Theory and Applications of Categories, Vol. 26, No. 10, 2012, pp. 281–303.

NOTES ON BIMONADS AND HOPF MONADS

BACHUKI MESABLISHVILI AND ROBERT WISBAUER

Abstract. For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a pre-Hopf monad by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the pre-Hopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered.

1. Introduction

The classical definitions of bialgebras and Hopf algebras over fields (or rings) heavily depend on constructions based on the tensor product. This may have been one of the reasons why first generalisations of this notions were formulated for monoidal categories, or even autonomous monoidal categories when the properties of finite dimensional Hopf algebras were in the focus. This was also the starting point for the definitions of Hopf monads by I. Moerdijk in [13]. McCrudden [8] suggested to call these functors opmonoidal monads and A. Bruguières and A. Virelizier just called them bimonads in [3, Section 2.3].

To be more precise, such a bimonad on a monoidal category (𝕍, ⊗, 𝕀) is a monad T = (T, m, e) on 𝕍 endowed with natural transformations χ : T ⊗ → T ⊗ T and a morphism θ : T (𝕀) → 𝕀 subject to certain (compatibility) conditions. These allow to define left and right fusion operators by

\begin{matrix}H_{V,W}^{l} : (T (V) ⊗ m_{W})χ_{V,T (W)} : T (V ⊗ T (W)) ⟶ T (V) ⊗ T (W), \\ H_{V,W}^{r} : (m_{V} ⊗ T (W))χ_{T (V),W} : T (T (V) ⊗ W) ⟶ T (V) ⊗ T (W). \end{matrix}

As a general form of the Fundamental Theorem for Hopf algebras it is described in [2, Theorem 4.6] under which conditions the opmonoidal monads induce an equivalence between the base (autonomous monoidal) category and the category of related bimodules.

It was observed in [11] (see also [1]) that the notions around Hopf algebras can be formulated for any category 𝔸 without referring to tensor products. For a bimonad on 𝔸 one requires simply a monad and a comonad structure whose compatibility is essentially expressed by distributive laws (e.g. [11, Definition 4.1]).

Received by the editors 2011-12-22 and, in revised form, 2012-03-03.

Transmitted by Giuseppe Rosolini. Published on 2012-06-05.

2000 Mathematics Subject Classification: 18A40, 16T15, 18C20.

Key words and phrases: Opmonoidal functors, bimonads, Hopf monads, Galois entwinings.

\begin{matrix}c \\ ◯ \end{matrix} Bachuki Mesablishvili and Robert Wisbauer, 2012. Permission to copy for private use granted.

As pointed out in [11, Section 2.2], the opmonoidal monads yield special cases of the entwining of a monad with a comonad on any category: Hereby the monad T is entwined with the comonad G_{T(𝕀)} = − ⊗ T (𝕀). In [12, Theorem 5.11] the above mentioned [2, Theorem 4.6] is formulated in terms of entwining functors.

In [3] an opmonoidal monad (bimonad) is called a Hopf monad provided the left and right fusion operators are isomorphisms and it is called a left (resp. right) pre-Hopf monad if, for any V ∈ 𝕍, the morphisms H_{𝕀,V}^{l} (resp. H_{V,𝕀}^{r}) is invertible.

In this paper we show that the right pre-Hopf monads T are just those for which the related entwining is G_{T(𝕀)}-Galois in the sense of [11, 3.13]. This leads to an improved version of [3, Theorem 6.11] which describes when a pre-Hopf monad on 𝕍 induces an equivalence between 𝕍 and the category of left Hopf T-modules (see Theorem 4.7).

In Section 1 we recall some basic notions and can use [3, Lemma 2.19] to improve some of our own results on Galois entwinings (see Theorem 2.12).

This is applied in Section 2 to find new properties of a bimonad in the sense of [11] to make it a Hopf monad, provided the base category is Cauchy complete.

In Section 3 opmonoidal monads T on (𝕍, ⊗, 𝕀) are investigated. In this case T (𝕀) is a comonoid in 𝕍 and we have an entwining between T and G_{T(𝕀)} = − ⊗ T (𝕀). As mentioned above, the main result in this section is Theorem 2.12 which tells us when pre-Hopf monads induce an equivalence between 𝕍 and 𝕍_{T}^{G_{T(𝕀})}. We also observe (in 4.2) that for any 𝕍-comonoid C = (C, δ, ε), T (C) also allows for a 𝕍-comonoid structure provided C allows for a group-like morphism g : 𝕀 → C. In this case we get functors from 𝕍 to 𝕍_{T}^{G_{T(C})} and the question arises under which conditions these induce an equivalence. It is shown in Theorem 4.8 that this is only the case if g : 𝕀 → C is an (comonad) isomorphism.

In the final section we consider applications to cartesian monoidal categories and provide examples of pre-Hopf functors for which the related comparison functor is not an equivalence.

2. Preliminaries

For a monad T = (T, m, e) on a category 𝔸, we write 𝔸_{T} for the Eilenberg-Moore category of T-modules and write

η_{T} , ε_{T} : ϕ_{T} ⊣ U_{T} : 𝔸_{T} → 𝔸

for the corresponding forgetful-free adjunction. Dually, if G = (G, δ, ε) is a comonad on 𝔸, we denote by 𝔸^{G} the Eilenberg-Moore category of G-comodules and by

η^{G}, ε^{G} : U^{G} ⊣ ϕ^{G} : 𝔸 → 𝔸^{G}

the corresponding forgetful-cofree adjunction.

For convenience we recall some notions and results from [12, Section 3].

2.1. Module functors. For a monad T = (T, m, e) on 𝔸, a (left) T-module consists of a functor R : 𝔹 → 𝔸, equipped with a natural transformation α : T R → R satisfying α · eR = 1_{R} and α · mR = α · T α.

According to [4, Proposition II.1.1], if (R, α) is a T-module, then the assignment

b ⟼ (R(b), α_{b})

extends uniquely to a functor \overline{R} : 𝔹 → 𝔸_{T} with U_{T} \overline{R} = R. This gives a bijection, natural in T, between left T-module structures on R : 𝔹 → 𝔸 and functors \overline{R} : 𝔹 → 𝔸_{T} with U_{T} \overline{R} = R.

It is also shown in [4] that, for any T-module (R : 𝔹 → 𝔸, α) admitting a left adjoint functor F : 𝔸 → 𝔹, the composite

t\overline{R} : \xymatrix{T \ar[r]^{T η} & T RF \ar[r]^{αF} & RF} ,

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.

2.2. Definition. ([1, 2.19]) A left T-module R : 𝔹 → 𝔸 with a left adjoint F : 𝔸 → 𝔹 is said to be T-Galois if the corresponding morphism t\overline{R} : T → RF of monads on 𝔸 is an isomorphism.

Expressing the dual of [10, Theorem 4.4] in the present situation gives:

2.3. Proposition. The functor \overline{R} is an equivalence of categories if and only if the functor R is T-Galois and monadic.

2.4. Comodule functors. Given a comonad G = (G, δ, ε) on 𝔸, a left G-comodule is a functor F : 𝔹 → 𝔸 equipped with a natural transformation β : F → GF satisfying εF · β = 1_{F} and δF · β = Gβ · β.

A left G-comodule structure on F : 𝔹 → 𝔸 is equivalent to the existence of a functor \overline{F} : 𝔹 → 𝔸^{G} (dual to [4, Proposition II.1.1]) with F = U^{G}\overline{F}.

If a G-comodule (F, β) admits a right adjoint R : 𝔸 → 𝔹, with counit σ : F R → 1, then there is a comonad morphism

\xymatrix{t\overline{F} : F R \ar[r]^{βR} & GF R \ar[r]^{Gσ} & G}

from the comonad generated by the adjunction F ⊣ R to the comonad G.

2.5. Definition. ([11, Definition 3.5]) A left G-comodule F : 𝔹 → 𝔸 with a right adjoint R : 𝔸 → 𝔹 is said to be G-Galois if the corresponding morphism t\overline{F} : F R → G of comonads on 𝔸 is an isomorphism.

Now [5, Theorem 2.7] (also [10, Theorem 4.4]) can be rephrased as follows:

2.6. Proposition. The functor \overline{F} is an equivalence of categories if and only if the functor F is G-Galois and comonadic.

Recall [12, Definition 1.19]:

2.7. Definition. Let T = (T, m, e) be a monad and G = (G, δ, ε) a comonad on 𝔸. If (G, α : T G → G) is a left T-module, then we say that (G, α) is T-Galois, if the composite

\xymatrix{γ^{G} : T G \ar[r]^{T δ} & T GG \ar[r]^{αG} & GG}

is an isomorphism.

Dually, if (T, β : T → GT) is a left G-comodule, then (T, β) is G-Galois, if the composite

\xymatrix{γ_{T} : T T \ar[r]^{βT} & GT T \ar[r]^{Gm} & GT}

is an isomorphism

2.8. Entwinings. Recall (for example, from [15]) that an entwining or mixed distributive law from a monad T = (T, m, e) to a comonad G = (G, δ, ε) on a category 𝔸 is a natural transformation λ : T G → T G with certain commutative diagrams (e.g. [14, 5.3]).

It is well-known (see [15]) that the following structures are in bijective correspondence:

• entwinings λ : T G → GT;

• comonads Ĝ = (Ĝ, δ̂, ε̂) on 𝔸_{T} that extend G in the sense that

U_{T} Ĝ = GU_{T}, U_{T} δ̂ = δU_{T} and U_{T} ε̂ = εU_{T};

• monads T̂ = (T̂, m̂, ê) on 𝔸^{G} that extend T in the sense that

U^{G}T̂ = T U^{G}, U^{G}m̂ = mU^{G} and U^{G}ê = eU^{G}.

For any entwining λ : T G → GT, (a, h_{a}) ∈ 𝔸_{T} and (a, θ_{a}) ∈ 𝔸^{G} (e.g. [14, Section 5]),

Ĝ(a, h_{a}) = (G(a), G(h_{a}) · λ_{a}), δ̂_{(a,h_{a})} = δ_{a}, ε̂_{(a,h_{a})} = ε_{a},
T̂(a, θ_{a}) = (T (a), λ_{a} · T (θ_{a})), m̂_{(a,θ_{a})} = m_{a}, ê_{(a,θ_{a})} = e_{a}.

We write 𝔸_{T}^{G}(λ) (or just 𝔸_{T}^{G} when λ is understood) for the category whose objects are triples (a, h_{a}, θ_{a}), where (a, h_{a}) ∈ 𝔸_{T} and (a, θ_{a}) ∈ 𝔸^{G}, with commuting diagram

\xymatrix{T (a) \ar[r]^{h_{a}} \ar[d]_{T (θ_{a})} & a \ar[r]^{θ_{a}} & G(a) \\ T G(a) \ar[rr]_{λ_{a}} &  & GT (a). \ar[u]_{G(h_{a})}}

The assignments (a, h_{a}, θ_{a}) → ((a, h_{a}), θ_{a})) and ((a, h_{a}), θ_{a})) → ((a, θ_{a}), h_{a}) yield isomorphisms of categories

𝔸_{T}^{G}(λ) ≃ (𝔸_{T})Ĝ ≃ (𝔸^{G})T̂.

We fix now an entwining λ : T G → GT and let K : 𝔸 → (𝔸^{G})T̂ be a functor satisfying ϕ^{G} = UT̂K. Writing α_{K} : T̂ϕ^{G} → ϕ^{G} for the corresponding T̂-module structure on ϕ^{G} (see 2.1), the natural transformation

U^{G}(α_{K}) : T G = T U^{G}ϕ^{G} = U^{G}T̂ϕ^{G} ⟶ U^{G}ϕ^{G} = G

provides a left T-module structure on G (see [12, Section 2]).

Similarly, if K : 𝔸 → (𝔸_{T})Ĝ is a functor with ϕ_{T} = U ĜK, then the natural transformation

U_{T} (β_{K}) : T = U_{T} ϕ_{T} ⟶ GT = GU_{T} ϕ_{T} = U_{T} Ĝϕ_{T} ,

where β_{K} : ϕ_{T} → Ĝϕ_{T} is the corresponding Ĝ-comodule structure on ϕ_{T} (see 2.4), induces a G-comodule structure on T (see again [12, Section 2]).

The following part of [3, Lemma 2.19] is of use for our investigation.

2.9. Lemma. Let τ : F U_{T} → F′U_{T} be a natural transformation, where F, F′ : 𝔸 → 𝔹 are arbitrary functors. If the natural transformation

τϕ_{T} : F T = F U_{T} ϕ_{T} ⟶ F′U_{T} ϕ_{T} = F′T

is an isomorphism, then so is τ.

2.10. Proposition. Suppose K : 𝔸 → (𝔸_{T})Ĝ to be a functor with U ĜK = ϕ_{T} and denote by β_{K} : ϕ_{T} → Ĝϕ_{T} the corresponding Ĝ-comodule structure on ϕ_{T}. Then (ϕ_{T} , β_{K}) is Ĝ-Galois if and only if (T, U_{T} (β_{K})) is G-Galois.

Proof. By 2.5, (ϕ_{T} , β_{K}) is Ĝ-Galois if the comonad morphism t_{K} : ϕ_{T} U_{T} → Ĝ, which is the composite

ϕ_{T} U_{T} ⟶^{β_{K}U_{T}} Ĝϕ_{T} U_{T} ⟶^{Ĝε^{T}} Ĝ,

is an isomorphism, while, by 2.7, (T, U_{T} (β_{K})) is G-Galois if the composite

\xymatrix{γ_{T} : T T \ar[r]^{U_{T} β_{K}T} & GT T \ar[r]^{Gm} & GT}

is an isomorphism. So, we have to show that t_{K} is an isomorphism if and only if γ_{T} is so. Since U_{T} Ĝ = GU_{T}, the natural transformation

U_{T} t_{K} : U_{T} ϕ_{T} U_{T} ⟶^{U_{T} β_{K}U_{T}} U_{T} Ĝϕ_{T} U_{T} ⟶^{U_{T} Ĝε^{T}} U_{T} Ĝ

can be rewritten as

T U_{T} ⟶^{U_{T} β_{K}U_{T}} GU_{T} ϕ_{T} U_{T} ⟶^{GU_{T} ε^{T}} GU_{T} .

Then U_{T} t_{K}ϕ_{T} is the composite

T T = T U_{T} ϕ_{T} ⟶^{U_{T} β_{K}U_{T} ϕ_{T}} GU_{T} ϕ_{T} U_{T} ϕ_{T} ⟶^{GU_{T} ε^{T} ϕ_{T}} GU_{T} ϕ_{T} ,

and since U_{T} ε^{T} ϕ_{T} = m : T T = U_{T} ϕ_{T} U_{T} ϕ_{T} → U_{T} ϕ_{T} = T, it follows that U_{T} t_{K}ϕ_{T} is just γ_{T} . Now, if t_{K} is an isomorphism, it is then clear that γ_{T} = U_{T} (t_{K})ϕ_{T} is also an isomorphism. Conversely, if γ_{T} is an isomorphism, then by Lemma 2.9, U_{T} t_{K} is also an isomorphism. But since U_{T} is conservative, t_{K} is an isomorphism too. This completes the proof.

Dually, one has

2.11. Proposition. Suppose that K : 𝔸 → (𝔸^{G})T̂ is a functor with UT̂K = ϕ^{G} and let α_{K} : T̂ϕ^{G} → ϕ^{G} be the corresponding T̂-module structure on ϕ^{G}. Then (ϕ^{G}, α_{K}) is T̂-Galois if and only if (G, U^{G}(α_{K})) is T-Galois.

In view of Propositions 2.10 and 2.11, we get from Propositions 2.3 and 2.6:

2.12. Theorem. In the situation of Proposition 2.10, the functor K : 𝔸 → (𝔸_{T})Ĝ is an equivalence of categories if and only if (T, U_{T} (β_{K})) is G-Galois and the monad T is of effective descent type (i.e. the functor ϕ_{T} : 𝔸 → 𝔸_{T} is comonadic.)

Dually, in the situation of Proposition 2.11, the functor K : 𝔸 → (𝔸^{G})T̂ is an equivalence if and only if (G, U^{G}(α_{K})) is T-Galois and the comonad G is of effective codescent type (i.e. the functor ϕ^{G} : 𝔸 → 𝔸^{G} is monadic).

2.13. Galois entwinings. Let T = (T, m, e) be a monad and G = (G, δ, ε) a comonad on a category 𝔸 with an entwining λ : T G → GT. If G has a group-like morphism g : 1 → G (in the sense of [12, Definition 3.1]), then T has two left G-comodule structures given by

gT : T → GT and \xymatrix{g̃ : T \ar[r]^{T g} & T G \ar[r]^{λ} & GT} ,

and it was shown in [12] that the equaliser (T^{g}, i) of these natural transformations admits the structure of a monad in such a way that i : T^{g} → T becomes a monad morphism. We write i^{∗} : 𝔸_{T} → 𝔸_{T^{g}} for the functor that takes an arbitrary T-algebra (a, h_{a}) ∈ 𝔸_{T} to the T^{g}-algebra (a, h_{a} · i_{a}) ∈ 𝔸_{T^{g}}. When the category 𝔸_{T} admits coequalisers of reflexive pairs (which is certainly the case if 𝔸 has coequalisers of reflexive pairs and the functor T preserves them), i^{∗} has a left adjoint i_{∗} : 𝔸_{T^{g}} → 𝔸_{T}. In this case, according to the results of [12], there is a comparison functor \overline{i} : 𝔸_{T^{g}} → (𝔸_{T})Ĝ yielding commutativity of the diagram

\xymatrix{𝔸 \ar[rr]^{K_{g,G}} \ar[dr]_{ϕ_{T}} \ar[r]^{ϕ_{T^{g}}} & 𝔸_{T^{g}} \ar[d]^{i_{∗}} \ar[r]^{\overline{i}} & (𝔸_{T})Ĝ \ar[dl]^{U Ĝ} \\  & 𝔸_{T}}  (1)

where U Ĝ : (𝔸_{T})Ĝ → 𝔸_{T} is the evident forgetful functor and K_{g,G} : 𝔸 → (𝔸_{T})Ĝ is the functor that takes a ∈ 𝔸 to ((T (a), m_{a}), g̃_{a}) ∈ (𝔸_{T})Ĝ (see [12, Section 3]).

Let us write G̃ for the comonad generated by the adjunction i^{∗} ⊣ i_{∗} and write

• S_{K_{g,G}} : U_{T} ϕ_{T} → Ĝ for the comonad morphism corresponding to the outer diagram in (1),

• S_{ϕ_{T^{g}}} : U_{T} ϕ_{T} → G̃ for the comonad morphism corresponding to the left triangle in • and S\overline{i} : G̃ → Ĝ for the comonad morphism corresponding to the right triangle in

(1) that exists according to [12, Proposition 1.20].

2.14. Definition. [12] Under the circumstances above, we call (T, G, λ, g) a Galois entwining if the comonad morphism S\overline{i} : G̃ → Ĝ is an isomorphism, or, equivalently, the functor i_{∗} is Ĝ-Galois. In this case g : 1 → G is said to be a Galois group-like morphism.

2.15. Theorem. [12] Let λ : T G → GT be an entwining from a monad T to a comonad G on a category 𝔸. Suppose that g : 1 → G is a group-like morphism such that the corresponding functor i^{∗} : 𝔸_{T} → 𝔸_{T^{g}} admits a left adjoint functor i_{∗} : 𝔸_{T^{g}} → 𝔸_{T}. Then the comparison functor \overline{i} : 𝔸_{T^{g}} → (𝔸_{T})Ĝ is an equivalence of categories if and only if (T, G, λ, g) is a Galois entwining and the functor i_{∗} is comonadic.

3. Bimonads

The preceding results allow to formulate new conditions which turn bimonads into Hopf monads. Recall from [11, Definition 4.1] that a bimonad H on any category 𝔸 is an endofunctor H : 𝔸 → 𝔸 with a monad structure \underline{H} = (H, m, e), a comonad structure \overline{H} = (H, δ, ε), and an entwining λ : HH → HH from the monad \underline{H} to the comonad \overline{H} inducing commutativity of the diagrams

\xymatrix{HH \ar[r]^{εH} \ar[r]_{Hε} \ar[d]_{m} & H \ar[d]^{ε} & 1 \ar[r]^{e} \ar[d]_{e} & H \ar[d]^{δ} & 1 \ar[r]^{e} \ar[dr]_{=} & H \ar[d]^{ε} \\ H \ar[r]^{ε} & 1, & H \ar[r]^{eH} \ar[r]_{He} & HH, &  & 1,}

\xymatrix{HH \ar[r]^{m} \ar[d]_{Hδ} & H \ar[r]^{δ} & HH \\ HHH \ar[rr]_{λH} &  & HHH. \ar[u]_{Hm}}

Given a bimonad H, one has the comparison functor

K_{H} : 𝔸 → 𝔸_{H}^{H} = 𝔸\overline{H} ; \underline{H}(λ), a ↦ (H(a), m_{a}, δ_{a})

with commutative diagrams

\xymatrix{𝔸 \ar[r]^{K_{H}} \ar[dr]_{ϕ\underline{H}} & 𝔸_{H}^{H} ≃ (𝔸\underline{H})ˆ\overline{H} \ar[d]^{U ˆ\overline{H}} & 𝔸 \ar[r]^{K_{H}} \ar[dr]_{ϕ\overline{H}} & 𝔸_{H}^{H} ≃ (𝔸\overline{H})ˆ\underline{H} \ar[d]^{U ˆ\underline{H}} \\  & 𝔸\underline{H}, &  & 𝔸\overline{H}.}

Writing K\underline{H} (resp. K\overline{H}) for the composite 𝔸 ⟶^{K_{H}} 𝔸_{H}^{H} ≃ (𝔸\underline{H})ˆ\overline{H} (resp. 𝔸 ⟶^{K_{H}} 𝔸_{H}^{H} ≃ (𝔸\overline{H})ˆ\underline{H}) and writing α_{K\underline{H}} (resp. α_{K\overline{H}}) for the ˆ \overline{H}-comodule (resp. ˆ \underline{H}-module) structure on ϕ\underline{H} (resp. ϕ\overline{H}) that exists by 2.4 (resp. 2.1), we know from [11, 4.3] that U\underline{H}(α_{K\underline{H}}) = δ : H → HH and that U\overline{H}(α_{K\overline{H}}) = m : HH → H. It then follows from 2.7 that γ\underline{H} : \underline{H} \underline{H} → \overline{H}\underline{H} is the composite

HH ⟶^{δH} HHH ⟶^{Hm} HH,

while γ\overline{H} : \underline{H}\overline{H} → \overline{H} \overline{H} is the composite

HH ⟶^{Hδ} HHH ⟶^{mH} HH.

Employing the notions considered above we have the following list of

3.1. Characterisations of Hopf monads. For a bimonad H on a Cauchy complete category 𝔸, the following are equivalent:

(a) (ϕ\underline{H}, α_{K\underline{H}}) is ˆ \overline{H}-Galois, i.e., the composite t_{K\underline{H}} : ϕ\underline{H}U\underline{H} ⟶^{α_{K\underline{H}} U\underline{H}} ˆ \overline{H}ϕ\underline{H}U\underline{H} ⟶^{ˆ\overline{H}ε\underline{H}} ˆ \overline{H} is an isomorphism;

(b) (ϕ\overline{H}, α_{K\overline{H}}) is ˆ \underline{H}-Galois, i.e., the composite t_{K\overline{H}} : ˆ \underline{H} ⟶^{ˆ\underline{H}η\overline{H}} ˆ \underline{H}ϕ\overline{H}U\overline{H} ⟶^{α_{K\overline{H}} U\overline{H}} ϕ\overline{H}U\overline{H} is an isomorphism;

(c) the unit e : 1 → H is a Galois group-like morphism;

(d) the functor K_{H} : 𝔸 → 𝔸_{H}^{H} (hence also K\underline{H} : 𝔸 → (𝔸\underline{H})ˆ\overline{H} and K\overline{H} : 𝔸 → (𝔸\overline{H})ˆ\underline{H}) is an equivalence of categories;

(e) (H, m) is \overline{H}-Galois, i.e., γ\underline{H} : HH ⟶^{δH} HHH ⟶^{Hm} HH is an isomorphism;

(f) (H, δ) is \underline{H}-Galois, i.e., γ\overline{H} : HH ⟶^{Hδ} HHH ⟶^{mH} HH is an isomorphism;

(g) H has an antipode, i.e., there exists a natural transformation S : H → H with

m · HS · δ = e · ε = m · SH · δ.

Proof. (a), (c) and (d) are equivalent by [12, 4.2], while (e), (f) and (g) are equivalent by [11, 5.5]. Moreover, (a)⇔(e) follows by Proposition 2.10 and (b)⇔(f) by Proposition 2.11.

3.2. Example. Let (𝕍, τ) be a lax braided monoidal category (see, for example, [3]) and A = (A, m, e, δ, ε) a bialgebra in 𝕍. We write H for the endofunctor A ⊗ − : 𝕍 → 𝕍. It is easy to verify directly, using the axioms of lax braidings, that the natural transformation \overline{τ} = τ_{A} ⊗ − : HH → HH is a local prebraiding (in the sense of [11]) and that

(H, \overline{m}, \overline{e}, \overline{δ}, \overline{ε}),

where \overline{m} = m ⊗ −, \overline{e} = e ⊗ −, \overline{δ} = δ ⊗ − and \overline{ε} = ε ⊗ −, is a τ-bimonad on 𝕍. Then, according to [11, Section 6], the composite τ̃ = \overline{m}H · H\overline{τ} · \overline{δ}H is an entwining from the monad (H, \overline{m}, \overline{e}) to the comonad (H, \overline{δ}, \overline{ε}) that makes (H, \overline{m}, \overline{e}, \overline{δ}, \overline{ε}) a bimonad on 𝕍. Writing 𝕍_{A}^{A} for the category 𝕍\overline{H} ; \underline{H}(τ̃), we get from Theorem 3.1 the following generalisation of [11, Theorem 6.12]:

3.3. Proposition. Let (𝕍, τ) be a lax braided category such that 𝕍 is Cauchy complete. If A is a bialgebra in 𝕍, then the comparison functor

K : 𝕍 → 𝕍_{A}^{A}, V ↦ (A ⊗ V, m ⊗ V, δ ⊗ V),

is an equivalence of categories if and only if A is a Hopf algebra, that is, A has an antipode.

4. Opmonoidal Monads

4.1. Pre-Hopf monads. Recall (for example, from [8]) that an opmonoidal functor from a monoidal category (𝕍, ⊗, 𝕀) to a monoidal category (𝕍′, ⊗′, 𝕀′) is a triple (S, χ, θ), where S : 𝕍 → 𝕍′ is a functor, χ : S⊗ → S ⊗′ S is a natural transformation, and θ : S(𝕀) → 𝕀′ is a morphism that are compatible with the tensor structures. Note that opmonoidal functors S take 𝕍-comonoids (i.e. comonoids in 𝕍) into 𝕍′-comonoids in the sense that if C = (C, δ, ε) is a 𝕍-comonoid, then the triple S(C) = (S(C), χ_{C,C} · S(δ), θ · S(ε)) is a 𝕍′-comonoid.

Recall also (again from [8]) that an opmonoidal monad on a monoidal category (𝕍, ⊗, 𝕀) is a monad T = (T, m, e) on the category 𝕍 whose functor part T is an opmonoidal endofunctor together with natural transformations

χ_{V,W} : T (V ⊗ W) → T (V) ⊗ T (W) for V, W ∈ 𝕍

and a morphism θ : T (𝕀) → 𝕀 that are compatible with the monad structure.

For example, it was pointed out in [2] that any bialgebra A = (A, μ, η, δ, ε) in a braided monoidal category (𝕍, ⊗, 𝕀) with braiding τ_{V,W} : V ⊗ W → W ⊗ V gives rise to an opmonoidal 𝕍-monad A ⊗ −, where the natural transformation χ_{V,W} : A ⊗ V ⊗ W → A ⊗ V ⊗ A ⊗ W is the composite

\xymatrix{A ⊗ V ⊗ W \ar[r]^{δ⊗V ⊗W} & A ⊗ A ⊗ V ⊗ W \ar[r]^{A⊗τ_{A,V} ⊗W} & A ⊗ V ⊗ A ⊗ W ,}

while θ : A → 𝕀 is just ε.

From now on we shall assume (actually without loss of generality by the coherence theorem in [7]) that all our monoidal categories are strict.

According to [3], an opmonoidal monad T = (T, m, e) on the monoidal category (𝕍, ⊗, 𝕀) is left pre-Hopf if, for any object V of 𝕍, the composite

H_{𝕀,V}^{l} : T T (V) = T (𝕀 ⊗ T (V)) ⟶^{χ_{𝕀,T (V )}} T (𝕀) ⊗ T T (V) ⟶^{T (𝕀)⊗m_{V}} T (𝕀) ⊗ T (V)

is an isomorphism, and T is right pre-Hopf provided

H_{V,𝕀}^{r} : T T (V) = T (T (V) ⊗ 𝕀) ⟶^{χ_{T (V ),𝕀}} T T (V) ⊗ T (𝕀) ⟶^{m_{V} ⊗T (𝕀)} T (V) ⊗ T (𝕀)

is an isomorphism. T is called a pre-Hopf monad if it is both left and right pre-Hopf. For for any (V, h_{V}) ∈ 𝕍_{T} and W ∈ 𝕍, consider the morphisms

ℍ_{V,W}^{r} : T (V ⊗ W) ⟶^{χ_{V,W}} T (V) ⊗ T (W) ⟶^{h_{V} ⊗T (W)} V ⊗ T (W),

and for any V ∈ 𝕍 and (W, h_{W}) ∈ 𝕍_{T}, define

ℍ_{V,W}^{l} : T (V ⊗ W) ⟶^{χ_{V,W}} T (V) ⊗ T (W) ⟶^{T (V)⊗h_{W}} T (V) ⊗ W.

It is shown in [3] that, for any V ∈ 𝕍, H_{−,V}^{r} (resp. H_{V,−}^{l}) is an isomorphism if and only if ℍ_{−,V}^{r} (resp. ℍ_{V,−}^{l}) is so. In particular, T is right (resp. left) pre-Hopf monad if and only if for any (V, h_{V}) ∈ 𝕍_{T}, the morphism ℍ_{V,𝕀}^{r} (resp. ℍ_{𝕀,V}^{l}) is an isomorphism.

4.2. Entwined modules. Let (𝕍, ⊗, 𝕀) be a monoidal category and let T = (T, m, e) be an opmonoidal monad on 𝕍. As the functor T is opmonoidal, for any 𝕍-comonoid C = (C, δ, ε), the triple T (C) = (T (C), χ_{C,C} · T (δ), θ_{𝕀} · T (ε)) is also a 𝕍-comonoid. In particular, the triple T (𝕀) = (T (𝕀), χ_{𝕀, 𝕀}, θ) is a 𝕍-comonoid corresponding to the trivial 𝕍-comonoid I = (𝕀, 1_{𝕀}, 1_{𝕀}). Given a 𝕍-comonoid C, we write G_{C} for the comonad on 𝕍 whose functor part is G_{C} = − ⊗ C.

The compatibility axioms for T ensure that the natural transformation

λ_{−}^{C} := H_{−,C}^{l} = (T (−) ⊗ m_{C}) · χ_{−, T (C)} : T (− ⊗ T (C)) → T (−) ⊗ T (C)

is a mixed distributive law (entwining) from the monad T to the comonad G_{T (C)} and the diagrams in 2.8 come out as

\xymatrix{ & V⊗ T (C) \ar[dl]_{e_{V ⊗T (C)}} \ar[d]^{e_{V} ⊗T (C)} & T (V⊗T (C)) \ar[r]^{T (V ⊗T (ε))} \ar[d]_{λ_{V,T (C)}^{C}} & T (V⊗T (𝕀)) \ar[dr]^{T (V ⊗θ)} \\ T (V⊗ T (C)) \ar[r]_{λ_{V,T (C)}^{C}} & T (V)⊗T (C), & T (V)⊗T (C) \ar[r]_{T (V)⊗T (ε)} & T (V)⊗T (𝕀) \ar[r]_{T (V)⊗θ} & T (V),}

\xymatrix{T (V ⊗ T (C)) \ar[dd]_{λ_{V,T (C)}^{C}} \ar[r]^{T (V ⊗T (δ))} & T (V ⊗ T (C ⊗ C)) \ar[r]^{T (V ⊗χ_{C,C})} & T (V ⊗ T (C) ⊗ T (C)) \ar[d]^{λ_{V ⊗T (C),T (C)}^{C}} \\  &  & T (V ⊗ T (C)) ⊗ T (C) \ar[d]^{λ_{V,T (C)}^{C}⊗T (C)} \\ T (V) ⊗ T (C) \ar[r]^{T (V)⊗T (δ)} & T (V) ⊗ T (C ⊗ C) \ar[r]^{T (V)⊗χ_{C,C}} & T (V) ⊗ T (C) ⊗ T (C),}

\xymatrix{T (T (V ⊗ T (C))) \ar[rr]^{T (λ_{V,T (C))}^{C}} \ar[dr]_{m_{V ⊗T (C)}} &  & T (T (V) ⊗ T (C)) \ar[r]^{λ_{T (V ),T (C)}^{C}} & T T (V) ⊗ T (C) \ar[d]^{m_{V} ⊗T (C)} \\  & T (V ⊗ T (C)) \ar[rr]^{λ_{V,T (C)}^{C}} &  & T (V) ⊗ T (C).}

The entwined T (C)-modules are objects V ∈ 𝕍 with a T -module structure h : T (V) → V and a T (C)-comodule structure ρ : V → V ⊗ T (C) inducing commutativity of the diagram

\xymatrix{T (V) \ar[r]^{h} \ar[d]_{T (ρ)} & V \ar[r]^{ρ} & V ⊗ T (C) \\ T (V ⊗ T (C)) \ar[r]_{χ_{V,T (C)}} & T (V) ⊗ T T (C) \ar[r]_{T (V)⊗m_{C}} & T (V) ⊗ T (C). \ar[u]_{h⊗T (C)}}

They form a category in an obvious way which we denote by 𝕍_{T}^{T (C)}. It is clear that 𝕍_{T}^{T (C)} is just the category 𝕍_{T}^{G_{T (C)}}(λ_{C}) = (𝕍_{T}){ \atop G_{T (C)}}.

When C = I is the trivial 𝕍-comonad, the entwined T (I)-modules are named right Hopf T -modules in [2, Section 4.2] (also [3, 6.5]).

There is another description of the category 𝕍_{T}^{T (C)}. Since T is opmonoidal, 𝕍_{T} is a monoidal category, and the functor ϕ_{T} : 𝕍 → 𝕍_{T} is also opmonoidal. Then, for any 𝕍-comonoid C, the triple

ϕ_{T} (C) = ((T (C), m_{C}), χ_{C,C} · T (δ), θ_{𝕀} · T (ε)))

is a 𝕍_{T} -comonoid and it is easy to see that the comonad { \atop G_{T (C)}} is just the comonad G_{ϕ_{T} (C)} and that the category 𝕍_{T}^{T (C)} is just the category (𝕍_{T})^{ϕ_{T} (C)}. In particular, if ϕ_{T} (I) = ((T (𝕀), m_{𝕀}) χ_{𝕀, 𝕀}, θ) is a 𝕍_{T} -comonoid corresponding to the trivial 𝕍-comonoid

I = (𝕀, 1_{𝕀}, 1_{𝕀}), then { \atop G_{T (I)}} = G_{ϕ_{T} (I)} and 𝕍_{T}^{T (I)} = (𝕍_{T})^{ϕ_{T} (I)}.

4.3. Remark. It follows from the results of [12, 5.13] that, for an arbitrary bialgebra A = (A, μ, η, δ, ε) in a braided monoidal category (𝕍, ⊗, 𝕀), the following are equivalent:

(i) the natural transformation λ_{−}^{I} = H_{−,𝕀}^{l} : A ⊗ − ⊗ A → A ⊗ − ⊗ A, corresponding to the opmonoidal 𝕍-monad A ⊗ −, is an isomorphism;

(ii) the morphism λ_{𝕀}^{I} = H_{𝕀,𝕀}^{l} : A ⊗ A → A ⊗ A is an isomorphism;

(iii) the composite A ⊗ A ⟶^{δ⊗A} A ⊗ A ⊗ A ⟶^{A⊗m} A ⊗ A is an isomorphism.

Recall (for example from [10]) that condition (iii) is in turn equivalent to saying that A has an antipode, i.e. A is a Hopf algebra. It follows from the equivalence (i)⇔(iii) that, for any V ∈ 𝕍, the natural transformation H_{−,V}^{l}, which is easily seen to be just the natural transformation H_{−,𝕀}^{l} ⊗ V, is an isomorphism, or equivalently, the monad A ⊗ − is left Hopf, if and only if A is a Hopf algebra. Moreover, if the monad A ⊗ − is left pre-Hopf (and hence, in particular, the morphism H_{𝕀,𝕀}^{l} is an isomorphism), then according to the equivalence (ii)⇔(iii), A is a Hopf algebra. Putting this information together and using that, quite obviously, any left Hopf monad is left pre-Hopf, we have proved that the following are equivalent:

(i) A is a Hopf algebra;

(ii) the monad A ⊗ − is left pre-Hopf;

(iii) the monad A ⊗ − is left Hopf.

This result may be compared with [3, Proposition 5.4(a)].

4.4. Group-like morphisms. Suppose now that the 𝕍-comonoid C allows for a grouplike element g : 𝕀 → C (see [10], [11]). Then direct inspection shows that \overline{g} : 𝕀 ⟶^{g} C ⟶^{e_{C}} T (C) is a group-like element for the 𝕍-comonoid T (C) implying that the natural transformation − ⊗ \overline{g} : 1 → − ⊗ T (C) is a group-like morphism. Thus the results of [10] apply. In particular, the composite

T (−) ⟶^{T (−⊗\overline{g})} T (− ⊗ T (C)) ⟶^{λ_{−}^{C}} T (−) ⊗ T (C)

gives the structure ϑ : ϕ_{T} → ϕ_{T} { \atop G_{T (C)}} of a { \atop G_{T (C)}}-comodule on the functor ϕ_{T} : 𝕍 → 𝕍_{T} . Since in the diagram

[figure: T (−); T (−⊗g); χ_{−,𝕀}; T (− ⊗ C); T (−⊗e_{C}); χ_{−,C}; T (− ⊗ T (C)); χ_{−,T (C)}; T (−) ⊗ T^{2}(C); T (−)⊗m_{C}; T (−) ⊗ T (𝕀); T (−)⊗T (g); T (−) ⊗ T (C); T (−)⊗T (e_{C}); T (−) ⊗ T (C)]

the rectangle and the top triangle commute by naturality of χ, while the bottom triangle commutes since e is the unit for the multiplication m, it follows that ϑ is just the natural transformation

T (−) ⟶^{χ_{−,𝕀}} T (−) ⊗ T (𝕀) ⟶^{T (−)⊗T (g)} T (−) ⊗ T (C).

It then follows that the assignment V ⟶ ((T (V), m_{V}), (T (V) ⊗ T (g)) · χ_{V,𝕀}) yields a functor

K_{g,C} := K_{g,G_{T (C)}} : 𝕍 → 𝕍_{T}^{T (C)} = (𝕍_{T}){ \atop G_{T (C)}}

with ϕ_{T} = U{ \atop G_{T (C)}}K_{g,C}.

One calculates that for any (V, h_{V}) ∈ V_{T}, the (V, h_{V})-component of the induced comonad morphism S_{K_{g,C}} : ϕ_{T} U_{T} → { \atop G_{T (C)}} is the composite

\xymatrix{T (V) \ar[r]^{χ_{V, 𝕀}} & T (V) ⊗ T (𝕀) \ar[r]^{T (V)⊗T (g)} & T (V) ⊗ T (C) \ar[r]^{h_{V} ⊗T (C)} & V ⊗ T (C).}

In particular, when C is the trivial 𝕍-comonoid I together with the evident group-like morphism 1_{𝕀} : 𝕀 → 𝕀, the morphism χ_{−,𝕀} : T (−) → T (−) ⊗ T (𝕀) gives the structure ϑ′ : ϕ_{T} → ϕ_{T} { \atop G_{T (I)}} of a { \atop G_{T (I)}}-comodule on the functor ϕ_{T} : 𝕍 → 𝕍_{T} , and then one has ϕ_{T} = U{ \atop G_{T (I)}}K_{1_{𝕀},I} with the comparison functor K_{1_{𝕀},I}(V) = ((T (V), m_{V}), χ_{V, I}). Moreover, for any (V, h_{V}) ∈ V_{T}, the (V, h_{V})-component of the induced comonad morphism S_{K_{1_{𝕀},I}} : ϕ_{T} U_{T} → { \atop G_{T (I)}} is the composite

\xymatrix{T (V) \ar[r]^{χ_{V, 𝕀}} & T (V) ⊗ T (𝕀) \ar[r]^{h_{V} ⊗T (𝕀)} & V ⊗ T (𝕀).}

Comparing now ϑ and ϑ′ gives

ϑ = (T (−) ⊗ T (g)) · ϑ′  (2)

while comparing S_{K_{g,C}} and S_{K_{e_{𝕀},𝕀}} and using that

(h_{V} ⊗ T (C)) · (T (V) ⊗ T (g)) = (V ⊗ T (g)) · (h_{V} ⊗ T (𝕀))

by bifunctoriality of the tensor product, gives

S_{K_{g,C}} = (− ⊗ T (g)) · S_{K_{e_{𝕀},𝕀}}.  (3)

It is easy to see that S_{K_{e_{𝕀},𝕀}} just the composite ℍ_{V,𝕀}^{r}. This yields in particular a fact proved in [3, Lemma 6.5]:

4.5. Lemma. ℍ_{−,𝕀}^{r} : T(−) → − ⊗ T(𝕀) is a morphism of comonads ϕ_{T}U_{T} → \begin{matrix} \\ G & T(I) \end{matrix}.

We already know (see 4.1) that T is a right pre-Hopf monad iff the natural transformation ℍ_{−,𝕀}^{r} (or, equivalently, the comonad morphism S_{K_{e_{𝕀},𝕀}}) is an isomorphism. It now follows from Proposition 2.10:

4.6. Proposition. An opmonoidal monad T on 𝕍 is a right pre-Hopf monad if and only if T is G_{T(I)}-Galois.

This allows us to present an improved version of [3, Theorem 6.11].

4.7. Theorem. For an opmonoidal monad T on a monoidal category (𝕍, ⊗, 𝕀), the following are equivalent:

(a) the functor K_{1_{𝕀},I} : 𝕍 → 𝕍_{T}^{T(I)} is an equivalence of categories;

(b) (i) T is G_{T(I)}-Galois, (ii) T is of effective descent type.

Proof. The assertion follows by Proposition 2.6.

4.8. Theorem. Let T = (T, m, e) be an opmonoidal monad on a monoidal category (𝕍, ⊗, 𝕀) and C = (C, δ, ε) a 𝕍-comonoid with a group-like element g : 𝕀 → C. The following are equivalent:

(a) the functor K_{g,C} : 𝕍 → (𝕍_{T})^{ϕ_{T}(C)} is an equivalence of categories;

(b) K_{1_{𝕀},I} : 𝕍 → (𝕍_{T})^{ϕ_{T}(I)} is an equivalence of categories and g is an isomorphism;

(c) (i) T is G_{T(I)}-Galois, (ii) T is of effective descent type, (iii) g is an isomorphism.

Proof. Note first that, being group-like morphisms, g and T (g) are split monomorphisms. Hence the natural transformation −⊗T (g) : G_{T (𝕀)} → G_{T (C)} is also a split monomorphism.

Now, if the functor K_{g,C} : 𝕍 → 𝕍_{T}^{T(C} = (𝕍_{T}){ \atop G_{T (C)}} is an equivalence of categories, then it follows from Proposition 2.6 that the monad T is of effective descent type and the comonad morphism S_{K_{g,C}} : ϕ_{T} U_{T} → { \atop G_{T (C)}} is an isomorphism. Since S_{K_{g,C}} = (− ⊗ T (g)) · S_{K_{e_{𝕀},𝕀}} by (3) and since the natural transformation − ⊗ T (g) : G_{T (I)} → G_{T (C)} is a split monomorphism, it follows that the natural transformations − ⊗ T (g) and S_{K_{e_{𝕀},𝕀}} are both isomorphisms. Then, in particular, T (g) is an isomorphism. Since T is of effective descent type, the functor T is conservative (see, [9, Proposition 3.11]). Thus g is also an isomorphism. Since T is of effective descent type and since S_{K_{e_{𝕀},𝕀}} is an isomorphism, it follows from Proposition 2.6 that the functor K_{e_{𝕀},𝕀} : 𝕍 → 𝕍_{T}^{T(I)} is an equivalence of categories. Hence (a) implies (b). Since (b) trivially implies (a), (a) and (b) are equivalent. Finally, (b) and (c) are equivalent by Theorem 4.7.

4.9. Galois group-like morphisms. We will assume from now on that our monoidal category (𝕍, ⊗, 𝕀) admits equalisers.

Let T = (T, m, e) be an opmonoidal monad on 𝕍, C a 𝕍-comonoid and g : 𝕀 → C a group-like element. Since 𝕍 has equalisers, one can consider the 𝕍-monad T^{−⊗\overline{g}}. We write T^{g} for this monad. Let us say that g : 𝕀 → C is a Galois group-like element if the induced group-like morphism − ⊗ \overline{g} : 1 → G_{T (C)} is Galois. In particular, the group-like element 1_{𝕀} : 𝕀 → 𝕀 is Galois if the group-like morphism − ⊗ \overline{1_{𝕀}} = − ⊗ e_{C} : 1 → G_{T (I)} is Galois.

Specialising now Theorem 2.15 to the present situation gives:

4.10. Theorem. Let T be an opmonoidal monad on a monoidal category (𝕍, ⊗, 𝕀) and C a 𝕍-comonoid. Suppose that g : 𝕀 → C is a group-like element such that the corresponding functor i^{∗} : 𝕍_{T} → 𝕍_{T^{g}} admits a left adjoint functor i_{∗} : 𝕍_{T^{g}} → 𝕍_{T}. Then the comparison functor \overline{i} : 𝕍_{T^{g}} → 𝕍_{T}^{T (C)} = (𝕍_{T}){ \atop G_{T (C)}} is an equivalence of categories if and only if g is a Galois group-like element and the functor i_{∗} is comonadic.

Direct inspection shows (see also [12, Section 5]) that T^{1_{𝕀}} is given by the equaliser

\xymatrix{T^{1_{𝕀}}(−) \ar[r] & T (−) \ar[r]^{T (−)⊗e_{𝕀}} \ar[r]_{χ_{−,𝕀}} & T (−) ⊗ T (𝕀),}

while T^{g} is given by the equaliser

\xymatrix{T^{g}(−) \ar[r] & T (−) \ar[r]^{T (−)⊗e_{𝕀}} \ar[r]_{χ_{−,𝕀}} & T (−) ⊗ T (𝕀) \ar[r]^{T (−)⊗T (g)} & T (−) ⊗ T (C).}

Since g, being a group-like morphism, is a split monomorphism, so too is the natural transformation T (−) ⊗ T (g). It follows that the monad T^{1_{𝕀}} can be identified with the monad T^{g}. Since g : 𝕀 → C is nothing but a comonoid morphism from the trivial 𝕍comonoid I to the 𝕍-comonoid C and since any opmonoidal functor preserves comonoid morphisms, T (g) : T (𝕀) → T (C) can be seen as a morphism of 𝕍-comonoids T (I) → T (C). It is then easy to see that the induced morphism of 𝕍-comonads −⊗T (g) : G_{T (𝕀)} → G_{T (C)} can be lifted to a morphism { \atop − ⊗ T (g)} : { \atop G_{T (I)}} → { \atop G_{T (C)}} of 𝕍_{T} -comonads. Using that ϑ = (T (−)⊗T (g))·ϑ′ by (2), it follows from [12, Lemma 3.9] that one has the commutative diagram

\xymatrix{i_{∗} \ar[rr]^{α} \ar[dr]_{α′} &  & { \atop G_{T (C)}} · i_{∗} \\  & { \atop G_{T (I)}} · i_{∗} \ar@{-->}[ur]_{{ \atop −⊗T (g)}·i_{∗}} & ,}

where i : T^{g} = T^{1_{𝕀}} → T is the canonical inclusion, while α (resp. α′) is a left { \atop G_{T (C)}}- comodule (resp. { \atop G_{T (I)}}-comodule) structure on i_{∗}. It then follows that one also has commutativity of

\xymatrix{{ \atop G_{T (C)}} \ar[rr]^{S\overline{i}} \ar@{-->}[dr]_{S\overline{i}} &  & { \atop G_{T (C)}} \\  & { \atop G_{T (I)}} \ar@{-->}[ur]_{{ \atop −⊗T (g)}} & ,}  (4)

where { \atop G_{T (C)}} denotes the comonad generated by the adjunction i^{∗} ⊢ i_{∗} (see 2.13).

4.11. Proposition. Let T be an opmonoidal monad on a monoidal category (𝕍, ⊗, 𝕀) and C a 𝕍-comonoid.

(1) A group-like element g : 𝕀 → C is Galois if and only if the group-like element 1_{𝕀} : 𝕀 → 𝕀 is Galois and the morphism T (g) : T (𝕀) → T (C) is an isomorphism.

(2) If the monad T is conservative, then any 𝕍-comonoid admitting a Galois group-like element is (isomorphic to) the trivial 𝕍-comonoid I.

Proof. To say that the group-like morphism g : 𝕀 → C (resp. 1_{𝕀} : 𝕀 → 𝕀) is Galois is to say that the comonad morphism S\overline{i} : { \atop G_{T (I)}} → { \atop G_{T (I)}} (resp. S\overline{i} : { \atop G_{T (C)}} → { \atop G_{T (C)}}) is an isomorphism. Now, since T (g) is a split monomorphism, the result follows from the commutativity of diagram (4). This proves (1). Recalling that a monad is called conservative provided that its functor part is conservative, one sees that (2) follows from (1).

Combining Theorem 4.10 and Proposition 4.11 gives:

4.12. Theorem. Let T be an opmonoidal monad on a monoidal category (𝕍, ⊗, 𝕀) such that the functor i^{∗} : 𝕍_{T} → 𝕍_{T^{1_{𝕀}}} admits a left adjoint functor i_{∗} : 𝕍_{T^{1_{𝕀}}} → 𝕍_{T} and C a 𝕍-comonoid. Then, for any group-like element g : 𝕀 → C, the following are equivalent:

(a) g : 𝕀 → C is a Galois group-like element and the functor i_{∗} is comonadic;

(b) the comparison functor \overline{i} : 𝕍_{T^{g}} → (𝕍_{T})^{ϕ_{T}(C)} is an equivalence of categories;

(c) 1_{𝕀} : 𝕀 → 𝕀 is a Galois group-like element, the functor i_{∗} is comonadic and the morphism T (g) : T (𝕀) → T (C) is an isomorphism;

(d) the comparison functor \overline{i} : 𝕍_{T^{1_{𝕀}}} → (𝕍_{T})^{ϕ_{T}(I)} is an equivalence of categories and the morphism T (g) : T (𝕀) → T (C) is an isomorphism.

It is easy to see that, in the case where the monad T^{1_{𝕀}} = T^{g} is (isomorphic to) the identity monad, the functor ϕ_{T^{1_{𝕀}}} = ϕ_{T^{g}} is (isomorphic to) the identity functor, the functor i_{∗} is (isomorphic to) the functor ϕ_{T}, while the functor \overline{i} is (isomorphic to) the comparison functor K_{g,C}. Using now that the monad T is conservative provided that the functor ϕ_{T} is so, in the light of Proposition 4.11, we get from Theorems 4.8 and 4.12:

4.13. Theorem. Let T be an opmonoidal monad on a monoidal category (𝕍, ⊗, 𝕀) such that the monad T^{1_{𝕀}} is (isomorphic to) the identity monad and C a 𝕍-comonoid. Then, for any group-like element g : 𝕀 → C, the following are equivalent:

(a) g : 𝕀 → C is a Galois group-like element and the functor ϕ_{T} is comonadic;

(b) the comparison functor K_{g,C} : 𝕍 → (𝕍_{T})^{ϕ_{T}(C)} is an equivalence of categories;

(c) 1_{𝕀} : 𝕀 → 𝕀 is a Galois group-like element, the functor ϕ_{T} is comonadic, and the morphism g : 𝕀 → C is an isomorphism;

(d) the comparison functor K_{g,I} : 𝕍 → (𝕍_{T})^{ϕ_{T}(I)} is an equivalence of categories and the morphism g : 𝕀 → C is an isomorphism; (e) (i) T is G_{T(I)}-Galois,

(ii) T is of effective descent type, (iii) g is an isomorphism.

4.14. Definition. We say that an opmonoidal monad T on a monoidal category 𝕍 is augmented if it is equipped with a monad morphism σ : T → 1_{𝕍}. In this case σ is said to be an augmentation.

4.15. Lemma. Suppose that T is an augmented right-Hopf opmonoidal monad on a monoidal category 𝕍 with an augmentation σ : T → 1_{𝕍}. Then, for any V ∈ 𝕍, the composite

\overline{σ}_{V} : T (V) ⟶^{χ_{V,𝕀}} T (V) ⊗ T (𝕀) ⟶^{σ_{V} ⊗T (𝕀)} V ⊗ T (𝕀)

is an isomorphism.

Proof. Just note that, since σ : T → 1_{𝕍} is a morphism of monads, for any V ∈ 𝕍, (V, σ_{V}) is an object of 𝕍_{T}.

4.16. Theorem. Let T = (T, m, e) be an augmented right-Hopf opmonoidal monad on a Cauchy complete monoidal category (𝕍, ⊗, 𝕀) with an augmentation σ : T → 1_{𝕍}. Then, for any group-like element g : 𝕀 → C, the following are equivalent:

(a) g : 𝕀 → C is a Galois group-like element;

(b) the comparison functor K_{g,C} : 𝕍 → (𝕍_{T})^{ϕ_{T}(C)} is an equivalence of categories;

(c) the comparison functor K_{g,I} : 𝕍 → (𝕍_{T})^{ϕ_{T}(I)} is an equivalence of categories and the morphism g : 𝕀 → C is an isomorphism;

(d) 1_{𝕀} : 𝕀 → 𝕀 is a Galois group-like element and the morphism g : 𝕀 → C is an isomorphism;

(e) (i) T is G_{T(I)}-Galois, (ii) g is an isomorphism.

Proof. Using naturality of χ, it is not hard to check that the diagram

\xymatrix{T (V) \ar[d]_{_{\overline{σ}_{V}}} \ar[r]^{T (V)⊗e_{𝕀}} \ar[r]_{χ_{V,𝕀}} & T (V) ⊗ T (𝕀) \ar[d]^{\overline{σ}_{V} ⊗T (𝕀)} \\ V ⊗ T (𝕀) \ar[r]^{V ⊗T (𝕀)⊗e_{𝕀}} \ar[r]_{V ⊗χ_{𝕀,𝕀}} & V ⊗ T (𝕀) ⊗ T (𝕀)}

commutes. Since \overline{σ}_{V} is an isomorphism by Lemma 4.15, it follows that T^{1_{𝕀}}(V) is (isomorphic to) the equaliser of the pair

\xymatrix{V ⊗ T (𝕀) \ar[r]^{V ⊗T (𝕀)⊗e_{𝕀}} \ar[r]_{V ⊗χ_{𝕀,𝕀}} & V ⊗ T (𝕀) ⊗ T (𝕀)} .

Using now that

• θ · e_{𝕀} = 1_{𝕀} and (θ ⊗ T (𝕀)) · χ_{𝕀,𝕀} = 1_{T (𝕀)}, since the monad T is opmonoidal, and

• (θ ⊗ T (𝕀)) · (T (𝕀) ⊗ e_{𝕀}) = e_{𝕀} · θ by naturality of composition,

one sees that the diagram

\xymatrix{𝕀 \ar[r]^{e_{𝕀}} & T (𝕀) \ar[l]^{θ} \ar[r]^{T (𝕀)⊗e_{𝕀}} \ar[r]_{χ_{𝕀,𝕀}} & T (𝕀) ⊗ T (𝕀) \ar[l]^{θ⊗T (𝕀)}} .

is a split equaliser. Since split equalisers are preserved by any functor the diagram

\xymatrix{V \ar[r]^{V ⊗e_{𝕀}} & V ⊗ T (𝕀) \ar[r]^{V ⊗T (𝕀)⊗e_{𝕀}} \ar[r]_{V ⊗χ_{𝕀,𝕀}} & V ⊗ T (𝕀) ⊗ T (𝕀)}

is a (split) equaliser. Thus the monad T^{1_{𝕀}} is (isomorphic to) the identity monad.

Next, as σ : T → 1_{𝕍} is a morphism of monads, σ · e = 1. Thus the unit of the monad T is a split monomorphism, and since the category 𝕍 is Cauchy complete by hypothesis, it follows from [9, Corollary 3.17] that T is of effective descent type, i.e. the functor ϕ_{T} is comonadic. Putting now this information together, the assertions follow by Theorem 4.13.

5. Applications

In this final section we outline some applications of the notions developed.

5.1. Monads on cartesian monoidal categories. Let 𝔸 be a category with finite products. Then 𝔸 is equipped with the (symmetric) monoidal structure (𝔸, ×, 1) (known as the cartesian monoidal structure), where a×b is some chosen product of a and b, and 1 is a chosen terminal object in 𝔸. For any object a ∈ 𝔸, we write !_{a} for the unique morphism a → 1. Given morphisms f : a → x and g : a → y in 𝔸, we write < f, g >: a → x × y for the unique morphism inducing commutativity of the diagram

\xymatrix{ & a \ar[dl]_{f} \ar[dr]^{g} \ar[d]|{<f,g>} \\ x & x × y \ar[l]^{p_{1}} \ar[r]_{p_{2}} & y.}

Any monad T on 𝔸 has a canonical structure of an opmonoidal monad given by

χ_{a,b} =< T (p_{1}), T (p_{2}) >: T (a × b) → T (a) × T (b), θ =!_{T (1)} : T (1) → 1.

Thus, for any monad T on 𝔸, the category 𝔸_{T} is also cartesian.

Since, for any a ∈ 𝔸, the projection p_{1} : a ≃ a × 1 → a is (isomorphic to) the identity morphism 1_{a} : a → a, while the projection p_{2} : a ≃ a × 1 → 1 is (isomorphic to) the morphism !_{a} : a → 1, χ_{a,1} : T (a) → T (a) × T (1) is just the morphism < 1_{T (a)}, T (!_{a}) >.

An arbitrary object a ∈ 𝔸 has a canonical 𝔸-comonoid structure given by the diagonal morphism Δ_{a} =< 1_{a}, 1_{a} >: a → a × a. Writing a for the corresponding 𝔸-comonoid, one has that 𝔸^{a} is (isomorphic to) the comma category 𝔸↓a (see, for example, [12]). Modulo this isomorphism, the forgetful functor U^{a} : 𝔸^{a} → 𝔸 corresponds to the functor

Σ_{a} : 𝔸↓a → 𝔸, (x → a) ⟶ x,

while its right adjoint ϕ_{a} : 𝔸 → 𝔸^{a} corresponds to the functor

a^{∗} : 𝔸 → 𝔸↓a, x ⟶ (p_{1} : a × x → a).

Suppose now T to be a monad on a cartesian category 𝔸 such that the category 𝔸_{T} admits equalisers. Then one can form the monad T^{1_{1}}. Moreover, modulo the isomorphism of categories (𝔸_{T})^{ϕ_{T} (1)} ≃ (𝔸_{T} ↓ ϕ_{T} (1)), one rewrites Diagram (1) from 2.13 as

\xymatrix{𝔸 \ar[rrr]^{K_{1_{1},1}} \ar[dr]_{ϕ_{T}} \ar[r]^{ϕ_{T^{1_{1}}}} & 𝔸_{T^{1_{1}}} \ar[d]^{i_{∗}} \ar[rr]^{\overline{i}} &  & (𝔸_{T}) ↓ ϕ_{T} (1) \ar[dll]^{Σ_{ϕ_{T} (1)}} \\  & 𝔸_{T} & .}  (5)

Note that, for any a ∈ 𝔸, K_{1_{1},1}(a) = ((T (a), m_{a}), T (!_{a})).

5.2. Remark. Obviously, for any (a, h_{a}) ∈ 𝔸_{T}, the (a, h_{a})-component of the natural transformation ℍ_{−,1}^{r} : T → − × T (1) is the composite

\xymatrix{T (a) \ar[r]^{<1_{T (a)},T (!_{a})>} & T (a) × T (1) \ar[r]^{h_{a}×T (1)} & a × T (1),}

which is the same as the morphism

T (a) ⟶^{<h_{a},T (!_{a})>} a × T (1).

If T (1) ≃ 1, then T (!_{a}) ≃!_{T (a)} and thus <h_{a}, T (!_{a})> can be identified with the morphism h_{a} : T (a) → a.

Now fix a monad T = (T, m, e) on a cartesian monoidal category 𝔸 with equalisers. Then, for any a ∈ 𝔸, T^{1_{1}}(a) can be calculated as the equaliser of the diagram

\xymatrix{T (a) \ar[r]^{<1_{T (a)},T (!_{a})>} \ar[r]_{1_{T (a)}×e_{1}} & T (a) × T (1).}

But since 1_{T (a)} × e_{1} can be identified with the morphism < 1_{T (a)}, e_{1}·!_{T (a)} >, the diagram

\xymatrix{T^{1_{1}}(a) \ar[r]^{i_{a}} & T (a) \ar[r]^{<1_{T (a)},T (!_{a})>} \ar[r]_{<1_{T (a)},e_{1}·!_{T (a)}>} & T (a) × T (1)}

is an equaliser if and only if so is

\xymatrix{T^{1_{1}}(a) \ar[r]^{i_{a}} & T (a) \ar[r]^{p_{2}·<1_{T (a)},T (!_{a})>} \ar[r]_{p_{2}·<1_{T (a)},e_{1}·!_{T (a)}>} & T (1)} .

As p_{2}· < 1_{T (a)}, T (!_{a}) >= T (!_{a}) and p_{2}· < 1_{T (a)}, e_{1}·!_{T (a)} >= e_{1}·!_{T (a)}, the diagram

\xymatrix{T^{1_{1}}(a) \ar[r]^{i_{a}} & T (a) \ar[r]^{T (!_{a})} \ar[r]_{e_{1}·!_{T (a)}} & T (1)}

is an equaliser. It follows that if T (1) ≃ 1, then i_{a} is an isomorphism. Conversely, if i_{a} is an isomorphism, then T (!_{a}) = e_{1}·!_{T (a)}. In particular, T (!_{1}) = e_{1}·!_{T (1)}. But T (!_{1}) = 1_{1}, implying that both e_{1} and !_{T (1)} are isomorphisms. Thus:

5.3. Lemma. Let T be a monad on a cartesian monoidal category (𝔸, ×, 1). Then the canonical inclusion i : T^{1_{1}} → T is an isomorphism if and only if the functor part T preserves the terminal object.

5.4. Proposition. Let (𝔸, ×, 1) be a cartesian monoidal category. For any monad T on 𝔸, whose functor part preserves the terminal object 1, the comparison functor

\overline{i} : 𝔸_{T^{1_{1}}} → (𝔸_{T})^{ϕ_{T} (1)}

is an equivalence. In particular, 1_{1} : 1 → 1 is a Galois group-like element (w.r.t. T).

Proof. Since T (1) ≃ 1, the monads T^{1_{1}} and T are isomorphic by Lemma 5.3. It then follows from commutativity of diagram (5) that \overline{i} is just the functor (ϕ_{T} (1))^{∗} : 𝔸_{T} → (𝔸_{T}) ↓ ϕ_{T} (1). But since T (1) ≃ 1, ϕ_{T} (1) is a terminal object in 𝔸_{T}. Thus the functor (ϕ_{T} (1))^{∗} (and hence also \overline{i}) is an isomorphism of categories.

Now the last assertion follows from Theorem 4.12.

Recall that a monad T = (T, m, e) on a category 𝔸 is said to be idempotent if the multiplication m : T T → T is a natural isomorphism.

5.5. Proposition. Let (𝔸, ×, 1) be a cartesian monoidal category. Any idempotent monad on 𝔸, whose functor part preserves the terminal object 1, is right pre-Hopf.

Proof. It is well-known that if T = (T, m, e) is an idempotent monad on a category 𝔸, then for any (a, h_{a}) ∈ 𝔸_{T}, the morphism h_{a} : T (a) → a is an isomorphism. Thus the result follows from Remark 5.2.

5.6. Example. Recall from [7] that a category 𝔸 with all finite products is called cartesian closed if for each object a ∈ 𝔸, the functor a × − : 𝔸 → 𝔸 has a right adjoint (−)^{a} : 𝔸 → 𝔸.

For any object a ∈ 𝔸, the endofunctor T_{a} = (−)^{a} can be made a monad with multiplication and unit

m_{x} = x^{Δ_{a}} : T_{a}T_{a}(x) = (x^{a})^{a} ≃ x^{a×a} → T_{a}(x) = x^{a}, e_{x} = x^{!_{a}} : x → x^{a} = T_{a}(x).

Let 𝔸 be a cartesian closed category such that the terminal object 1 has a nontrivial proper subobject u ↣ 1 (for example, let 𝔸 be the category of set-valued sheaves on a nontrivial topological space). Since u × u ≃ u, the diagonal Δ_{u} : u → u × u is an isomorphism, whence the monad T_{u} is idempotent. Since 1^{u} = 1, the functor (−)^{u} preserves the terminal object and it follows from Proposition 5.5 that the opmonoidal monad T_{U} is right pre-Hopf.

Note that by Proposition 5.4, the comparison functor K_{1_{1},1} : 𝕍 → (𝕍_{T_{u}})^{ϕ_{_{T}_{u}} (1)} is not an equivalence of categories. Thus T_{u} is an example of an opmonoidal monad which is right pre-Hopf, but the corresponding comparison functor K_{1_{1},1} is not an equivalence of categories.

5.7. Example. Recall that the covariant power set functor P : Set → Set is defined by

P(X) = Sub(X), P(f : X → Y) = P(X) ⟶^{P(f)} P(Y),

where Sub(X) is the set of all subsets of X and for each U ∈ Sub(X), P(f)(U) is the image f(U) of U under f. P is actually the functor part of a monad (P, e, m) with

e_{X} : X → P(X) the singleton map, e_{X} : x → {x}, and
m_{X} : PP(X) → P(X) the union, m_{X}({X_{α}}) = ⋃_{α} X_{α}.

It is well-known that the Eilenberg-Moore category of P-algebras is isomorphic to the category CSLat of complete (join-)semilattices. Recall that the category CSLat has as its objects partially ordered sets (X, ≤) which admit arbitrary suprema, and as its morphisms f : X → Y maps which preserve suprema. We write 2 for the two-element semilattice ϕ_{P}(1) = {0 ≤ 1}.

It is not hard to check that P^{1_{1}} is just the proper power set functor P^{+}, where P^{+}(X) = P(X) \ {∅}. It is also well-known (see, for example, [6, Problem 1.3.3.]) that the Eilenberg-Moore category of P^{+}-algebras is isomorphic to the category ACSLat of almost complete (join-)semilattices, i.e. partially ordered sets (X, ≤) such that the suprema of all non-empty subsets of X exists. Morphisms f : (X, ≤) → (Y, ≤) of ACSLat are non-empty suprema preserving maps.

Writing i : P^{+} → P for the canonical inclusion, it is not hard to see that the functor

i^{∗} : Set_{P} = CSLat → Set_{P^{+}} = ACSLat

just forgets about the bottom element, while

i_{∗} : ACSLat → CSLat

takes an object X ∈ ACSLat to the complete semilattice \overline{X} obtained from X by adding a bottom element 0_{X}. It then follows in particular that the endofunctor i_{∗}i^{∗} : CSLat → CSLat takes a complete semilattice X to the complete semilattice \overline{X} obtained from X by adding a new bottom element 0\overline{X} < 0_{X}. Direct inspection shows that, for any X ∈ CSLat, the X-component of the comonad morphism S\overline{i} : G_{i} → G_{ϕ_{P} (1)} is the map ω : \overline{X} → X × 2 defined by

ω(x) = \{ \begin{matrix}(x, 1) & if x ≠ o\overline{X} \\ (0_{X}, 0) & if x = o\overline{X}. \end{matrix}

It is clear that ω is not an isomorphism. Thus 1_{1} : 1 → 1 is not a Galois group-like element (w.r.t. the monad P), and hence, by Theorem 4.12, the comparison functor

\overline{i} : Set_{P^{+}} = ACSLat → (Set_{P} ↓ ϕ_{P}(1)) = (CSLat ↓ 2),

which sends an object X ∈ ACSLat to (ω : \overline{X} → 2) ∈ (CSLat ↓ 2) with

ω(x) = \{ \begin{matrix}1 & if x ≠ o\overline{X} \\ 0 & if x = o\overline{X}, \end{matrix}

is not an equivalence of categories. According to [12, 1.4], \overline{i} admits a right adjoint r: for any (ω : \overline{X} → 2) ∈ (CSLat ↓ 2), r(ω) = (ω)^{−1}(1). It is now easy to see that r\overline{i} ≃ 1. Thus ACSLat is a full coreflective subcategory of (CSLat ↓ 2).

Note finally that P^{+}(1) = 1. Now it follows from Proposition 5.4 that 1_{1} : 1 → 1 is a Galois group-like element w.r.t. the monad P^{+}.

Acknowledgments. These investigations were started during a visit of the first author at the Department of Mathematics of the Heinrich Heine University of Düsseldorf. He wants to thank the colleagues there for the kind hospitality and appreciates the financial support by the German Research Foundation (DFG).

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A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 2, University st., Tbilisi 0186, Georgia, and Tbilisi Centre for Mathematical Sciences, Chavchavadze Ave. 75, 3/35, Tbilisi 0168, Georgia,

Department of Mathematics of HHU, 40225 Düsseldorf, Germany

Email: bachi@rmi.ge
      wisbauer@math.uni-duesseldorf.de

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