Monads and comonads on module categories

MONADS AND COMONADS ON MODULE CATEGORIES GABRIELLA BÖHM, TOMASZ BRZEZIŃSKI, AND ROBERT WISBAUER Abstract. Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor − ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor − ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodules (or coalgebras) of − ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA (B, −) and HomA (C, −) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of HomA (B, −)-comodules is isomorphic to the category of B-modules, while the category of HomA (C, −)-modules (called Ccontramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of C-comodules and HomA (C, −)-modules are equivalent provided C is a coseparable coring. Furthermore, we describe equivalences between categories of HomA (C, −)-modules and comodules over a coring D in terms of new Galois properties of bicomodules. Finally, we characterise Hopf algebras H over a commutative ring R by properties of the functor HomR (H, −) and the category of mixed HomR (H, −)-bimodules. This generalises in particular the fact that a finite dimensional vector space H is a Hopf algebra if and only if the dual space H ∗ is a Hopf algebra. Contents 1. Introduction 2. Categorical framework 3. Rings and corings in module categories 4. Functors between co- and contramodules 5. Galois bicomodules 6. Contramodules and entwining structures 7. Bialgebras and bimodules References 1 3 11 14 17 22 24 29 1. Introduction The purpose of this paper is to present a categorical framework for studying problems in the theories of rings and modules, corings and comodules, bialgebras and Date: April 2009. 2000 Mathematics Subject Classification. Primary 16D90; Secondary 16W30. 1 2 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER (mixed) bimodules and Hopf algebras and Hopf modules. The usefulness of this framework is illustrated by analysing the structure of the category of contramodules and the bearing of this structure on the properties of corings and bialgebras. It is well-known that for a right module V over an R-algebra A, the dual R-module V ∗ = HomR (V, R) is a left module over A. It is equally well-known that for a right comodule V of an R-coalgebra C, in general V ∗ is not a C-comodule (left or right). It has already been realised in [10, Chapter IV.5] that to a coalgebra C two (different) representation categories can be associated: the familiar category of C-comodules and the category of C-contramodules introduced therein. If V is a C-comodule, then V ∗ is a C-contramodule. While comodules of coalgebras (and corings) have been intensively studied, contramodules seem to have been rather neglected. Yet the category of contramodules is as fundamental as that of comodules, and both categories are complementary to each other. To substantiate this claim, one needs to resort to the categorical point of view on corings. An A-coring can be defined as an A-bimodule C such that the tensor endofunctor − ⊗A C on the category of right A-modules MA is a comonad or a cotriple. Right C-comodules are the same as comodules (or coalgebras in category theory terminology) of the comonad − ⊗A C. On the other hand, the tensor functor − ⊗A C has a right adjoint, the Hom-functor HomA (C, −), which we denote by [C, −]. By purely categorical arguments (see Eilenberg and Moore [11, Proposition 3.1]), the functor −⊗A C is a comonad if and only if its right adjoint [C, −] is a monad. Thus, C is an A-coring if and only if [C, −] is a monad on MA ; right C-contramodules are simply modules (or algebras in category theory terminology) of this monad. This categorical interpretation explains the way in which contramodules complement comodules (e.g. 3.6, 3.7). Again purely categorical considerations (see [11]) explain that, while there are two categories of representations of a coring, there is only one category of representations of a ring – the familiar category of modules. More precisely, a ring morphism A → B can be equivalently described as the monad structure of the tensor functor − ⊗A B on MA associated to an A-bimodule B. With this interpretation, right B-modules are simply modules of the monad − ⊗A B. The right adjoint functor HomA (B, −) is a comonad on MA and the category of comodules of HomA (B, −) is isomorphic to the category of modules of the monad − ⊗A B. Consequently, there is only one type of representation categories for rings – the category of right (or left) modules over a ring. The above comments illustrate how the categorical point of view can give significant insight into algebraic structures. There are many constructions developed in category theory that are directly applicable to ring theoretic situations but they seem not to be sufficiently explored. Contramodules of a coring are a good example of this. On one hand, from the category point of view, they are as natural as comodules, on the other hand, their structure was not analysed properly until very recently, when their important role in semi-infinite homology was outlined by Positselski [24]. The main motivation of our paper is a study of contramodules of corings. This aim is achieved by placing it in a broader context: we revisit category theory, more specifically the theory of adjoint comonad-monad pairs, in the context of rings and modules. We begin by summarising the categorical framework, and then apply it first to rings in module categories, next to corings. In the latter case, we concentrate on properties MONADS AND COMONADS ON MODULE CATEGORIES 3 of the less-known category of contramodules, and derive consequences of the categorical formulation in this context. We analyse functors between categories of comodules and contramodules, and introduce the notion of a [C, −]-Galois bicomodule. We then relate the equivalences between categories of C-contramodules and D-comodules to the existence of [C, −]-Galois bicomodules. In particular we prove that the existence of the latter is a necessary condition for such an equivalence; see 5.8 and 5.9. We also derive the characterisation of entwining structures as liftings of Hom-functors to module, comodule and contramodule categories; see 6.1. Finally, we study contramodules over corings associated to bialgebras and provide new extensions of the Fundamental Theorem of Hopf algebras (see 7.9). These are achieved by investigating properties of the category of contramodules over the corings associated to a bialgebra B. Again this can be seen as B being a Galois comodule with respect to the Hom-functors of the associated corings, and thus indicates the role which is played by adjoint functors (that are not tensor functors) in the description of Hopf algebra dualities. 2. Categorical framework Our main concern is to apply abstract categorical notions to special situations in module categories. We begin by recalling some basic definitions and properties (e.g. from [11]) to fix notation, and then develop a categorical framework which is later applied to categories of (co)modules. Throughout, the composition of functors is denoted by juxtaposition, and the usual composition symbol ◦ is reserved for natural transformations and morphisms. Given functors F , G and a natural transformation ϕ, F ϕG denotes the natural transformation, which, evaluated at an object X gives a morphism obtained by applying F to a morphism provided by the natural transformation ϕ evaluated at the object GX. By A ≃ B we denote equivalences between categories and A ∼ = B is written for their ∼ isomorphisms. The symbol = is also used to denote isomorphisms between objects in any category, in particular isomorphisms of modules and (natural) isomorphisms of functors. 2.1. Adjoint functors. A pair (L, R) of functors L : A → B and R : B → A between categories A, B is called an adjoint pair if there is a natural isomorphism ∼ = MorB (L(−), −) −→ MorA (−, R(−)). This can be described by natural transformations unit η : IA → RL and counit ε : LR → IB satisfying the triangular identities εL ◦ Lη = IL and Rε ◦ ηR = IR . e R) e 2.2. Natural transformations for adjoints. For two adjunctions (L, R) and (L, between A and B, with respective units η, ηe and counits ε, εe, there is an isomorphism between the natural transformations (cf. [17], [20]) e → Nat(R, e R), f 7→ f¯ := Re e ◦ η R, e Nat(L, L) ε ◦ Rf R with the inverse map e R) → Nat(L, L), e Nat(R, e ◦ Lf¯L e ◦ Le f¯ 7→ f := εL η. e R). e For natural We say that f and f¯ are mates under the adjunctions (L, R) and (L, transformations f : L1 → L2 and g : L2 → L3 between left adjoint functors, naturality and the triangle identities imply g ◦ f = f¯◦ḡ. In particular, f is a natural isomorphism 4 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER if and only if its mate f¯ is a natural isomorphism. Moreover, if for an adjunction (L, R), the composites LL1 (and hence LL2 ) are meaningful, then Lf = f¯R. Similarly, if the composites L1 L (and thus L2 L) are meaningful then f L = Rf¯. 2.3. Monads on A. A monad on the category A is a triple F = (F, m, i), where F : A → A is a functor with natural transformations m : F F → F and i : IA → F satisfying associativity and unitality conditions. A morphism of monads (F, m, i) → (F ′ , m′ , i′ ) is a natural transformation ϕ : F → F ′ such that m′ ◦ ϕF ′ ◦ F ϕ = ϕ ◦ m and ϕ ◦ i = i′ . An F -module is a pair consisting of A ∈ Obj(A) and a morphism ̺A : F A → A satisfying ̺A ◦ mA = ̺A ◦ F ̺A and ̺A ◦ iA = IA . Morphisms between F -modules f : A → A′ are morphisms in A with ̺A′ ◦ F f = f ◦ ̺A and the Eilenberg-Moore category of F -modules is denoted by AF . For any object A of A, F A is an F -module and this yields the free functor φF : A → AF , A 7→ (F A, mA), which is left adjoint to the forgetful functor UF : AF → A by the isomorphism MorAF (φF A, B) → MorA (A, UF B), f 7→ f ◦ iA. The full subcategory of AF consisting of all free F -modules (i.e. the full subcategory of AF generated by the image of φF ) is called the Kleisli category of F and is denoted eF . by A 2.4. Comonads on A. A comonad on A is a triple G = (G, d, e), where G : A → A is a functor with natural transformations d : G → GG and e : G → IA satisfying coassociativity and counitality conditions. A morphism of comonads is a natural transformation that is compatible with the coproduct and counit. A G-comodule is an object A ∈ A with a morphism ̺A : A → GA compatible with d and e. Morphisms between G-comodules g : A → A′ are morphisms in A with ′ ̺A ◦ g = Gg ◦ ̺A and the Eilenberg-Moore category of G-comodules is denoted by AG . For any A ∈ A, GA is a G-comodule yielding the (co)free functor φG : A → AG , A 7→ (GA, dA) which is right adjoint to the forgetful functor U G : AG → A by the isomorphism MorAG (B, φG A) → MorA (U G B, A), f 7→ eA ◦ f. The full subcategory of AG consisting of all (co)free G-comodules (i.e. the full subcategory of AG generated by the image of φG ) is called the Kleisli category of G and e G. is denoted by A 2.5. (Co)monads related to adjoints. Let L : A → B and R : B → A be an adjoint pair of functors with unit η : IA → RL and counit ε : LR → IB . Then RεL η F = (RL, RLRL −→ RL, IA −→ RL) is a monad on A. Similarly a comonad on B is defined by LηR ε G = (LR, LR −→ LRLR, LR −→ IB ). As already observed by Eilenberg and Moore in [11], the monad structure of an endofunctor induces a comonad structure on any adjoint endofunctor. MONADS AND COMONADS ON MODULE CATEGORIES 5 2.6. Adjoints of monads and comonads. Let L : A → A and R : A → A be an adjoint pair of functors. (1) L is a monad if and only if R is a comonad. In this case the Eilenberg-Moore categories AL and AR are isomorphic. (2) L is a comonad if and only if R is a monad. e L and A e R are isomorphic to each other. In this case the Kleisli categories A Proof. (1) The first claim is proven in [11, Proposition 3.1], here it follows from 2.2 (see also [20]). The isomorphism of AL and AR is stated in [29, p. 3935]. (2) The first claim is proved similarly to the first claim in (1). The isomorphism of L e and A e R was observed in [18] and is also stated in [29, p. 3935]. It is provided by A the canonical isomorphisms for A, A′ ∈ A, MorAL (φL A, φL A′ ) ∼ = MorA (LA, A′ ) ∼ = MorA (A, RA′ ) ∼ = MorAR (φR A, φR A′ ).  2.7. Relative projectivity and injectivity. An object A of a category A is said to be projective relative to a functor F : A → B (or F -projective in short) if MorA (A, f ) : MorA (A, X) → MorA (A, Y ) is surjective for all those morphisms f in A, for which F f is a split epimorphism in B. Dually, A ∈ A is said to be injective relative to F (or F -injective) if MorA (f, A) : MorA (Y, A) → MorA (X, A) is surjective for all such morphisms f in A, for which F f is a split monomorphism in B. For an adjunction (L : A → B, R : B → A), with unit η and counit ε, an object A ∈ A is L-injective if and only if ηA is a split monomorphism in A. Dually, B ∈ B is R-projective if and only if εB is a split epimorphism in B. Recall (e.g. [5, Section 6.5],[14, Chapter 2]) that the Cauchy completion, also called Karoubian closure, of any category A is the smallest (unique up to equivalence) category A that contains A as a subcategory and in which idempotent morphisms split (i.e. can be written as a composite of an epimorphism and its section). 2.8. Equivalent subcategories. Let A be a category in which idempotent morphisms split. Then for any comonad (L, d, e) on A with right adjoint monad (R, m, i), there is an equivalence E : ALinj → Aproj R , where ALinj denotes the full subcategory of AL whose objects are injective relative to denotes the full subcategory of AR whose the forgetful functor U L : AL → A and Aproj R objects are projective relative to the forgetful functor UR : AR → A. Explicitly, for (A, ̺A ) ∈ ALinj , the object E(A, ̺A ) is given by the equaliser of the parallel morphisms R̺A and ω := mLA ◦ RηA : RA → RLA, where η is the unit of the adjunction (L, R). Proof. By 2.6(2), the Kleisli categories ÃL and ÃR are isomorphic and this isomorphism extends to their Karoubian closures. The Karoubian closure of ÃR is equivalent to the full subcategory of UR -projective objects of AR (see [16], [27, Theorem 2.5]). Dually, the Karoubian closure of ÃL is equivalent to the full subcategory of U L -injective objects of AL . This proves the equivalence ALinj ≃ Aproj R . 6 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER The explicit form of the equivalence functor is obtained by computing the composite of the isomorphism between the Karoubian closures of the Kleisli categories with the equivalences in [27, Theorem 2.5]. This (straightforward) computation yields the equaliser E(A, ̺A ) → RA of the identity morphism IRA and the idempotent morphism Rν A ◦ ω : RA → RA, where ν A is a retraction of η(A, ̺A ) = ̺A in AL . This equaliser exists by the assumption that idempotents split in A. Since ω ◦ Rν A ◦ ω = R̺A ◦ Rν A ◦ ω and Rν A ◦ R̺A = IRA , E(A, ̺A ) → RA is also an equaliser of ω and R̺A .  Recall from [23], [26] that a functor F : B → A is said to be separable if and only if the transformation MorB (−, −) → MorA (F (−), F (−)), f 7→ F f, is a split natural monomorphism. Separable functors reflect split epimorphisms and split monomorphisms. Questions related to 2.9(1) are also discussed in [6, Proposition 6.3]. 2.9. Separable monads and comonads. Let A be a category. (1) For a monad (R, m, i) on A, the following are equivalent: (a) m has a natural section m b such that Rm ◦ mR b =m b ◦ m = mR ◦ Rm; b (b) the forgetful functor UR : AR → A is separable. (2) For a comonad (L, d, e) on A, the following are equivalent: (a) d has a natural retraction db such that b ◦ Ld = d ◦ db = Ldb ◦ dL; dL (b) the forgetful functor U L : AL → A is separable. Proof. (1) By Rafael’s theorem [26, Theorem 1.2], UR is separable if and only if the counit εR of the adjunction (φR , UR ) (see 2.3) is a split natural epimorphism. (1) (a)⇒(b). A section ν : IAR → φR UR of εR is given by a morphism ν(X, ̺X ) : X iX / RX mX b / RRX R̺X / RX, for any (X, ̺X ) in AR . By naturality and the properties of m b required in (a), ν(X, ̺X ) is an R-module morphism, i.e. mX ◦ Rν(X, ̺X ) = ν(X, ̺X ) ◦ ̺X . Since m b is a section of m, ν(X, ̺X ) is a section of εR (X, ̺X ) = ̺X . In order to see that, use also associativity and unitality of the R-action ̺X . The morphism ν is natural, i.e. for f : (X, ̺X ) → (Y, ̺Y ) in AR , Rf ◦ ν(X, ̺X ) = ν(Y, ̺Y ) ◦ f . This follows by definition of an R-module morphism and naturality. (b)⇒(a). A section ν : IAR → φR UR of εR induces a section of m = UR εR φR by putting m b := UR νφR . It obeys the properties in (a) by naturality. (2) The proof is symmetric to (1).  2.10. Separability of adjoints. Let L, R : A → A be an adjoint pair of endofunctors with unit η : IA → RL and counit ε : LR → IA . MONADS AND COMONADS ON MODULE CATEGORIES 7 If (L, d, e) is a comonad with corresponding monad (R, m, i), then there are pairs of adjoint (free and forgetful) functors (see 2.3, 2.4): A AL φR / AR , AR A, A UL / UR φL / / A, AL , with unit ηR and counit εR , and with unit η L and counit εL . (1) φL is separable if and only if φR is separable. (2) U L is separable if and only if UR is separable. If the properties in part (2) hold, then any object of AL is injective relative to U L and every object of AR is projective relative to UR . Proof. (1) By Rafael’s theorem [26, Theorem 1.2], φL is separable if and only if εL = e is a split natural epimorphism, while φR is separable if and only if ηR = i is a split natural monomorphism. By construction, i and e are mates under the adjunction (L, R) and the trivial adjunction (IA , IA ). That is, e = ε ◦ Li equivalently, i = Re ◦ η. Hence a natural transformation bi : R → IA is a retraction of i if and only if its mate eb := biL ◦ η under the adjunctions (IA , IA ) and (L, R) is a section of e. (2) Since d and m are mates under the adjunctions (L, R) and (LL, RR), a natural transformation m b satisfies the properties in 2.9(1)(a) if and only if its mate db satisfies the properties in 2.9(2)(a). Thus the claim follows by 2.9. It remains to prove the final claims. Following 2.7, an L-comodule (A, ̺A ) is U L injective if and only if η L (A, ̺A ) = ̺A is a split monomorphism in AL . Since ̺A is split in A (by eA) and U L , being separable, reflects split monomorphisms, any (A, ̺A ) ∈ AL is U L -injective. UR -projectivity of every object of AR is proven by a symmetrical reasoning.  2.11. Lifting of endofunctors. For a monad F , a comonad G and an endofunctor T on the category A, consider the diagrams with Eilenberg-Moore and Kleisli categories T A F _ _ _/ A F UF UF  A T /  A, Tb AG _ _ _/ AG UG UG  A T /  A, A T / A φF φF   e eF , e F _ _T _/ A A A T / A φG φG  ≈  T e G, e G _ _ _/ A A where the U ’s denote the forgetful functors and the φ’s the free functors. If there ≈ exist T , Tb, Te or T making the corresponding diagram commutative, they are called liftings of T . By Power and Watanabe’s observation [25], liftings of endofunctors A → A arise as images under a strict monoidal functor, from the monoidal category (vertical subcategory at A of the 2-category) of (co)monad morphisms, to the monoidal category of endofunctors on A. By their result, liftings to (co)monads are in bijective correspondence with (co)monads in the monoidal category of (co)monad morphisms, that is, with various distributive laws [4]: 2.12. Monad distributive laws. For monads F and T on a category A, the liftings of T to a monad T on AF are in bijective correspondence with monad distributive eT . laws λ : F T → T F , and also with the liftings of F (along φT ) to a monad Fe on A 8 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER A monad distributive law λ : F T → T F induces a canonical monad structure on T F and T F -modules are equivalent to T -modules. For more details we refer, e.g., to [4, p. 120], [32, 4.4], [19]. 2.13. Comonad distributive laws. If G and T are comonads on A, a natural transformation ϕ : T G → GT is called a comonad distributive law provided T can be lifted to a comonad Tb (in 2.11), equivalently, F can be lifted along φT to a comonad ≈ e T . In this case ϕ : T G → GT induces a comonad structure on T G and F on A T G-comodules are equivalent to Tb-comodules. For more details see [1], [4], [32], [22]. 2.14. Mixed distributive laws. Let F = (F, m, i) be a monad and T = (T, d, e) a comonad on A. A natural transformation λ : F T → T F is called a mixed distributive law, or an entwining, from F to T , provided T λ ◦ λT ◦ F d = dF ◦ λ, eF ◦ λ = F e, T m ◦ λF ◦ F λ = λ ◦ mT, λ ◦ iT = T i. These conditions are equivalent to the existence of a comonad lifting T : AF → AF or also a monad lifting Fb : AT → AT (see [4, p. 133], [32, 5.3, 5.4]). We call a natural transformation ψ : T F → F T a mixed distributive law, or an entwining, from T to F , provided (e.g. [20, 2.4]) ψT ◦ T ψ ◦ dF = F d ◦ ψ, F e ◦ ψ = eF, mT ◦ F ψ ◦ ψF = ψ ◦ T m, ψ ◦ T i = iT. eF → A eF These conditions are equivalent to the existence of a comonad lifting Te : A ≈ eT → A e T (see [25, Sec 8]). or a monad lifting F : A 2.15. Mixed bimodules. With the notation above, let λ : F T → T F be a mixed distributive law. Mixed bimodules or λ-bimodules are defined as those A ∈ Obj(A) with morphisms h k /A / TA FA such that (A, h) is an F -module and (A, k) is a T -comodule satisfying the pentagonal law FA h / A k / TA O Th Fk  λA / T F A. FTA Morphisms between two λ-bimodules, called bimodule morphisms, are both F module and T -comodule morphisms. These notions yield the category of λ-bimodules denoted by ATF . It can also be considered as the category of T -comodules for the comonad T : AF → AF , or the category of Fb-modules for the monad Fb : AT → AT . 2.16. Distributive laws for adjoint functors. Let (L, R) be an adjoint pair of endofunctors on a category A with unit η and counit ε, and F be an endofunctor on A. Consider a natural transformation ψ : LF → F L and set ψ̃ = RF ε ◦ RψR ◦ ηF R : F R → RF. MONADS AND COMONADS ON MODULE CATEGORIES 9 (1) If L and F are monads, then ψ is a monad distributive law if and only if ψ̃ is a mixed distributive law (or entwining). (2) If L is a monad and F is a comonad, then ψ is a mixed distributive law (or entwining) if and only if ψ̃ is a comonad distributive law. (3) If L is a comonad and F is a monad, then ψ is a mixed distributive law (or entwining) if and only if ψ̃ is a monad distributive law. (4) If L and F are comonads, then ψ is a comonad distributive law if and only if ψ̃ is a a mixed distributive law (or entwining). Proof. All these claims are easily checked by using that the structure maps of the adjoint monad-comonad (or comonad-monad) pair (L, R) are mates under adjunctions, together with naturality and the triangle identities. Details are left to the reader.  Combining the correspondences in 2.16(1) and (2) with the isomorphism of module and comodule categories in 2.6(1), further isomorphisms, between categories of mixed bimodules, can be derived. The following was obtained in cooperation with Bachuki Mesablishvili. 2.17. Modules and distributive laws. Let L be a monad with right adjoint comonad R on a category A. (1) Let G be a comonad with a mixed distributive law λ : LG → GL. Then the category of mixed (L, G)-bimodules AG L is isomorphic to the category of GRcomodules AGR for the composite comonad (see 2.13) defined by the associated comonad distributive law λ̃ : GR → RG (see 2.16(2)). (2) Let F be a monad with a mixed distributive law τ : F R → RF . Then the category of mixed (F, R)-bimodules AR F is isomorphic to the category of F Lmodules AF L for the composite monad defined by the monad distributive law τ̃ : LF → F L (see 2.16(1)). Proof. (1) The mixed distributive law λ : LG → GL yields a lifting of G to a comonad G on the category AL of L-modules. Moreover, it determines a comonad distributive b on the category law λ̃ : GR → RG which is equivalent to a lifting of G to a comonad G AR of R-comodules. By 2.6(1), there is an isomorphism K : AL → AR , and this b and G. That is, G b = KGK −1 isomorphism obviously ‘intertwines’ the comonads G R as comonads. Thus the isomorphism K : AL → A lifts to an isomorphism between b The lifted isomorphism has the the categories of G-comodules and G-comodules. object map and morphism map b (A, ̺A : A → GA) 7→ (KA, K̺A : KA → KGA = GKA)
′
f Kf ′ (A, ̺A ) −→ (A′ , ̺A ) 7→ (KA, K̺A ) −→ (KA′ , K̺A ) . b The inverse functor has the same form in terms of K −1 . By characterisation of Gcomodules as comodules for the composite comonad GR, and characterisation of Gcomodules as mixed (L, G)-bimodules, we obtain the isomorphism claimed. (2) is shown similarly to (1).  2.18. F -actions on functors. Let A and B be categories. Given a monad F = (F, m, i) on A, composition by F on the left yields a monad on the category whose 10 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER objects are the functors B → A and whose morphisms are natural transformations. A module (R, ϕ) for this monad is called a (left) F -module functor. Explicitly, this means a functor R : B → A and a natural transformation ϕ : F R → R satisfying associativity and unitality conditions (see [20, 3.1]). By 2.3, for any functor R : B → A, (F R, mR) is an F -module functor. 2.19. F -Galois functors. For a monad F on a category A and any functor R : B → A, consider the diagram L R AO F _ _ _/ B _ _ _/ AF φF UF   L _  R / A. A _ _/ B There exists some functor R making the right square commutative if and only if R has an F -module structure ϕ : F R → R (see 2.18). If R has a left adjoint L : A → B with unit η, then there is a monad morphism (see [17, Proposition 3.3]), Fη ϕL can : F −→ F RL −→ RL. We call an F -module functor R an F -Galois functor if it has a left adjoint and can is an isomorphism. Consider an F -module functor R : B → A with F -action ϕ, a left adjoint L, unit η and counit ε of the adjunction. If B admits coequalisers of the parallel morphisms L̺X and εLX ◦ LϕLX ◦ LF ηX : LF X → LX, for any object (X, ̺X ) in AF , then this coequaliser yields the left adjoint L(X, ̺X ) of R (see left square). By uniqueness of the adjoint, LφF ∼ = L. Denoting the coequaliser natural epimorphism LUF → L by p, the unit of the adjunction (L, R) is the unique natural morphism η : IAF → R L such that UF η = Rp ◦ ηUF . The counit is the unique natural morphism ε : L R → IB , such that ε ◦ pR = ε. If B has coequalisers of all parallel morphisms, then the following are equivalent (dual to [20, Theorem 3.15]): (a) R is an F -Galois functor; (b) the unit of (L, R) is an isomorphism for (i) all free F -modules (i.e. modules in the Kleisli category of F), or (ii) all UF -projective F -modules. From [3], [9] and [2, Theorem 3.14] we recall a result of central importance in our setting in the form it can be found in [15, Theorem 1.7]. 2.20. Beck’s theorem. Consider a monad F on a category A and an F -module functor R : B → A. Then the induced lifting R : B → AF in 2.19 is an equivalence if and only if the following hold: (i) R is an F -Galois functor, (ii) R reflects isomorphisms, (iii) B has coequalisers of R-contractible coequaliser pairs and R preserves them. The Galois property of a functor also transfers to its adjoint functor. MONADS AND COMONADS ON MODULE CATEGORIES 11 2.21. Proposition. Consider an adjoint pair (F, G) of endofunctors on a category A. Let T : B → A be a functor which has both a left adjoint L and a right adjoint R. (1) If F is a monad (equivalently, G is a comonad), then T is an F -Galois functor as in 2.19 if and only if it is a G-Galois functor (as in [20, Definition 3.5]). (2) If F is a comonad (equivalently, G is a monad), then R is a G-Galois functor as in 2.19 if and only if L is an F -Galois functor (as in [20, Definition 3.5]). Proof. Denote the unit of the adjunction (F, G) by η and its counit by ε. Denote furthermore the unit and counit of the adjunction (L, T ) by ηL and εL , respectively, and for the unit and counit of the adjunction (T, R) write ηR and εR , respectively. (1) A bijective correspondence between F -actions ϕT and G-coactions ϕT on T is given by ϕT := GϕT ◦ ηT . The comonad morphism corresponding to ϕT comes out as ηT R GϕT R GεR /G. / GT R / GF T R f : TR can Comparing it with the canonical monad morphism can : F → T L in 2.19, they are easily seen to be mates under the adjunctions (F, G) and (T L, T R). That is, f = GεR ◦ GT εL R ◦ GcanT R ◦ ηT R. can f is an isomorphism if and only if can is an isomorphism. Thus can (2) A bijective correspondence between G-actions ϕR : GR → R and F -coactions ϕL : L → F L is given by ϕL := εL F L ◦ LεR T F L ◦ LGϕR T F L ◦ LT GηR F L ◦ LT ηL ◦ LηL . f : LT → F corresponding to ϕL , and the The canonical comonad morphism can canonical monad morphism can : G → RT corresponding to ϕR , turn out to be mates under the adjunctions (LT, RT ) and (F, G). That is, f = εL F ◦ LεR T F ◦ LT canF ◦ LT η. can f is an isomorphism, as stated.  Thus can is a natural isomorphism if and only if can 3. Rings and corings in module categories Let A be an associative ring with unit. We first study the relationship between ring extensions of A and monads on the category MA of right A-modules. 3.1. A-rings. A ring B is said to be an A-ring provided there is a ring morphism ι : A → B. Equivalently, B is an A-bimodule with A-bilinear multiplication µ : B ⊗A B → B and unit ι : A → B subject to associativity and unitality conditions. A right B-module is a right A-module M with an A-linear map ̺M : M ⊗A B → M satisfying the associativity and unitality conditions. B-module morphisms f : M → N are A-linear maps with f ◦ ̺M = ̺N ◦ (f ⊗A IB ). The category of right B-modules is denoted by MB . It is isomorphic to the module category over the ring B and thus is an abelian category with B as a projective generator. As an endofunctor on MA , − ⊗A B is left adjoint to the endofunctor HomA (B, −). 3.2. Monad-comonad. For an A-bimodule B, the following are equivalent: (a) (B, µ, ι) is an A-ring; (b) − ⊗A B : MA → MA is a monad; (c) HomA (B, −) : MA → MA is a comonad. 12 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER Proof. (a)⇔(b) is obvious and (b)⇔(c) follows by 2.6.  Adjointness of the free and forgetful functors for the monad − ⊗A B is just the isomorphism HomB (− ⊗A B, N ) → HomA (−, N ), f 7→ f ◦ (− ⊗A ι). We write [B, −] = HomA (B, −) for short. Comultiplication and counit of the comonad [B, −] in 3.2(c) are denoted by [µ, −] and [ι, −], respectively. The following was pointed out in [11, p. 397], see [2, Section 3.7]. It is a special case of 2.6(1). 3.3. B-modules are HomA (B, −)-comodules. For any A-ring B, the category of right B-modules is isomorphic to the category of HomA (B, −)-comodules, that is, there exists an isomorphism ∼ = MB −→ M[B,−] . Next we investigate the relationship between a comonad and its right adjoint monad. 3.4. A-Corings. An A-coring is an A-bimodule C with A-bilinear maps, the coproduct ∆ : C → C ⊗A C and the counit ε : C → A, subject to coassociativity and counitality conditions. Similar to the characterisation of A-rings in 3.2, we derive from 2.6: 3.5. Comonad-monad. For an A-bimodule C, the following are equivalent: (a) (C, ∆, ε) is an A-coring; (b) − ⊗A C : MA → MA induces a comonad; (c) HomA (C, −) : MA → MA induces a monad. Writing HomA (C, −) = [C, −], the related monad is ([C, −], [∆, −], [ε, −]). In the rest of this section C will be an A-coring. We first recall properties of the category of comodules (e.g. [8]). 3.6. The category MC . The comodules for the comonad − ⊗A C : MA → MA are called right C-comodules and their category is denoted by MC . (1) MC is an additive category with coproducts and cokernels. (2) The (co)free functor − ⊗A C is right adjoint to the forgetful functor. (3) For any monomorphism f : X → Y in MA , f ⊗A IC : X ⊗A C → Y ⊗A C is a monomorphism in MC . (4) C is a flat left A-module if and only if monomorphisms in MC are injective maps. Left comodules of an A-coring C are defined symmetrically to the right comodules in 3.6, as comodules of the comonad C ⊗A − on the category A M. Furthermore, if C is an A-coring and D is a B-coring, then we can consider the (composite) comonad C ⊗A − ⊗B D on the category of (A, B)-bimodules. Its comodules are called (C, D)bicomodules. Equivalently, a (C, D)-bicomodule is a left C-comodule and a right Dcomodule such that the right D-coaction is a left C-comodule map. The category of (C, D)-bicomodules is denoted by C MD . While C-comodules are well studied in the literature (e.g. [8]), [C, −]-modules have not attracted so much attention so far. They were addressed by Eilenberg-Moore in [10] and [11] as C-contramodules and reconsidered recently by Positselski [24] in the context of semi-infinite cohomology. MONADS AND COMONADS ON MODULE CATEGORIES 13 3.7. The category M[C,−] . The modules for the monad [C, −] : MA → MA are right A-modules N with some A-linear map [C, N ] → N subject to associativity and unitality conditions. Their category is denoted by M[C,−] . (1) M[C,−] is an additive category with products and kernels. (2) The (free) functor [C, −] : MA → M[C,−] is left adjoint to the forgetful functor. (3) For any epimorphism h : X → Y in MA , [C, h] : [C, X] → [C, Y ] is an epimorphism (not necessarily surjective) in M[C,−] . (4) C is a projective right A-module if and only if epimorphisms in M[C,−] are surjective maps. Proof. The proofs are similar to the comodule case. Some of the assertions can also be found in [24].  3.8. Right and left contramodules. In 3.7, modules of the monad [C, −] ≡ Hom−,A (C, −) on the category MA of right A-modules are considered. Symmetrically, an A-coring C determines a monad HomA,− (C, −) also on the category A M of left A-modules. Modules for the monad [C, −] ≡ Hom−,A (C, −) on MA are called right C-contramodules, and modules for the monad HomA,− (C, −) on A M are called left C-contramodules. If not specified otherwise, we mean by contramodules right contramodules, throughout. Following a long-established (co)module-theoretic tradition, we often do not write explicitly the structure morphism αM : [C, M ] → M for a contramodule (M, αM ). In the same way, the set (or abelian group) of all [C, −]-module maps (M, αM ) → (N, αN ) is denoted by Hom[C,−] (M, N ). We saw in 3.3 that for any A-ring B, the categories MB and M[B,−] are isomorphic (see also 2.6(1)). In view of the asymmetry of assertions (1) and (2) in 2.6, the corresponding statement for corings is no longer true and we will come back to this question in 4.6. So far we know from 2.6(2): 3.9. Related Kleisli categories. For any A-coring C, the Kleisli categories of −⊗A C and [C, −] are isomorphic by the isomorphisms for X, Y ∈ MA , HomC (X ⊗A C, Y ⊗A C) ∼ = HomA (X ⊗A C, Y ) ∼ = HomA (X, [C, Y ]) ∼ = Hom[C,−] ([C, X], [C, Y ]). Recall that for any A-coring C, the right dual C ∗ = HomA (C, A) has a ring structure by the convolution product for f, g ∈ C ∗ , f ∗ g = f ◦ (g ⊗A IC ) ◦ ∆ (convention opposite to [8, 17.8]). Similarly a product is defined for the left dual ∗ C. The relation between C-comodules and modules over the dual ring of C is well studied (see e.g. [8, Section 19]). 3.10. The comonads − ⊗A C and [∗ C, −]. The comonad morphism α : − ⊗A C → HomA (∗ C, −), − ⊗ c 7→ [f 7→ −f (c)], ∗ yields a faithful functor Gα : MC → M[ C,−] ∼ = M∗ C , and the following are equivalent: (a) αN is injective for each N ∈ MA ; (b) Gα is a full functor; (c) C is a locally projective left A-module. 14 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER If these conditions are satisfied, MC is equal to σ[C∗ C ], the full subcategory of M∗ C subgenerated by C. Similar to 3.10, C-contramodules can be related to C ∗ -modules. 3.11. The monads [C, −] and − ⊗A C ∗ . The monad morphism β : − ⊗A C ∗ → HomA (C, −), − ⊗ f 7→ [c 7→ −f (c)], yields a faithful functor Fβ : M[C,−] → MC ∗ , and the following are equivalent: (a) β is surjective for all M ∈ MA ; (b) Fβ is an isomorphism; (c) C is a finitely generated and projective right A-module. C In general, C is not a [C, −]-module and [C, A] is not a C-comodule. In fact, [C, A] ∈ M holds provided C is finitely generated and projective as a right A-module. 4. Functors between co- and contramodules Categories of comodules and contramodules have complementary features. Therefore, it is of interest to find A-corings C and B-corings D (over possibly different base rings) such that the category of D-comodules and that of [C, −]-modules are equivalent. As we will see in 4.4, functors between these categories are provided by bicomodules. It turns out that the question, when they provide an equivalence, fits the standard problem in (categorical) descent theory. Since comodules for the trivial B-coring B are simply B-modules, our considerations include the particular case when the category of [C, −]-modules is equivalent to the category of B-modules. Dually, when the coring C is trivial (i.e. equal to A), the problem reduces to a study of equivalences between A-module and D-comodule categories. This question is already discussed in the literature, see e.g. [30], [15]. Throughout this section C is an A-coring and D a B-coring for rings A and B. The following observation was made in [24, 5.1.2]. 4.1. [C, −]-modules induced by C-comodules. Let N be a (C, D)-bicomodule with left C-coaction N̺. For any Q ∈ MD , there is an isomorphism ϕ : HomA (C, HomD (N, Q)) → HomD (C ⊗A N, Q), h 7→ [c ⊗ m 7→ h(c)(m)], (see e.g. [8, 18.11]). Then the right A-module N Q := HomD (N, Q) is a [C, −]-module by αN Q : HomA (C, HomD (N, Q)) ϕ / HomD (C ⊗A N, Q) HomD (N̺,Q) / HomD (N, Q). Thus there is a bifunctor HomD (−, −) : (C MD )op × MD → M[C,−] , (N, Q) 7→ (N Q , αN Q ), (f, g) 7→ HomD (f, g). In a symmetric way, for any left D-comodule Q and a (D, C)-bicomodule N (with C-coaction ̺N : N → N ⊗A C), HomD (N, Q) is a left C-contramodule by HomD (̺N , Q). If N is just a left C-comodule we tacitly assume D = B = EndC (N ) to apply the preceding notions and results. 4.2. Corollary. MONADS AND COMONADS ON MODULE CATEGORIES 15 (1) Let N be a left C-comodule with B = EndC (N ). For any subring B ′ ⊂ B and Q ∈ MB ′ , HomB ′ (N, Q) is a [C, −]-module. (2) For any Q ∈ MC , HomC (C, Q) is a [C, −]-module. 4.3. Contratensor product. For any (C, D)-bicomodule N , the construction in 4.1 yields a functor HomD (N, −) : MD → M[C,−] , inducing the commutative diagram of (right adjoint) functors MD HomD (N,−) M[C,−] / U[C,−] MD HomD (N,−) /  MA . Since HomD (N, −) : MD → MA has the left adjoint − ⊗A N and MD has coequalisers, it follows from 2.19 that HomD (N, −) : MD → M[C,−] also has a left adjoint which comes out as follows (see [24]). For any (C, D)-bicomodule (N, N̺, ̺N ) and [C, −]-module (M, αM ), the contratensor product, M ⊗[C,−] N is defined as the coequaliser HomA (C, M ) ⊗A N / / M ⊗A N / M ⊗[C,−] N, where the coequalised maps are f ⊗ n 7→ (f ⊗A IN ) ◦ N̺(n) and αM ⊗A IN . Projection of an element m ⊗ n to M ⊗[C,−] N is denoted by m⊗[C,−] n. As a coequaliser of right D-comodule maps, M ⊗[C,−] N is a right D-comodule, and thus defines a functor −⊗[C,−] N : M[C,−] → MD . Note that this coequaliser splits in MD provided (M, αM ) is U[C,−] -projective. 4.4. Functors between comodules and contramodules. Any (C, D)-bicomodule N induces an adjoint pair of functors − ⊗[C,−] N : M[C,−] → MD , HomD (N, −) : MD → M[C,−] , that is, for M ∈ M[C,−] and P ∈ MD , there is an isomorphism HomD (M ⊗[C,−] N, P ) ∼ = Hom[C,−] (M, HomD (N, P )). Conversely, any right adjoint functor F : MD → M[C,−] is naturally isomorphic to HomD (N, −), for an appropriate (C, D)-bicomodule N . Proof. In view of the discussion in 2.19, for [C, −]-modules M , the unit of the adjunction is given by ηM : M → HomD (N, M ⊗[C,−] N ), m 7→ [n 7→ m⊗[C,−] n]. Also by 2.19, the counit of the adjunction comes out (and is in particular well defined) as εQ : HomD (N, Q)⊗[C,−] N → Q, f ⊗[C,−] n 7→ f (n), for all right D-comodules Q. Conversely, assume that F : MD → M[C,−] has a left adjoint. Then so does the composite F ′ := U[C,−] ◦ F : MD → MA , in light of 3.7. Hence it follows by [30, Theorem 3.2] that there exists an (A, D)-bicomodule N such that F ′ is naturally 16 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER isomorphic to HomD (N, −). Moreover, by construction, for any Q ∈ MD , HomD (N, Q) is a [C, −]-module via some action κQ : HomA (C, HomD (N, Q)) → HomD (N, Q), and for q ∈ HomD (Q, Q′ ), HomD (N, q) is a morphism of [C, −]-modules. This amounts to saying that κ(−) is a natural transformation HomD (C ⊗A N, −) → HomD (N, −). Therefore, it follows by the Yoneda Lemma that there is an (A, D)-bicomodule map τ : N → C⊗A N , such that κQ = HomD (τ, Q), for Q ∈ MD . Unitality and associativity of the action κQ , for any Q ∈ MD , imply counitality and coassociativity of the left C-coaction τ , respectively.  Consider a (C, D)-bicomodule N , over an A-coring C and a B-coring D. A [C, −]module map g : (L, αL ) → (M, αM ) is said to be [C,−] N -pure provided the sequence / 0 ker g ⊗[C,−] N / L⊗[C,−] N g⊗[C,−] IN / M ⊗[C,−] N , is exact (in MB ). 4.5. Some tensor relations. Let (N, N̺) be a left C-comodule. Then: (1) For any right A-module X, HomA (C, X)⊗[C,−] N ∼ = X ⊗A N. M (2) If (M, ̺ ) is right C-comodule for which the map γ : HomA (C, M ) → HomA (C, M ⊗A C), f 7→ ̺M ◦ f − (f ⊗A IC ) ◦ ∆, is [C,−] N -pure, then HomC (C, M )⊗[C,−] N is isomorphic to the cotensor product M ⊗C N. Proof. (1) This is outlined in [24, 5.1.1]. (2) Consider the commutative diagram in MB (for B = EndC (N )). 0 0 / HomC (C, M )⊗ [C,−] N  ϑ   / M ⊗C N / HomA (C, M )⊗[C,−] N γ⊗[C,−] IN / HomA (C, M ⊗A C)⊗[C,−] N ∼ = ∼ =  / M ⊗A N  / M ⊗A C ⊗A N, Since γ is [C,−] N -pure, and HomC (C, M ) = ker γ, the top row is exact. The bottom row is the defining exact sequence of the cotensor product (see e.g. [8, 21.1]). The vertical isomorphisms are obtained from part (1). Thus there is an isomorphism ϑ : HomC (C, M ⊗[C,−] N ) → M ⊗C N extending the diagram commutatively.  From previous considerations we obtain the following result by Positselski. 4.6. Correspondence of categories. (1) For any A-coring C, there is an adjoint pair of functors −⊗[C,−] C : M[C,−] → MC , HomC (C, −) : MC → M[C,−] . (2) On the classes of free objects, the functors in (1) restrict to the following maps. For any X ∈ MA , X ⊗A C 7→ HomC (C, X ⊗A C) ∼ = HomA (C, X), HomA (C, X) 7→ HomA (C, X) ⊗[C,−] C ∼ = X ⊗A C. Thus the functors in part (1) restrict to inverse isomorphisms between the Kleisli subcategories of MC and M[C,−] . MONADS AND COMONADS ON MODULE CATEGORIES 17 (3) There is an equivalence HomC (C, −) : MCinj → Mproj [C,−] , where MCinj denotes the full subcategory of MC of objects relative injective to the forgetful functor MC → MA , and Mproj [C,−] the full subcategory of M[C,−] of objects relative projective to the forgetful functor M[C,−] → MA . Proof. This is shown in [24, Theorem in 5.3]. Here (1) follows by putting D = C in 4.4 and considering C as a (C, C)-bicomodule, see 4.2. Claim (2) (cf. 3.9) is obtained by applying 2.6(2) to the adjoint comonad-monad functor pair (− ⊗A C, HomA (C, −)). Part (3) follows from 2.8. Note that the equaliser in the more general situation of 2.8 yields here the equivalence functor E(M, ̺M ) = HomC (C, M ), for any (M, ̺M ) ∈ MCinj , as stated.  Recall that an A-coring C is said to be a coseparable coring if its coproduct is a split monomorphism of C-bicomodules. Equivalently, there is an A-bimodule map δ : C ⊗A C → A such that δ ◦ ∆ = ε and (IC ⊗A δ) ◦ (∆ ⊗A IC ) = (δ ⊗A IC ) ◦ (IC ⊗A ∆). Such a map δ is called a cointegral (e.g. [8, 26.1]). Equivalently, coseparable corings can be described by separable functors as follows. 4.7. Coseparable corings. For C the following are equivalent. (a) C is a coseparable coring; (b) the forgetful functor U C : MC → MA is separable; (c) the forgetful functor U[C,−] : M[C,−] → MA is separable. If these assertions hold then, in particular, any C-comodule is U C -injective and any [C, −]-module is U[C,−] -projective. Proof. Equivalence (a)⇔(b) is quoted from [8, 26.1]. It can be derived alternatively from 2.9(2). Equivalence (b)⇔(c) and the final claims follow by 2.10(2).  Combining 4.7 with 4.6 we obtain: 4.8. Comodules and contramodules of coseparable corings. For a coseparable coring C, the category MCinj coincides with MC and Mproj [C,−] is equal to M[C,−] . Thus there is an equivalence HomC (C, −) : MC → M[C,−] . This equivalence between comodules and contramodules for coseparable corings plays an important role in the characterisation of categories of Hopf (contra)modules in 7.6. 5. Galois bicomodules In this section we analyse when a comodule category is equivalent to a contramodule category. Any such equivalence is necessarily given by functors associated to a bicomodule. The latter must possess additional properties. 18 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER 5.1. [C, −]-Galois bicomodules. For a (C, D)-bicomodule (N, N̺, ̺N ), the commutative diagram in 4.3 yields a canonical monad morphism (by 2.19) canN : HomA (C, −) → HomD (N, − ⊗A N ), f 7→ (f ⊗A IN ) ◦ N̺. Let η denote the unit of the adjunction (− ⊗[C,−] N, HomD (N, −)) in 4.4. The following statements are equivalent: (a) The natural transformation canN is an isomorphism; (b) ηHomA (C,Q) is an isomorphism, for all Q ∈ MA ; (c) ηM is an isomorphism, for all U[C,−] -projective M ∈ M[C,−] . If these conditions hold, then HomD (N, −) : MD → MA is a [C, −]-Galois functor and we call N ∈ C MD a [C, −]-Galois bicomodule. If N is just a left C-comodule we tacitly take D = B = EndC (N ) and call N a [C, −]-Galois left comodule. Symmetrically to the above considerations, any right adjoint functor from the category of left comodules of a B-coring D to the category of left contramodules of an A-coring C is naturally isomorphic to HomD (N, −), for some (D, C)-bicomodule N . In analogy with 5.1, also HomA,− (C, −)-Galois (D, C)-bicomodules and in particular HomA,− (C, −)-Galois right C-comodules can be defined. Studying [C, −]-Galois bicomodules we are on the one side interested in there own structural properties and on the other side also in conditions which make the related functors fully faithful. 5.2. − ⊗[C,−] N fully faithful. Let N be a (C, D)-bicomodule. Then the functor − ⊗[C,−] N : M[C,−] → MD is fully faithful if and only if (i) N is a [C, −]-Galois bicomodule and (ii) for any [C, −]-module M , the functor HomD (N, −) : MD → MA preserves the coequaliser HomA (C, M ) ⊗A N / / M ⊗A N / M ⊗[C,−] N , defining the contratensor product (cf. 4.3). Proof. Since in MD any parallel pair of morphisms has a coequaliser, the claim follows by (the dual version of) [15, Theorem 2.6].  5.3. Corollary. Let N ∈ C MD be a [C, −]-Galois bicomodule. If the functor HomD (N, −) : MD → MA preserves coequalisers, then − ⊗[C,−] N : M[C,−] → MD is fully faithful and C is a projective right A-module. Proof. It follows immediately by 5.2 that − ⊗[C,−] N : M[C,−] → MD is fully faithful. The left adjoint functor − ⊗A N always preserves cokernels and HomD (N, −) does so by hypothesis. Thus their composite HomD (N, − ⊗A N ) : MA → MA preserves cokernels, i.e. epimorphisms. Since canN in 5.1 is assumed to be an isomorphism, we conclude that also the functor HomA (C, −) preserves epimorphisms, i.e. C is projective as a right A-module.  For our investigation it is of interest to extend the notion of Galois comodules from [31, 4.1] to bicomodules. MONADS AND COMONADS ON MODULE CATEGORIES 19 5.4. D-Galois bicomodules. For any (C, D)-bicomodule N , the left adjoint functor − ⊗[C,−] N : M[C,−] → MB is a left − ⊗B D-comodule functor (in the sense of [20, 3.3]) by the coaction − ⊗[C,−] ̺N : − ⊗[C,−] N → − ⊗[C,−] N ⊗B D. We call N a D-Galois bicomodule if − ⊗[C,−] N : M[C,−] → MB is a − ⊗B D-Galois functor (in the sense of [20, 3.3]), that is, if the comonad morphism HomB (N, −) ⊗[C,−] N IHomB (N,−) ⊗[C,−] ̺N / HomB (N, −) ⊗[C,−] N ⊗B D ε⊗B ID/ − ⊗B D is an isomorphism. For a right D-comodule N , one can put C = A = EndD (N ). In this way we re-obtain the usual notion of a D-Galois right comodule in [31, 4.1]. The D-Galois property of a (D, C)-bicomodule is defined symmetrically by the Galois property of the induced functor between the category of left B-modules and the category of left C-contramodules. In the particular case of a left D-comodule N , it reduces to the usual notion of a D-Galois left comodule in [8] by putting A = C = EndD (N ). 5.5. HomD (N, −) fully faithful. Let N be a (C, D)-bicomodule. Then the functor HomD (N, −) : MD → M[C,−] is fully faithful if and only if (i) N is a D-Galois bicomodule and (ii) the functor − ⊗[C,−] N : M[C,−] → MB preserves the equaliser D Hom (N, Q) / HomB (N, Q) HomB (N,̺Q ) ω / / HomB (N, Q ⊗B D), for any right D-comodule (Q, ̺Q ), where ω(f ) = (f ⊗B ID ) ◦ ̺N . Proof. This follows again by [15, Theorem 2.6].  5.6. Corollary. Let N ∈ C MD be a D-Galois bicomodule. If the functor − ⊗[C,−] N : M[C,−] → MB preserves equalisers, then HomD (N, −) : MD → M[C,−] is fully faithful and D is a flat left B-module. Proof. (Compare with [31, 4.8]). The first assertion follows immediately from 5.5. In particular, this means that N is a generator in MD . Moreover, there is a natural isomorphism − ⊗B D ∼ = HomB (N, −) ⊗[C,−] N , where the right adjoint functor HomB (N, −) always preserves kernels and by assumption so does − ⊗[C,−] N . This implies that − ⊗B D : MB → MB preserves kernels, i.e. monomorphisms, hence D is flat as a left B-module.  Recall from [8, 19.19] that, for a left C-comodule (N, N̺) which is finitely generated and projective as a left A-module, the left dual ∗ N = HomA,− (N, A) carries a canonical right C-comodule structure, via ∗ N −→ HomA,− (N, C) ∼ = ∗ N ⊗A C, g 7→ (IC ⊗A g) ◦ N̺. In what follows, [C, −]-Galois and C-Galois properties of a finitely generated projective comodule are compared. 20 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER 5.7. [C, −]-Galois comodules and C-Galois comodules. Let N be a left C-comodule finitely generated and projective as a left A-module. The following assertions are equivalent. (a) N is a Hom−,A (C, −)-Galois left comodule; (b) N is a C-Galois left comodule; (c) ∗ N is a HomA,− (C, −)-Galois right comodule; (d) ∗ N is a C-Galois right comodule. Proof. (b)⇔(d) is proven in [7, p. 514]. (a)⇔(d). Put B = EndC (N ) and consider the (A, B)-bimodule N and the (B, A)bimodule ∗ N . The stated equivalence follows by applying 2.21(2) to the adjoint comonad-monad pair (−⊗A C, HomA (C, −)) and the functor −⊗A N ∼ = HomA (∗ N, −) : MA → MB , possessing the right adjoint HomB (N, −) and the left adjoint − ⊗B ∗ N . (b)⇔(c) is proven similarly to (a)⇔(d).  Sufficient and necessary conditions for the equivalence between a comodule and a contramodule category are obtained by applying Beck’s theorem; see 2.20. 5.8. Equivalences. For an A-coring C and a B-coring D, the following assertions are equivalent. (a) The categories M[C,−] and MD are equivalent; (b) there exists a (C, D)-bicomodule N with the properties: (i) N is a [C, −]-Galois bicomodule, (ii) the functor HomD (N, −) : MD → MA reflects isomorphisms, (iii) the functor HomD (N, −) : MD → MA preserves HomD (N, −)-contractible coequalisers. (c) there exists a (C, D)-bicomodule N with the properties: (i) N is a D-Galois bicomodule, (ii) the functor − ⊗[C,−] N : M[C,−] → MB reflects isomorphisms, (iii) the functor − ⊗[C,−] N : M[C,−] → MB preserves − ⊗[C,−] N -contractible equalisers. Proof. (a)⇔(b). By 4.4, any equivalence functor MD → M[C,−] is naturally isomorphic to HomD (N, −), for some (C, D)-bicomodule N . By Beck’s theorem 2.20, HomD (N, −) : MD → M[C,−] is an equivalence if and only if the conditions in part (b) hold. (a)⇔(c) is shown with similar arguments.  5.9. Equivalence for abelian categories. For a (C, D)-bicomodule N , the following are equivalent. (a) HomD (N, −) : MD → M[C,−] is an equivalence, C is a projective right A-module and D is a flat left B-module; (b) D is flat as a left B-module and N is a [C, −]-Galois bicomodule and a projective generator in MD ; (c) C is projective as a right A-module and N is a D-Galois bicomodule and the functor − ⊗[C,−] N : M[C,−] → MB is left exact and faithful. MONADS AND COMONADS ON MODULE CATEGORIES 21 Proof. (a)⇒(b). By Theorem 5.8, N is a [C, −]-Galois bicomodule. Being an equivalence, HomD (N, −) : MD → M[C,−] is faithful. Since the forgetful functor from M[C,−] to MA (or to MZ or Set) is faithful, so is the composite HomD (N, −) : MD → Set. This proves that N is a generator in MD . Finally, U[C,−] : M[C,−] → MA is right exact by 3.7(4). Since HomD (N, −) : MD → M[C,−] is an equivalence, this implies that also HomD (N, −) : MD → MA is right exact, by commutativity of the diagram in 4.3. By flatness of D as a left B-module, this implies projectivity of N (cf. [8, 18.20]). (b)⇒(a). By the hypothesis, the functor HomD (N, −) : MD → MA preserves coequalisers and reflects isomorphisms. Thus HomD (N, −) : MD → M[C,−] is an equivalence by Theorem 5.8 and C is a projective right A-module by Corollary 5.3. (a)⇒(c). If HomD (N, −) : MD → M[C,−] is an equivalence, then so is its left adjoint −⊗[C,−] N . Thus N is a D-Galois bicomodule by 5.8. The functor −⊗[C,−] N : M[C,−] → MB is equal to the composite of the equivalence − ⊗[C,−] N : M[C,−] → MD and the forgetful functor MD → MB . The forgetful functor is faithful and also left exact by the flatness of the left B-module D. Thus the functor − ⊗[C,−] N : M[C,−] → MB is also faithful and left exact. (c)⇒(a). Since C is a projective right A-module, M[C,−] is abelian. Hence faithfulness of − ⊗[C,−] N : M[C,−] → MB implies that it reflects isomorphisms. Since it also preserves equalisers by assumption, it follows by Theorem 5.8 that − ⊗[C,−] N : M[C,−] → MD is an equivalence, with inverse HomD (N, −). The left B-module D is flat by Corollary 5.6.  In the rest of the section we study the particular case of a trivial B-coring D = B. That is, the situation when the category of contramodules of a coring C is equivalent to that of modules over a ring B. 5.10. Lemma. [12, Proposition 2.5] Let N be an (A, B)-bimodule which is finitely generated and projective as an A-module. Consider the comatrix coring C := N ⊗B ∗ N and denote by T the ring of endomorphisms of N as a left C-comodule. Then N ∼ = N ⊗B T via the right T -action on N . The next result may be seen as a counterpart to the Galois comodule structure theorem [8, 18.27], [30, Corollary 3.7]. 5.11. Theorem. Let N ∈ C M be a [C, −]-Galois comodule over an A-coring C, put T = EndC (N ) and assume T to be a B-ring for some ring B. Assume that N is a projective generator of right B-modules. Then the following hold. (1) − ⊗[C,−] N : M[C,−] → MB is an equivalence. (2) C is a projective right A-module. (3) N is a finitely generated and projective left A-module. (4) C is isomorphic to the comatrix A-coring N ⊗B ∗ N . (5) B is isomorphic to T . (6) If, in addition, C is a generator of right A-modules, then N is a faithfully flat left A-module. Proof. Assertions (1) and (2) are immediate by 5.9. (3) Since − ⊗[C,−] N is an equivalence, it has a left adjoint HomB (N, −) : MB → M[C,−] . The free functor HomA (C, −) has a left adjoint −⊗[C,−] C : M[C,−] → MA by 4.3. 22 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER Hence also the composite functor, that is naturally isomorphic to −⊗A N : MA → MB by 4.5(1), has a left adjoint. This proves that N is a finitely generated and projective left A-module. (4) By part (3), HomB (N, − ⊗A N ) ∼ = HomB (N, HomA (∗ N, −)) ∼ = HomA (N ⊗B ∗ N, −). Composing this natural isomorphism with the canonical monad morphism canN (at D = B), it yields a monad isomorphism HomA (C, −) ∼ = HomA (N ⊗B ∗ N, −). By ∗ Yoneda’s Lemma this proves C ∼ = N ⊗B N . (5) The composite of the forgetful functor HomT (T, −) : MT → MB and HomB (N, −) : MB → MA is naturally isomorphic to HomB (N, HomT (T, −)) ∼ = HomT (N ⊗B T, −) ∼ = HomT (N, −), where the last isomorphism follows by part (4) and Lemma 5.10. The forgetful functor MT → MB reflects isomorphisms. Since N is a generator in MB by assumption, the (fully faithful) functor HomB (N, −) : MB → MA reflects isomorphisms too. Hence also the composite HomT (N, −) : MT → MA reflects isomorphisms. The forgetful functor MT → MB has a right adjoint (the coinduction functor HomB (T, −)) hence it preserves coequalisers. Since N is a projective right B-module by assumption, HomB (N, −) : MB → MA preserves coequalisers too. Hence also the composite HomT (N, −) : MT → MA preserves coequalisers. The equivalence functor − ⊗[C,−] N : M[C,−] → MB factorises through − ⊗[C,−] N : M[C,−] → MT and the forgetful functor MT → MB . Thus the forgetful functor is full (and obviously faithful). This implies that − ⊗[C,−] N : M[C,−] → MT is fully faithful, hence the corresponding canonical monad morphism HomA (C, −) → HomT (N, − ⊗A N ), f 7→ [ n 7→ (f ⊗A IN ) ◦ N ̺(n)], is a natural isomorphism by Theorem 5.2. So we conclude by Theorem 5.8 that − ⊗[C,−] N : M[C,−] → MT is an equivalence and so is the forgetful functor MT → MB . This proves the isomorphism of algebras T ∼ = B. (6) N is a flat left A-module by part (3). Hence it suffices to show that, under the assumptions made, − ⊗A N : MA → MB is a faithful functor, so it reflects both monomorphisms and epimorphisms. Recall that, by 4.5(1), − ⊗A N : MA → MB is naturally isomorphic to the composite of the free functor HomA (C, −) : MA → M[C,−] and the equivalence − ⊗[C,−] N : M[C,−] → MB . By assumption, HomA (C, −) : MA → MA is faithful. Then also HomA (C, −) : MA → M[C,−] is faithful, what completes the proof.  Note that C is a generator of right A-modules (as required in Theorem 5.11 (6)), for example, provided the counit of C is an epimorphism. 6. Contramodules and entwining structures As recalled in 2.14, lifting of a monad F on a category A to a monad on the category AG for a comonad G, or lifting of a comonad G to a comonad on the category AF for a monad F, are both equivalent to the existence of a mixed distributive law (entwining) between F and G. Combining this general fact with properties of module categories, we obtain a description of entwinings between A-rings and A-corings (A MONADS AND COMONADS ON MODULE CATEGORIES 23 is an associative ring with unit). Recall that a (left) entwining map between an A-ring B and an A-coring C is an A-bimodule morphism ψ : B ⊗A C → C ⊗A B which respects (co)multiplications and (co)units. Similarly, (right) entwining maps λ : C ⊗A B → B ⊗A C are defined (e.g. [8, Chapter 5]). Note that a left entwining structure is the same as a mixed distributive law in the bicategory of algebras – bimodules – bimodule maps, in the same sense as distributive laws in any bicategory discussed in [28, Section 6]. A right entwining structure can be described as a mixed distributive law in the bicategory with opposite horizontal composition. The following theorem is a consequence of monoidal equivalences between the category of A-bimodules; the category of left adjoint endofunctors on the category of left A-modules; and the category of right adjoint endofunctors on the category of right A-modules, the latter considered with the opposite composition of natural transformations. 6.1. Entwining maps. For all A-rings B and A-corings C, the following assertions are equivalent. (a) There is an entwining map ψ : B ⊗A C → C ⊗A B; (b) the monad B ⊗A − on A M has a lifting to a monad on C M; (c) the comonad C ⊗A − on A M has a lifting to a comonad on B M; (d) the monad HomA (C, −) on MA has a lifting to a monad on MB ; (e) the comonad HomA (B, −) on MA has a lifting to a comonad on M[C,−] . Proof. (a)⇔(b) and (a)⇔(c). An entwining map ψ determines a mixed distributive law Ψ := ψ ⊗A − : B ⊗A C ⊗A − → C ⊗A B ⊗A −. Conversely, if Ψ : B ⊗A C ⊗A − → C ⊗A B ⊗A − is a mixed distributive law, then ψ := ΨA is an entwining map. (a)⇔(d) and (a)⇔(e). An entwining map ψ determines a mixed distributive law e Ψ: ∼ HomA (C, HomA (B, −)) ∼ = = HomA (B, HomA (C, −)) HomA (C ⊗A B, −) HomA (ψ,−) −→ HomA (B ⊗A C, −). e : HomA (C ⊗A On the other hand, by the Yoneda Lemma, any mixed distributive law Ψ B, −) → HomA (B ⊗A C, −) is of this form.  By [8, 18.28], part (c) of 6.1 implies that under the equivalent conditions of 6.1, C ⊗A B is a B-coring, cf. [8, 32.6]. Its contramodules can be described as follows. 6.2. [C ⊗A B, −]-modules. Let B be an A-ring and C an A-coring with an entwining map ψ : B ⊗A C → C ⊗A B. Then the following structures on a right B-module M are equivalent. (a) A module structure map ̺M : HomB (C ⊗A B, M ) → M ; P ψ (b) a B-linear module structure map αM : HomA (C, M ) → M (where
ψ, Pf b = fψ(− )b for f ∈ Hom(C, M ), b ∈ B, hence B-linearity means αM (f )b = αM f (− )bψ , P with notation ψ(b ⊗A c) = cψ ⊗A bψ ); (c) a module structure for the monad HomA (C, −) on MB ; (d) a comodule structure for the comonad HomA (B, −) on M[C,−] . 24 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER Proof. (a)⇔(b). The isomorphism HomB (C ⊗A B, M ) ∼ = HomA (C, M ) of right Amodules induces an isomorphism ξ : HomA (HomA (C, M ), M ) → HomA (HomB (C ⊗A B, M ), M ). As easily checked, ξ(αM ) belongs to HomB (HomB (C ⊗A B, M ), M ) if and only if αM satisfies the B-linearity condition in (b). Associativity and unitality of a [C ⊗A B, −]action ξ(αM ) are equivalent to analogous properties of the [C, −]-action αM . Equivalences (b)⇔(c) and (b)⇔(d) follow by 2.15 (cf. [32, 5.7]).  In light of 6.1, the following describes a special case of 2.16 and 2.17. 6.3. Distributive laws for rings and corings. Let B be an A-ring and C an A-coring over any ring A. (1) λ : C ⊗A B → B ⊗A C is an entwining map if and only if λ̃ : HomA (B, −) ⊗A C → HomA (B, − ⊗A C), f ⊗ c 7→ (f ⊗A IC ) ◦ λ(c ⊗ −), is a comonad distributive law. Then HomA (B, −) ⊗A C is a comonad on MA and the category of its comodules is isomorphic to the category of λ-bimodules (i.e. usual entwined modules, cf. 2.15). (2) ψ : B ⊗A C → C ⊗A B is an entwining map if and only if ψ̃ : HomA (C, −) ⊗A B → HomA (C, − ⊗A B), g ⊗ a 7→ (g ⊗A IB ) ◦ ψ(a ⊗ −), is a monad distributive law. Then HomA (C, − ⊗A B) is a monad on MA and the category of its modules is isomorphic to the category of ψ̃-bimodules (cf. 2.15). Note that for a commutative ring R, any R-algebra A and R-coalgebra C are entwined by the twist maps C ⊗R A → A ⊗R C and A ⊗R C → C ⊗R A. Applying 6.3 to these particular entwinings, we conclude that the canonical natural transformations HomR (A, −) ⊗R C → HomR (A, − ⊗R C), HomR (C, −) ⊗R A → HomR (C, − ⊗R A), f ⊗ c 7→ f (−) ⊗ c, g ⊗ a 7→ g(−) ⊗ a, and yield a comonad distributive law and a monad distributive law, respectively. 7. Bialgebras and bimodules There are many equivalent characterisations of bialgebras and Hopf algebras. A bialgebra over a commutative ring R can be seen as an R-module that is both an algebra and a coalgebra entwined in a certain way. In category theory terms, bialgebra is defined as an R-module such that the tensor functor − ⊗R B is a bimonad on MR . Associated to a bialgebra B, there is a category of Hopf modules, whose objects are B-modules with a compatible B-comodule structure. A Hopf algebra can be characterised as a bialgebra B such that the functor −⊗R B is an equivalence between the categories of R-modules and Hopf B-modules. In this section we supplement this description of bialgebras and Hopf algebras by the equivalent description in terms of properties of the Hom-functor [B, −], and hence in terms of contramodules. Throughout, R is a commutative ring. The unit element of a (bi)algebra B is denoted by 1B . For the coproduct ∆ of a bialgebra B, if applied to an element b ∈ B, we use Sweedler’s index notation ∆(b) = b1 ⊗ b2 , where implicit summation is understood. MONADS AND COMONADS ON MODULE CATEGORIES 25 7.1. Bialgebras. Let B be an R-module which is both an R-algebra µ : B ⊗R B → B, ι : R → B, and an R-coalgebra ∆ : B → B ⊗R B, ε : B → R. Based on the canonical twist tw : B ⊗R B → B ⊗R B, we obtain the R-module maps ψr = (IB ⊗R µ) ◦ (tw ⊗R IB ) ◦ (IB ⊗R ∆) : B ⊗R B → B ⊗R B, ψl = (µ ⊗R IB ) ◦ (IB ⊗R tw) ◦ (∆ ⊗R IB ) : B ⊗R B → B ⊗R B. Evaluated on elements, ψr (a ⊗ b) = b1 ⊗ ab2 and ψl (a ⊗ b) = a1 b ⊗ a2 . To make B a bialgebra, µ and ι must be coalgebra maps (equivalently, ∆ and ε are to be algebra maps) with respect to the obvious product and coproduct on B ⊗R B (induced by tw). The compatibility between multiplication and comultiplication can be expressed by commutativity of the diagram B ⊗R B µ / B ∆ B ⊗O R B / ∆⊗R IB µ⊗R IB  IB ⊗R ψr B ⊗R B ⊗R B B ⊗R B ⊗R B, / or, equivalently, by its symmetrical counterpart (IB ⊗R µ) ◦ (ψl ⊗R IB ) ◦ (IB ⊗R ∆) = ∆ ◦ µ. Given an R-bialgebra B, we sometimes write B when our focus is on the algebra structure and B when focussing on the coalgebra part. For a bialgebra B, both maps ψr and ψl are (right, respectively, left) entwining maps between the algebra B and the coalgebra B. Going to the functor level it turns out that ψr yields a mixed distributive law from the monad − ⊗R B to the comonad − ⊗R B (equivalently, from the comonad B ⊗R − to the monad B ⊗R −), while ψl induces a mixed distributive law from the monad B ⊗R − to the comonad B ⊗R − (equivalently, from the comonad − ⊗R B to the monad − ⊗R B). The entwining maps ψr : B ⊗R B → B ⊗R B and ψl : B ⊗R B → B ⊗R B determine B-corings B ⊗R B, denoted by B ⊗rR B and B ⊗lR B, respectively. The following is obtained by applying 2.15 to the mixed distributive law − ⊗R ψr from the monad − ⊗R B to the comonad − ⊗R B. 7.2. B-Hopf modules. Let B be an R-bialgebra and consider the B-coring B ⊗rR B. The following structures on a right B-module M are equivalent: (a) A right B ⊗rR B-comodule structure map ̺M : M → M ⊗B (B ⊗rR B); (b) a right B-linear B-comodule structure map αM : M → M ⊗R B, (where Blinearity means commutativity of the diagram M ⊗R B αM / M αM / M ⊗O R B αM ⊗R IB αM ⊗R IB  M ⊗R B ⊗R B IM ⊗R ψr / M ⊗R B ⊗R B, where αM : M ⊗R B → M denotes the B-action on M ); (c) a comodule structure for the comonad − ⊗R B on MB ; (d) a module structure for the monad − ⊗R B on MB . A right B-module M with these equivalent properties is called a B-Hopf module. Morphisms of B-Hopf modules are B ⊗rR B-comodule maps. Equivalently, they are 26 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER B-module as well as B-comodule maps. The category of right B-Hopf modules is B⊗rR B . denoted by MB B . By the above considerations, it is isomorphic to M Based on the mixed distributive law ψl ⊗R − from the monad B⊗R − to the comonad B ⊗R −, left B-Hopf modules are defined symmetrically. Note that a bialgebra B is both a left and a right B-Hopf module. From 6.2 we obtain: 7.3. [B, −]-Hopf modules. Let B be an R-bialgebra and consider the B-coring B ⊗lR B. Then the following structures on a right B-module M are equivalent. (a) A [B ⊗lR B, −]-module structure map ̺M : HomB (B ⊗lR B, M ) → M ; (b) a B-linear [B, −]-module structure
map αM : HomR (B, M ) → M P (i.e. αM (f )b = αM f (b1 −)b2 for f ∈ HomR (B, M ), b ∈ B); (c) a module structure for the monad HomR (B, −) on MB ; (d) a comodule structure for the comonad HomR (B, −) on M[B,−] . A right B-module M with these equivalent properties is called a [B, −]-Hopf module or right Hopf contramodule for B. Morphisms of [B, −]-Hopf modules are B ⊗lR Bcontramodule maps. Equivalently, they are B-module as well as B-contramodule [B,−] maps. The category of [B, −]-Hopf modules is denoted by M[B,−] . By the above considerations, it is isomorphic to M[B⊗lR B,−] . Based on ψr , left Hopf contramodules for B are defined symmetrically. Applying 6.3, the following alternative description of Hopf modules is obtained. 7.4. Distributive laws for bialgebras. Let B be an R-bialgebra. Then: (1) The entwining ψr in 7.1 induces a comonad distributive law X f ((−)1 ) ⊗ b(−)2 . HomR (B, −) ⊗R B → HomR (B, − ⊗R B), f ⊗ b 7→ Hence HomR (B, −) ⊗R B is a comonad on MR . The category of its comodules is isomorphic to the category of B-Hopf modules. (2) The entwining ψl in 7.1 induces a monad distributive law X f (b1 −) ⊗ b2 . HomR (B, −) ⊗R B → HomR (B, − ⊗R B), f ⊗ b 7→ Hence HomR (B, − ⊗R B) is a monad on MR . The category of its modules is isomorphic to the category of [B, −]-Hopf modules. 7.5. Hopf algebras. An R-bialgebra (H, µ, ι, ∆, ε) is said to be a Hopf algebra if there is an R-module map S : H → H, called the antipode, such that µ ◦ (IH ⊗R S) ◦ ∆ = ι ◦ ε = µ ◦ (S ⊗R IH ) ◦ ∆. If the antipode exists, then it is unique and it is an anti-algebra and anti-coalgebra map. The unit ι : R → H of an R-bialgebra (H, µ, ι, ∆, ε) determines the Sweedler Hcoring H ⊗R H; see [8, 25.1]. A bialgebra H is known to be a Hopf algebra if and only if the H-coring map X H ⊗R H → H ⊗rR H, a ⊗ b 7→ ab1 ⊗ b2 , MONADS AND COMONADS ON MODULE CATEGORIES 27 is an isomorphism, and if and only if the H-coring map X H ⊗R H → H ⊗lR H, a ⊗ b 7→ a1 ⊗ a2 b, is an isomorphism; see e.g. [8, 15.5]. Thus in particular, for a Hopf algebra H, the H-corings H ⊗rR H and H ⊗lR H are mutually isomorphic. 7.6. Hopf algebras and coseparability. Let H be an R-Hopf algebra. (1) The H-coring H ⊗rR H is coseparable. (2) The following functor is an equivalence: r r HomH⊗R H (H ⊗rR H, −) : MH⊗R H → M[H⊗rR H,−] . (3) The category of H-Hopf modules (in 7.2) and the category of [H, −]-Hopf modules (in 7.3) are equivalent. Proof. (1) For any R-bialgebra (H, µ, ι, ∆, ε), the inclusion ι : R → H is split by the R-(bi)module map ε. Consequently, the corresponding Sweedler coring H ⊗R H is coseparable; see [8, 26.9]. Since, for a Hopf algebra H, the Sweedler H-coring H ⊗R H is isomorphic to H ⊗rR H (see 7.5), the assertion (1) follows. (2) In view of (1), this is a special case of 4.8. r (3) The category of H-Hopf modules is isomorphic to MH⊗R H and the category of [H, −]-Hopf modules is isomorphic to M[H⊗lR H,−] . So the claim follows by coring isomorphism H ⊗rR H ∼  = H ⊗lR H in 7.5 and part (2). The final aim of this section is to characterise Hopf algebras via their induced (co)monads. The following notions were introduced in [32] and [20]. Note that these terms have different meanings in Moerdijk [21] and Bruguières-Virelizier [6]. 7.7. Bimonads and Hopf monads. A bimonad on a category A is a functor F : A → A with a monad structure F = (F, m, i) and a comonad structure F = (F, d, e) subject to the compatibility conditions (i) e is a monad morphism F → IA ; (ii) i is a comonad morphism IA → F ; (iii) there is a mixed distributive law Ψ : F F → F F , satisfying d ◦ m = F m ◦ ΨF ◦ F d. A bimonad (F, m, i, d, e) is called a Hopf monad if there exists a natural transformation S : F → F , called the antipode, such that m ◦ SF ◦ d = i ◦ e = m ◦ F S ◦ d. A class of examples of bimonads is provided by the following construction in [20, Proposition 6.3] (see also [13]). Let F be a functor A → A allowing a monad structure F = (F, m, i) as well as a comonad structure F = (F, d, e). Consider a double entwining τ , i.e. a natural transformation F F → F F , which is an entwining both in the sense F F → F F and also F F → F F . The functor F is called a τ -bimonad provided that the above conditions (i) and (ii) hold and in addition mF ◦ F F m ◦ F τ F ◦ dF F ◦ F d = d ◦ m. By [20, Proposition 6.3], a τ -bimonad F is a bimonad with respect to the mixed distributive law Ψ := mF ◦ F τ ◦ dF . 28 G. BÖHM, T. BRZEZIŃSKI, AND R. WISBAUER A τ -bimonad with an antipode is called a τ -Hopf monad. As described in [20], if a τ -bimonad F has a left or right adjoint G, then the mates under the adjunction of the structure maps of the monad and comonad F , equip G with a comonad and a monad structure, respectively. Moreover, the mate τ̄ of τ under the adjunction is a double entwining for G, and G is a τ̄ -bimonad. If F is a τ -Hopf monad, then G is a τ̄ -Hopf monad. 7.8. The bimonad − ⊗R B. For an R-bialgebra (B, µ, ι, ∆, ε), the functor − ⊗R B : MR → MR is a tw-bimonad, hence a bimonad with respect to the mixed distributive law (− ⊗R IB ⊗R µ) ◦ (− ⊗R tw ⊗R IB ) ◦ (− ⊗R IB ⊗R ∆) = − ⊗R ψr . By duality, HomR (B, −) is a tw-bimonad, with coproduct [µ, −] and counit [ι, −] in 3.2, product [∆, −] and unit [ε, −] in 3.5, where tw : HomR (B, HomR (B, −)) → HomR (B, HomR (B, −)) is given by switching the arguments. Thus HomR (B, −) : MR → MR is a bimonad with respect to the mixed distributive law HomR (ψl , −): ∼ HomR (B, HomR (B, −)) ∼ = = HomR (B, HomR (B, −)) HomR (B ⊗R B, −) HomR (ψl ,−) −→ HomR (B ⊗R B, −). A motivating example of a (tw-)Hopf monad in [20] is the functor − ⊗R H : MR → MR , induced by a Hopf algebra H. Summarising the preceding observations we obtain the following. 7.9. Characterisations of Hopf algebras. For an R-bialgebra (H, µ, ι, ∆, ε), the following assertions are equivalent. (a) H is a Hopf algebra; ∆⊗ I I⊗ µ R R H H ⊗R H ⊗R H −→ H ⊗R H is an isomorphism; (b) the map γ : H ⊗R H −→ (c) H is an H ⊗rR H-Galois right (equivalently, left) comodule; (d) H is an H ⊗lR H-Galois right (equivalently, left) comodule; (e) − ⊗R H is a tw-Hopf monad on MR ; (f) for the tw-bimonad [H, −] = HomR (H, −), the natural transformation [H,[µ,−]] [∆,[H,−]] [γ, −] : [H, [H, −]] −→ [H, [H, [H, −]]] −→ [H, [H, −]] is an isomorphism; (g) HomR (H, −) is a tw-Hopf monad on MR ; (h) − ⊗R H : MR → MH H is an equivalence; [H,−] (i) HomR (H, −) : MR → M[H,−] is an equivalence; (j) H is a Hom−,H (H ⊗rR H, −)-Galois left comodule (equivalently, a HomH,− (H ⊗rR H, −)-Galois right comodule); (k) H is a Hom−,H (H ⊗lR H, −)-Galois left comodule (equivalently, a HomH,− (H ⊗lR H, −)-Galois right comodule). MONADS AND COMONADS ON MODULE CATEGORIES 29 Proof. (a)-(d) and (h) are standard equivalent characterisations of Hopf algebras, see e.g. [8, 15.2 and 15.5]. (a)⇔(e)⇔(f)⇔(g) is proven in [20], (c)⇔(j) and (d)⇔(k) follow by 5.7, while (i)⇒(k) follows by 5.8 (a)⇒(b)(i). (h)⇒(i). There is a sequence of equivalences, [H,−] ∼ H⊗rR H ≃ M[H⊗r H,−] ∼ MH = M[H⊗lR H,−] ∼ = M[H,−] , H = M R cf. 7.2, 7.6, 7.5 and 7.3 (note that (h)⇒(a)). Combining this composite with the equivalence in part (h), we obtain an equivalence functor l [H,−] HomH⊗R H (H ⊗lR H, − ⊗R H) : MR → M[H,−] . We claim that the functor in part (i) is naturally isomorphic to this equivalence, hence it is an equivalence, too. The equivalence in part (h) gives rise to an R-module isomorphism r HomH⊗R H (H ⊗rR H, M ⊗R H) → HomR (H, M ), Ψ 7→ (IM ⊗R ε) ◦ Ψ (− ⊗R ι), for any R-module M , that is natural in M . Using the coring isomorphism H ⊗rR H ∼ = H ⊗lR H in 7.5, we can transfer it to a natural isomorphism l βM : HomH⊗R H (H ⊗lR H, M ⊗R H) → HomR (H, M ), Φ 7→ (IM ⊗R ε) ◦ Φ ◦ ∆. An easy computation shows that βM is a morphism of [H ⊗lR H, −]-modules, what completes the proof.  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Research Institute for Particle and Nuclear Physics, Budapest, H-1525 Budapest 114, P.O.B. 49, Hungary E-mail address: G.Bohm@rmki.kfki.hu Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. E-mail address: T.Brzezinski@swansea.ac.uk Department of Mathematics, Heinrich-Heine University, D-40225 Düsseldorf, Germany E-mail address: wisbauer@math.uni-duesseldorf.de