Bimonads and Hopf monads on categories

BIMONADS AND HOPF MONADS ON CA TEGORIES BAC HUKI MESABLISHVI LI , TBILISI AND ROBER T W ISBAUER, D ¨ USSELDOR F Abstra ct. The purp ose of this paper is to dev elop a theory of bimonads and H opf monads on arbitrary categ ories th us p ro viding the p ossibility to transf er the essential s of th e theory of Hopf alg ebras in vector spaces to mor e general settings. There are sev eral extensions of this theory to m onoidal categories whic h in a certain sense follo w the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on an y category A and b y this w e retain some of the combinatorial complexity whic h makes th e theory so interesting. As a basic t ool w e use distributive laws b etw een monads and comonads (en tw inings) on A : w e d eﬁne a bimonad on A as an endofunctor B which is a monad and a comonad with an ent wining λ : B B → B B satisfying certain conditions. This λ is also emplo yed to deﬁne the category A B B of (mixed) B -bimo dules. In the classical situation, an ent wining λ is derived from the tw ist map for vector sp aces. H ere this need n ot b e th e case b u t there may ex ist sp ecial distributive la ws τ : B B → B B satisfying the Y ang-Baxter eq uation ( lo c al pr ebr aidings ) whic h ind u ce an entw ining λ and lead to an extension of th e theory of br aide d Hopf algebr as . An antipo de is deﬁn ed as a natural transformation S : B → B with sp ecial prop erties and for categori es A with limits or colimits and bimonads B preserving t hem, t he existence of an antip ode is equiv alen t to B inducing an equiv alence b etw een A and the category A B B of B -bimo dules. This is a general form of th e F undamental The or em of Hopf algebras. Finally we observe a nice symmetry: If B is an endofunct or with a right adjoin t R , then B is a (Hopf ) bimonad if and only if R is a (Hopf ) bimonad. Thus a k -vector space H is a Hopf algebra if and only if H om k ( H , − ) is a Hopf bimonad. This provides a rich source for Hopf monads not deﬁned by ten sor pro ducts and generalises the we ll-known fact that a ﬁnite dimensional k - vector sp ace H is a Hopf algebra if and only if its du al H ∗ = Hom k ( H , k ) is a Hopf algebra. Moreo ver, we obtain that any set G is a group if and only if the functor Map(G , − ) is a Hopf monad on the category of sets. Contents 1. In tro duction 1 2. Distributiv e la ws 3 3. Actions on fun ctors and Galois functors 6 4. Bimonads 12 5. An tip o de 15 6. Lo cal prebraidings for Hopf m on ad s 18 7. Adjoin ts of bimonad s 28 References 32 1. Introduction The theory of algebras (monads ) as w ell as of coal gebras (comonads) is well und er s too d in v arious ﬁelds of mathematis as algebra (e.g. [8]), universal algebra (e.g . [13]), logic or op erational seman tics (e.g. [31]), theoretical computer science (e.g. [23]). The relationship b et ween monads and comonads is cont rolled b y distributive laws in tro duced in the s even ties 1 2 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF b y Bec k (see [2]). In algebra one of the fundamental n otions emerging in this conte xt are the Hopf alg ebras. The d eﬁ nition is m aking hea vy use of the tensor pro duct and thus generalisatio ns of this theory were mainly considered f or monoidal c ate gories . Th ey allo w readily to transfer formalisms from the ca tegory of v ector spaces to the more general sett ings (e.g. Bespalo v and Braban t [3] and [21]). A Hopf algebra is an algebra as well as a coalgebra. T h u s one w a y of generalisation is to consider distinct alge bras and coalg eb ras and some r elationship b et ween them. This leads to the theory of entwining structur es and c orings o v er asso ciativ e rings (e.g. [8]) and one ma y ask how to formulate this in more general categories. The d eﬁnition of bimonads on a monoidal category as monads whose functor part is comonoidal b y Brugui ` eres and Virelizier in [7, 2.3] may b e seen as going in this direction. Suc h fu n ctors are called Hopf monads in Moerd ijk [22] and opmonoidal monads in McCrud d en [18, Example 2.5]. In 2.2 we giv e more d etails of this notion. Another extension of th e theory of corings are the gener alise d bialgebr as in Lo day in [17]. These are Sc hur functors (on vecto r spaces) with a monad stru cture (op erads) and a sp eciﬁed coalgebra structure satisfying certain compatibilit y conditions [17, 2.2.1]. While in [17] us e is made of the canonical t wist map, it is stressed in [7] that the theory is bu ilt up without reference to an y br aiding. More commen ts on these constructions are giv en in 2.3. The purp ose of the presen t p ap er is to formulate th e essentia ls of the classical theory of Hopf algebras for any (n ot necessarily monoidal) categ ory , th us making it accessible to a wide ﬁeld of applications. W e also emp lo y the fact that the category of endofun ctors (with the Go d emen t p r o duct as comp osition) alw ays has a tensor pr o duct giv en b y comp osition of n atural transformations but no tensor pro du ct is requ ired for the base category . Compatibilit y b et ween m on ad s and comonads are formulat ed as distribu tive la ws wh ose prop erties are recalled in S ection 2. In Section 3, general categ orical notions are presen ted and Galois functors are deﬁned and inv estigated, in particular equiv ale nces induced for related categories (relativ e injectiv es). As suggested in [33, 5.13], we deﬁne a bimona d H = ( H , m, e, δ, ε ) on any category A as an endofunctor H with a monad and a comonad structur e satisfying compatibilit y conditions (en twining) (see 4.1). T he latter do n ot refer to an y braiding b ut in sp ecial cases they can b e d eriv ed from a lo c al pr ebr aid ing τ : H H → H H (see 6.3). In this case the bimonad sh ows the characte ristics of br aide d b i algebr as (Section 6). Related to a bimonad H there is the (Eilenb erg-Moore) category A H H of bimo du les with a comparison functor K H : A → A H H . An antip o de is deﬁn ed as a natural tr an s formation S : H → H satisfying m · S H · δ = e · ε = m · H S · δ . It exists if and only if the natural transformation γ := H m · δ H : H H → H H is an isomorphism. If the category A is Cauc h y complete and H preserv es limits or colimits, the existence of an an tip o de is equiv alen t to the comparison functor b eing an equiv alence (see 5.6). This is a general form of the F u ndamenta l Theorem for Hopf algebras. An y generalisation of Hopf algebras should oﬀer an extension of this imp ortant r esult. Of course, bialgebras and Hopf algebras o ver comm utativ e r ings R pro vid e the protot yp es for this theory: on R -Mo d, the categ ory of R -mo dules, one considers the endofun ctor B ⊗ R − : R -Mo d → R -Mo d where B is an R -mo du le with algebra and coalg eb ra structures, and an ent wining deriv ed from the twist map (braiding) M ⊗ R N → N ⊗ R M (e.g. [5, Section 8]). More generally , for a comonad H , the en t winin g λ : H H → H H ma y b e derive d from a lo c a l pr eb r aiding τ : H H → H H (see 6.7) and then results s im ilar to those known f or braided Hopf algebras are obtained. In particular, the comp osition H H is again a bimonad (see 6.8) and, if τ 2 = 1, an opp osite bimonad can b e d eﬁned (see 6.10). BIMONADS AND HO PF MONADS ON CA TEGORIES 3 In case a bimonad H on A has a r ight (or left) adjoint endofunctor R , then R is again a bimonad and has an an tip o de (or lo cal prebraiding) if and only if so do es H (see 7.5). In particular, for R -mo du les B , the fu nctor Hom R ( B , − ) is right adjoin t to B ⊗ R − and hence B is a Hopf algebra if and only if Hom R ( B , − ) is a Hopf monad. This provides a ric h source for examples of Hopf m onads n ot d eﬁned by a tensor pro du ct and extends a symmetry prin ciple k n o wn for ﬁnite dimensional Hopf algebras (see 7.8). W e close with the obs er v ation that a set G is a group if and only if the endofunctor Map(G , − ) is a Hopf monad on the catgeo ry of sets (7.9). Note that the pattern of our deﬁnition of bimonads resembles the deﬁnition of F r ob enius monads on any category b y Street in [27 ]. Those are monads T = ( T , µ, η ) with natural transformations ε : T → I and ρ : T → T T , sub j ect to s uitable conditions, w hic h ind uce a comonad structure δ = T µ · ρT : T → T T and pr o duct an d copro d uct on T satisfy the compatibilit y condition T µ · δT = δ · µ = µT · T δ . 2. Distributive la ws Distributiv e la ws b et ween endofun ctors w ere stud ied by Bec k [2 ], Barr [1] and others in the sev en ties of the last cen tury . They are a fu n damen tal to ol for us and we recall some facts needed in th e sequel. F or more d etails and references we refer to [33]. 2.1 . En t w ining from monad to comonad. Let T = ( T , m, e ) b e a monad and G = ( G, δ , ε ) a comonad on a category A . A natural transformation λ : T G → GT is called a mixe d distributive law or entwining from th e monad T to the comonad G if th e diagrams G eG ~ ~ | | | | | | | | Ge ! ! C C C C C C C C T G λ / / GT , T G T ε ! ! C C C C C C C C λ / / GT εT } } { { { { { { { { T T G λ   T δ / / T GG λG / / GT G Gλ   and T T G mG   T λ / / T GT λT / / GT T Gm   GT δT / / GGT T G λ / / GT are commutat iv e. It is sh o wn in [34] that f or an arb itrary mixed distrib utiv e la w λ : T G → GT from a monad T to a comonad G , the triple b G = ( b G, b δ , b ε ), is a comonad on the category A T of T -mo dules (also called T -algebras), where for an y ob ject ( a, h a ) of A T , • b G ( a, h a ) = ( G ( a ) , G ( h a ) · λ a ); • ( b δ ) ( a,h a ) = δ a , and • ( b ε ) ( a,h a ) = ε a . b G is called the lifting of G corresp ond ing to the mixed d istributive law λ . F urthermore, the trip le b T = ( b T , b m, b e ) is a monad on the category A G of G -comodu les, where for an y ob ject ( a, θ a ) of the catego ry A G , • b T ( a, θ a ) = ( T ( a ) , λ a · T ( θ a )); • ( b m ) ( a,θ a ) = m a , and • ( b e ) ( a,θ a ) = e a . 4 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF This monad is called the lifting of T corresp onding to the mixed distribu tiv e law λ . One has an isomorphism of categories ( A G ) b T ≃ ( A T ) b G , and w e write A G T ( λ ) for this catego ry . An ob ject of A G T ( λ ) is a triple ( a, h a , θ a ), where ( a, h a ) ∈ A T and ( a, θ a ) ∈ A G with commuting diagram (2.1) T ( a ) h a / / T ( θ a )   a θ a / / G ( a ) T G ( a ) λ a / / GT ( a ) . G ( h a ) O O W e consider t wo examples of en t winings wh ich ma y (also ) b e considered as generalisations of Hopf algebras. They are d iﬀeren t from our approac h and we will not refer to them later on. 2.2 . Opmonoidal functors. Let ( V , ⊗ , I ) b e a strict monoidal category . F ollo wing Mc- Crudd en [18 , Example 2.5], one ma y call a m onad ( T , µ, η ) on V opmono idal if there exist morphisms θ : T ( I ) → I and χ X,Y : T ( X ⊗ Y ) → T ( X ) ⊗ T ( Y ), the latter n atural in X , Y ∈ V , whic h are compatible w ith the tensor structure of V and the monad stru cture of T . Suc h f u nctors can also b e c haracterised by the condition that the tensor pro duct of V can b e lifted to the category of T -mo du les (e.g. [33, 3.4]). They w ere introd uced and n amed Hopf monads by Mo erdij k in [22, Deﬁnition 1.1] and called bimonads by Brugui` e res and Virelizier in [7, 2.3]. It is men tioned in [7, Example 2.8] that Szlac h´ an yi’s bialgebroids in [29] ma y b e interpreted in terms of su c h ”bimonads”. I t is pr eferable to us e the terminology from [18] since these functors are n either b im on ad s nor Hopf monad s in a strict sense bu t rather an ent wining (as in 2.1) b et w een the monad T and the comonad T ( I ) ⊗ − on V : Indeed, the compatibilit y conditions requ ired in the deﬁn itions ind uce a copr o duct χ I , I : T ( I ) → T ( I ) ⊗ T ( I ) with counit θ : T ( I ) → I . Moreo v er, the relation b et ween χ and µ (e.g. (15) in [7, 2.3]) lead to the commuta tiv e diagram (usin g X ⊗ I = X ) T T ( X ) µ / / T ( χ I ,X )   T ( X ) χ I ,X / / T ( I ) ⊗ T ( X ) T ( T ( I ) ⊗ T ( X )) χ T ( I ) ,T ( X ) / / T T ( I ) ⊗ T T ( X ) µ I ⊗ T T ( X ) / / T ( I ) ⊗ T T ( X ) T ( I ) ⊗ µ X O O This sh o ws that T ( X ) is a mixed ( T , T ( I ) ⊗ − )-bimodu le for the en twining map λ = ( µ I ⊗ T ( − )) ◦ χ T ( I ) , − : T ( T ( I ) ⊗ − ) → T ( I ) ⊗ T ( − ) . The antip o de of a classical Hopf algebra H is deﬁn ed as a sp ecial endomorp hism of H . Since opmonoidal monads T relat e tw o d istinct functors it is not surpr ising that the notion of an an tip o de can not b e tr an s ferred easily to this situation and th e attempt to do so leads to an ”apparen tly complicated deﬁnition” in [7, 3.3 and Remark 3.5]. Hereby the b ase category C is required to b e autonomous . 2.3 . Generalized bialgebras and Hopf op erads. The gener alise d bialgebr as o v er ﬁelds as d eﬁned in L o da y [17, Section 2.1] are similar to the m ixed bimo d ules (see 2.1): they are v ector spaces wh ich are mo dules ov er some op er ad A (Sch ur functors with multiplicat ion and u nit) and como dules o v er some coalgebras C c , whic h are linear duals of some op erad C . BIMONADS AND HO PF MONADS ON CA TEGORIES 5 Similar to the opmonoidal monads the coalgebraic structure is based on the tensor pro du ct (of vect or spaces). The Hyp othesis (H0) in [7] r esem bles the role of the ent wining λ in 2.1. Th e Hyp othesis (H1) requires that the free A -algebra is a ( C C , A )-bialgebra: this is similar to the condition on an A -coring C , A an asso ciativ e algebra, to h a ve a C -como dule structure (equiv alen tly the existence of a group-like element, e.g. [8, 28.2]). The cond ition (H2iso) pla ys the role of the canonical isomorph ism deﬁning Galois c orings and the Galois Coring Structur e The or em [8, 28.19] ma y b e compared with the Rigidity The or em [17, 2.3.7]. The latter can b e considered as a generalisation of th e Hopf-Borel Theorem (see [17, 4.1.8]) and of the Cartier-Milnor-Mo ore Th eorem (see [17, 4.1.3]). In [17, 3.2], Hopf op er ad s are deﬁned in the sense of Mo erdijk [22] and thus the coalg ebraic part is d ep endent on th e tensor pro d uct. This is only a sk etc h of the similarities b et ween Lo day’s setting and our approac h h ere. It will b e in teresting to work out the relationship in more detail. Similar to 2.1 w e will also need the notion of mixed d istributiv e la ws from a comonad to a monad. 2.4 . E n twining from comonad to monad. A natural transformation λ : GT → T G is a mixe d distributive law from a comonad G to a monad T , also called an entwining of G and T , if the d iagrams G Ge ~ ~ | | | | | | | | eG ! ! D D D D D D D D GT εT B B B B B B B B λ / / T G T ε ~ ~ | | | | | | | | GT λ / / T G , T GT T Gm   λT / / T GT T λ / / T T G mG   GGT Gλ / / GT G λG / / T GG GT λ / / T G, GT δT O O λ / / T G T δ O O are commutiv e. F or conv enience w e r ecall the distribu tiv e laws b et ween t wo monads and b et wee n tw o comonads (e.g. [2], [1], [33, 4.4 and 4.9]). 2.5 . Monad distributive. Let F = ( F, m, e ) and T = ( T , m ′ , e ′ ) b e monads on the category A . A natural transformation λ : F T → T F is said to b e monad distributive if it induces the comm utativ e diagrams T e T ~ ~ | | | | | | | | T e ! ! B B B B B B B B F T λ / / T F , F F e ′ } } { { { { { { { { e ′ F ! ! C C C C C C C C F T λ / / T F . F F T m T / / F λ   F T λ   F T F λ F / / T F F T m / / T F , F T T F m ′ / / λ T   F T λ   T F T T λ / / T T F m ′ F / / T F . In this case λ : F T → T F induces a canonical monad structure on T F . 2.6 . Comonad distributive. Let G = ( G, δ , ε ) and T = ( T , δ ′ , ε ′ ) b e comonads on the catego ry A . A n atural transformation ϕ : T G → GT is said to b e c omonad distributive if it induces th e comm u tativ e d iagrams 6 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF T G T ε ! ! C C C C C C C C ϕ / / GT ε T } } { { { { { { { { T , T G ε ′ G ! ! C C C C C C C C ϕ / / GT Gε ′ } } { { { { { { { { G , T G T δ / / ϕ   T GG ϕ G / / GT G Gϕ   GT δ T / / GGT , T G δ ′ G / / ϕ   T T G T ϕ / / T GT ϕ T   GT Gδ ′ / / GT T . In this case ϕ : T G → GT induces a canonical comonad structure on T G . 3. Actions on fun ctors and Galois funct ors The language of mo d ules o v er rings can also b e used to describ e actions of monads on functors. Doing this w e deﬁne Galois functors and to c h aracterise those we inv estigate the relationships b et w een categories of r elativ e inj ectiv e ob jects. 3.1 . T -actions on functors. Let A and B b e categories. Giv en a monad T = ( T , m, e ) on A and any fun ctor L : A → B , we sa y that L is a (right) T -mo dule if there exists a natural transformation α L : LT → L such th at the diagrams (3.1) L A A A A A A A A A A A A A A A A Le / / LT α L   L, LT T Lm / / α L T   LT α L   LT α L / / L comm ute. It is easy to see that ( T , m ) and ( T T , T m ) b oth are T -mo d u les. Similarly , giv en a comonad G = ( G, δ, ε ) on A , a fun ctor K : B → A is a left G -c omo dule if there exists a n atural transformation β K : K → GK for whic h the diagrams K C C C C C C C C C C C C C C C C β K / / GK εK   K, K β K / / β K   GK δK   GK Gβ K / / GGK comm ute. Giv en t wo T -mo d ules ( L, α L ), ( L ′ , α L ′ ), a n atural transf orm ation g : L → L ′ is called T -line ar if the d iagram (3.2) LT g T / / α L   L ′ T α L ′   L g / / L ′ comm utes. 3.2. Lemma. L et ( L, α L ) b e a T -mo dule. If f , f ′ : T T → L ar e T -line ar morphisms fr om the T -mo dule ( T T , T m ) to the T -mo dule ( L, α L ) such that f · T e = f ′ · T e , then f = f ′ . BIMONADS AND HO PF MONADS ON CA TEGORIES 7 Pro of. Sin ce f · T e = f ′ · T e , we ha ve α L · f T · T eT = α L · f ′ T · T eT . Moreo ver, sin ce f and f ′ are b oth T -linear, we ha v e the comm utative diagrams T T T T m   f T / / LT α L   T T f / / L, T T T T m   f ′ T / / LT α L   T T f ′ / / L. Th us α L · f T = f · T m and α L · f ′ T = f ′ · T m , and we ha ve f · T m · T eT = f ′ · T m · T eT . It f ollo ws - s ince T m · T eT = 1 - th at f = f ′ . ⊔ ⊓ 3.3 . Left G -comodule functors. Let G b e a comonad on a category A , let U G : A G → A b e the forgetful functor and wr ite φ G : A → A G for th e cofree G -como dule f u nctor. Fix a functor F : B → A , and consider a fun ctor F : B → A G making the diagram (3.3) B F / / F   > > > > > > > A G U G ~ ~ } } } } } } } } A comm utativ e. Then F ( b ) = ( F ( b ) , α F ( b ) ) for some α F ( b ) : F ( b ) → GF ( b ). Consid er the natural transformation (3.4) ¯ α F : F → GF, whose b -comp onen t is α F ( b ) . I t sh ould b e p ointe d out that ¯ α F mak es F a left G -como d ule, and it is easy to s ee that there is a one to one corresp ondence b et wee n fun ctors F : B → A G making the diagram (3.3) comm ute and natural transform ations ¯ α F : F → GF making F a left G -como dule. The follo wing is an immediate consequence of (the du al of ) [10, Prop ositions I I,1.1 and I I,1.4]: 3.4. Theorem. Supp ose that F has a right adjoint R : A → B with unit η : 1 → RF and c ounit ε : F R → 1 . Then the c omp osite t F : F R ¯ α F R / / GF R Gε / / G. is a morphism fr om the c omo nad G ′ = ( F R, F η R, ε ) gener ate d by the adjunction η , ε : F ⊣ R : A → B to the c omona d G . Mor e over, the assignment F − → t F yields a one to one c orr esp ond e nc e b etwe en functors F : B → A G making the diagr am (3.3) c ommutative and morphisms of c omonads t F : G ′ → G . 3.5. Deﬁnition. W e sa y that a left G -como dule F : B → A with a r ight adj oin t R : B → A is G -Galois if the corresp onding morphism t F : F R → G of comonads on A is an isomorphism. As an example, consider an A -coring C , A an asso ciativ e r ing, and any r ight C -como dule P with S = End C ( P ). T hen there is a natural transformation ˜ µ : Hom A ( P , − ) ⊗ S P → − ⊗ A C and P is called a Galois c omo dule provided ˜ µ X is an isomorphism for any r igh t A -mo dule X , that is, the fu nctor − ⊗ S P : M S → M C is a − ⊗ A C -Galois como dule (see [32, Deﬁniton 4.1]). 8 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF 3.6 . Righ t adjoin t functor of F . When the catego ry B has equalisers, the functor F h as a right adjoin t, wh ic h can b e d escrib ed as follo ws: W riting β R for the comp osite R ηR / / RF R Rt F / / RG, it is not h ard to see that the equaliser ( R, e ) of the f ollo wing diagram RU G RU G η G / / β R U G / / RGU G = R U G φ G U G , where η G : 1 → φ G U G is the unit of the adjunction U G ⊣ φ G , is righ t adj oin t to F . 3.7 . Adjoints and monads. F or categ ories A , B , let L : A → B b e a functor with right adjoin t R : B → A . Let T = ( T , m, e ) b e a monad on A and sup p ose there exists a functor R : B → A T yielding the comm u tativ e d iagram B R / / R   @ @ @ @ @ @ @ @ A T U T } } | | | | | | | | A . Then R ( b ) = ( R ( b ) , β b ) for some β b : T R ( b ) → R ( b ) and the collection { β b , b ∈ B } con- stitutes a natural tran s formation β R : T R → R . It is p r o ved in [10] that the natural transformation t R : T T η / / T RL β L / / RL is a morphism of monads . By the dual of [21, Theorem 4.4], we obtain: The functor R i s an e quivalenc e of c ate gories i ﬀ the functor R is monadic and t R is an isomorph ism of monad s. In view of the c haracterisation of Galois fun ctors we ha ve a closer lo ok at some related classes of relativ e injectiv e ob j ects. Let F : B → A b e an y fu n ctor. Rec all (from [14]) th at an ob ject b ∈ B is said to b e F - inje ctive if for an y diagram in B , b 1 g   f / / b 2 h   b with F ( f ) a split m on omorp hism in A , there exists a morphism h : b 2 → b such that hf = g . W e write Inj ( F , B ) for the full su b category of B with ob jects all F -in j ectiv es. The follo w ing r esult fr om [26] will b e needed. 3.8. Prop osition. L et η , ε : F ⊣ R : A → B b e an adjunction. F or any obje ct b ∈ B , the fol lowing assertions ar e e quivalent: (a) b is F - inje ctive; (b) b is a c or etr act for some R ( a ) , with a ∈ A ; (c) the b - c omp o nent η b : b → R F ( b ) of η is a split monom orphism. 3.9. Remark. F or any a ∈ A , R ( ε a ) · η R ( a ) = 1 b y one of the triangular identit ies for the adjunction F ⊣ R . Th us, R ( a ) ∈ Inj ( F , B ) f or all a ∈ A . Moreo v er, since the comp osite of coretracts is again a coretract, it follo ws f rom (b) th at Inj ( F , B ) is closed u n der coretracts. BIMONADS AND HO PF MONADS ON CA TEGORIES 9 3.10 . F unctor b et ween injectives. Let K G ′ : B → A G ′ b e the comparison functor (nota- tion as in 3.4). If b ∈ B is F -injectiv e, then K G ′ ( b ) = ( F ( b ) , F ( η b )) is U G ′ -injectiv e, since by the fact that η b is a s p lit monomorphism in B , ( η G ′ ) φ G ′ ( b ) = F ( η b ) is a sp lit monomorp hism in A G ′ ( G ′ as in 3.4). Th u s the f u nctor K G ′ : B → A G ′ yields a functor Inj ( K G ′ ) : Inj ( F , B ) → Inj ( φ G ′ , A G ′ ) . When B has equalisers, this fu n ctor is an equiv alence of categories (see [26]). W e sh all henceforth assume that B has equalisers. 3.11. Prop osition. The functor R : A G → B r estricts to a functor R ′ : Inj ( U G , A G ) → Inj ( F , B ) . Pro of. Let ( a, θ a ) b e an arbitrary ob ject of Inj ( U G , A G ). Th en, by Prop osition 3.8, there exists an ob j ect a 0 ∈ A such that ( a, θ a ) is a coretraction of φ G ( a 0 ) = ( G ( a 0 ) , δ a 0 ) in A G , i.e., there exist morp hisms f : ( a, θ a ) → ( G ( a 0 ) , δ a 0 ) and g : ( G ( a 0 ) , δ a 0 ) → ( a, θ a ) in A G with g f = 1. Since f and g are morp hisms in A G , the diagram G ( a 0 ) g   ( δ G ) a 0 / / GG ( a 0 ) G ( g )   a f O O θ a / / G ( a ) G ( f ) O O comm utes. By n aturalit y of β R , the diagram RG ( a 0 ) R ( g )   ( β R ) G ( a 0 ) / / RGG ( a 0 ) RG ( g )   R ( a ) R ( f ) O O ( β R ) a / / RG ( a ) RG ( f ) O O also comm utes. Consider n o w the follo wing comm u tativ e diagram (3.5) R ( a 0 )   β a 0 / / RG ( a 0 ) R ( g )   ( β R ) G ( a 0 ) / / R (( δ G ) a 0 ) / / RGG ( a 0 ) RG ( g )   R ( a, θ a ) O O e ( a,θ a ) / / R ( a ) R ( f ) O O ( β R ) a / / R ( θ a ) / / RG ( a ) . RG ( f ) O O It is not hard to see that the top row of this diagram is a (split) equaliser (see, [12]), and since the b ottom r o w is an equ aliser by the ve ry deﬁnition of e , it f ollo ws fr om the comm utativit y of the diagram that R ( a, θ a ) is a coretract of R ( a 0 ), and thus is an ob ject of Inj ( F , B ) (see Remark 3.9). It m eans that th e functor R : A G → B can b e restricted to a functor R ′ : Inj ( U G , A G ) → Inj ( F , B ). ⊔ ⊓ 3.12. Proposition. Supp ose that for any b ∈ B , ( t F ) F ( b ) is an isomorphism. Then the functor F : B → A G c an b e r estricte d to a fu nctor F ′ : Inj ( F , B ) → Inj ( U G , A G ) . 10 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF Pro of. Let δ ′ denote the com ultiplication in the comonad G ′ (see 3.4), i.e., δ ′ = F η R. Then for any b ∈ B , F ( RF ( b )) = A t F ( φ G ′ ( U F ( b ))) = A t F ( F RF ( b ) , F η RF ( b ) ) = A t F ( G ′ F ( b ) , δ ′ F ( b ) ) = ( G ′ F ( b ) , ( t F ) G ′ F ( b ) · δ ′ F ( b ) ) . Consider n o w the diagram G ′ F ( b ) ( t F ) F ( b ) / / δ ′ F ( b )   GF ( b ) δ F ( b )   G ′ G ′ F ( b ) (1) ( t F ) F ( b ) . ( t F ) F ( b ) ' ' N N N N N N N N N N N N N N N N N N N N N N N N ( t F ) G ′ F ( b )   GG ′ F ( b ) G (( t F ) F ( b ) ) / / GGF ( b ) , in wh ic h the triangle commutes by the deﬁnition of the comp osite ( t F ) F ( b ) . ( t F ) F ( b ) , w h ile the diagram (1) commutes sin ce t F is a morp hism of comonads. The commutati vit y of the outer diagram sho w s that ( t F ) F ( b ) is a morphism from the G -coalg ebra F ( RF ( b )) = ( G ′ F ( b ) , ( t F ) G ′ F ( b ) · δ ′ F ( b ) ) to the G -coalge bra ( GF ( b ) , δ F ( b ) ). Moreo v er, ( t F ) F ( b ) is a n isomor- phism by our assu mption. Thus, for an y b ∈ B , F ( RF ( b )) is isomorp h ic to the G -coalgebra ( GF ( b ) , δ F ( b ) ), wh ich is of course an ob ject of the category I nj ( U G , A G ). No w, since any b ∈ Inj ( F , B ) is a co r etract of RF ( b ) (see Remark 3.9), an d since an y fun ctor tak es coretracts to coretracts, it f ollo ws that, for any b ∈ I nj ( F , B ), F ( b ) is a coretract of the G -coalg ebra ( GF ( b ) , δ F ( b ) ) ∈ I nj ( U G , A G ), and thus is an ob ject of the category Inj ( U G , A G ) again by Remark 3.9. This completes the pro of. ⊔ ⊓ The follo w ing tec hnical obs er v ation is n eeded for th e next pr op osition. 3.13. Lemma. L et ι, κ : W ⊣ W ′ : Y → X b e an adjunction of any c ate gories. If i : x ′ → x and j : x → x ′ ar e morphisms in X suc h that j i = 1 and if ι x is an isomorphism, then ι x ′ is also an i somorphism. Pro of. Since j i = 1, the diagram x ′ i / / x 1 / / ij / / x is a split equaliser. T hen the diagram W ′ W ( x ′ ) W ′ W ( i ) / / W ′ W ( x ) 1 / / W ′ W ( ij ) / / W ′ W ( x ) is also a split equaliser. No w consid er in g the follo wing comm utativ e diagram x ′ ι x ′   i / / x κ x   1 / / ij / / x κ x   W ′ W ( x ′ ) W ′ W ( i ) / / W ′ W ( x ) 1 / / W ′ W ( ij ) / / W ′ W ( x ) BIMONADS AND HO PF MONADS ON CA TEGORIES 11 and r ecalling that the ve rtical t wo morphisms are b oth isomorphism s by assump tion, w e get that the morph ism ι x ′ is also an isomorp hism. ⊔ ⊓ 3.14. Prop osition. In the situation of P r op osition 3.12, Inj ( F , B ) is (isomorphic to) a c or eﬂe c tive sub c ate gory of the c ate gory Inj ( U G , A G ) . Pro of. By Pr op osition 3.11, th e fu nctor R r estricts to a f u nctor R ′ : Inj ( U G , A G ) → Inj ( F , B ) , while according to Prop osition 3.12 , the functor F restricts to a fun ctor F ′ : Inj ( F , B ) → Inj ( U G , A G ) . Since • F is a left adjoint to R , • Inj ( F , B ) is a full su b category of B , and • Inj ( U G , A G ) is a f ull sub category of A G , the functor F ′ is left adjoint to th e f u nctor R ′ , and the unit η ′ : 1 → R ′ F ′ of the adjun ction F ′ ⊣ R ′ is the r estriction of η : F ⊣ R to th e sub catego ry Inj ( F , B ), while the counit ε ′ : F ′ R ′ → 1 of this adju nction is the restriction of ε : F R → 1 to the s u b category Inj ( U G , A G ). Next, since the top of the diagram 3.5 is a (split) equaliser, R ( G ( a 0 ) , δ a 0 ) ≃ R ( a 0 ). In particular, taking ( GF ( b ) , δ F ( b ) ), we see th at RF ( b ) ≃ R ( GF ( b ) , δ F ( b ) ) = R F ( U F ( b )) . Th us, the RF ( b )-co mp onent η ′ RF ( b ) of the unit η ′ : 1 → R ′ F ′ of the adjunction F ′ ⊣ R ′ is an isomorph ism. It no w follo ws from Lemma 3.13 - since an y b ∈ Inj ( F , B ) is a coretraction of RF ( b ) - that η ′ b is an isomorp hism for all b ∈ Inj ( F, B ) proving that the u nit η ′ of the adjunction F ′ ⊣ R ′ is an isomorphism. T hus Inj ( F , B ) is (isomorph ic to) a coreﬂecti v e sub category of the category Inj ( U G , A G ). ⊔ ⊓ 3.15. C orollary . In the situation of Pr op osition 3.12 , supp ose that e ach c omp onent of the unit η : 1 → R F is a split monomorp hism. Then the c ate gory B is (isomorphic to) a c or eﬂe c tive sub c ate gory of Inj ( U G , A G ) . Pro of. When ea ch co mp onent of the u nit η : 1 → RF is a sp lit monomorp hism, it follo ws from Prop osition 3.8 that every b ∈ B is F -injectiv e; i.e. B = Inj ( F , B ). The assertion no w follo ws f r om Prop osition 3.14. ⊔ ⊓ 3.16 . Cha racterisation of G-Galois como dules. Assume B to admit e qualisers, let G b e a c omonad on A , and F : B → A a functor with right adjoint R : A → B . If ther e exists a functor F : A → A G with U G F = F , then the fol lowing ar e e quiv alent: (a) F is G -Galois, i.e. t F : G ′ → G is an isomorphism ; (b) the fol lowing c omp osite is an isomorph ism: F R η G F R / / φ G U G F R = φ G F R φ G ε / / φ G ; (c) the f u nctor F : B → A G r estricts to an e quivalenc e of c ate gories Inj ( F , B ) → Inj ( U G , A G ); (d) for any ( a, θ a ) ∈ Inj ( U G , A G ) , the ( a, θ a ) -c omp on ent ε ( a,θ a ) of the c ounit ε of the adjunction F ⊣ R , is an isomorphism; 12 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF (e) for any a ∈ A , ε φ G ( a ) = ε ( G ( a ) ,δ a ) is an isomorp hism. Pro of. Th at (a) and (b) are equiv alen t is pro ved in [9 ]. By the pro of of [12, Th eorem of 2.6], for an y a ∈ A , ε φ G ( a ) = ε ( G ( a ) ,δ a ) = ( t F ) a , thus (a) and (e) are equiv alen t. By Remark 3.9, (d) implies (e). Since B admits equalisers by our assu mption on B , it follo ws from P rop osition 3.10 that the fun ctor Inj ( K G ′ ) is an equiv alence of catego ries. No w, if t F : G ′ → G is an isomorphism of comonads, then the fun ctor A t F is an isomorphism of categ ories, and thus F is isomorph ic to the comparison functor K G ′ . It now follo ws from Prop osition 3.10 that F r estricts to the functor Inj ( F , B ) → Inj ( U G , A G ) wh ic h is an equiv alence of categories. Thus (a) ⇒ (c). If the functor F : B → A G restricts to a functor F ′ : Inj ( F , B ) → Inj ( U G , A G ) , then one can pr o ve as in the pro of of Prop osition 3.9 th at F ′ is left adjoin t to R ′ and that the counit ε ′ : F ′ R ′ → 1 of this adjun ction is the restriction of the counit ε : F R → 1 of the adjun ction F ⊣ R to the su b category Inj ( U G , A G ). No w, if F ′ is an equiv alence of catego ries, th en ε ′ is an isomorph ism. T h u s, for an y ( a, θ a ) ∈ Inj ( U G , A G ), ε ′ ( a,θ a ) is an isomorphism pro v in g that (c) ⇒ (d). ⊔ ⊓ 4. Bimonads The follo wing deﬁnition w as suggested in [33, 5.13]. F or mon oidal catego r ies similar conditions were considered by T ake uc h i [30, Deﬁnition 5.1] and in [21]. Notice that the term b imonad is used with a d iﬀerent meaning in by Brugui` e res and Virelizier (see 2.2). 4.1. Deﬁnition. A bimonad H on a category A is an endofunctor H : A → A whic h has a monad stru cture H = ( H , m, e ) and a comonad structur e H = ( H , δ, ε ) suc h that (i) ε : H → 1 is a morphism f rom the m onad H to the id en tity monad; (ii) e : 1 → H is a morphism fr om the identit y comonad to the comonad H ; (iii) there is a mixed d istributiv e la w λ : H H → H H fr om the monad H to the comonad H yielding the comm utativ e diagram (4.1) H H m / / H δ   H δ / / H H H H H λH / / H H H , H m O O Note that the conditions (i), (ii) just m ean commutat ivit y of the diagrams (4.2) H H H ε / / m   H ε   H ε / / 1 , 1 e / / e   H δ   H eH / / H H , 1 e / / =   ? ? ? ? ? ? ? ? H ε   1 . 4.2 . Hopf mo dules. Giv en a b imonad H = ( H , H , λ ) on A , th e ob jects of A H H ( λ ) are called mixe d H - bimo dules or H -Hopf mo dules . By 2.1, they are triples ( a, h a , θ a ), where ( a, h a ) ∈ A H and ( a, θ a ) ∈ A H with commuting diagram BIMONADS AND HO PF MONADS ON CA TEGORIES 13 (4.3) H ( a ) h a / / H ( θ a )   a θ a / / H ( a ) H H ( a ) λ a / / H H ( a ) . H ( h a ) O O The morphism s in A H H ( λ ) are m orphisms in A w hic h are H -monad as w ell as H -comonad morphisms, Recall that a morph ism q : a → a in a category A is an idemp otent w hen q q = q , and an idemp oten t q is said to split if q has a factoriza tion q = i · ¯ q with ¯ q · i = 1. This happ ens if and only if the equaliser i = Eq(1 a , q ) exists or - equiv alen tly - th e co equaliser ¯ q = C o eq(1 a , q ) exists (e.g. [6 , Pr op osition 1]). The catgeory A is called Cauchy c omplete pr o vided every idemp oten t in A splits. 4.3 . Comparison functors. Giv en a bimon ad H = ( H = ( H, m, e ) , H = ( H , δ, ε ) , λ ) on a catego ry A , the mixed d istr ibutiv e la w λ ind u ces f unctors K H : A → ( A H ) b H , a 7→ (( H ( a ) , m a ) , δ a ) , K H : A → ( A H ) b H , a 7→ (( H ( a ) , δ a ) , m a ) , where b H is the lifting of the comonad H and b H is the lifting of the monad H by the mixed distributive la w λ . W e know that ( A H ) b H ≃ ( A H ) b H and d enote this category by A H H ( λ ) (see 2.1). Th ere are comm utative diagrams (4.4) A K H / / φ H A A A A A A A A A A A A A A A A A A ( A H ) b H U b H   A H , A K H / / φ H A A A A A A A A A A A A A A A A A A ( A H ) b H U b H   A H . (i) The functor φ H . The forgetful functor U H : A H → A is right adjoin t to th e free functor φ H and the unit η H : 1 → U H φ H of this adjun ction is the natural transform ation e : 1 → H . S ince ε : H → 1 is a morphism from the monad H to th e iden tity m on ad , ε · e = 1, thus e is a sp lit monomorphism. The adjunction φ H ⊣ U H generates th e comonad φ H U H on A H . Recall that for an y ( a, h a ) ∈ A H , φ H U H ( a, h a ) = ( H ( a ) , m a ) and b H ( a, h a ) = ( H ( a ) , H ( h a ) · λ a ) . As p oin ted out in [21], for an y ob ject b of A , K H ( b ) = ( H ( b ) , α H ( b ) ) for some α : H ( b ) → H H ( b ), thus indu cing a n atural transformation α K H : φ H → b H φ H , whose comp onen t at b ∈ A is α H ( b ) , w e m ay c ho ose it to b e just δ b , and w e ha ve a m orp hism of comonads t K H : φ H U H α K H U H / / b H φ H U H b H ε H / / b H , 14 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF where ε H is the counit of the adjunction φ H ⊣ U H , and since ( ε H ) ( a,h a ) = h a , w e see that for all ( a, h a ) ∈ A H , ( t K H ) ( a,h a ) is the comp osite (4.5) H ( a ) δ a / / H H ( a ) H ( h a ) / / H ( a ) . (ii) The functor φ H . The cofree H -comod ule functor φ H has the forgetful fun ctor U H : A H → A as a left adjoin t. The unit η : 1 → φ H U H and counit σ : U H φ H → 1 of the adjunction U H ⊣ φ H are give n b y the formulas: η ( a, θ a ) = θ a : ( a, θ a ) → φ H U H ( a, θ a ) = ( H ( a ) , δ a ) and σ a = ε a : H ( a ) = U H φ H ( a ) → a. Since ε is a sp lit epimorphism, it follo ws from C orollary 3.17 of [20] that, when A is Cauch y complete, the fu n ctor φ H is m onadic. Since K H ( a ) = (( H ( a ) , δ a ) , m a ), it is easy to see that the a -comp onent of α K H : b H K H → K H is j ust the m orphism m a : H H ( a ) → H ( a ) , and we h a ve a monad morphism t K H : b H b H η / / H φ H U H α K H U H / / φ H U H . It f ollo ws that f or any ( a, θ a ) ∈ A H , ( t K H ) ( a, θ a ) is the comp osite (4.6) H ( a ) H ( θ a ) / / H H ( a ) m a / / H ( a ) . 4.4 . The comparison functor as an equiv alence. L et A b e a Cauchy c omplete c ate gory. F or a bi monad H = ( H = ( H , m, e ) , H = ( H, δ, ε ) , λ ) , the f ol lowing ar e e quivalent: (a) K H : A → A H H ( λ ) , a → ( H ( a ) , δ a , m a ) , is an e quiv alenc e of c ate gories; (b) t K H : φ H U H → b H is an isomorphism of c omonads; (c) for any ( a, h a ) ∈ A H , the c omp osite H ( h a ) · δ a is an isomorphism ; (d) t K H : b H → φ H U H is an isomorphism of monads; (e) for any ( a, θ a ) ∈ A H , the c omp osite m a · H ( θ a ) is an isomorphism. Pro of. W e ma y identify the functors K H , K H and K H . (a) ⇔ (b) Sin ce A is Cauch y complete and since the u nit η H : 1 → U H φ H of the adju nction φ H ⊣ U H is a split monomorp hism, the functor φ H is comonadic by the dual of [19, Theorem 6]. No w, b y [21, Theorem 4.4.], K H is an equiv alence if and only if t K H is an isomorp hism. (b) ⇔ (c) and (d) ⇔ (e). By 4.3, the morp hisms in (b) come out as the m orphisms in (c), and the morphisms in (d) are just those in (e). (a) ⇔ (d) Since ε is a s plit epimorp hism, it follo ws f rom [20, C orollary 3.17] that (since A is C auc hy complete) the fun ctor φ H is monadic and hence K is an equiv alence by 3.7. ⊔ ⊓ BIMONADS AND HO PF MONADS ON CA TEGORIES 15 5. Antipode W e consider a bimonad H = ( H, m, e, δ, ε, λ ) on any catgeory A . 5.1 . Canonical maps. Deﬁne th e comp osites (5.1) γ : H H δH / / H H H H m / / H H , γ ′ : H H H δ / / H H H mH / / H H . In the diagram H H H δH H / / H m   H H H H H mH / / H H m   H H H H m   H H δH / / H H H H m / / H H , the le ft square commutes by n aturalit y of δ , while the right square comm utes b y associativit y of m . F rom this w e see that γ is left H -linear as a morphism fr om ( H H , H m ) to itself. A similar d iagram sho ws that γ ′ is r ight H -linear as a morp hism from ( H H , mH ) to itself. Moreo v er, in the d iagram H H e / / δ   H H δH / / H H H H m   H H H H e 6 6 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l H H the top triangle commute s b y fu n ctorialit y of comp osition, while the b ottom triangle com- m u tes b ecause m · H e = 1. Dra win g a similar diagram f or H δ and mH , we obtain (5.2) γ · H e = δ, γ ′ · eH = δ. 5.2. Deﬁnition. A natural transformation S : H → H is said to b e • a left antip o de if m · ( S H ) · δ = e · ε ; • a right antip o de if m · ( H S ) · δ = e · ε ; • an antip o de if it is a left and a r igh t ant ip o de. A b imonad H is said to b e a H opf monad pro vided it has an ant ip o de. F ollo wing the pattern of the pro of of [8, 15.2] we obtain: 5.3. Prop osition. We r efer to the notation in 5.1. (1) If γ has an H -line ar left inverse, then H has a left antip o de. (2) If γ ′ has an H - line ar left inverse, then H has a right antip o de. Pro of. (1) Supp ose there exists an H -linear morphism β : H H → H H with β · γ = 1. Consider th e comp osite S : H H e / / H H β / / H H εH / / H . 16 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF W e claim that S is a left an tip o de of H . Indeed, in the diagram H δ / / H H H eH / / P P P P P P P P P P P P P P P P P P P P P P P P P P P P H H H (1) β H / / H m   H H H (2) H m   εH H / / H H m   H H β / / H H εH / / H , the triangle commutes since e is the u nit for the monad H , r e ctangle (1) comm utes by H -linearit y of β , and r e ctangle (2) commutes by naturalit y of ε . Th u s m · S H · δ = m · εH H · β H · H eH · δ = εH · β · δ, and u s ing (5.2), we ha v e εH · β · δ = εH · β · γ · H e = εH · H e = e · ε. Therefore S is a left ant ip o de of H . (2) Denoting the left in v erse of γ ′ b y β ′ , it is sho w n along the same lines that S ′ = H ε · β ′ · eH is a righ t antipo d e. ⊔ ⊓ 5.4. Lemma. Supp ose that γ is an epimorphism. If f , g : H → H ar e two natur al tr ansfor- mations such that m · f H · δ = m · g H · δ or m · H f · δ = m · H g · δ, then f = g . Pro of. Assume m · f H · δ = m · g H · δ . Since γ · H e = δ by (5.2), we ha v e m · f H · γ · H e = m · g H · γ · H e, and, since γ is also H -linear, it follo ws b y Lemma 3.2 that m · f H · γ = m · g H · γ . But γ is an epimorp hism by our assump tion, th u s m · f H = m · g H . By n aturalit y of e : 1 → H , we ha v e the comm utativ e d iagrams H H e   f / / H H e   H H f H / / H H , H H e   g / / H H e   H H g H / / H H . Th us, sin ce m · H e = 1, f = m · H e · f = m · f H · H e = m · g H · H e = m · H e · g = g . If m · H f · δ = m · H g · δ similar arguments app ly . ⊔ ⊓ 5.5 . Characterising Hopf monads. L et H = ( H , m, e, δ, ε, λ ) b e a bi monad. (1) The f ol lowing ar e e qui valent: (a) γ = H m · δ H : H H → H H is an isomorphism; (b) γ ′ = mH · H δ : H H → H H is an isomorphism; (c) H has an antip o de. BIMONADS AND HO PF MONADS ON CA TEGORIES 17 (2) If H has an antip o de and A admits e qualisers, then the c omp arison functor (se e 4.3) K H : A → A H H ( λ ) makes A (isomorphic to) a c or eﬂe ctive sub c ate gory of the c ate gory A H H ( λ ) . Pro of. (1) (c) ⇒ (a) The pro of for [21, P rop osition 6.10] applies almost literally . (a) ⇒ (c) W rite β : H H → H H for the in v erse of γ . Since γ is H -linear, it follo ws that β also is H -linear. Then, b y P rop osition 5.3, S = εH · β · H e is a left an tip o d e of H . W e sho w that S is also a right an tip o de of H . In the diagram H δ / / δ ! ! B B B B B B B B B B B B B B B B B H H (1) δH / / H H H (2) H S H / / H H H (3) mH / / H m   H H m   H H H δ < < y y y y y y y y y y y y y y y y y y H ε / / H H e / / H H m / / H . • (1) commutes by coasso ciativit y of δ ; • (2) commutes b ecause S is a left antipo d e of H ; • (3) commutes by asso ciativit y of m . Since m · H e = 1 = m · eH an d H ε · δ = 1 = εH · δ, it follo ws that m · ( m · H S · δ ) H · δ = m · mH · H S H · δH · δ = m · H e · H ε · δ = m · eH · εH · δ = m · (( e · ε ) H ) · δ. γ b eing an epimorphism, Lemma 5.4 imp lies m · H S · δ = e · ε , p ro ving that S is also a r igh t an tip o de of H . (b) ⇔ (c) can b e sho w n in a similar wa y . (2) Since • to say that γ is an isomorphism is to sa y that ( t K H ) ( H ( a ) ,m a ) is an isomorp hism for all a ∈ A ; • ( H ( a ) , m a ) = φ H ( a ); • the unit η H : 1 → φ H U H of the adjunction φ H ⊣ U H is just e : 1 → H , wh ich is a split monomorph ism, w e can apply Corollary 3.15 to get the desired result. ⊔ ⊓ Com b ining 5.5 and 4.4, we get: 5.6 . Antip o de and equiv a le nce. L et H = ( H, m, e, δ, ε, λ ) b e a bimonad on a c ate gory A and assume that A admits c olimits or limits and H pr eserves them. Then the f ol lowing ar e e quivalent: (a) H has an antip o de; (b) γ = H m · δ H : H H → H H is an isomorphism; (c) γ ′ = mH · H δ : H H → H H is an isomorphism; (d) K H : A → A H H ( λ ) , a → ( H ( a ) , δ a , m a ) , is an e quiv alenc e. Pro of. (a) ⇔ (b) ⇔ (c) (in any category) is shown in 5.5. (b) ⇔ (d) Since H preserves colimits, the category A H admits colimits and the fun ctor U H : A H → A creates them (see, for example, [24]). Thus • the f u nctor φ H U H preserve s colimits; 18 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF • an y functor L : B → A H preserve s colimits if and only if the comp osite U H L do es; s o, in particular, the fu nctor b H preserv es colimits, since U H b H = H U H and since the functor H U H , b eing the comp osite of t w o colimit-preserving fu n ctors, is colimit-preserving. The full sub category of A H giv en by the free H -mo d ules is dense and since the functors φ H U H and b H b oth preserve colimits, it f ollo ws from [24, Th eorem 17.2.7] that the n atural transformation (see 4.4) t K H : φ H U H → b H is an isomorphism if and only if its restriction to the free H -mo dules is so; i.e. if ( t K H ) φ H ( a ) is an isomorp hism for all a ∈ A . But since φ H ( a ) = ( H ( a ) , m a ), t K H is an isomorp hism if and only if the comp osite H H ( a ) δ H ( a ) / / H H H ( a ) H ( m a ) / / H H ( a ) is an isomorphism f or all a ∈ A , th at is, the isomorph ism γ : H H δH / / H H H H m / / H H . (c) ⇔ (d) Since the f unctor H p reserv es limits, the catego ry A H admits and the functor U H creates limits. Since φ H , b eing righ t adjoint , p reserv es limits, the functor φ H U H also preserve s limits. Moreo v er, since th e mon ad b H is a lifting of the monad H along the functor U H , U H b H = H U H , imp lying that the functor b H also p reserv es limits. No w , sin ce the fu ll sub category of A H spanned b y cofree H -comod ules is codense, it follo ws f rom the d u al of [24, Theorem 17.2.7] that the natural transf orm ation t K H (see 4.4) is an isomorphism if and only if its restriction to f r ee H -comod ules is so. But for any a ∈ A , ( t K H ) ( H ( a ) ,δ a ) = m H ( a ) · H ( δ a ). Th us t K H is an isomorphism if and only if the comp osite γ ′ is an isomorp hism. ⊔ ⊓ 6. Local preb raidings for Hopf monad s F or any category A w e n o w ﬁx a s y s tem H = ( H , m, e, δ, ε ) consisting of an endofunctor H : A → A and natural tr an s formations m : H H → H , e : 1 → H , δ : H → H H and ε : H → 1 suc h that the trip le H = ( H , m, e ) is a monad and the triple H = ( H , δ, ε ) is a comonad on A . 6.1 . Double ent winings. A natural transformation τ : H H → H H is called a double entwining if (i) τ is a mixed distribu tiv e la w fr om the monad H to the comonad H ; (ii) τ is a mixed distr ibutiv e la w from the comonad H to the m onad H . These conditions are ob viously equ iv alent to (iii) τ is a monad distr ib utiv e la w for the monad H ; (iv) τ is a comonad distrib u tiv e law for the comonad H . Explicitely (i) enco des the iden tities (6.1) H e = τ · eH (6.2) H ε = εH · τ (6.3) δ H · τ = H τ · τ H · H δ (6.4) τ · mH = H m · τ H · H τ , BIMONADS AND HO PF MONADS ON CA TEGORIES 19 and (ii) is equiv alen t to the identitie s (6.5) eH = τ · H e (6.6) εH = H ε · τ (6.7) H δ · τ = τ H · H τ · δ H (6.8) τ · H m = mH · H τ · τ H 6.2 . τ -bimonad. L et τ : H H → H H b e a double ent wining. Then H is called a τ -b imonad pro vided the diagram (6.9) H H δδ   m / / H δ / / H H H H H H H τ H / / H H H H mm O O is commutativ e, that is δ · m = mm · H τ H · δδ = H m · mH H · H τ H · H H δ · δ H , and also the follo wing d iagrams comm ute (6.10) H H H ε / / m   H ε   H ε / / 1 , 1 e / / e   H δ   H eH / / H H , 1 e / / =   ? ? ? ? ? ? ? ? H ε   1 . 6.3. Prop osition. L et H b e a τ -bimonad. Then the c omp osite ˜ τ : H H δH / / H H H H τ / / H H H mH / / H H is a mixe d distributive law fr om the monad H to the c omonad H . Thus H is a bimonad (as in 4.1) with mixe d distributive law ˜ τ . Pro of. W e ha v e to sho w that ˜ τ satisﬁes (6.11) H e = ˜ τ · eH (6.12) H ε = εH · ˜ τ (6.13) δ H · ˜ τ = H ˜ τ · ˜ τ H · H δ (6.14) ˜ τ · mH = H m · ˜ τ H · H ˜ τ Consider the diagram H (1) eH / / eH   H H (2) τ / / eH H   H H eH   F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F H H δH / / H H H H τ / / H H H mH / / H H , whic h is comm u tativ e since sq u ar e (1) comm u tes by (6.10); squar e (2) comm utes by functo- rialit y of comp osition; the triangle comm u tes since e is the identit y of the monad H . Th us ˜ τ · eH = mH · H τ · δ H · eH = τ · eH , and (6.1) implies ˜ τ · eH = H e , s ho win g (6.11). 20 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF Consider now the diagram H H δH / / H H H H τ / / H H ε % % K K K K K K K K K K εH H   H H H (1) mH / / H εH   H H εH   H H εH / / H H H (2) H ε 4 4 i i i i i i i i i i i i i i i i i i i i in whic h squar e (1) commutes b ecause ε is a morp hism of monads and thus ε · m = ε · H ε ; the triangle comm utes b ecause of (6.2 ), diagr am (2) commutes b ecause of functorialit y of comp osition. Th us εH · ˜ τ = εH · mH · H τ · δ H = H ε · εH H · δ H = H ε , sho wing (6.12). Constructing su itable comm utativ e diagram we can show ˜ τ · mH = m H · H τ · δ H · mH = mH · H H m · H mH H · H H τ H · H τ H H · H H H τ · δ δ H , H m · ˜ τ H · H ˜ τ = H m · mH H · H τ H · δ H H · H m H · H H τ · H δ H = mH · H H m · H mH H · H H τ H · H τ H H · H H H τ · δ δ H . Comparing this t wo id entities we get the condition (6.14). T o sho w that (6.13) also holds, consider the diagram H H H δH H / / δH H   3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 (1) H H H H H τ H / / H δ H H   H H H H mH H / / H H δ H   (3) H H H H δ H   H H H H H (2) H H τ H   H H H H H H H H τ   mH H H / / (4) H H H H H H τ   H H δδ / / H δ O O H H H H δH H H > > | | | | | | | | | | | | | | | | | H H H H H H τ H H = = z z z z z z z z z z z z z z z z z H H H H H mmH ) ) S S S S S S S S S S S S S S S mH H H / / H H H H H mH   H H H , in whic h the triangles and diagr ams (1) and (3) comm u te b y functorialit y of comp osition; diagr am (2) comm u tes by (6.7); diagr am (4) comm utes by naturalit y of m . Finally we construct the diagram H H δδ   δH / / H H H (1) H H δ t t j j j j j j j j j j j j j j j j H τ / / H H H (2) H δ H   mH / / H H δH   H H H H (3) H τ H / / δH H H   H H H H (4) δH H   H H τ / / H H H H δH H H   H H H H H H H H H H τ H / / H H H H H H τ H H * * T T T T T T T T T T T T T T T H H H τ / / H H H H H H τ H H / / (5) H H H H H mmH O O H H H H H H H H τ 4 4 j j j j j j j j j j j j j j j BIMONADS AND HO PF MONADS ON CA TEGORIES 21 in which diagr am (1) comm utes by (6.3); diagr am (2) comm utes b y (6.9) b ecause δ H H H · H δ H = δ δ H ; the triangle and diagr ams (3), (4) and (5) comm u te by functorialit y of com- p osition. It now follo ws from the comm utativit y of th ese diagrams that δ H · ˜ τ = δ H · mH · H τ · δ H = mmH · H H H τ · H τ H H · H H τ H · δ H H H · δ δ = ( H mH · H H τ · H δ H ) · ( mH H · H τ H · δ H H ) · H δ = H ˜ τ · ˜ τ H · H δ . Therefore ˜ τ satisﬁes the conditions (6.11)-(6.14 ) and hence is a mixed distr ibutiv e la w fr om the m on ad H to the comonad H . ⊔ ⊓ 6.4. Corollary . In the situation of the pr evious pr op osition, if ( a, θ a ) ∈ A H , then ( H ( a ) , θ H ( a ) ) ∈ A H , wher e θ H ( a ) is the c omp osite H ( a ) H ( θ a ) / / H H ( a ) δ H ( a ) / / H H H ( a ) H τ a / / H H H ( a ) m H ( a ) / / H H ( a ) . Pro of. W rite b H for the monad on the category A H that is the lifting of H corresp onding to the m ixed d istributiv e law ˜ τ . Since θ H ( a ) = ˜ τ a · H ( θ a ), it follo w s that ( H ( a ) , θ H ( a ) ) = b H ( a, θ a ) , and thus ( H ( a ) , θ H ( a ) ) is an ob j ect of the category A H . ⊔ ⊓ 6.5 . τ -Bimo dules. Giv en the conditions of Prop osition 6.3, we ha ve the co mm utativ e diagram (see (4.1)) H H m / / H δ   H δ / / H H H H H ˜ τ H / / H H H , H m O O and th us H is a b imonad b y the en twining ˜ τ and the mixed bimo du les are ob jects a in A with a mo du le structure h a : H ( a ) → a and a como d ule structure θ a : a → H ( A ) with a comm utativ e diagram H ( a ) H ( θ a )   h a / / a θ a / / H ( a ) H H ( a ) ˜ τ a / / H H ( a ) . H ( h a ) O O By d eﬁnition of ˜ τ , commutativit y of this d iagram is equiv alen t to the comm tativit y of (6.15) H ( a ) H ( θ a ) y y r r r r r r r r r r h a / / a θ a / / H ( a ) H H ( a ) δ H ( a ) & & L L L L L L L L L L H H ( a ) H ( h a ) f f L L L L L L L L L L H H H ( a ) H ( τ a ) / / H H H ( a ) m H ( a ) 9 9 r r r r r r r r r r . A morp hism f : ( a, h a , θ a ) → ( a ′ , h a ′ , θ a ′ ) is a morphism f : a → a ′ suc h that f ∈ A H and f ∈ A H . W e d enote the category A H H ( ˜ τ ) by A H H . 22 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF 6.6 . An tip o de of a τ -bimonad. L et H = ( H, m, e, δ, ε ) b e a τ -bi monad with an antip o de S wher e τ : H H → H H is a double entwining. Then (6.16) S · m = m · S S · τ and δ · S = τ · S S · δ . If τ · H S = S H · τ and τ · S H = H S · τ , then S : H → H is a monad as wel l as a c omonad morphism . Pro of. Sin ce ( H H , H τ H · δ , εε ) is a comonad and ( H , m, e ) is a monad , the collectio n Nat( H H , H ) of all n atural transformations from H H to H form s a semigroup with unit e · εε and with pro d uct f ∗ g : H H δδ / / H H H H H τ H / / H H H H f g / / H H m / / H . Consider n o w the diagram H H m   H ε w w o o o o o o o o o o o o o δδ / / H H H H (2) H τ H / / H H H H mH H   H (1) ε   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? H ε   δ ( ( R R R R R R R R R R R R R R R R H H H S H H   H m u u k k k k k k k k k k k k k k k H H (3) (4) S H ) ) S S S S S S S S S S S S S S S S H H H H m   I e / / H H H m o o in whic h th e d iagrams (1),(2) and (3) comm ute b ecause H is a bimonad, while diagram (4) comm utes by naturalit y . It follo ws that m · H m · S H H · mH H · H τ H · δ δ = e · ε · H ε = εε · e. Th us S · m = m − 1 in Nat( H H , H ). F urtherm ore, by (a s omewh at tedious) computation we can sh o w m · H m · H H S · H S H · H τ · mH H · H τ H · δδ = e · ε · H ε = e · εε. This shows that m · S S · τ = m − 1 in Nat( H H , H ). Thus m · S S · τ = S · m . T o prov e the form ula for the copro duct consider Nat( H , H H ) as a monoid with unit ee · ε and the con volutio n pro duct for f , g ∈ Nat( H, H H ) giv en b y f ∗ g : H δ / / H H f H / / H H H H H g / / H H H H mm / / H H . By computation we get ( δ · S ) ∗ δ = eH · e · ε = ee · ε, δ ∗ ( τ · S S · δ ) = H e · e · ε = ee · ε. Th us ( δ · S ) ∗ δ = 1 and δ ∗ ( τ · S S · δ ) = 1, and hence δ · S = τ · S S · δ. No w assume τ · H S = S H · τ and τ · S H = H S · τ . Then w e h av e S S · τ = S H · H S · τ = S H · τ · S H = τ · H S · S H = τ · S S, thus S · m = m · S S · τ = m · τ · S S = m ′ · S S. Moreo v er, sin ce m · H e = 1 , we hav e S · e = m · H e · S · e nat = m · S H · H e · e ( 6.10 ) = m · S H · δ · e an tip. = e · ε · e ( 6.10 ) = e . Hence S is a monad morph ism from ( H , m, e ) to ( H , m · τ , e ). BIMONADS AND HO PF MONADS ON CA TEGORIES 23 F or the copro du ct, S S · τ = τ · S S imp lies δ · S = τ · S S · δ = S S · τ · δ = S S · δ ′ . F urthermore, ε · S = ε · S · H ε · δ nat = ε · H ε · S H · δ ( 6.10 ) = ε · m · S H · δ an tip. = ε · e · ε (6.10) = ε. This sh o ws that S is a comonad morph ism from ( H , δ, ε ) to ( H , τ · δ, ε ) . ⊔ ⊓ It is readily c hec ked that for a bimon ad H , the comp osite H H is again a comonad as well as a monad. Ho w ev er, the compatibilit y b et wee n these tw o structures n eeds an additional prop erty of the double ent wining τ . T his will also help to construct a bimonad ”opp osite” to H . 6.7 . Lo cal prebraiding. Let τ : H H → H H b e a natur al tr ansformation. τ is said to satisfy the Y ang-Baxter e quation (YB ) if it indu ces comm u tativit y of the diagram H H H τ H / / H τ   H H H H τ / / H H H τ H   H H H τ H / / H H H H τ / / H H H . τ is called a lo c al pr ebr aiding p r o vided it is a double ent wining (see 6.1) and s atisﬁes the Y ang-Baxter equation. 6.8 . Doubling a bimonad. L et H = ( H , m, e, δ, ε ) b e a τ -bi monad wher e τ : H H → H H is a lo c al pr ebr aiding. Then HH = ( H H , ¯ m, ¯ e, ¯ δ , ¯ ε ) is a ¯ τ -bimonad with ¯ e = ee , ¯ ε = εε , ¯ m : H H H H H τ H / / H H H H mm / / H H , ¯ δ : H H δδ / / H H H H H τ H / / H H H H and double entwining ¯ τ : H H H H H τ H / / H H H H τ H H / / H H H H H H τ / / H H H H H τ H / / H H H H . Pro of. W e already know that ( H H , ¯ m, ¯ e ) is a monad and that ( H H , ¯ δ , ¯ ε ) is a comonad. First w e ha ve to sho w that ¯ τ is a mixed distribu tiv e la w from the monad ( H H , ¯ m, ¯ e ) to the comonad( H H , ¯ δ , ¯ ε ), that is H H ¯ e = ¯ τ · ¯ eH H , H H ¯ ε = ¯ εH H · ¯ τ , H H ¯ m · ¯ τ H H · H H ¯ τ = ¯ τ · ¯ mH H , H H ¯ τ · ¯ τ H H · H H ¯ δ = ¯ δ H H · ¯ τ . The ﬁ rst tw o equalities can b e veriﬁed by p lacing the comp osites in suitable comm utativ e diagrams. Th e second tw o identiti es are obtained by lengthy standard computations (as kno w n for classical Hopf algebras). It remains to sh o w that ( H H , ¯ m, ¯ e, ¯ δ , ¯ ε ) satisﬁes the conditions of Deﬁnition 4.1 with resp ect to ¯ τ . Again ¯ ε · ¯ m = ε · H ε · H H εε = ¯ ε · H H ¯ ε, and ¯ δ · ¯ e = H H ee · H e · e = H H ¯ e ¯ e are sh o wn by stand ard computations and ¯ ε ¯ e = ε · εH · eH · e (4.2) = ε · e = 1 . 24 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF T o sho w that ( H H , ¯ m, ¯ e, ¯ δ , ¯ ε, ¯ τ ) s atisﬁes (4.1), consider the diagram H 4 (1) H 2 δH   H τ H / / H 4 H δ H 2 ' ' N N N N N N N N N N N N N mH 2 / / H 3 δH H ' ' N N N N N N N N N N N N N H m / / H 2 (3) δH / / H 3 H 2 δ   H 5 (4) (2) τ H 3 ' ' N N N N N N N N N N N N N H 4 H 2 m 7 7 o o o o o o o o o o o o o H 3 δ   H 5 (5) H 4 δ   H τ H 2 / / H 5 (6) H 2 τ H 7 7 p p p p p p p p p p p p p H 4 δ   τ H 3 / / H 5 (7) H 4 δ   H 2 τ H / / H 5 (8) H mH 2 7 7 p p p p p p p p p p p p p H 4 δ   H 5 (9) (10) H 2 τ H 2 ' ' O O O O O O O O O O O O O H 4 H τ H / / H 4 H 5 (11) (12) H τ H 2 / / H 3 m O O H 5 H 3 m O O H 6 H 3 τ H   H τ H 3 / / H 6 (14) H 3 τ H   (13) τ H 4 / / H 6 (15) H 3 τ H   H 2 τ H 2 / / H 6 H mH 3 @ @                   H 3 τ H / / H 6 H mH 3 7 7 o o o o o o o o o o o o o H 2 τ H 2 / / H 6 (16) H τ H 3 / / H 6 H 2 mH 2 O O H 6 H τ H 3 / / H 6 τ H 4 / / H 6 H 2 τ H 2 / / H 6 H 3 τ H > > } } } } } } } } } } } } } } } } } H τ H 3 / / H 6 , H 3 τ H > > } } } } } } } } } } } } } } } } } in wh ic h diagr am (1) comm utes b ecause τ is a mixed d istributiv e la w and th us H τ · τ H · H δ = δ H · τ ; the diagr ams (2) and (9) comm ute by (4.1); the diagr ams (3)-(8), (10), (11), (13), (14) and (16) comm u te by naturalit y; diagr am (12) commutes b ecause τ is a mixed d istributiv e law (hence H m · τ H · H τ = τ · mH ); diagr am (15) comm u tes b y 6.7. By comm utativit y of the whole d iagram, ¯ δ · ¯ m = H τ H · H 2 δ · δ H · H m · mH 2 · H τ H = H 2 m · H 2 mH 2 · H 3 τ H · H τ H 3 · H 2 τ H 2 · τ H 4 · H τ H 3 · H 3 τ H · H 4 δ · H 2 δ H = H H ¯ δ · ¯ τ H H · H H ¯ m, and h en ce HH = ( H H , ¯ m, ¯ e, ¯ δ , ¯ ε ) is a ¯ τ -bimonad. ⊔ ⊓ 6.9 . O pp osite monad and comonad. L et τ : H H → H H b e a natur al tr ansformation satisfying the Y ang- B axter e quation. (1) If ( H , m, e ) is a monad and τ is monad distributive, then ( H , m · τ , e ) is also a monad and τ is monad distributive for i t. (2) If ( H , δ, ε ) i s a c omo nad and τ is c omona d distributive, then ( H, τ · δ , ε ) is also a c omonad and τ is c omona d distributive for it. BIMONADS AND HO PF MONADS ON CA TEGORIES 25 Pro of. (1) T o show that m · τ is asso ciativ e constru ct the d iagram H H H τ H / / H τ   (1) H H H m H / / H τ   (2) H H τ   H H H τ H   H H H τ H / / H m   (3) H H H H τ / / H H H H m / / mH   (4) H H m   H H τ / / H H m / / H , where the r e ctangle (1) is comm utativ e by th e YB-condition, (2) and (3) are comm utativ e b y the monad d istributivit y of τ , and the squar e (4) is commutati v e by asso ciativit y of m . No w comm utativit y of the outer d iagram shows asso ciativit y of m · τ . F rom 2.5 w e kno w that τ · e H = H e and τ · H e = e H and this implies that e is also the unit for ( H , m · τ , e ). The t wo p en tagons for monad distribu tivit y of τ for ( H , m · m, e ) can b e read from the ab o ve diagram b y com binin g the t wo top rectangles as well as the tw o left h an d rectangles. (2) The pro of is dual to the pr o of of (1). ⊔ ⊓ 6.10 . Opp osite bimonad. L et τ : H H → H H b e a lo c al pr ebr aiding with τ 2 = I and let H = ( H , m, e, δ, ε ) b e a τ -bimonad on A . Then: (1) H ′ = ( H , m · τ , e, τ · δ , ε ) is also a τ -bimonad. (2) If H has an antip o de S with τ · H S = S H · τ and τ · S H = H S · τ , then S is a τ -bimonad morphism b etwe en the τ - bimonads H and H ′ . In this c ase S is an antip o de for H ′ . Pro of. (1) By (1), (2) in 6.9, τ is a (co)monad d istributiv e law fr om the (co)monad H to the (co)monad H ′ , and ε ′ · e ′ = ε · e = 1 by (6.10). Moreo ve r, ε ′ · m ′ = ε · m · τ ( 6.10 ) = ε · H ε · τ 2.4 = ε · εH = ε · H ε = ε ′ · H ε ′ , and δ ′ · e ′ = τ · δ · e ( 6.10 ) = τ · eH · e 2.1 = H e · e = eH · e = e ′ H · e ′ . T o pro v e compatibilit y for H ′ w e hav e to sh o w the comm utativit y of the diagram (6.17) H H m ′ / / δ ′ δ ′   H δ ′ / / H H H H H H H τ H / / H H H H . m ′ m ′ O O F or this standard computations (from Hopf algebras) apply . (2) By 6.6, S is a τ -bim on ad morphism from the τ -bimon ad H to the τ -bimonad H ′ . T o sho w that S is an an tip o de for H ′ w e need the equalities m ′ · S H · δ ′ = e ′ · ε ′ = e · ε and m ′ · H S · δ ′ = e ′ · ε ′ = e · ε. Since τ · S H = H S · τ , we h a ve m ′ · S H · δ ′ = m · τ · S H · τ · δ = m · H S · τ · τ · δ τ 2 =1 = m · H S · δ = e · ε. Since τ · H S = S H · τ , we ha ve m ′ · H S · δ ′ = m · τ · H S · τ · δ = m · S H · τ · τ · δ τ 2 =1 = m · S H · δ = e · ε. 26 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF ⊔ ⊓ As w e ha ve seen in Theorem 5.6, the existence of an an tip o de f or a bimonad H on a catego ry A is equiv alen t to the comparison fun ctor b eing an equiv alence pro vided A is Cauc hy complete and H p reserv es colimits. Giv en a lo cal p r ebraiding the latter condition on H can b e r eplaced by cond itions on th e antipo de (compare [3, Theorem 3.4], [4, Lemma 4.2] for the situation in braided m onoidal category). 6.11 . Antip o de and equiv alence. L et τ : H H → H H b e a lo c al pr eb r aiding and let H = ( H , m, e, δ, ε ) b e a τ - bimonad on a c ate gory A in which idemp otents split. Consider the c ate gory of bimo dules A H H = A H H ( ˜ τ ) , wher e ˜ τ = mH · H τ · δ H (se e 6.3, 6.5). If H has an antip o de S such that τ · S H = H S · τ and τ · H S = S H · τ , then the c omp ariso n functor K H : A → A H H is an e quivalenc e of c ate gories. Pro of. W e k n o w that the functor K H has a righ t adjoin t if for eac h ( a, h a , θ a ) ∈ A H H , the equ aliser of th e ( a, h a , θ a ) − comp onent of the pair of fun ctors (6.18) U H U b H U H U b H e b H / / β U H U b H / / U H b H U b H = U H U b H φ b H U b H exists. Here e b H : 1 → φ b H U b H is the unit o f the adj unction U b H ⊣ φ b H and β U H is the comp osite U H e H U H / / U H φ H U H U H ( t K H ) / / U H b H . Using th e fact that for any ( a, h a ) ∈ A H , ( t K H ) ( a,h a ) = H ( h a ) · δ a and H ( h a ) · δ a · e a = H ( h a ) · H ( e a ) · e a = e a , it is not h ard to show that the ( a, h a , θ a )-comp onen t of Diagram 6.18 is the p air a θ a / / e a / / H ( a ) . Th us, K H has a r igh t adjoin t if for ea c h ( a, H a , θ a ) ∈ A H H , th e equaliser of th e pair of morphisms ( e a , θ a ) exists. Supp ose now th at H has an an tip o de S : H → H . F or eac h ( a, h a , θ a ) ∈ A H H , consider the comp osite q a = h a · S a · θ a : a → a . By a (tedious) s tand ard computation - applying 6.15 , 6.6, 2.4 - on e can sho w e a · q a = θ a · q a and q a · q a = q a . 6.12. Remark. Dually , one can prov e that for eac h ( a, h a , θ a ) ∈ A H H , q a · ε a = q a · h a , thus i a · ¯ q a · ε a = i a · ¯ q a · h a , and since i a is a (split) monomorph ism, it follo w s that ¯ q a · ε a = ¯ q a · h a . Since idemp oten ts split in A , there exist morp hisms i a : ¯ a → a and ¯ q a : a → ¯ a such that ¯ q a · i a = 1 a and i a · ¯ q a = q a . Since ¯ q a is a (sp lit) epimorp hism and sin ce e a · i a · ¯ q a = e a · q a = θ a · q a = θ · i a · ¯ q a , it follo ws th at (6.19) e a · i a = θ a · i a . BIMONADS AND HO PF MONADS ON CA TEGORIES 27 Using th is equalit y it is s traigh tforward to sho w th at the d iagram (6.20) ¯ a i a / / a ¯ q a t t e a / / θ a / / H ( a ) h a · S a s s is a split equaliser. Hence for any ( a, h a , θ a ) ∈ A H H , the equaliser of the pair of morp hisms ( e a , θ a ) exists, whic h implies that the fu nctor K H has a righ t adjoin t R H : A H H → A whic h is given by R H ( a, H a , θ a ) = ¯ a. Since for an y ( a, h a , θ a ) ∈ A H H , • δ a · e a = e H ( a ) · e a and ε a · e a = 1 by 6.2; • ε H ( a ) · δ a = 1, since ( H , ε, δ ) is a comonad; • e a · ε a = ε H ( a ) · e H ( a ) b y naturalit y , w e get a sp lit equaliser diagram a e a / / H ( a ) ε a s s e H ( a ) / / δ a / / H 2 ( a ) H ( ε a ) r r . This is preserved by any fun ctor, and sin ce R H ( H ( a ) , m a , δ a ) is the equaliser of the pair of morphisms ( e H ( a ) , δ a ), in particular a ≃ R H ( H ( a ) , m a , δ a ) = R H ( K H ( a )). Thus R H K H ≃ 1. F or an y ( a, h a , θ a ) ∈ A H H , write α a for the comp osite h a · H ( i a ) : H (¯ a ) → a. W e claim that α a is a morphism in A H H from K H (¯ a ) = ( H ( ¯ a ) , m ¯ a , δ ¯ a ) to ( a, h a , θ a ). In deed, w e hav e α a · m ¯ a = h a · H ( i a ) · m ¯ a naturalit y = h a · m a · H 2 ( i a ) ( a, h a ) ∈ A H = h a · H ( h a ) · H 2 ( i a ) = h a · H ( H ( h a ) · i a ) = h a · H ( α a ) , and this just means that α a is a morphism in A H from ( H ( ¯ a ) , m ¯ a ) to ( a, h a ). Next - using 6.15, 6.19 - we compute θ a · α a = H ( α a ) · δ ¯ a . Th us, α a is a morp hism in A H from ( H (¯ a ) , δ ¯ a ) to ( a, δ a ), and h ence α a is a morp hism in A H H from K H (¯ a ) = ( ¯ a, m ¯ a , δ ¯ a ) to ( a, h a , θ a ). Similarly it is pr o ve d that the comp osite β a = H ( ¯ q a ) · θ a : a → H (¯ a ) is a morp hism in A H from ( a, h a , δ a ) to ( H (¯ a ) , m ¯ a , δ ¯ a ) and a fu rther calculation yields α a · β a = 1 a and β a · α a = 1 H (¯ a ) . Hence w e hav e pr o ve d that for an y ( a, h a , θ a ) ∈ A H H , α a is an isomorphism in A H H , and using the fact that th e same argument as in Remark 2.4 in [12] sho ws that α a is the counit of the adjunction K H ⊣ R H , one concludes that K H R H ≃ 1 . Thus the functor K H is an equiv alence of catego ries. T his completes the p ro of. ⊔ ⊓ F or an example, let V = ( V , ⊗ , I , σ ) b e a b raided mon oidal ca tegory and H = ( H, m, e, δ, ε ) a bialgebra in V . T hen ( H ⊗ − , m ⊗ − , e ⊗ − , δ ⊗ − , ε ⊗ − , τ = σ H,H ⊗ − ) is a bimonad on V , and it is easy to see that the catego ry V H H of Hopf mo dules is just the catego ry V H ⊗− H ⊗− ( ¯ τ ) = V H ⊗− H ⊗− . 6.13 . Theorem. L et V = ( V , ⊗ , I , σ ) b e a br aide d monoidal c ate gory such that idemp otents split i n V . Then for any bialgebr a H = ( H , m, e, δ, ε ) in V , the fol lowing ar e e qui v alent: 28 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF (a) H has an antip o de; (b) the c omp arison functor K H : V → V H H , V 7→ ( H ⊗ V , m ⊗ V , δ ⊗ V ) , f 7→ H ⊗ f , is an e quivalenc e of c ate gories. 7. Adjoints of bimonad s This section deals with the trans fer of pr op erties of monads and comonads to adj oin t (endo-)functors. The relev ance of this in terp la y was already observ ed by Eilen b erg and Mo ore in [11]. An eﬀectiv e formalism to handle this wa s develo p ed f or adjunctions in 2- catego ries and is nicely presente d in Kelly and Street [16]. F or our p urp ose we only need this for th e 2-category of catego ries and for conv enience w e recall the basic facts of this situation here. 7.1 . Adjunctions. Let L : A → B , R : B → A b e an adjoin t pair of functors with u nit and counit η , ε , and L ′ : A ′ → B ′ , R ′ : B ′ → A ′ b e an adjoin t p air of functors with unit and counit η ′ , ε ′ . Give n an y functors F : A → A ′ and G : B → B ′ , there is a bijection b et ween natural transformations α : L ′ F → GL and α : F R → R ′ G where α is obtained as th e comp osite F R η ′ F R − → R ′ L ′ F R R ′ αR − → R ′ GLR R ′ Gε − → R ′ G, and α is giv en as the comp osite L ′ F L ′ F η − → L ′ F RL L ′ αL − → L ′ R ′ GL ε ′ GF − → GL. In this situation, α and α are calle d mates un der the giv en adjunction and this is denoted b y a ⊣ α . It is nicely d ispla yed in th e diagram A L / / F   B R / / G   A F   α z  | | | | | | | | | | | | | | A ′ α : B } } } } } } } } } } } } } } L ′ / / B ′ R ′ / / A ′ . Giv en fur ther (i) adjunctions ˜ L : C → A , ˜ R : A → C and ˜ L ′ : C ′ → A ′ , ˜ R ′ : A ′ → C ′ and a functor H : C → C ′ , or (ii) an adjunction L ′′ : A ′′ → B ′′ , R ′′ : B ′′ → A ′′ and functors F ′ : A ′ → A ′′ and G ′ : B ′ → B ′′ , we get the diagram C ˜ L / / H   A L / / F   B R / / G   A F   α y  { { { { { { { { { { { { { { { { ˜ R / / C γ y  { { { { { { { { { { { { { { H   C ′ ˜ R ′ / / γ : B | | | | | | | | | | | | | | A ′ α 9 A | | | | | | | | | | | | | | | | L ′ / / F ′   B ′ R ′ / / G ′   A ′ F ′   β y  | | | | | | | | | | | | | | | | ˜ R ′ / / C ′ A ′′ L ′′ / / β : B } } } } } } } } } } } } } } B ′′ R ′′ / / A ′′ , BIMONADS AND HO PF MONADS ON CA TEGORIES 29 yielding the mates ( M 1) L ′′ F ′ F β F − → G ′ L ′ F G ′ α − → G ′ GL ⊣ F ′ F G F ′ α − → F ′ R ′ G β G − → R ′′ G ′ G, ( M 2) L ′ ˜ L ′ H L ′ β − → L ′ F ˜ L α ˜ L − → LG ˜ L ⊣ H ˜ RR β G − → ˜ R ′ R ′ G ˜ R ′ β − → ˜ R ′ R ′ G. 7.2 . Prop erties of mates. L et L, L ′ : A → B b e functors with right adjoints R , R ′ , r esp e ctively, and α : L ′ → L a natur al tr ansform ation. (i) If L ′′ : A → B is a functor with right adjoint R ′′ and β : L ′′ → L ′ a natur al tr ansfo r- mation, then α · β ⊣ β · α. (ii) If ˜ L : C → A i s a fu nctor with right adjoint ˜ R , then ( α L ′ : L ′ ˜ L → L ˜ L ) ⊣ ( ˜ R α : ˜ RR → ˜ RR ′ ) . (iii) If L o : B → C is a functor with right adjoint R o , then ( L o α : L o L ′ → L o L ) ⊣ ( αR o : RR o → R ′ R o ) . Pro of. (i) is a sp ecial case of 7.1(M1). (ii) follo ws from 7.1(M2) by p utting A ′ = A , B ′ = B , C ′ = C and H ′ = H . (iii) is derive d by applyin g 7.1 to the diagram A L / / B L o / / C R o / / B I {                R / / A α z  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A ˜ L ′ / / α ; C               B I ; C               L o / / C R o / / B R ′ / / A . ⊔ ⊓ As o bserve d b y Eilen b er g and Mo ore in [11 , Section 3] , for a left adjoin t endofunctor whic h is a monad, the r igh t adjoint (if it exists) is a comonad (and vice v ersa). The tec hniques outlined ab o ve pro v id e a con v enien t and eﬀectiv e wa y to describ e this transition and to pro v e related prop erties. Recall that for any endofunctor L : A → A with righ t adjoin t R , for a p ositiv e intege r n , the p o w ers L n ha ve the righ t adjoints R n . 7.3 . Adjoin ts of monads and comonads. L et L : A → A b e an endofunctor with right adjoint R . (1) If L = ( L, m L , e L ) is a monad, then R = ( R , δ R , ε R ) is a c omonad, wher e δ R , ε R ar e the mates of m L , e L in the diagr ams A L / / A R / / A ε R {  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A e L ; C               I / / A I / / A , A L / / A R / / A δ R z  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A m L ; C               H H / / A RR / / A . (2) If L = ( L, δ L , ε L ) is a c omonad, then R = ( R , m R , e R ) is a monad wher e m R , e R ar e the mates of δ L , ε L in the diagr ams A I / / A I / / A e R {  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A ε L ; C               L / / A R / / A , A LL / / A RR / / A m R z  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A δ L ; C               L / / A R / / A . 30 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF Pro of. (1) Since e L ⊣ ε R and m L ⊣ δ R , it follo ws f rom 7.2 (ii) and (iii) that Le L ⊣ ε R R, e L L ⊣ Rε R , m L L ⊣ Rδ R , Lm L ⊣ δ R R. Applying 7.2 (i) no w yields m L · Le L ⊣ ε R R · δ R , m L · e L L ⊣ Rε R · δ R , m L · m L L ⊣ Rδ R · δ R , m L · Lm L ⊣ δ R R · δ R . Since L is a monad w e ha ve m L · e L L = m L · Le L = I and m L · m L L = m L · Lm L , implying ε R R · δ R = R ε R · δ R = I and R δ R · δ R = δ R R · δ R . This sh o ws that R = ( R, δ R , ε R ) is a comonad. The pr o of of (2) is similar. ⊔ ⊓ The metho ds un d er consideration also app ly to the natur al transformations LL → LL whic h were b asic for the deﬁnition and in vestiga tion of bimon ad s in pr evious sections. T he follo wing results were obtained in co op er ation with Gabriella B¨ ohm and T omasz Brzezi´ n ski. 7.4 . Adjoin tness and distributiv e la ws. L et L : A → A b e an endofunctor with right adjoint R and a natur al tr ansfor mation λ L : LL → LL . This yields a mate λ R : RR → RR in the diagr am A LL / / A RR / / A λ R {                A λ L ; C               LL / / A RR / / A with the fol lowing pr op erties: (1) Lλ L ⊣ λ R R and λ L L ⊣ Rλ R . (2) λ L satisﬁes the Y ang-Baxter e qu ation if and only if λ R do es. (3) λ 2 L = I if and only if λ 2 R = I . (4) If L = ( L, m L , e L ) is a monad and λ L is mona d distributive, then λ R is c omonad distributive f or the c omonad R = ( R, δ R , ε R ) . (5) If L = ( L, δ L , ε L ) is a c omonad and λ L is c omonad distributive, then λ R is monad distributive f or the c omonad R = ( R , m R , e R ) . Pro of. (1) follo w s f rom 7.2, (ii) and (iii). The remainin g assertions follo w by (1) and the id en tities in the pr o of of 7.3 . ⊔ ⊓ Recall fr om Deﬁnition 4.1 that a bi monad H is a monad and a co monad with compatibilit y conditions inv olving an ent wining λ H : H H → H H . 7.5 . Adjoin ts of bimonads. L et H b e a monad H = ( H , m H , e H ) and a c omonad H = ( H , δ H , ε H ) on the c ate gory A . Then a right adjoint R of H induc es a monad R = ( R , m R , e R ) and a c omonad R = ( R, δ R , ε R ) (se e 7.3) and (1) H = ( H , H ) is a bimonad with entwining λ H : H H → H H if and only if R = ( R, R ) is a bimonad with entwining λ R : RR → RR . (2) H = ( H , H ) is a bimonad with entwining λ ′ H : H H → H H if and only if R = ( R, R ) is a bimonad with entwining λ ′ R : R R → RR . (3) If H = ( H , H , λ H ) is a bimonad with antip o d e, then R = ( R, R, λ R ) is a bimonad with antip o de (Hopf monad). BIMONADS AND HO PF MONADS ON CA TEGORIES 31 Pro of. (1) With arguments similar to th ose in th e pro of of 7.4 we get that λ R is an en twining from R to R . It remains to s ho w the prop er ties required in Deﬁnition 4.1. F rom 7.2(i) w e kno w that ε H · H ε H ⊣ e R R · e R , ε H · m H ⊣ δ R · e R , δ H · e H ⊣ ε R · m R , e H H · e H ⊣ ε R · Rε R , ε H · e H ⊣ ε R · e R . Th us the equalities ε H · H ε H = ε H · m H , δ H · e H = ε H H · e H , ε H · e H = I hold if and only if e R R · e R = δ R · e R , ε R · m R = ε R · Rε R , ε R · e R = I . The transfer of the compatibilit y b et ween p ro duct and copro duct 4.1 is seen from the corresp onding d iagrams H H m H / / H δ H   H δ H / / H H H H H λ H H / / H H H , H m H O O RR R δ R o o RR m R o o δ R R   RRR m R R O O RRR . Rλ R o o The pr o of of (2) is similar. (3) By 5.5, the existence of an anti p o de is equiv alen t to the bijectivit y of the morph ism γ H = H m H · δ H H : H H → H H . Since δ H H ⊣ R m R and H m H ⊣ δ R R , γ H is an isomorphism if and only if γ R = R m R · δ R R is an isomorphism. ⊔ ⊓ F unctors w ith right (resp. left) adj oin ts preserve colimits (resp. limits) and thus 5.6 and 7.5 imp ly: 7.6 . Hopf monads with adjoints. Assu me the c ate gory A to admit limits or c olimits. L et H = ( H, m H , e H , δ H , ε H , λ H ) b e a bimonad on A with a right adjoint bimonad R = ( R, m R , e R , δ R , ε R , λ R ) . Then the fol lowing ar e e qu ivalent: (a) the c omp arison functor K H : A → A H H ( λ H ) is an e qui valenc e; (b) the c omp arison functor K R : A → A R R ( λ R ) is an e quivalenc e; (c) H has an antip o de; (d) R has an antip o de. Finally we observ e that lo cal prebr aidings are also tranferred to the adjoin t fu nctor. 7.7 . Adjointness of τ -bimonads. L et H b e a monad H = ( H , m H , e H ) and a c omonad H = ( H , δ H , ε H ) on the c ate gory A with a right adjoint R . If H = ( H , H ) i s a bimonad with double e ntwining τ H : H H → H H , then R = ( R, R ) is a bimonad with double entwining τ R : RR → RR . Mor e over, τ H satisﬁes the Y ang-Baxter e quation i f and only if so do es τ R . Pro of. Most of the assertions follo w immediately from 7.4 and 7.5. It remains to verify the compatibilit y condition 6.9. F or this observe that f rom 7.2(i) w e get δ H δ H ⊣ m H m H , H τ H H ⊣ Rτ R R, m H m H ⊣ δ R δ R , and h en ce m H m H · τ H H · δ H δ H ⊣ m R m R · Rτ R R · δ R δ R and δ H · m M ⊣ δ R · m R . 32 BACHUKI M ESABLISHVILI, TBILISI AND ROBER T WISBAUER, D ¨ USSELDORF It f ollo ws that H satiﬁes 6.9 if and only if so d o es R . ⊔ ⊓ 7.8 . Dual Hopf algebras. Let B b e a mo du le o v er a comm utative ring R . B is a Hopf algebra if and on ly if the endofu n ctor B ⊗ R − on the catgeory of R -mo d ules is a Hopf monad. By 7.5, B ⊗ R − is a bimonad (with an tip o de) if and only if its right adjoin t fun ctor Hom R ( B , − ) is a bimonad (with an tip o de). This s ituation is consid ered in more detail in [5]. If B is ﬁn itely generated and pro jectiv e as an R -mo du le and B ∗ = Hom R ( B , R ), th en Hom R ( B , − ) ≃ B ∗ ⊗ R − and we obtain the familiar result that B is a Hopf algebra if and only if B ∗ is. 7.9 . C haracterisations of groups. 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Bimonads and Hopf monads on categories

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