Galois functors and entwining structures Bachuki Mesablishvili, Robert Wisbauer

Galois functors and entwining structures

Bachuki Mesablishvili, Robert Wisbauer

Abstract

Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition ofGalois functorsover some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context.

Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of agrouplike natural transformationg:I→Ggeneralising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to opmonoidal monads with antipode on autonomous monoidal categories (named Hopf monadsby Brugui`eres and Virelizier) which can be understood as an entwining of two related functors.

As well-known, for any setGthe productG×−defines an endofunctor on the category of sets and this is a Hopf monad if and only ifGallows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading toGalois objectsin the sense of Chase and Sweedler.

Key Words: Corings, (Galois) comodules, Galois functors, relative injectives (projec- tives), equivalence of categories.

AMS classification: 18A40, 16T15.

Contents

1 Galois comodule and module functors 3

2 Entwinings 11

3 Grouplike morphisms 15

4 Bimonads 23

5 Opmonoidal monads 25

6 Categories with finite products and Galois objects 30

Introduction

Anentwining of an algebraA and a coalgebra C over a commutative ringR is given by an R-linear mapλ: A⊗RC → C⊗RA satisfying certain conditions (Brzezi´nski and Majid in [8]). The corresponding entwined modules are defined as R-modules M which allow for an A-module structure%M :A⊗RM →M and aC-comodule structure%^M :M →C⊗RM with a compatibility condition expressed by the commutativity of the diagram (e.g. [9, 32.4])

A⊗_RM ^%M //

IA⊗%^M

M ^%

M //C⊗M

A⊗C⊗M ^λ⊗IM //C⊗A⊗M.

I_C⊗%M

OO

An entwining structure (A, C, λ) makesC:=C⊗RAto anA-coring and the entwined modules are just the left comodules for the coringC.

In [27], a leftC-comoduleP withS= End^C(P). is called a Galois comoduleprovided the natural transformation HomA(P,−)⊗SP → C ⊗A−is an isomorphism. Such modules can be characterised by properties of the (C, A)-injective comodules [27, 4.1].

If A itself is a C-comodule, that is, it allows for a grouplike element, then C is called a Galois coringprovidedA is a Galois comodule.

As a special case, if anR-moduleB has entwined algebra and coalgebra structures, then B⊗_RB is aB-coring. In this situation,B is aB⊗_RB-Galois comodule, that is, B⊗_RB is a Galois coring, if and only ifB is a Hopf algebra overR.

Since the tensor product is fundamental for these notions, generalisations to monoidal categories were investigated, e.g. in McCrudden [17], Brugui`eres, and Virelizier [7], Loday [16], Mesablishvili [18], and others.

In this paper we are concerned with the extension of these formalisms to endofunctors on arbitrary categories. The key to this is the following observation. TheR-algebra Ainduces a monadA⊗R−, and theR-coalgebraCyields a comonadC⊗R−on the (monoidal) category of R-modules. Thus the entwining (A, C, λ) becomes a special case of the entwining of a monad with a comonad on any category which is known as mixed distributive law from early papers of Barr [1], Beck [2], van Osdol [26], and others (see [28] for more references). The theory of the related entwined modules is well understood in this context.

Yet, the additional constructions and notions for (A, C, λ) mentioned above are only partly transferred to monads and comonads on arbitrary categories, e.g. in G´omez-Torrecillas [14], B¨ohm and Menini [5], B¨ohm, Brzezi´nski and Wisbauer [4], and [21]. The purpose of this article is to continue these investigations.

A basic notion for this approach is the following (e.g. [14], [21]). Given a comonad G = (G, δ, ε) on a category A, a functor F : B → A is called a G-comodule if there is a natural transformation ¯α: F → GF makingF a G-comodule in an obvious sense. Such a functor is said to be G-Galoisprovided F has a right adjoint R : A→ B, and the induced comonad morphism F R → G is an isomorphism (Definition 1.3). In Section 1 we continue the investigation of these functors, in particular of their behaviour towards relative injective modules. Dually, given a monad T = (T, m, e) on A, a functor R : B → A is said to be a T-module if there is a natural transformation α : T R → R making R a T-module in an obvious way. Such a functor is called T-Galois if the induced monad morphism T → RF, where F :A→B is a left adjoint to R, is an isomorphism (Definition 1.16). These functors show a special behaviour toward relative projectives and this is outlined in the last part of Section 1.

Section 2 is concerned withG-comodule andT-module functors considered in the context of mixed distributive laws λ:T G→GT. In particular, the relevance of the Galois property for making the comparison functor K:A→(A^G)

Tb an equivalence is of interest.

As mentioned before, in an entwining structure (A, C, λ), the grouplike elements allow for a C⊗RA-comodule structure on A. In Section 3 we introduce, for a comonad (G, δ, ε) on A, grouplike morphisms g :I →G requiring suitable properties. For a monad Fon Awith a mixed distributive law λ : F G → GF, the grouplike element g induces two G-comodule structures on the functor F, namely gF : F → GF and ˜g : F ^{λ◦F g}−→ GF. The equaliser F^g −→^iF F of these two structure maps can be seen as monad morphism. Properties of the resulting functors are investigated and eventually conditions are given to obtain an equivalence betweenAF^gand the category (AF)^Ge(see 3.14). This generalises the characterisation of Galois corings in module categories (e.g. [9, 28.18]).

In Section 4, the preceding results are applied to the case of an endofunctor H, which is a monad (H, m, e) as well as a comonad (H, δ, ε) subject to some compatibility conditions.

Such functors are calledbimonads in [21]. Under mild conditions on the base category A, it follows thatH is a Hopf monad (has an antipode) if and only if (H, m, e) is an (H, δ, ε)-Galois comodule or - equivalently - (H, δ, ε) is an (H, m, e)-Galois module.

In Section 5 we consideropmonoidal monads T= (T, m, e) on a strict monoidal category

(V,⊗,I) (see [17]), calledbimonadsin [7]. HerebyT(I) has the structure of a coalgebra inV, and, as pointed out in [21, 2.2], their theory can be understood as an entwining between the monad T and the comonad− ⊗T(I) onV. Thus our theory applies and results from [7] are reconsidered from this point of view. This leads to an improvement of [7, Theorem 4.6] which may be seen as an extended version of the Fundamental Theorem of Hopf algebras for right autonomous strict monoidal categories.

In the final section we generalise known properties of the endofunctors G× − on the category of sets,G any set, to categories with finite products. This relates our notions with Galois objectsin the sense of Chase and Sweedler [11] (in the category opposite to commutative algebras) and we obtain a more general form of their Theorem 12.5 by replacing the condition on the Hopf algebra to be finitely generated and projective over the base ring by flatness without finiteness condition.

1 Galois comodule and module functors

LetAand Bdenote any categories. By Ia,I_Aor just by I we denote the identity morphism of an objecta∈A, respectively the identity functor of a categoryA.

Recall (e.g. from [13]) that amonad Ton A is a triple (T, m, e) where T : A →A is a functor with natural transformations m : T T → T, e : I → T satisfying associativity and unitality conditions. A T-module is an objecta∈Awith a morphism ha :T(a)→asubject to associativity and unitality conditions. The (Eilenberg-Moore) category of T-modules is denoted byAT and there is a free functorφT :A→AT, a7→(T(a), ma) which is left adjoint to the forgetful functorUT :AT →A.

Dually, acomonadGonAis a triple (G, δ, ε) whereG:A→Ais a functor with natural transformationsδ:G→GG,ε:G→I, andG-comodulesare objectsa∈Awith morphisms ρ_a :a →G(a). Both notions are subject to coassociativity and counitality conditions. The (Eilenberg-Moore) category of G-comodules is denoted byA^G and there is a cofree functor φ^G:A→A^G, a7→(G(a), δ_a) which is right adjoint to the forgetful functorU^G:A^G →A.

For convenience we recall some notions from [21, Section 3].

1.1. G-comodule functors. Given a comonad G = (G, δ, ε) onA, a functor F : B →A is a left G-comoduleif there exists a natural transformation β :F →GF with commutative diagrams

F BB BB BB BB

BB BB BB BB

β //GF

εF

F,

F ^β //

β

GF

δF

GF Gβ //GGF.

(1.1)

Obviously (G, δ) and (GG, δG) both are leftG-comodules.

AG-comodule structure onF :B→Ais equivalent to the existence of a functorF :B→ A^G (dual to [12, Proposition II.1.1]) leading to a commutative diagram

B

F //

FAAAAAA AA A^G

U^G

A.

Indeed, ifF is such a functor, thenF(b) = (F(b), βb) for some morphismβb :F(b)→GF(b) and the collection{βb, b∈B} constitutes a natural transformationβ :F →GF makingF a G-comodule. Conversely, if (F, β :F →GF) is aG-module, then F :B→A^G is defined by F(b) = (F(b), βb).

If aG-comodule (F, β) admits a right adjoint R:A→B, with counitσ:F R→I, then the composite

t_F :F R ^βR //GF R ^Gσ //G

is a comonad morphism from the comonad generated by the adjunctionF aRto the comonad G.

1.2 Proposition. ([18, Theorem 4.4]) The functor F is an equivalence of categories if and only if the functorF is comonadic andt_F is an isomorphism of comonads.

1.3. Definition. ([21, Definition 3.5]) A left G-comodule F : B → A with a right adjoint R:A→Bis said to be G-Galois if the corresponding morphism t_F :F R→Gof comonads onAis an isomorphism.

Thus,F is an equivalence if and only ifF isG-Galois and comonadic.

1.4. Right adjoint for F. If the category Bhas equalisers of coreflexive pairs, the functor F has a right adjoint.

Proof. This can be described as follows (see [12]): With the composite γ: R ^ηR //RF R

Rt_F

//RG,

a right adjoint toF is the equaliser (R, e) of the diagram RU^G

RU^Gη^G //

γU^G

//RGU^G=RU^Gφ^GU^G,

withη^G:I→φ^GU^G the unit ofU^Gaφ^G.

An easy inspection shows that for any (a, θa)∈ A^G, the (a, θa)-component of the above diagram is

R(a)

R(θ_a) //

γa //RG(a).

Now, for anya∈A, (R(F))(a) can be seen as the equaliser (R(F))(a)

e_F(a)

//RF(a)

R(βa) //

γ_F(a) //RGF(a).

Thus, writingP for the monad on Agenerated by the adjunctionF aR, the diagram P ^e //RF

Rβ //

γF //RGF

is an equaliser diagram. tu

In view of the characterisation of Galois functors we have a closer look at some related classes of relative injective objects.

Let F : B → A be any functor. Recall (from [25]) that an object b ∈ B is said to be F-injective if for any diagram inB,

b1 g

f //b2

 h

b

withF(f) a split monomorphism inA, there exists a morphismh:b2→bsuch that hf =g.

We write Inj(F,B) for the full subcategory ofBwith objects allF-injectives.

The following result from [25] will be needed.

1.5 Proposition. Let η, ε : F a R : A → B be an adjunction. For any object b ∈ B, the following assertions are equivalent:

(a) b isF-injective;

(b) b is a coretract for some R(a), witha∈A;

(c) the b-component ηb:b→RF(b) ofη is a split monomorphism.

1.6 Remark. For any a ∈ A, R(εa)·η_R(a) =I by one of the triangular identities for the adjunction F a R. Thus, R(a) ∈Inj(F,B) for all a ∈ A. Moreover, since the composite of coretracts is again a coretract, it follows from (b) that Inj(F,B) is closed under coretracts.

1.7. Functor between injectives. Let F : B → A be a G-module with a right adjoint R:A→Band unitη:I→RF. WriteG⁰for the comonad onAgenerated by the adjunction F a R and consider the comparison functor K_G⁰ : B → A^G

0. If b ∈ B is F-injective, then K_G⁰(b) = (F(b), F(η_b)) isU_G⁰-injective, since by the fact thatη_b is a split monomorphism in B, (η_G⁰)_φG0

(b) = F(η_b) is a split monomorphism in A^G

0. Thus the functor K_G⁰ : B → AG⁰

yields a functor

Inj(KG⁰) : Inj(F,B)→Inj(U^G0,A^G

0).

WhenBhas equalisers, this functor is an equivalence of categories (see [25]).

We shall henceforth assume thatBhas equalisers.

1.8 Proposition. With the data given in 1.7, the functor R:A^G→Brestricts to a functor R⁰: Inj(U^G,A^G)→Inj(F,B).

Proof. Let (a, θ_a) be an arbitrary object of Inj(U^G,A^G). Then, by Proposition 1.5, there exists an objecta0∈Asuch that (a, θa) is a coretraction ofφ^G(a0) = (G(a0), δa₀) inA^G, i.e., there exist morphisms

f : (a, θa)→(G(a0), δa₀) andg: (G(a0), δa₀)→(a, θa) in A^G withgf=I. Sincef andgare morphisms inA^G, the diagram

G(a0)

g

(δ_G)_a0

//GG(a0)

G(g)

a

f

OO

θ_a //G(a)

G(f)

OO

commutes. By naturality of γ(see 1.4), the diagram RG(a0)

R(g)

γ_G(a

0 ) //RGG(a0)

RG(g)

R(a)

R(f)

OO

γa //RG(a)

RG(f)

OO

also commutes. Consider now the following commutative diagram R(a0)

γa0 //RG(a0)

R(g)

γ_{G(a0 )}

//

R((δG)a0)

//RGG(a0)

RG(g)

R(a, θ_a)

OO

e_(a,θa) //R(a)

R(f)

OO

γ_a //

R(θ_a) //RG(a).

RG(f)

OO (1.2)

It is not hard to see that the top row of this diagram is a (split) equaliser (see [14]), and since the bottom row is an equaliser by the very definition ofe, it follows from the commutativity of the diagram that R(a, θa) is a coretract of R(a0), and thus is an object of Inj(F,B) (see Remark 1.6). It means that the functor R : A^G → B can be restricted to a functor R⁰ :

Inj(U^G,A^G)→Inj(F,B). tu

1.9 Proposition. With the data given in 1.7, suppose that for any b ∈ B, (t_F)F(b) is an isomorphism. Then the functorF :B→A^G can be restricted to a functor

F⁰ : Inj(F,B)→Inj(U^G,A^G).

Proof. Letδ⁰ denote the comultiplication in the comonadG⁰ (see 1.7). Recall from [18]

that F =At_F ·KG⁰, where At_F is the functorA^G

0 →A^G induced by the comond morphism t_F :G⁰→G. Then for anyb∈B,

F(RF(b)) =At_F(KG⁰(U F(b))) =At_F(F RF(b), F ηRF(b))

=At_F(G⁰F(b), δ_F(b)⁰ ) = (G⁰F(b),(t_F)_G⁰_F(b)·δ⁰_F(b)).

Consider now the diagram

G⁰F(b) ^(tF)F(b) //

δ⁰_F(b)

GF(b)

δ_F(b)

G⁰G⁰F(b)

(1)

(t_F)_F(b).(t_F)_F(b)

**U

UU UU UU UU UU UU UU UU U

(t_F)_G0F(b)

GG⁰F(b)

G((t_F)_F(b))

//GGF(b),

in which the triangle commutes by the definition of the composite (t_F)_F(b).(t_F)_F(b), while the diagram (1) commutes since t_F is a morphism of comonads. The commutativity of the outer diagram shows that (t_F)_F(b) is a morphism from the G-coalgebra F(RF(b)) = (G⁰F(b),(t_F)_G0F(b)·δ_F⁰ _(b)) to theG-coalgebra (GF(b), δ_F(b)). Moreover, (t_F)_F(b)is an isomor- phism by our assumption. Thus, for any b∈ B, F(RF(b)) is isomorphic to the G-coalgebra (GF(b), δ_F(b)), which is of course an object of the category Inj(U^G,A^G). Now, since any b∈Inj(F,B) is a coretract ofRF(b) (see Remark 1.6), and since any functor takes coretracts to coretracts, it follows that, for any b ∈ Inj(F,B), F(b) is a coretract of the G-coalgebra (GF(b), δ_F(b)) ∈ Inj(U^G,A^G), and thus is an object of the category Inj(U^G,A^G), again by

Remark 1.6. This completes the proof. tu

The following technical observation is needed for the next proposition.

1.10 Lemma. Let ι, κ:W aW⁰ :Y→Xbe an adjunction of any categories. If i:x⁰ →x and j:x→x⁰ are morphisms inXsuch that ji=I and if ι_x is an isomorphism, thenι_x0 is also an isomorphism.

Proof. Sinceji=I, the diagram

x^{0 i} //x ^I //

ij //x is a split equaliser. Then the diagram

W⁰W(x⁰) ^W

0W(i) //W⁰W(x) ^I //

W⁰W(ij)

//W⁰W(x)

is also a split equaliser. Now considering the following commutative diagram x⁰

ι_x0

i //x

κ_x

I //

ij //x

κ_x

W⁰W(x⁰)

W⁰W(i)

//W⁰W(x) ^I //

W⁰W(ij)

//W⁰W(x)

and recalling that the vertical two morphisms are both isomorphisms by assumption, we get

that the morphismιx⁰ is also an isomorphism. tu

1.11 Proposition. In the situation of Proposition 1.9, Inj(F,B) is (isomorphic to) a core- flective subcategory of the category Inj(U^G,A^G).

Proof. By Proposition 1.8, the functorR restricts to a functor R⁰: Inj(U^G,A^G)→Inj(F,B),

while according to Proposition 1.9, the functorF restricts to a functor F⁰ : Inj(F,B)→Inj(U^G,A^G).

Since

• F is a left adjoint toR,

• Inj(F,B) is a full subcategory ofB, and

• Inj(U^G,A^G) is a full subcategory ofA^G,

the functorF⁰ is left adjoint to the functorR⁰, and the unitη⁰ :I→R⁰F⁰ of the adjunction F⁰ a R⁰ is the restriction of η : F a R to the subcategory Inj(F,B), while the counit ε⁰ : F⁰R⁰→I of this adjunction is the restriction ofε:F R→I to the subcategory Inj(U^G,A^G).

Next, since the top of the diagram 1.2 is a (split) equaliser, R(G(a0), δa₀) 'R(a0). In particular, taking (GF(b), δ_F(b)), we see that

RF(b)'R(GF(b), δF(b)) =R F(U F(b)).

Thus, theRF(b)-componentη⁰_RF(b) of the unitη⁰ :I→R⁰F⁰ of the adjunction F⁰aR⁰ is an isomorphism. It now follows from Lemma 1.10 - since any b ∈Inj(F,B) is a coretraction of RF(b) - thatη⁰_b is an isomorphism for allb∈Inj(F,B), proving that the unitη⁰ of the adjunc- tionF⁰ aR⁰ is an isomorphism. Thus Inj(F,B) is (isomorphic to) a coreflective subcategory

of the category Inj(U^G,A^G). tu

1.12 Corollary. In the situation of Proposition 1.9, suppose that each component of the unit η :I →RF is a split monomorphism. Then the category Bis (isomorphic to) a coreflective subcategory ofInj(U^G,A^G).

Proof. When each component of the unitη:I→RF is a split monomorphism, it follows from Proposition 1.5 that every b ∈ Bis F-injective; i.e. B= Inj(F,B). The assertion now

follows from Proposition 1.11. tu

1.13. Characterisation of G-Galois comodules. Assume Bto admit equalisers, letGbe a comonad on A, and F : B→A a functor with right adjoint R :A →B. If there exists a functorF :B→A^G withU^GF =F, then the following are equivalent:

(a) F is G-Galois, i.e. t_F :G⁰→Gis an isomorphism;

(b) the following composite is an isomorphism, F R

η_GF R//φ^GU^GF R=φ^GF R ^φ

Gε //φ^G ; (c) the functorF :B→A^G restricts to an equivalence of categories

Inj(F,B)→Inj(U^G,A^G);

(d) for any (a, θ_a) ∈ Inj(U^G,A^G), the (a, θ_a)-component ε_(a,θa) of the counit ε of the ad- junction F aR, is an isomorphism;

(e) for any a∈A,ε_φG(a)=ε_(G(a),δa) is an isomorphism.

Proof. That (a) and (b) are equivalent is proved in [12]. By the proof of [14, Theorem of 2.6], for anya∈A,ε_φG(a)=ε_(G(a),δa)= (t_F)_a, thus (a) and (e) are equivalent.

By Remark 1.6, (d) implies (e).

Since B admits equalisers by our assumptions, it follows from Proposition 1.7 that the functor Inj(K_G⁰) is an equivalence of categories. Now, ift_F :G⁰ →Gis an isomorphism of comonads, then the functor A^t_F is an isomorphism of categories, and thus F is isomorphic to the comparison functor K_G⁰. It now follows from Proposition 1.7 that F restricts to the functor Inj(F,B)→Inj(U^G,A^G) which is an equivalence of categories. Thus (a)⇒(c).

If the functorF :B→A^G restricts to a functor

F⁰ : Inj(F,B)→Inj(U^G,A^G),

then one can prove, as in the proof of Proposition 1.11, thatF⁰ is left adjoint toR⁰ and that the counitε⁰:F⁰R⁰→I of this adjunction is the restriction of the counit ε:F R→Iof the adjunctionF aRto the subcategory Inj(U^G,A^G). Now, ifF⁰ is an equivalence of categories, then ε⁰ is an isomorphism. Thus, for any (a, θa) ∈ Inj(U^G,A^G), ε⁰_(a,θ

a) is an isomorphism

proving that (c)⇒(d). tu

1.14. T-module functors. Given a monadT= (T, m, e) on A, a functorR:B→Ais said to be a(left) T-moduleif there exists a natural transformationα:T R→R with commuting diagrams

R BB BB BB BB

BB BB BB BB^eR //T R

α

R,

T T R ^mR //

T α

T R

α

T R _α //R.

(1.3)

It is easy to see that (T, m) and (T T, mT) both are leftT-modules.

AT-module structure onRis equivalent to the existence of a functorR:B→AT inducing a commutative diagram (see [12, Proposition II.1.1])

B

R //

RAAAAAA AA AT

UT

A.

Indeed (compare [12]), ifR is such a functor, thenR(b) = (R(b), αb) for some morphism αb : T R(b) → R(b) and the collection {αb, b ∈ B} constitutes a natural transformation α : T R → R making R a T-module. Conversely, if (R, α: T R → R) is a T-module, then R:B→AT is defined byR(b) = (R(b), αb).

For any T-module (R : B → A, α) admitting a left adjoint functor F : A → B, the composite

t_R: T ^{T η} //T RF ^αF //RF ,

where η : I → RF is the unit of the adjunction F a R, is a monad morphism from T to the monad on A generated by the adjunction F a R. Dual to [18, Lemma 4.3], we have a commutative diagram

B

KR//

RBBBBBB!!

BB ARF AtR

AT,

with the comparison functorKR:B→ARF, b7→(R(b), R(εb)), where εis the counit of the adjunction FaR. As the dual of [18, Theorem 4.4], we have

1.15 Proposition. The functorRis an equivalence of categories if and only if the functorR is monadic (i.e. K_R is an equivalence) andt_R is an isomorphism of monads.

Similar to 1.3 one defines ([21, Definition 3.5], [4, 2.19])

1.16. Definition. A left T-moduleR :B →Awith a left adjoint F :A→B is said to be T-Galois if the corresponding morphismt_R:T →RF of monads onAis an isomorphism.

Given a functor R : B → A, we write Proj(R,B) for the full subcategory of B given by R-projective objects. The following is dual to 1.13.

1.17. Characterisation of T-Galois modules. Assume the categoryBto have equalisers.

Let T= (T, m, e)be a monad onA, andR:B→Aa left T-module functor with left adjoint F :A→B(and unit η, counitε). If there exists a functor R:B→AT with U_TR=R, then the following are equivalent:

(a) R isT-Galois;

(b) the following composition is an isomorphism:

φT

φ_Tη //φTRF =φTUTRF ^εTRF//RF ;

(c) the functor R : B → AT restricts to an equivalence between the categories Proj(R,B) andProj(U_T,AT);

(d) for any (a, h_a) ∈ Proj(U_T,AT), the (a, h_a)-component of the unit η of the adjunction LaR, is an isomorphism;

(e) for any a∈A,η_φ

T(a)=η_{(T(a), m}

a)is an isomorphism.

Dual to 1.4 we observe:

1.18. Left adjoint for R. If B admits coequalisers of reflexive pairs, then the functor R admits a left adjoint.

Proof. Let (R, α:T R→R) be a leftT-module with a left adjointF :B→A. Consider the composite

β: F T

F t_R

//F RF ^εF //F ,

where ε:F R→Iis the counit ofF aR. It is easy to check that (F, β) is a rightT-module.

According to [12, Theorem A.1], when a coequaliser (R, i) exists for the diagram of functors F U_Tφ_TU_T =F T U_T ^{F UTεT} //

βUT

//F U_T, (1.4)

where ε_T :φ_TU_T →I is the counit of φ_T aU_T, then R is left adjoint toR: B→AT. It is easy to see that for any (a, h_a)∈AT, the (a, h_a)-component in the diagram 1.4 is the pair

F T(a)

F(ha) //

βa

//F(a) (1.5)

which is a reflexive pair sinceβa·F(ea) =F(ha)·F(ea) =I. This proves our claim tu So far we have dealt with (co)module structures on functors. It is also of interest to consider the corresponding relations between monads and comonads.

1.19. Definitions. Let T= (T, m, e) be a monad and G= (G, δ, ε) a comonad on A. We say that GisT-Galois, if there exists a leftT-module structureα:T G→Gon the functor Gsuch that the composite

γ^G:T G ^{T δ} //T GG ^αG //GG is an isomorphism.

Dually,TisG-Galois, if there is a leftG-comodule structureβ:T →GT on the functor T such that the composite

γT :T T ^βT //GT T ^Gm //GT is an isomorphism.

We need the following (dual of [24, Lemma 21.1.5])

1.20 Proposition. Let η, ε:F aR:C→Aand η⁰, ε⁰ :F⁰ aR⁰:C→B be adjunctions and let

A

F?????

??^X //B

F⁰

C

be a diagram of categories and functors with F⁰X=F. Writeαfor the composition XR ^η

0XR//R⁰F⁰XR=R⁰F R ^R

0ε //R⁰.

Then the natural transformation SX =F⁰α:F R=F⁰XR→F⁰R⁰ is a morphism of comon- ads.

Note that for the commutative diagram (see 1.1) B

F //

F@@@@@@

@@ A^G

U^G

A,

where F has a right adjoint R, the related comonad morphism S_F : F R → G is just the comonad morphismt_F :F R→G.

From the proof of [24, Theorem 21.1.10(b)] we obtain:

1.21 Proposition. Let F a R : D → A, F⁰ a R⁰ : D → B and F⁰⁰ a R⁰⁰ : D → C be adjunctions and let

A

F??????

? ^X //B

F⁰

Y //C

F⁰⁰



D

be a commutative diagram of categories and functors. Write SX for the comonad morphism F R→F⁰R⁰,SY for the comonad morphismF⁰R⁰ →F⁰⁰R⁰⁰ andSY X for the comonad mor- phism F R→F⁰⁰R⁰⁰ that exist according to the previous proposition. Then SY X =SYSX.

2 Entwinings

2.1. Entwinings. We fix a mixed distributive law, also called anentwining, λ:T G→GT from the monadT = (T, m, e) to the comonad G= (G, δ, ε), and writeTb = (T ,b m,b be) for a monad on A^G liftingT, andGb = (G,b bδ,bε) for a comonad onAT liftingG(e.g. [28, Section 5]).

It is well-known that for any object (a, h_a) ofAT,

• G(a, hb a) = (G(a), G(ha)·λa), • (bδ)_(a,ha)=δa, • (bε)_(a,ha)=εa, while for any object (a, θa) of the categoryA^G,

• T(a, θb a) = (T(a), λa·T(θa)); • (m)b _(a,θa)=ma, • (be)_(a,θa)=ea, and that there is an isomorphism of categories

(A^G)

Tb'(AT)^Gb.

We writeA^GT(λ) (or justA^GT,when the mixed distributive lawλis understood) for the category whose objects are triples (a, h_a, θ_a), where (a, h_a) ∈ AT and (a, θ_a) ∈ A^G with commuting diagram

T(a) ^ha //

T(θa)

a ^θa //G(a)

T G(a)

λa

//GT(a).

G(ha)

OO (2.1)

LetK:A→(A^G)

Tb be a functor inducing a commutative diagram A

K //

φD^GDDDDDD""

DD (A^G)

Tb U

Tb

A^G.

(2.2)

WriteαK :T φb ^G →φ^G for the correspondingT-module structure onb φ^G (see 1.14). SinceTb is the lifting of Tcorresponding toλ,U^GTb=T U^G and one has the natural transformation

α=U^G(αK) :U^GT φb ^G=T U^Gφ^G =T G−→U^Gφ^G=G.

It is easy to see thatαprovides a leftT-module structure onGwith commutative diagram T G ^α //

T δ

G ^δ //GG

T GG λG //GT G.

Gα

OO (2.3)

Conversely, a natural transformation

α:U^GT φb ^G=T U^Gφ^G=T G−→U^Gφ^G=G

makingGa left T-module, can be lifted to a leftT-module structure onb φ^G if and only if for every a∈A, αa :T G(a)→G(a) is a morphism inA^G from the G-coalgebra (T G(a), λ_G(a)· T(δa)) to theG-coalgebra (G(a), δa), which is just to say that thea-component of the diagram (2.3) commutes. Thus we have proved:

2.2 Proposition. With the data given in 2.1, the assignment K:A→(A^G)

Tb 7−→ U^G(αK) :T G→G,

yields a bijection between functors K making the diagram (2.2) commute and left T-module structuresα:T G→Gon Gfor which the diagram (2.3) commutes.

Now letK⁰:A→(AT)^Gb be a functor inducing a commutative diagram A

K⁰ //

φDTDDDDDD!!

DD (AT)^Gb

U^Gb

A_T.

(2.4)

Writeβ_K⁰ :φ_T →Gφb _T for the correspondingG-comodule structure onb φ_T (see 1.1). One has the natural transformation

β=UT(βK⁰) :UTφT =T →UTGφb T =GUTφT =GT which induces a G-comodule structure onT with commutative diagram

T T ^m //

T β

T ^β //GT

T GT λT //GT T.

Gm

OO (2.5)

From this we obtain:

2.3 Proposition. In the situation described above, the assignment K⁰ :A→(AT)^Gb 7−→ UT(βK⁰) :T →GT,

yields a bijection between functorsK⁰ making the diagram (2.4) commute and leftG-comodule structuresβ :T →GT on the functorT for which the diagram (2.5) commutes.

To give a functorK⁰:A→(AT)^Gb making the diagram (2.4) commute is to give a natural transformationα:UT →UTGbmakingUT a rightG-comodule (see [14, Proposition 2.1]). Forb any (a, ha)∈AT,G(a, hb a) = (G(a), G(ha)·λa), the (a, ha)-componentα_(a,ha) is a morphism a→G(a) inAwith commutative diagrams

a AA AA AA AA A

AA AA AA AA A

α_(a,ha)

//G(a)

ε_a

a

α_(a,ha)

α_(a,ha)

//G(a)

δ_G(a)

a, G(a)

G(α_(a,ha)) //GG(a),

and the corresponding comonad morphismt_K⁰ :φ_TU_T →Gbis the composite φ_TU_T ^φTα //φTUTGb ^εTGb //

G .b

Then, since for any (a, ha)∈AT, (εT)_(a,ha)=ha, the component (tK⁰)_(a,ha) is the composite T(a) ^T(α(a,ha)) //T G(a) ^λa //GT(a) ^G(ha)//G(a).

Now it follows from Proposition 1.2:

2.4 Theorem. In the situation described above, the functorK⁰ is an equivalence of categories if and only if for any (a, ha)∈ AT, the composite G(ha)·λa·T(α(a,h_a)) is an isomorphism and the functorφT is comonadic.

For the dual situation, letK : A → (A^G)

Tb be a functor inducing commutativity of the diagram (2.2). Since the functorφ^G has a left adjointU^G :A^G→A, it follows from [14] that to give such a functor is to give a rightTb-module structureα:U^GTb→U^G onU^G.

For any (a, θ_a) ∈ A^G, Tb(a, θ_a) = (T(a), λ_a ·T(θ_a)), the (a, θ_a)-component α_(a,θa) is a morphismT(a)→ainAwith commutative diagrams

a AA AA AA AA A

AA AA AA AA

A ^ea //T(a)

α_(a,θa)

T T(a)

ma

T(α_(a,θa))

//T(a)

α_(a,θa)

a, T(a) _α

(a,ha)

//a,

and the corresponding monad morphismtK :φ^GU^G→Tbis the composite

Tb

η^GTb //φ^GU^GTb ^φ

Gα //φ^GU^G.

Now, since for any (a, θ_a)∈A^G, (η^G)_(a,θa)=θ_a, the component (t_K)_(a,θa)is the composite T(a) ^T(θa) //T G(a) ^λa //GT(a) ^G(α(a,ha)) //G(a).

As a consequence we get from Proposition 1.15:

2.5 Theorem. In the situation described above, the functor Kis an equivalence of categories if and only if for any (a, θa) ∈A^G, the composite G(α_(a,ha))·λa·T(θa) is an isomorphism and the functorφ^G is monadic.

The following observation is probably known but we are not aware of a suitable reference.

Recall that a functori:C→Awith Ca small category isdense, if the functor ei:A^op→[C,Set], a7→Mor_A(i(−), a),

is full and faithful.

2.6 Lemma. Let i:C→Abe a dense functor. Given two adjunctions F aU, F⁰ aU⁰:A→B,

and a natural transformationτ:F →F⁰, thenτ is an isomorphism of functors if and only if τ i:F i→F⁰i is so.

Proof. Writeτ⁰ :U⁰ →U for the natural transformation corresponding to τ, that is τ andτ⁰ are mates, denoted byτ aτ⁰ (e.g. [21, 7.1], [4, 2.2]). Thenτ is an isomorphism if and only ifτ⁰ is so. So it is enough to show thatτ⁰ is an isomorphism. Sinceτaτ⁰, the diagram

Mor_B(F⁰i(a), b)

α⁰_{i(a), b}

//

Mor_B(τ_i(a), b)

Mor_A(i(a), U⁰(b))

Mor_A(i(a),τ_b⁰)

Mor_B(F i(a), b) _α

i(a), b

//Mor_A(i(a), U(b)),

where α (resp. α⁰) is the bijection corresponding to the adjunction F aU (resp. F⁰ aU⁰), commutes for all a, b ∈A. Since τ_i(a) is an isomorphism by our assumption onτ, it follows that the natural transformation Mor_A(i(a), τ_b⁰) is an isomorphism, implying - sinceiis dense

- that τ⁰:U⁰ →U is an isomorphism. tu

2.7 Proposition. With the data given in 2.1, letK⁰:A→(AT)^Gbbe a functor with U^GbK⁰ = φ_T andβ_K⁰ : φ_T →Gφb _T the corresponding G-comodule structure onb φ_T (see 1.1). Suppose that

(i) A admits equalisers of coreflexive pairs and bothT andGhave right adjoints, or (ii) A admits small colimits and bothT andGpreserve them.

Then(φ_T, β_K⁰)isG-Galois if and only ifb (T, U_T(β_K⁰))isG-Galois.

Proof. For any a ∈ A, the φ_T(a) = (T(a), m_a)-component of t_K⁰ : φ_TU_T → Gb is just (γ_T)_a (see 1.19). Thus it is enough to show that t_K⁰ is an isomorphism if and only if its restriction to freeT-modules is.

(i) IfT has a right adjoint, there exists a comonadHinducing an isomorphism of categories A^T ' A^H; this implies that the functor UT is comonadic and hence has a right adjoint. It follows that the compositeGUT also has a right adjoint. Next, sinceGb is the lifting ofG, we have the commutative diagram

AT Gb //

UT

AT UT

A _G //A. Since

• GUT has a right adjoint,

• the functorU_T is comonadic, and

• AT admits equalisers of coreflexive pairs (sinceAdoes so),

it follows from the dual of [12, Theorem A.1] that the functorGb has a right adjoint.

Now, since the full subcategory of AT given by free T-modules is dense in AT, it follows from Lemma 2.6 thattK⁰ :φTUT →Gb is an isomorphism if and only if its restriction to free T-modules is.

(ii) SinceT preserves colimits, the categoryAT admits colimits and the functorU_T :AT → Acreates them. Thus

• the functorφ_TU_T preserves colimits;

• any functorL:B→AT preserves colimits if and only if the compositeUTLdoes; so, in particular, the functorGbpreserves colimits, sinceUTGb=T UT andT UT is the composite of two colimit-preserving functors.

The full subcategory of AT given by the free T-modules is dense and since the functors φTUT and Gb both preserve colimits, it follows from [24, Theorem 17.2.7] that the natural transformation

tK⁰ :φTUT →Gb

is an isomorphism if and only if its restriction to the free T-modules is so; i.e. if (tK⁰)_φT(a)is an isomorphism for alla∈A. This completes the proof. tu

Dually, one has

2.8 Proposition. With the data given in 2.1, letK:A→(A^G)

Tbbe a functor withU

TbK=φ^G andαK :T φb ^G→φ^G the correspondingT-module structure onb φ^G. Suppose that

(i) A admits coequalisers of reflexive pairs and bothT andGhave left adjoints, or (ii) A admits all small limits and bothT andGpreserve them.

Then(φ^G, αK)isT-Galois if and only ifb (G, U^G(αK))isT-Galois.

The results of the preceding two propositions may be compared with B¨ohm and Menini’s [5, Theorem 3.3].

3 Grouplike morphisms

In this section we extend the theory of Galois coringsCover a ringAto entwinings of a monad F and a comonadGon general categories. For this we extend the notion of a grouplike element in C(e.g. [9, 28.1]) to the notion of a grouplike natural transformation I→G.

3.1. Definition. LetG= (G, δ, ε) be a comonad on a categoryA. A natural transformation g:I→Gis called agrouplike morphismprovided it induces commutative diagrams

I ^g //

=>>>>>

>> G

ε

I,

I ^g //

ggCCCCCC!!

CC G

δ

GG.

Comonads with grouplike morphisms are calledcomputational in [6] (see also [23]). The next result transfers Proposition 5.1 in [18].

3.2. Grouplike morphisms and comodule structure. Let F = (F, m, e) be a monad and G= (G, δ, ε) a comonad on a category Awith an entwining λ:F G→GF. If Ghas a grouplike morphismg:I→G, thenF has two leftG-comodule structures (see 1.1) given by

(1) g˜:F^{F g} //F G ^λ //GF and (2) gF :F →GF.

Proof. (1) In the diagram

F ^{F g} //

=BBBBBB!!

BB F G

F ε

λ //GF

εF

F ₌ //F,

the triangle is commutative by the grouplike properties of g and the square is commutative by the properties of the entwiningλ. In the diagram

F ^{F g} //

F g

F gg

##G

GG GG GG

GG F G ^λ //

F δ

GF

δF

F G

λ

F GG

λG

GF ^{GF g}//GF G ^Gλ //GGF,

the right rectangle is commutative by properties of entwinings, the triangle is commutative by properties of the grouplike morphismg, and the pentagon is commutative by naturality of composition. This shows that ˜g makesF a leftG-comodule.

(2) To say that (F, gF :F →GF) is a leftG-comodule is to say that the diagrams F ^gF //

=BBBBBB!!

BB GF

εF

F,

F ^gF //

gF

GF

δF

GF GgF//GGF.

are commutative. Using the fact that

GgF·gF =ggF,

the commutativity of these diagrams follows from the definition of a grouplike morphism. tu The pattern of the proof of [18, Proposition 5.3] also yields:

3.3. F as mixed bimodule. With the data given in 3.2,(F, m,˜g)is a mixed(F, G)-bimodule.

Proof. We need to show commutativity of the diagram F F ^m //

Fg˜

F ^˜g //GF

F GF ^λF //GF F.

Gm

OO

However, by the definition of ˜g, we get the diagram F F ^m //

F F g

F ^{F g} //F G ^λ //GF

F F G

F λ //

mG

55k

kk kk kk kk kk kk kk kk

F GF ^λF //GF F,

Gm

OO

in which the right pentagon is commutative sinceλis an entwining and the triangle is com- mutative by naturality of composition. This proves our claim. tu Combining 2.3, 3.2 and 3.3 yields the existence of a functorKg :A→(A^F)^Gb making the diagram

A

K_g

//

φDFDDDDDD!!

DD (AF)^Gb

U^Gb

AF

commute. Note thatKg(a) = ((F(a), ma),g˜a).

Now assume thatA admits equalisers. Then the category of endofunctors of A also has equalisers and we have the

3.4. Equaliser functor. With the data given in 3.2, define a functorF^g as an equaliser of functors

F^{g iF} //F ^gF //

F gDDDDDDD!!

D GF

F G.

λ

<<

xx xx xx xx

ThenF^g is a monad onA andiF :F^g→F is a monad morphism.

Proof. We adapt the proof of [18, 5.2]. The following two diagrams are commutative by naturality of composition,

I ^e //

g

F

gF

G Ge //GF,

I ^e //

g

F

F g

G

eG //F G.

Sinceλ·eG=Ge, it follows that

λ·F g·e=λ·eG·g=Ge·g=gF·e.

Thus there exists a unique morphisme⁰ :I→F^g yielding a commutative diagram F^{g iF} //F

I

e⁰

OO

e

>>

||

||

||

|| .

Observe that

(α) the diagrams F F ^{F F g}//

m

F F G

mG

F _{F g} //F G,

F F ^{gF F}//

m

GF F

Gm

F gF //GF commute by naturality of composition,

(β) λ·mG=Gm·λF·F λ, sinceλis an entwining;

(γ) λ·F g·i_F =gF ·i_F, sincei_F is an equaliser ofgF andλ·F g, (δ) iFiF =iFF·F^giF =F iF·iFF^g, by naturality of composition.

Hence we have

λ·F g·m·i_Fi_F =_(α) λ·mG·F F g·i_Fi_F

=_(β) Gm·λF ·F λ·F F g·i_Fi_F

=_(δ) Gm·λF ·F λ·F F g·F i_F·i_FF^g

=_(γ) Gm·λF ·F gF·F i_F·i_FF^g

=_(δ) Gm·λF ·F gF·iFF·F^giF

=_(γ) Gm·gF F·iFF·F^giF

=_(δ) Gm·gF F·iFiF

=_(α) gF ·m·iFiF. Considering now the diagram

F^gF^g

m⁰

i_Fi_F

//F F

m

gF F

''

F gF //F GF ^λF //GF F

Gm

F^g _i

F

//F

gF

''F g //F G ^λ //GF ,

one sees that there exists a unique morphismm⁰ :F^gF^g→F^g making the left square of the diagram commute. The result now follows from [3, Lemma 3.2]. tu As we have seen, the morphismβ = ˜g: F →GF makesF a left G-comodule. Consider the related functorK_g :A→(AF)^Gband writet:φ_FU_F →Gbfor the corresponding morphism of comonads onAF. It is easy to see that for any (a, h_a)∈AF,t_(a,ha) is the composite

F(a) ^F(ga)//F G(a) ^λa //GF(a) ^G(ha)//G(a). (3.1) SinceF g·e=eG·gby naturality of composition andλ·eG=Ge, the (a, h_a)∈AF-component of the morphism

γ:UF

η^FUF//UFφFUF

U_Ft //U_FGb

is just the morphism g_a :a→G(a). It follows that the monad generated by the functor K_g and its right adjointR_g is given by the equaliser of the diagram

F

gF //

˜g=λ·F g //GF.

ThusF^g is just the monad onAgenerated by the adjunctionKgaRg.

Since any functor with a right adjoint is full and faithful if and only if the unit of the adjunction is an isomorphism, we have the

3.5 Proposition. Let g :I →G be a grouplike morphism. Then the corresponding functor Kg:A→(AF)^Gbis full and faithful if and only if the functorF^g is (isomorphic to) the identity monad on A.

For an entwiningλ:T G→GT and a grouplike morphismg:I→G, for any (a, ha)∈AF, the (a, h_a)-component t_(a,ha) of the comonad morphismt:φ_FU_F →G, corresponding to theb functorKg:A→(AF)^Gb, is given in (3.1). Consider the diagram

F(a)

(1) F(ea) //

F(g_a)

F F(a)

(2) F(g_F(a))

F(a)

e_F(a)

oo

g_F(a)

F G(a)

(3) F G(e_a) //

λ_a

F GF(a)

(4) λ_F(a)

GF(a)

e_GF(a)

oo

G(e_F(a))

vvmmmmmmmmmmmmm

GF(a) ^GF(ea) //

(5)

QQ QQ QQ QQ QQ QQ Q

QQ QQ QQ QQ QQ QQ

Q GF F(a)

G(ma)

(6)

GF(a) GF(a), in which

• diagram (1) is commutative by naturality ofg:I→G;

• diagram (2) is commutative by naturality of composition;

• diagram (3) is commutative by naturality ofλ:F G→GF;

• diagram (4) is commutative sinceλis an entwining, and

• diagrams (5) and (6) are commutative sinceF is a monad.

It follows from the commutativity of this diagram that the diagram F(a)

e_F(a)

//

F(ea) //F F(a)

F(g_F(a))

F GF(a)

λ_F(a)

GF F(a)

G(m_a)

F(a)

g_F(a)

//

λ_a·F(ga) //GF(a)

(3.2)

is serially commutative.

3.6 Proposition. Let λ : T G → GT be an entwining and g : I → G be a grouplike mor- phism. If the monad F is of descent type (that is, the free F-algebra functorφ_F :A→AF is precomonadic) and if the monad F isG-Galois w.r.t. theG-coaction eg :F →GF (see 3.2), then the monadF^g is (isomorphic to) the identity monad.

Proof. To say thatF is of descent type is to say that the diagram a ^ea //F(a)

e_F(a)

//

F(e_a) //F F(a)