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Galois functors and generalised Hopf modules

As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the characterisation of opmonoidal monads on monoidal categories as right Hopf monads in the sense of Bruguieres and Virelizier. Hereby the central part is to describe conditions under which a comparison functor between the base category and the category of Hopf modules becomes an equivalence (Fundamental Theorem). For monoidal categories, Aguiar and Chase extended the setting by replacing the base category by a comodule category for some comonoid and considering a comparison functor to generalised Hopf modules. For duoidal categories, Bohm, Chen and Zhang investigated a comparison functor to the Hopf modules over a bimonoid induced by the two monoidal structures given in such categories. In both approaches fundamental theorems are proved and the purpose of this paper is to show that these can be derived from the theory of Galois functors.


Introduction
Bialgebras A over a commutative ring R induce an endofunctor A ⊗ R − on the category M R of R-modules which has a monad and a comonad structure subject to some compatibility conditions. To make the bialgebra A a Hopf algebra the comparison functor from M R to the category of Hopf modules M A A induced by A ⊗ R − has to be an equivalence (e.g. [5, 7.9]).
Since all these constructions are based on the tensor product in M R , one may try to extend the notions to monads T = (T, m, e) on (strictly) monoidal categories (V, ⊗, I). To ensure that the Eilenberg-Moore category V T is again monoidal, T has to be an opmonoidal monad (e.g. [11]). Such functors consist of two parts: the monad T and a comonad − ⊗ T (I) on V induced by the coalgebra T (I) which are related by a mixed distributive law (entwining) (e.g. [16,Section 5]). Then a comparison functor between V and the category of the entwined modules determined by this entwining (called right Hopf T -modules in [9,Section 4.2]) may be considered. [17,Theorem 4.7] gives a necessary and sufficient condition for this comparison functor to be an equivalence of categories.
In [16], an entwining of a monad T = (T, m, e) and a comonad G = (G, δ, ε) on any category A is considered, that is, a natural transformation λ : T G → GT subject to certain commutativity conditions (e.g. [19, 5.3]). Then the comonad G on A can be lifted to a comonad G and the λ-entwining modules are just the Gcomodules in A T (see 1.8). For a comparison functor K : A → (A T ) G one requires commutativity of the diagram where φ T denotes the free T -module functor and U G the forgetful G-comodule functor. In [16,17] conditions are given which make K an equivalence.
This setting comprises the opmonoidal monads outlined above and it also applies to the bimonads on arbitrary categories introduced in [19, 5.13], [15,Definition 4.1].
To subsume the generalised Hopf modules studied by Aguiar and Chase in [1], one has to add an adjunction L ⊣ R : B → A to the picture and observe that the resulting adjunction φ T L ⊣ RU T generates a comonad on A T . Now the results from [16] can be applied to the diagram This is outlined in Section 2 leading to the Fundamental Theorem of generalised Hopf modules from [1].
Having made this extension, also the A-Hopf modules of a bimonoid A in a duoidal category (D, •, I, * , J) and the related comparison functor considered by Böhm, Chen and Zhang in [6] can be handled in our setting: roughly speaking, for a bimonoid A, − • A defines a monad while − * A is a comonad on D and the two functors are related by an entwining. Now it is fairly obvious how our techniques apply and at the end of Section 3 we obtain the Fundamental Theorem for A-Hopf modules from [6]. for the corresponding forgetful-free adjunction; • A T for the Kleisli category for T (as a full subcategory of A T , e.g. [5]) and φ T ⊣ u T : A T → A for the corresponding Kleisli adjunction.
Dually, if G = (G, δ, ε) is a comonad on A, we write A G for the Eilenberg-Moore category of G-coalgebras and for the corresponding forgetful-cofree adjunction.

Comodule functors.
Consider an adjunction η, σ : F ⊣ R : A → B and a comonad G = (G, δ, ε) on A. The functor F : B → A is called a left G-comodule (e.g. [15,Section 3]) if there exists a natural transformation κ F : F → GF inducing commutativity of the diagrams

There exist bijective correspondences between
(ii) left G-comodule structures κ F : F → GF on F ; (iii) comonad morphisms t K : F R → G from the comonad generated by the adjunction F ⊣ R to G.
These bijections are constructed as follows: Given a functor K making the diagram (1.2(i)) commute, K(X) = (F (X), κ X ) for some morphism κ X : F (X) → GF (X) and the collection {κ b , b ∈ B} constitutes a natural transformation κ F : is a comonad morphisms from the comonad generated by F ⊣ R to the comonad G. On the other hand, for any comonad morphism t : F R → G, the composite This gives a bijection, natural in T , between left T -module structures on R : B → A and the functors where η : 1 → RF is the unit of the adjunction F ⊣ R, is a monad morphism from T to the monad on A generated by the adjunction F ⊣ R.
Expressing the dual of [13,Theorem 4.4] in the present situation gives:

) is an equivalence of categories if and only if
(i) the functor R is monadic and (ii) R is a T -Galois module functor.
1.6. Mixed distributive laws. Let T = (T, m, e) be a monad and G = (G, δ, ε) a comonad on a category A.
A mixed distributive law or entwining from T to G is a natural transformation λ : T G → T G with certain commutative diagrams (e.g. [20], [19, 5.3

]).
A lifting of G to A T is a comonad G = ( G, δ, ε) on A T for which GU T = U T G, U T δ = δU T and U T ε = εU T .
The following is a version of [20, Theorem 2.2]: 1.7. Theorem. Let T = (T, m, e) be a monad and G = (G, δ, ε) a comonad on a category A. Then there is a one-to-one correspondence between To obtain a lifting G from a distributive law λ, one defines for (X, h) ∈ A T , G(X, h) as the T -module Conversely, if one has a lifting comonad G, one defines λ : 1.8. λ-bimodules. We write A G T (λ) (or just A G T when λ is understood) for the category whose objects are triples (X, h, θ), where (X, h) ∈ A T and (X, θ) ∈ A G , with commuting diagram (e.g. [19, 5.7]) The assignment (X, h, θ) → ((X, h), θ) yields an isomorphism of categories . Generalised Galois functor. With the data as given in Theorem 1.7, let λ : T G → GT be a mixed distributive law. Given an adjunction ν, ς : Write G ′ for the comonad on the category A T generated by the adjunction The following proposition gives a necessary and sufficient condition for the functor φ T L to be G-Galois (generalising [17, Proposition 2.10]).
1.11. Proposition. In the setting of 1.9, φ T L is G-Galois if and only if the natural transformation t K φ T is an isomorphism.
Proof. One direction is clear, so suppose that t K φ T is an isomorphism. Let κ : φ T L → Gφ T L be the left G-comodule structure on φ T L corresponding to the diagram (1.1). Then, by 1.
and, using U T G = GU T , rewrite it as By [10,Lemma 2.19], if U T t K φ T is an isomorphism, then U T t K is so. But since U T is conservative, t K is an isomorphism, too. This completes the proof.
⊔ ⊓ In view of Theorem 1.10, it is desirable to find sufficient conditions for the composite φ T L to be comonadic. The next proposition gives two such conditions. 1.12. Proposition. In the setting of 1.9, suppose that A is Cauchy complete and L is comonadic. Then the composite φ T L is comonadic under any of the conditions (i) the unit e : I → T is a split monomorphism, i.e. there is a natural transformation e ′ : T → I with e ′ e = 1; (ii) the monad T is of effective descent type (φ T : A → A T is comonadic) and A has and LR and LRLR preserve equalisers of coreflexive T -split pairs.

Proof. (i) Since
• A is Cauchy complete, • e : 1 → T is a split monomorphism, and • e can be seen as the unit of the adjunction φ T ⊣ U T , it follows from [14, Proposition 3.14] that any φ T -split pair is part of a split equaliser in A, and thus the functor φ T creates equalisers of φ T -split pairs as split equalisers in A, from which it follows that φ T L is comonadic whenever L is so.
(ii) Since the functor L is assumed to be comonadic, to say that A has and LR and LRLR preserve equalisers of coreflexive T -split pairs is to say that the functor L creates equalisers of those pairs whose image under L is part of a T -split equaliser (see, for example, [8,Proposition 4.3.2]). Since A has and T preserves equalisers of coreflexive T -split pairs if and only if A has and φ T preserves equalisers of coreflexive φ T -split pairs ([14, Proposition 3.11]), and since T is of effective descent type by hypothesis, it follows that B has and the composite φ T L preserves equalisers of coreflexive φ T L-split pairs. Using now the fact that φ T L, being a composite of two conservative functors, is conservative, the result follows from the dual of Beck's monadicity theorem (see [12]).

⊔ ⊓
For later use (in 3.7) we prove the following technical observation.
1.13. Proposition. Let A be Cauchy complete and L ⊣ R : A → B an adjunction whose unit is a split monomorphism. Then, in any commutative (up to isomorphism) diagram (ii) any coreflexive F -split pair of morphisms has a split equaliser in A; (iii) the functor F is comonadic if and only if it has a right adjoint.
Proof. (i) Since A is Cauchy complete and since the unit of the adjunction is a split monomorphism, the functor R is comonadic (e.g., [14,Proposition 3.16]) and, in particular, conservative. This implies -since KF is isomorphic to R -that F is conservative, too.
Then the morphisms F (f ) and F (g) have a split equaliser in E, so that the pair (F (f ), F (g)) is contractible (see [12]). Since contractible pairs, being equationally defined, are preserved by any functors, the pair (R(f ), R(g)), being isomorphic to the pair (KF (f ), KF (g)), is also contractible. But since A is Cauchy complete and since the unit of the adjunction is a split monomorphism (which just means that the functor R is 1 A -separable), it follows from [14,Proposition 3.8] that the pair (f, g) is contractible, too. Then, A being Cauchy complete, f and g have a split equaliser (e.g. [3]) and this equaliser is clearly preserved by F .
(iii) follows from the fact that split equalisers are preserved by any functor. ⊔ ⊓

Generalised Hopf modules
In [1], Aguiar and Chase studied generalised Hopf modules in monoidal categories and proved a Fundamental Theorem for them. In this section we show how this result can be obtained as a special case of our approach. We first recall elementary facts about modules and comodules in a monoidal category (e.g. [13], [18]).

2.1.
Monoids and comonoids in monoidal categories. Let (V, ⊗, I, a, l, r) be a monoidal category, where a, l, r are the associativity, left identity, and right identity isomorphisms for the monoidal structure on V.
A monoid in V consists of an object A of V endowed with a multiplication m : A ⊗ A → A and a unit morphism e : I → A such that the usual identity and associative conditions are satisfied. A monoid morphism f : A → A ′ is a morphism in V preserving m and e. The category of monoids in V is denoted by Mon(V).
For any monoid A in V, the left A-modules are the objects of a category A V.
A morphism of (A, B)-bimodules is a morphism in V which is a morphism of both the left A-modules and right B-modules. Write A V B for the corresponding category.
Comonoids and (left, right, bi-) comodules in V can be defined as monoids and left (right, bi-) modules in the opposite monoidal category (V op , ⊗, I, a −1 , l −1 , r −1 ). The resulting categories are denoted by Comon(V), C V, V C and C V C ′ , C and C ′ being comonoids in V.

Tensor product of modules. If
is defined as the object part of the coequaliser (if this exists) Given another left A-module (W ′ , ρ W ′ ) for which V ⊗ A W ′ exists, and a mor- • the left square commutes by naturality of a, and • the middle square commutes because f is a morphism of left A-modules; from this one sees that there ia a unique morphism Moreover, if these functors also preserve the equaliser defining V ⊗ A W ′ , then V ⊗ A f also becomes a left B-module morphism.
Recall (for example, from [18]) that the forgetful functor (1) for any N ∈ V A , the tensor product N ⊗ A M exists and is isomorphic to N⊗V ; Proof. Everything follows from 2.2 and the fact that the equaliser defining the tensor product N ⊗ A M is split and thus is preserved by any functor.
⊔ ⊓ 2.4. Remark. The full subcategory of A V generated by the left A-modules of the form A ⊗ V , V ∈ V, is just the Kleisli category V A T of the monad A T (e.g. [5, 2.4]). Hence Lemma 2.3 may be alternatively stated as follows: Let N ∈ B V A . Then, for any X ∈ V A T , the tensor product N⊗ A X exists and has the structure of a left B-module. So the assignment X → N ⊗ A X yields a functor
In [11], an opmonoidal monad on a monoidal category (V, ⊗, I) is defined as a monad T = (T, m, e) on V such that the functor T and the natural transformations m and e are opmonoidal. Such monads are also called bimonads (e.g. in [1, The basic property of opmonoidal monads T is that they lead to a monoidal structure on the Eilenberg-Moore category V T of T -modules in such a way that the forgetful functor U T : V T → V is strictly monoidal. Explicitly, for T -modules (V, h) and (W, g), the tensor product (V, h) ⊗ (W, g) is given by and the unit object of V T is the T -module (I, ξ : T (I) → I). The unitary and associativity isomorphisms for V T are inherited from V.

V-categories. A left V-category is a category A equipped with a bifunctor
called the action of V on A, and invertible natural transformations called the associativity and unit isomorphisms, respectively, satisfying two coherence axioms (see Bénabou [4]). Note that V has a canonical (left) action on itself, given by taking V ⋄ W = V ⊗ W , α = a, and λ = l.
Given a left V-category A and a monoid (A, e A , m A ) in V, one has a monad T l A on A defined on any X ∈ A by , and we write A A for the Eilenberg-Moore category A T l A of T l A -modules. For the canonical left action of V on itself, A A is just the category A V of (left) A-modules.
Dually, for any V-coalgebra (C, ε C , δ C ), the endofunctor C ⋄ − : A → A is the functor-part of a comonad G l C on A and one has the corresponding Eilenberg-Moore category C A = A G l C ; for A = V this is just the category C V of (left) C-comodules. We sometimes write A φ and C φ for the functors φ T l A and φ G l C , respectively. Symmetrically, one has the monad T r A = − ⋄ A (resp. comonad G r C = − ⋄ C) on A, the corresponding Eilenberg-Moore category A A (resp. A C ) of T r A -modules (resp. G r C -comodules), and the functor φ A : A → A A (resp. φ C : A → A C ). There is a left action of the monoidal category V T (with the monoidal structure from 2.5) on the category A S : given a T -module (V, f ) and an S-module (X, h), • Z = ((Z, σ), δ, ε) is a T -module-comonoid.
Since Z is a comonoid in the monoidal category V T and since V T acts from the left on A S , one has the A S -comonad G l Z . Moreover, since Z 0 = U T (Z) is a comonoid in the monoidal category V, one has the A-comonad G l Z0 , and it is not hard to check that G l Z is a lifting of G l Z0 to A S . It follows from Theorem 1.7 that there is a mixed distributive law λ from the A-monad S to the A-comonad G l Z0 . 2.10. Proposition. With the data considered in 2.8, λ is the composite Proof. By 2.8, for any (X, h) ∈ A T , In the diagram the rectangle commutes by naturality of χ, while m S ·e S = 1 implies commutativity of the triangle; it follows that λ X = (σ ⋄ S(X)) · χ Z,X . ⊔ ⊓ 2.11. Generalised Hopf modules. 2.12. Assumption 2. We henceforth suppose that (C, δ, ε) is a comonoid in V and that Z = T (C) is the corresponding T -module-comonoid (see 2.6).

2.13.
Lemma. In the situation considered above, the assignment yields a functor yielding commutativity of the diagram Proof. To show that (X, ϑ) ∈ Z0 A is to show commutativity of the diagrams S(X) where ε = ξ · T (ε) and δ = ω C,C · T (δ) are the counit and the comultiplication for the V T -module-comonoid Z = T (C) (see 2.6). In the diagram • the triangle commutes since (X, θ) ∈ C A, • the top rectangle commutes by naturality of χ, • the bottom rectangle commutes since S is a T -comodule-monad (see diagram (3.6) in [1]); it follows that diagram (I) is commutative. To show that (II) is also commutative, consider the diagram (2) and (3) commute by naturality of χ, • rectangle (4) commutes since S is a T -comodule-monad (see diagram (3.5) in [1]); therefore the outer square (and hence (II)) is commutative. Thus, (X, ϑ) ∈ Z0 A, and since (S(X), (m S ) X ) ∈ A S , in order to show that (S(X), (m S ) X , ϑ) ∈ Z (A S ), we need commutativity of the diagram since the rectangle commutes by naturality of m S , while the trapezoid commutes since S is a T -comodule-monad (see diagram (3.10) in [1]) the outer paths commute, too.

⊔ ⊓
As an immediate consequence we obtain from 1.2: Write G for the comonad on the category A S generated by the adjunction it follows from 1.2 that the morphism we obtain by Corollary 2.14 that κ C φ(X) is the composite In the diagram ' ' P P P P P P P P P P P P is an isomorphism for all X ∈ A, or, equivalently, φ S C U : C A → A S is a G l Z -Galois comodule functor. In view of Proposition 1.12(i), the preceding theorem implies: 2.18. Theorem. Assume that A is Cauchy complete and that e S : I → S is a split monomorphism. Under the assumptions 2.9, 2.12, the functor K : is an equivalence of categories if and only if the Since the functor S is conservative and the category A admits -and the functor S preserves -equalisers of coreflexive φ S -split pairs, it follows from the dual of Beck's monadicity theorem (see [12]) that the functor φ S : A → A S is comonadic, or equivalently, the monad S is of effective descent type. Since any φ S -split pair is automatically S-split, we may apply Proposition 1.12(ii) to deduce that the functor φ S C U : C A → A S is comonadic. The result now follows from Theorem 2.17.

Bimonoids in duoidal categories
In [2], Aguiar and Mahajan generalised bialgebras over fields to bimonoids in duoidal categories, that is, categories with two monoidal structures * and •. Any object A in such a category induces endofunctors − * A and − • A and for A being a bimonoid these functors have to be a comonad and a monad, respectively, related by a mixed distributive law ([2, Definition 6.25], compare [15,Proposition 6.3]). In [6], Böhm, Chen and Zhang studied which structures are required to define Hopf monoids in such categories. In this section we outline how their Fundamental Theorem for Hopf modules (see [6,Section 3.4]) can be seen as special case of the results in the Sections 1 and 2.
Recall from [2] that duoidal categories D are equipped with two monoidal structures (D, •, I) and (D, * , J), along with a natural transformation A morphism of bimonoids is a morphism of the underlying monoids and comonoids.
Recall [2,Proposition 6.27] that the tensor units I and J carry a unique bimonoid structure and that the morphism τ : I → J is a morphism of bimonoids. . It is shown in [7] that T r A is an opmonoidal monad on the monoidal category (D, * , J), with the structure morphisms We write D A A for the category ( corresponding to the mixed distributive law λ. This is called the category of A-Hopf modules in [6, Section 3]. Thus, an A-Hopf module is a triple (X, h, ϑ) such that (X, h : X • A → A) is a right A-module in the monoidal category (D, •, I), (X, ϑ : X → X * A) is a right A-comodule in the monoidal category (D, * , J) with commutative diagram Since (I, ∆, τ ) is a comonoid in the monoidal category (D, * , J), one has the category D I of I-comodules on this monoidal category. Recall that D I is the Eilenberg-Moore category of G r I -comodules. Now Lemma 2.13 implies: Write G for the comonad on the category D A generated by the adjunction

3.4.
Proposition. For any (X, h) ∈ D A , the (X, h)-component of the comonad morphism t K : G → G r A induced by the diagram in 3.3, is the composite is commutative, implying that the outer diagram is also commutative. Since D is assumed to be Cauchy complete, so also is D I . Now apply Proposition 1.13 to conclude that the composite φ A U I is conservative, and that any coreflexive (φ A U I )split pair of morphisms has a split equaliser in D I .
Next, since ι φ A = φ A , where ι : D T l A → D A is the canonical embedding, the composite φ A U I is conservative if and only if ι φ A U I = φ A U I is conservative, and a pair of morphisms in D I is φ A U I -split if and only if it is φ A U I -split. Thus, φ A U I is conservative and any φ A U I -split pair of morphisms has a split equaliser in D I . It follows -since the composite φ A U I admits as a right adjoint the composite φ I U A -that φ A U I is comonadic.