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Entwining Structures in Monoidal Catrgories

Interpreting entwining structures as special instances of J. Beck's distributive law, the concept of entwining module can be generalized for the setting of arbitrary monoidal category. In this paper, we use the distributive law formalism to extend in this setting basic properties of entwining modules.


Introduction
The important notion of entwining structures has been introduced by T. Brzeziński and S. Majid in [4]. An entwining structure (over a commutative ring K) consists of a K-algebra A, a K-coalgebra C and a certain K-homomorphism λ : C ⊗ K A → A ⊗ K C satisfying some axioms. Associated to λ there is the category M C A (λ) of entwining modules whose objects are at the same time A-modules and C-comodules, with compatibility relation given by λ.
The algebra A can be identified with the monad T = − ⊗ K A : Mod K → Mod K whose Eilenberg-Moore category of algebras, (Mod K ) T , is (isomorphic to) the category of right Amodules. Similarly, C can be identified with the comonad G = − ⊗ K C : Mod K → Mod K , and the corresponding Eilenberg-Moore category of coalgebras with the category of Ccomodules. It turns out that to give an entwining structure C ⊗ K A → A ⊗ K C is to give a mixed distributive law T G → GT from the monad T to the comonad G in the sense of J. Beck [2], which are in bijective corresondence with liftings (or extensions) G of the comonad G to the category (Mod K ) T ; or, equivalently, liftings T of the monad T to the category (Mod K ) G . Moreover, the categories M C A (λ) , ((Mod K ) T ) G and ((Mod K ) G ) T are isomorphic. Thus, the (mixed) distributive law formalism can be used to study entwining structures and the corresponding category of modules. In this article -based on this formalism-we extend in the context of monoidal categories some of basic results on entwining structures that appear in the literature (see, for example, [5], [6], [11]).
The paper is organized as follows. After recalling the notion of Beck's mixed distributive law and the basic facts about it, we define in Section 3 an entwining structure in any monoidal category. In Section 4, we prove some categorical results that are needed in the next section, but may also be of independent interest. Finally, in the last section we present our main results.

Mixed distributive laws
Let T = (T, η, µ) be a monad and G = (G, ε, δ) a comonad on a category A. A mixed distributive law from T = (T, η, µ) to G = (G, ε, δ) is a natural transformation λ : TG → GT for which the diagrams Given a monad T = (T, η, µ) on A, write A T for the Eilenberg-Moore category of Talgebras, and write F T ⊣ U T : A T → A for the corresponding forgetful-free adjunction. Dually, if G = (G, ε, δ) is comonad on A, then write A G for the category of G-coalgebras, and write F G ⊣ U G : A G → A for the corresponding forgetful-cofree adjunction.
• IfḠ = (Ḡ,ε,δ) is a comonad on A T extending the comonad G = (G, ε, δ), then the corresponding distributive law λ : TG → GT is given by • IfT = (T ,η,μ) is a monad on A G extending T = (T, η, µ), then the corresponding mixed distributive law is given by It follows from this theorem that if

Entwining structures in monoidal categories
Let V = (V, ⊗, I) be a monoidal category with coequalizers such that the tensor product preserves the coequalizer in both variables. Then for all algebras A = (A, e A , m A ) and B = (B, e B , m B ) and all M ∈ V A , N ∈ A V B and P ∈ B V, the tensor product M ⊗ A N exists and the canonical morphism (M ⊗ A N)⊗ B P → M ⊗ A (M ⊗ B P ) is an isomorphism. Using MacLane's coherence theorem (see, [9], XI.5), we may assume without loss of generality that V is strict. It is well known that every algebra and that V T A is (isomorphic to) the category V A of right A-modules.
Dually, if C = (C, ε C , δ C , ) is a coalgebra (=comonoid) in V, then one defines a comonad G C on V by and V G C is (isomorphic to) the category V C of right C-comodules.
Quite obviously, if λ is a mixed distributive law from T A to G C , then the morphism makes the following diagrams commutative: Conversely, if λ ′ : C ⊗ A → A ⊗ C is a morphism for which the above diagrams commute, then the natural transformation is a mixed distributive law from the monad T A to the comonad G C . It is easy to see that λ ′ = (− ⊗ λ ′ ) I . When I is a regular generator in V and the tensor product preserves all colimits in both variables, it is not hard to show that λ ≃ − ⊗ λ I . When this is the case, then the correspondences λ → λ I and λ ′ → − ⊗ λ ′ are inverses of each other.

3.1.
Definition. An entwining structure (C, A, λ) consists of an algebra A = (A, e A , m A ) and a coalgebra C = (C, ε C , δ C ) in V and a morphism λ : is a mixed distributive law from the monad T A to the comonad G C .
Let be (C, A, λ) be an entwining structure and letḠ = (Ḡ,ε,δ) be the comonad on V A that extends G = G C . Then we know that, for any (V, In particular, since (A, m A ) ∈ V A , A ⊗ C is a right A-module with right action Clearly they are also morphisms of left A-modules with the obvious left A-module structures arising from the multi- . Conversely, let A = (A, e A , m A ) be an algebra and C = (C, ε C , δ C ) a coalgebra in V, and suppose that A ⊗ C has the structure ξ A⊗C of a right A-module such that the triple is an A-coring. Then it is easy to see that the comonad G A⊗C on V A extends the comonad G C on V, and thus defines an entwining structure λ A⊗C : Then there exists a bijection between right A-module structures ξ A⊗C making (A⊗C, m A ⊗ C, ξ A⊗C ) an A-bimodule for which the triple (1) is an A-coring and entwining structures (C, A, λ), given by: with inverse given by

Some categorical results
Let G = (G, ε, δ) be a comonad on a category A, and let U G : A G → A be the forgetful functor. Fix a functor F : B → A, and consider a functorF : It is proved in [7] that: 4.1. Theorem. Suppose that F has a right adjoint R : A → B with unit η : 1 → F U and counit ε : F U → 1. Then the composite is a morphism from the comonad G ′ = (F U, ε, F ηU) generated by the adjunction η, ε : yields a one to one correspondence between functorsF : B → A G making the diagram (2) commutative and morphisms of comonads tF : G ′ → G.
Write β U for the composite U ηU / / UF U U tF / / UG .
LetF : B → A G be a functor making (2) commutative and let tF : G ′ → G be the corresponding morphism of comonads. Consider the following composition • A tF is the functor induced by the morphism of comonads tF : G ′ → G.

Lemma. The diagram
is commutative.
and since by naturality ofᾱ F , the diagram Thus which just means that A tF · K G ′ =F .
We are now ready to prove the following 4.4. Theorem. Let G be a comonad on a category A, η, ε : F ⊣ U : B → A an adjunction andF : B → A G a functor with U G ·F = F . Then the following are equivalent: (i) The functorF is an equivalence.
(ii) The functor F is comonadic and the morphism of comonads Proof. Suppose thatF is an equivalence of categories. Then F is isomorphic to the comonadic functor U G and thus is comonadic. Hence the comparison functor K G ′ : B → A G ′ is an equivalence and it follows from the commutative diagram (4) that A tF is also an equivalence, and since the diagram Suppose now that tF : G ′ → G is an isomorphism of comonads and F is comonadic. Then • K G ′ is an equivalence, since F is comonadic.
• A tF is an equivalence, since tF is an isomorphism.
And it now follows from the commutative diagram (4) thatF is also an equivalence. Thus (ii) =⇒ (i). This completes the proof of the theorem.

Remark.
In [8], J. Gómez-Torrecillas has proved thatF is an equivalence of categories iff tF is an isomorphism of comonads, F is conservative, and for any (X, x) ∈ A G , F preserves the equalizer of the pair of parallel morphisms When tF is an isomorphism of comonads, to say that F preserves the equalizer of the pair of morphisms (5) is to say that F preserves the equalizer of the pair of morphisms which we can rewrite as Since tF is an isomorphism of comonds, A tF is an equivalence of categories, and thus each object (X, where (X, x) ∈ A G . It follows that when tF is an isomorphism of comonds, to say that F preserves the equalizer of (5) for each (X, x) ∈ A G is to say that F preserves the equalizer of (6) for each (X, x ′ ) ∈ A G ′ . Thus, when tF is an isomorphism of comonds,F is an equivalence of categories iff F is conservative and preserves the equalizer of (6) for each (X, x ′ ) ∈ A G ′ , which according to (the dual of) Beck's theorem (see [9]), is to say that the functor F is comonadic. Hence our theorem 4.4 is equivalent to Theorem 1.7 of [8].

Some applications
Let (C, A, λ) be an entwining structure in a monoidal category V = (V, ⊗, I), and let g : I → C be a group-like element of C. (Recall that a morphism g : I → C is said to be a group-like element of C if the following diagrams

Proposition.
If C has a group-like element g : I → C, then A is a right C-comodule through the morphism

Proof. Consider the diagram
The triangle is commutative by (1) of the definition of g and the square is commutative by the definition of λ (see the second commutative diagram in the definition of entwining structures). Now, we have to show that the following diagram is also commutative, which it is since

Proposition. A C = (A, g A ) C is an algebra in V and i A : A C → A is an algebra morphism.
Proof. Consider the diagram Since is a natural transformation, the diagram is commutative. Similarly, since e A ⊗− : 1 V = I ⊗− → C ⊗− is a natural transformation, the following diagram is also commutative: Thus there exists a unique morphism e A : I → A C for which i A · e A C = e A . Since is commutative by naturality of g ⊗ −; • Thus the morphism m A · (i A ⊗ i A ) equalizes the morphisms λ · (g ⊗ A) and A ⊗ g, and hence there is a unique morphism commutes. It is now straightforward to show that the triple (A C , e A C , m A C ) is an algebra in V; moreover, the triangle of the diagram (7) and the diagram (8) show that i A is an algebra morphism.

Proposition. (A, m
it only remains to show that the following diagram is commutative: By the definition of g A , we can rewrite it as But this diagram is commutative, since • the middle square commutes because of naturality of g ⊗ −; • the right square commutes because of the definition of λ.
The algebra morphism i A : A C → A makes A an A C -A C -bimodule and thus induces the extension-of-scalars functor and the forgetful functor which is right adjoint to F i A . The corresponding comonad on V A makes A ⊗ A C A into an A-coring with the following counit and comultiplication: (where q is the canonical morphism) and We write A ⊗ A C A for this A-coring.

Lemma.
For any X ∈ V A C , the triple is an object of the category V C A (λ).
Moreover, by (9), the following diagram . The lemma shows that the assignment → V A is the underlying functor. It now follows from Theorem 3.1 that the composite We write can for this morphism. We say that A is (C, g)-Galois if can is an isomorphism of A-corings.
Applying Theorem 4.4 the commutative diagram we get: 5.5. Theorem. Let (C, A, λ) be an entwining structure, and let g : I → C be a group-like element of C. Then the functorF • faithfully flat, if the functor − ⊗ A M : is conservative and flat (equivalently, preserves and reflects equalizers); 5.6. Theorem. Let (C, A, λ) be an entwining structure, and let g : I → C be a group-like element of C. If C is flat, then the following are equivalent is an equivalence of categories.
(ii) A is (C, g)-Galois and A C A is faithfully flat.
Proof. Since any left adjoint functor that is conservative and preserves equalizers is comonadic by a simple and well-known application (of the dual of) Beck's theorem, one direction is clear from Therem 5.5; so suppose thatF is an equivalence of categories. Then, by Theorem 4.5, A is (C, g)-Galois and the functor F i A is comonadic. Since any comonadic functor is conservative, F i A is also conservative. Thus, it only remains to show that A C A is flat. Since C is flat by our assumption, A (A ⊗ C) is also flat. It follows that the underlying functor of the comonad G (A⊗C) λ on V A preserves equalizers. We recall (for example, from [3]) that if G = (G, ε G , δ G ) is a comonad on a category A, and if A has some type of limits preserved by G, then the category A G has the same type of limits and these are preserved by the underlying functor U G : A G → A. Thus the functor U (A⊗C) λ : V A (A⊗C) λ → V A preserves equalizers, and sinceF is an equivalence of categories, the functor F i A = −⊗ A C A also preserves equalizers, which just means that A C A is flat. This completes the proof.
From now on we suppose at all times that our V is a strict braided monoidal category with braiding σ X,Y : X ⊗ Y → Y ⊗ X. Then the tensor product of two (co)algebras in V is again a (co)algebra; the multiplication m A⊗B and the unit e A⊗B of the tensor product of two algebras A = (A, e A , m A ) and B = (B, e B , m B ) are given through e A⊗B = e A ⊗ e B . Recall that for any bialgebra H, the category V H is monoidal: The tensor product (X, δ X ) ⊗ (Y, δ Y ) of two right H-comodules (X, δ X ) and (Y, δ Y ) is their tensor product X ⊗ Y in V with the coaction The unit object for this tensor product is I with trivial H-comodule structure e H : I → H. is also an object of V H , and it follows from Theorem 1.1 that λ (A,α A ) is the composite Consider now the following diagram

Since in this diagram
• the triangle commutes because ε H is the counit for δ H ; • the left square commutes by naturality of σ; • the right square commutes because − ⊗ − is a bifunctor, it follows that Note that the morphism e H : I → H is a group-like element for the coalgebra H = (H, ε H , δ H ). Proof. We have to show that It now follows from Proposition 5.3 that Recall that for any (X, α X ) ∈ V H , the algebra X H = (X, α X ) H is the equalizer of the morphisms is an equivalence of categories iff the extension-of-scalars functor is comonadic and A is H-Galois (in the sense that the canonical morphism is an isomorphism).
is an equivalence of categories.  Proof. It follows from Propositions 5.16 and 5.17 that H is H-Galois, and according to Theorem 5.12, the functor V → V H H is an equivalence iff the functor − ⊗ H : V → VH is comonadic. But since the morphism e H : I → H is a split monomorphism in V, the unit of the adjunction F e H ⊣ U e H is a split monomorphism, and it follows from 3.16 of [10] that F e H is comonadic. This completes the proof.