain a k - linear stro ng mono ida l eq uiv alence Θ : E → R mo d R ≃ R e mo d . In particula r E is a ﬁnite k - linear ab elian catego ry . The category End ( C ) is ab elian as a c a tegory of functor s to an ab elian categor y , Ω is exact (the tensor pro duct o f C being exa ct in each v ar iable), and the inclusio n E ֒ → End( C ) is fully faithful, so HOPF M ONADS ON MONOIDAL CA TEGORIES 45 ω is exact. It is a well-known fact that a right (resp. left) exa ct k - linear functor betw een ﬁnite k - linea r ab elian categories admits a left (resp. rig h t) adjoint. Thu s ω ha s a left adjoint F , as well as a r igh t adjoint R . By Theorem 7 .7, we conclude that the co monoidal adjunction ( F , ω ) is monadic and its monad T = ω F is a Ho pf monad. Mo reov er T is k - linea r and has a right adjoint ω R . Now we trans port T along the k - linear monoidal equiv alence Θ : E → R mo d R . Pick a quasi- in verse Θ ∗ of Θ. The a djunction ( F Θ ∗ , Θ ω ) is a mo nadic Hopf ad- junction. Its monad T ′ is a k - linear Hopf mona d on R mo d R with a r ig h t adjoint Θ ω R Θ ∗ . By Theor em 7.1, T is of the form T A for some bialgebroid A with base R , which is by deﬁnition a Hopf alg ebroid. Monadicity ensures that the co mparison functor C → ( R mo d R ) T = A mo d is a k - linear mono idal equiv alence of c ategories. This concludes the pro of of Theo rem 7.6.  References [BW85] M. 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