Regular pairings of functors and weak (co)monads

For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B, MorB(L(A), B) ↔ MorA(A, R(B)). (L, R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a...

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Дата:	2013
Автор:	Wisbauer, R.
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Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2013
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	Regular pairings of functors and weak (co)monads / R. Wisbauer // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 127–154. — Бібліогр.: 23 назв. — англ.

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spelling	irk-123456789-1522582019-06-10T01:25:47Z Regular pairings of functors and weak (co)monads Wisbauer, R. For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B, MorB(L(A), B) ↔ MorA(A, R(B)). (L, R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category B, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors (L, R, α, β) with α = α ⋅ β ⋅ α and β = β ⋅ α ⋅ β. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on A gives rise to a regular pairing between A and the category of compatible (co)modules. 2013 Article Regular pairings of functors and weak (co)monads / R. Wisbauer // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 127–154. — Бібліогр.: 23 назв. — англ. 1726-3255 2010 MSC:18A40, 18C20, 16T15. http://dspace.nbuv.gov.ua/handle/123456789/152258 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B, MorB(L(A), B) ↔ MorA(A, R(B)). (L, R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category B, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors (L, R, α, β) with α = α ⋅ β ⋅ α and β = β ⋅ α ⋅ β. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on A gives rise to a regular pairing between A and the category of compatible (co)modules.
format	Article
author	Wisbauer, R.
spellingShingle	Wisbauer, R. Regular pairings of functors and weak (co)monads Algebra and Discrete Mathematics
author_facet	Wisbauer, R.
author_sort	Wisbauer, R.
title	Regular pairings of functors and weak (co)monads
title_short	Regular pairings of functors and weak (co)monads
title_full	Regular pairings of functors and weak (co)monads
title_fullStr	Regular pairings of functors and weak (co)monads
title_full_unstemmed	Regular pairings of functors and weak (co)monads
title_sort	regular pairings of functors and weak (co)monads
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/152258
citation_txt	Regular pairings of functors and weak (co)monads / R. Wisbauer // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 127–154. — Бібліогр.: 23 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT wisbauerr regularpairingsoffunctorsandweakcomonads
first_indexed	2025-07-13T02:40:33Z
last_indexed	2025-07-13T02:40:33Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 15 (2013). Number 1. pp. 127 – 154 c© Journal “Algebra and Discrete Mathematics” Regular pairings of functors and weak (co)monads Robert Wisbauer Abstract. For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B, MorB(L(A), B) α // MorA(A,R(B)) β oo . (L,R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category B, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors (L,R, α, β) with α = α · β · α and β = β · α · β. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on A gives rise to a regular pairing between A and the category of compatible (co)modules. Contents. 1. Introduction; 2. Pairings of functors; 3. Monads and modules; 4. Comonads and comodules; 5. Entwining monads and comonads; 6. Lifting of endofunctors to modules and comodules; 7. Mixed entwinings and liftings. 2010 MSC: 18A40, 18C20, 16T15. Key words and phrases: pairing of functors; adjoint functors; weak monads and comonads; r-unital monads; r-counital comonads; lifting of functors; distributive laws. 128 Regular pairings 1. Introduction Similar to the unit of an algebra, the existence of a unit of a monad is essential for (most of) the interesting properties of the related structures. Yet, there are numerous applications for which the request for a unit of a monad is too restrictive. Dropping the unit completely makes the theory fairly poor and the question was how to weaken the conditions on a unit such that still an effective theory can be developed. The interest in these questions was revived, for example, by the study of weak Hopf algebras by G. Böhm et al. in [6] and weak entwining structures by S. Caenepeel et al. in [9] (see also [1], [8]). To handle this situation the theory of weak monads (and comonads) was developed and we refer to [5] for a recent account on this. On any category, monads are induced by a pair of adjoint functors and, on the other hand, any monad (F, µ, η) induces an adjoint pair of functors, the free functor φF : A → AF and the forgetful functor UF : AF → A, where AF denotes the category of unital F -modules. This is all shown in Eilenberg-Moore [10]. In this correspondence the unitality of the monad is substantial and the purpose of the present paper is to exhibit a similar relationship between weak (co)monads and generalised forms of adjunctions. To this end, for functors L : A → B and R : B → A between categories A and B, we consider maps MorB(L(A), B) α // MorA(A,R(B)) β oo , required to be natural in A ∈ A and B ∈ B. We call this a pairing of functors, or a full pairing if we want to stress that we have maps in both directions. Such a pairing is said to be regular provided α and β are regular maps, more precisely, α = α · β · α and β = β · α · β. In Section 2, regular pairings of functors are defined and some of their general properties are described. Motivated by substructures showing up in pairings of functor, in Section 3.1, q-unital monads (F, µ, η) on A are defined as endofunc- tors F : A → A with natural transformations µ : FF → F and η : IA → F (quasi-unit) and the sole condition that µ is associative. (Non- unital) F -modules are defined by morphisms ̺ : F (A) → A satisfying R. Wisbauer 129 ̺ ◦ µA = ̺ ◦ F (̺), and the category of all F -modules is denoted by A −→F . For these data the free and forgetful functors, φF : A → A −→F and UF : A −→F → A. give rise to a full pairing. From this we define regularity of η and compati- bility for the F -modules. The q-unital monad (F, µ, η) is said to be r-unital (short for regular-unital) provided η is regular and µ is compatible as an F -module. Now the free functor φF : A → AF with the forgetful functor UF : AF → A form a regular pairing, where AF denotes the (sub)category of compatible F -modules. The dual notions for (non-counital) comonads are outlined in Section 4 and at the end of the section the comparison functors for a regular pairing (L,R, α, β) are considered (see 4.10). In Section 5 we study the lifting of functors between categories to the corresponding categories of compatible modules or compatible comodules, respectively. This is described by generalising Beck’s distributive laws (see [2]), also called entwinings, and it turns out that most of the diagrams are the same as for the lifting to unital modules (e.g. [22]) but to compensate the missing unitality extra conditions are imposed on the entwining natural transformation (e.g. Proposition 5.2). In this context we obtain a generalisation of Applegate’s lifting theorem for (co)monads to weak (co)monads (Theorem 5.4, 5.8). Lifting an endofunctor T of A to an endofunctor T of AF leads to the question when T is a weak monad (TF allows for the structure of a weak monad) and in Section 6 we provide conditions to make this happen. The final Section 7 is concerned with weak monads (F, µ, η) and weak comonads (G, δ, ε) on any category A and the interplay between the respective lifting properties. Hereby properties of the lifting G to AF and the lifting F̂ to A G are investigated (see Theorems 7.9 and 7.10) which generalise observations known for weak bi-algebras (and weak Hopf algebras). In our setting, notions like pre-units, pre-monads, weak monads, demi- monads, pre-A-corings, weak corings, weak Hopf algebras from the litera- ture (e.g. [1], [3], [7], [4], [21]) find their natural environment. In the framework of 2-categories weak structures are investigated by Böhm et al. in [3], [4], [5] and an extensive list of examples of weak structures is given there. 130 Regular pairings 2. Pairings of functors Throughout A and B will denote arbitrary categories. By IA, A or just by I, we denote the identity morphism of an object A ∈ A, IF or F stands for the identity natural transformation on the functor F , and IA means the identity functor of a category A. We write F−,− for the natural transformation of bifunctors determined by the maps FA,A′ : MorA(A,A′) → MorB(F (A), F (A′)) for A,A′ ∈ A. Before considering regularity for natural transformations we recall basic properties of 2.1. Regular morphisms. Let A,A′ be any objects in a category A. Then a morphism f : A → A′ is called regular provided there is a morphism g : A′ → A with fgf = f . Clearly, in this case gf : A → A and fg : A′ → A′ are idempotent endomorphisms. Such a morphism g is not necessarily unique. In particular, for gfg we also have f(gfg)f = fgf = f , and the identity (gfg)f(gfg) = gfg shows that gfg is again a regular morphism. If idempotents split in A, then every idempotent morphism e : A → A determines a subobject of A, we denote it by eA. If f is regular with fgf = f , then the restriction of fg is the identity morphism on fgA′ and gf is the identity on gfA. Examples for regular morphisms are retractions, coretractions, and isomorphisms. For modules M,N over any ring, a morphism f : M → N is regular if and only if the image and the kernel of f are direct summands in N and M , respectively. This notion of regularity is derived from von Neumann regularity of rings. For modules (and in preadditive categories) it was considered by Nicholson, Kasch, Mader and others (see [14]). We use the terminology also for natural transformations and functors with obvious interpretations. 2.2. Pairing of functors. (e.g. [19, 2.1]) Let L : A → B and R : B → A be covariant functors. Assume there are morphisms, natural in A ∈ A and B ∈ B, α : MorB(L(A), B) → MorA(A,R(B)), β : MorA(A,R(B)) → MorB(L(A), B). These maps correspond to two natural transformations between functors A op × B → Set. The quadruple (L,R, α, β) is called a (full) pairing (of functors). Given such a pairing, the morphisms, for A ∈ A, B ∈ B, R. Wisbauer 131 ηA := αA,L(A)(I) : A → RL(A) and εB := βR(B),B(I) : LR(B) → B correspond to natural transformations η : IA → RL, ε : LR → IB, which we call quasi-unit and quasi-counit of (L,R, α, β), respectively. From these the transformations α and β are obtained by αA,B : L(A) f −→ B 7−→ A ηA −→ RL(A) R(f) −→ R(B), βA,B : A g −→ R(B) 7−→ L(A) L(g) −→ LR(B) εB −→ B. Thus a pairing (L,R, α, β) is also described by a quadruple (L,R, η, ε). Naturality of ε and η induces an associative product and a quasi-unit for the endofunctor RL : A → A, RεL : RLRL → RL, η : IA → RL, and a coassociative coproduct and a quasi-counit for the endofunctor LR : B → B, LηR : LR → LRLR, ε : LR → IB. By the Yoneda Lemma we can describe compositions of α and β by the images of the identity transformations of the respective functors. 2.3. Composing α and β. Let (L,R, α, β) be a pairing with quasi-unit η and quasi-counit ε. The descriptions of α and β in 2.2 yield, for the identity transformations IL : L → L, IR : R → R, α(IL) = IA η −→ RL, β · α(IL) = L Lη −→ LRL εL −→ L, α · β · α(IL) = IA η −→ RL RLη −→ RLRL RεL −→ RL, β(IR) = LR ε −→ IB, α · β(IR) = R ηR −→ RLR Rε −→ R, β · α · β(IR) = LR LηR −→ LRLR LRε −→ LR ε −→ IB. The following morphisms will play a special role in what follows. 132 Regular pairings 2.4. Natural endomorphisms. With the notions from 2.2, we define the natural transformations ϑ := R(βα(IL)) : RL RLη // RLRL RεL // RL, ϑ := αβ(R(IL)) : RL ηRL // RLRL RεL // RL, γ := L(αβ(IR)) : LR LηR // LRLR LRε // LR, γ := βα(L(IR)) : LR LηR // LRLR εLR // LR, which have the properties RεL ·RLϑ = ϑ ·RεL, RεL · ϑRL = ϑ ·RεL, ϑ · ϑ = ϑ · ϑ; LRγ · LηR = LηR · γ, γLR · LηR = LηR · γ, γ · γ = γ · γ. 2.5. Definitions. Let (L,R, α, β) be a pairing (see 2.2). We call α regular if α · β · α = α; α symmetric if ϑ = ϑ. β regular if β · α · β = β; β symmetric if γ = γ; (L,R, α, β) regular if α = α · β · α and β = β · α · β. The following properties are easy to verify: (i) If α is regular, then β · α(IL), ϑ, and ϑ are idempotent and ϑ · η = η = ϑ · η; furthermore, for β′ := β · α · β, (L,R, α, β′) is a regular pairing. (ii) If β is regular, then α · β(IR), γ, and γ are idempotent and ε · γ = ε = ε · γ; furthermore, for α′ := α · β · α, (L,R, α′, β) is a regular pairing. Any pairing (L,R, α, β) with β · α = I or α · β = I is regular. The second condition defines the semiadjoint functors in Medvedev [16]. With manipulations known from ring theory one can show how pair- ings with regular components can be related with adjunctions provided idempotents split. 2.6. Related adjunctions. Let (L,R, α, β) be a pairing (with quasi-unit η, quasi-counit ε) and assume α to be regular. If the idempotent h := β · α(IL) : L Lη −→ LRL εL −→ L splits, that is, there are a functor L : A → B and natural transformations R. Wisbauer 133 p : L → L, i : L → L with i · p = h and p · i = IL, then the natural transformations η : IA η // RL Rp // RL , ε : LR iR // LR ε // IB , as quasi-unit and quasi-counit, define a pairing (L,R, α, β) with β ·α = I. If α · β = I, then (L̂, R, α, β) is an adjunction. In case the natural transformation β is regular, similar constructions apply if we assume that the idempotent α · β(IR) : R ηR −→ RLR Rε −→ R splits. The properties of the (RL,RεR, η) and (LR,LηR, ε) mentioned in 2.2 motivate the definitions in the next section. 3. Monads and modules 3.1. q-unital monads and their modules. We call (F, µ) a functor with product (or non-unital monad) provided F : A → A is an endofunctor on a category A and µ : FF → F is a natural transformation satisfying the associativity condition µ · Fµ = µ · µF . For (F, µ), a (non-unital) F -module is defined as an object A ∈ A with a morphism ̺ : F (A) → A in A satisfying ̺ · F̺ = ̺ · µA. Morphisms between F -modules (A, ̺), (A′, ̺′) are morphisms f : A → A′ in A with ̺′ ·F (f) = f ·̺. The set of all these is denoted by MorF (A,A′). With these morphisms, (non-unital) F -modules form a category which we denote by A −→F . By the associativity condition on µ, for every A ∈ A, (F (A), µA) is an F -module and this leads to the free functor and the forgetful functor, φF : A → A −→F , A 7→ (F (A), µA), UF : A −→F → A, (A, ̺) 7→ A. A triple (F, µ, η) is said to be a q-unital monad on A provided (F, µ) is a functor with product and η : IA → F is any natural transformation, called a quasi-unit (no additional properties are required). One always can define natural transformations ϑ : F Fη −→ FF µ −→ F, ϑ : F ηF −→ FF µ −→ F. Note that for any A ∈ A, ϑA is an F -module morphism whereas ϑA is not necessarily so. 134 Regular pairings Given q-unital monads (F, µ, η), (F ′, µ′, η′) on A, a natural transfor- mation h : F → F ′ is called a morphism of q-unital monads if µ′ · hh = h · µ and η′ = h · η. The existence of a quasi-unit allows the following generalisation of the Eilenberg-Moore construction for (unital) monads. 3.2. q-unital monads and pairings. For a q-unital monad (F, µ, η) we obtain a pairing (φF , UF , αF , βF ) with the maps, for A ∈ A, (B, ̺) ∈ A −→F , αF : MorF (φF (A), B) → MorA(A,UF (B)), f 7→ f · ηA, βF : MorA(A,UF (B)) → MorF (φF (A), B), g 7→ ̺ · F (g). The quasi-unit η is called regular if αF is regular, that is, IA η −→ F = IA η −→ F Fη −→ FF µ −→ F, and we say η is symmetric if αF is so, that is, ϑ = ϑ. An F -module ̺ : F (A) → A in A −→F is said to be compatible (with η) if βFαF (̺) = ̺, that is F (A) ̺ −→ A = F (A) FηA −→ FF (A) µA −→ F (A) ̺ −→ A. In particular, the natural transformation µ : FF → F is compatible if FF µ −→ F = FF FηF −→ FFF µF −→ FF µ −→ F. It is easy to see that this implies FF ϑϑ −→ FF µ −→ F = FF µ −→ F. Let AF denote the full subcategory of A −→F made up by the compatible F -modules. If µ is compatible, the image of the free functor φF lies in AF and (by restriction or corestriction) we get the functor pair (keeping the notation for the functors) φF : A → AF , UF : AF → A, and a pairing (φF , UF , αF , βF ) between A and AF . Since for (A, ̺) in A −→F , βF (IUF (A)) = ̺, the compatibility condition on ̺ implies that β · α · β(̺) = β(̺), i.e., β is regular in (φF , UF , αF , βF ) when restricted to AF . R. Wisbauer 135 3.3. Definition. A q-unital monad (F, η, µ) is called r-unital if η is regular and µ is compatible; weak monad if (F, η, µ) is r-unital and η is symmetric. Summarising the observations from 3.2 we have: 3.4. Proposition. Let (F, µ, η) be a q-unital monad. (1) The following are equivalent: (a) (F, µ, η) is an r-unital monad; (b) (φF , UF , αF , βF ) is a regular pairing of functors between A and AF . (2) The following are equivalent: (a) (F, µ, η) is a weak monad; (b) (φF , UF , αF , βF ) is a regular pairing between A and AF with αF symmetric. A quasi-unit η that is regular and symmetric is named pre-unit in the literature (e.g. [11, Definition 2.3]); for the notion of a weak monad (also called demimonad) see e.g. [5]. In case η is a unit, q-unital monads, r-unital monads and weak monads all are (unital) monads. In (non-unital) algebras over commutative rings, r-unital monads are obtained from idempotents while weak monads correspond to central idempotents (see 3.7). 3.5. Properties of weak monads. Let (F, µ, η) be a weak monad. (i) ϑ : F → F is a morphism of q-unital monads; (ii) for any (A,ϕ) ∈ AF , F (A) ϕ −→ A = F (A) ϕ −→ A ηA −→ F (A) ϕ −→ A and A ηA −→ F (A) ϕ −→ A is an idempotent F -morphism. In a q-unital monad (F, µ, η), if η is regular, a compatible multiplication for F can be found. More precisely one can easily show: 3.6. Proposition. Let (F, µ, η) be a q-unital monad. (1) If η is regular, then, for µ̃ := µ · Fµ · µFηF : FF → F , (F, µ̃, η) is an r-unital monad. (2) If µ is compatible, then, for η̃ := µ · Fη · η : IA → F , (F, µ, η̃) is an r-unital monad. 136 Regular pairings (3) If (F, µ, η) is an r-unital monad, then for µ̂ : FF ηFFη −→ FFFF µFF −→ FFF µF −→ FF µ −→ F, (F, µ̂, η) is a weak monad. As a special case, we consider q-unital monads on the category RM of modules over a commutative ring R with unit. In the terminology used here this comes out as follows. 3.7. Non-unital algebras. A q-unital R-algebra (A,m, u) is a non-unital R-algebra (A,m) with some R-linear map u : R → A. Put e := u(1R) ∈ A. Then: (1) u is regular if and only if e is an idempotent in A. (2) u is regular and symmetric if and only if e is a central idempotent (then Ae is a unital R-subalgebra of A). (3) µ is compatible if and only if ab = aeb for all a, b ∈ A. (4) If u is regular, then m̃(a⊗ b) := aeb, for a, b ∈ A, defines an r-unital algebra (A, m̃, u) (m̃ and u are regular). (5) If u is regular, then m̂(a ⊗ b) := eaebe, for a, b ∈ A, defines an r-unital algebra (A, m̂, u) with u symmetric. Clearly, the q-unital algebras (A,m, u) over R correspond to the q- unital monads given by (A⊗R −,m⊗ −, u⊗ −) on RM. For an A-module ̺ : A⊗M → M , writing as usual ̺(a⊗m) = am, the compatibility condition comes out as am = aem for all a ∈ A, m ∈ M . 3.8. Monads acting on functors. Let T : A → B be a functor and (G,µ′, η′) a q-unital monad on B. We call T a left G-module if there exists a natural transformation ̺ : GT → T such that GGT G̺ −→ GT ̺ −→ T = GGT µ′T −→ GT ̺ −→ T, and we call it a compatible G-module if in addition GT ̺ −→ T = GT Gη′ −→ GGT µ′T −→ GT ̺ −→ T. 3.9. Proposition. Let T : A → B be a functor and (G,µ′, η′) a weak monad on B. Then the following are equivalent: (a) there is a functor T : A → BG with T = UGT ; (b) T is a compatible G-module. R. Wisbauer 137 Proof. (b)⇒(a) Given T as a compatible G-module with ̺ : GT → T , a functor with the required properties is T : A → BG, A 7→ (T (A), ̺A : GT (A) → T (A)). (a)⇒(b) For any A ∈ A, there are morphisms ρA : GT (A) → T (A) and we claim that these define a natural transformation ρ : GT → T . For this we have to show that, for any morphism f : A → Â, the middle rectangle is commutative in the diagram GGT (A) µ′ T (A) // GT (A) GT (f) �� ρA zz GT (A) Gη′TA OO GT (f) �� ρA // T (A) T (f) �� GT (Â) Gη′T Â �� ρ Â // T (Â) GGT (Â) µ′ T (Â) // GT (Â). ρ Â dd The top and bottom diagrams are commutative by compatibility of the G-modules, the right trapezium is commutative since T (f) is a G-mor- phism, and the outer paths commute by symmetry of η′. Thus the inner diagram is commutative showing naturality of ρ. ⊔⊓ 4. Comonads and comodules In this section we sketch the transfer of the constructions for monads to comonads. 4.1. q-counital comonads and their comodules. A functor with coproduct (or non-counital comonad) is a pair (G, δ) where G : A → A is an endofunctor and δ : G → GG is a natural transformation subject to the coassociativity condition Gδ · δ = δG · δ. For (G, δ), a (non-counital) G-comodule is defined as an object A ∈ A with a morphism υ : A → G(A) in A such that Gυ · υ = δA · υ. Morphisms between G-comodules (A, υ), (A′, υ′) are morphisms g : A → A′ in A satisfying υ′ · g = G(g) ·υ, and the set of all these is denoted by MorG(A,A′). With these morphisms, (non-counital) G-comodules form 138 Regular pairings a category which we denote by A −→ G. For this there are the obvious free and forgetful functors φG : A → A −→ G, UG : A −→ G → A. A triple (G, δ, ε) is said to be a q-counital comonad provided (G, δ) is a functor with coproduct and ε : G → IA is any natural transformation, called a quasi-counit. One can always define natural transformations γ : G δ −→ GG Gε −→ G, γ : G δ −→ GG εG −→ G. Morphisms of q-counital comonads are defined in an obvious way (dual to 3.1). 4.2. q-counital comonads and pairings. For (G, δ, ε) as above, the functors φG and UG allow for a pairing (UG, φG, αG, βG) where, for A ∈ A and (B, υ) ∈ A −→ G, αG : MorA(UG(B), A) → MorG(B,φG(A)), f 7→ G(f) · υ, βG : MorG(B,φG(A)) → MorA(UG(B), A), g 7→ εA · g. The quasi-counit ε is called regular if βG is regular, that is, G ε −→ IA = G δ −→ GG Gε −→ G ε −→ IA, and we say η is symmetric provided φG is so, that is γ = γ. A (non-counital) G-comodule (B, υ) is said to be compatible (with ε) provided αGβG(υ) = υ, that is B υ −→ G(B) = B υ −→ G(B) δB −→ GG(B) GεB −→ G(B). In particular, δ is compatible if G δ −→ GG = G δ −→ GG δG −→ GGG GεG −→ GG. This obviously implies G δ −→ GG = G δ −→ GG γγ −→ GG. By A G we denote the full subcategory of A −→ G whose objects are compatible G-comodules. R. Wisbauer 139 If δ is compatible, the image of the free functor φG lies in A G and (by restriction and corestriction) we obtain the functor pairing (keeping the notation for the functors) φG : A → A G, UG : AG → A, leading to a pairing (UG, φG, αG, βG) between A and A G. Since for (B, υ) in A −→ G,αG(IUG(B)) = υ, the compatibility condition on υ implies that αG ·βG ·αG(υ) = αG(υ), i.e., α is regular in (UG, φG, αG, βG) when restricted to A G. 4.3. Definition. A q-counital comonad (G, δ, ε) is called r-counital if ε is regular and δ is compatible; weak comonad if it is r-counital and ε is symmetric. From the constructions above we obtain: 4.4. Proposition. Let (G, δ, ε) be a q-counital comonad. (1) The following are equivalent: (a) (G, δ, ε) is an r-counital comonad; (b) (UG, φG, αG, βG) is a regular pairing of functors between A and A G. (2) The following are equivalent: (a) (G, δ, ε) is a weak comonad; (b) (UG, φG, αG, βG) is a regular pairing of functors between A and A G with βG symmetric. Similar to the situation for modules, for any (counital) comonad (G, δ, ε), all non-counital G-comodules are compatible (i.e., A −→ G = A G). 4.5. Properties of weak comonads. Let (G, δ, ε) be a weak comonad. (i) γ : G → G is an idempotent morphism of q-counital comonads; (ii) for any (B, υ) ∈ A G, B υ −→ G(B) = B υ −→ G(B) εB −→ B υ −→ G(B) and B υ −→ G(B) εB −→ B is an idempotent G-morphism. Properties of pairings can improved in the following sense. 140 Regular pairings 4.6. Proposition. Let (G, δ, ε) be a q-counital comonad. (1) If ε is regular, then, for δ̃ : G δ −→ GG Gδ −→ GGG GεG −→ GG, (G, δ̃, ε) is an r-counital comonad. (2) If δ is compatible, then, for ε̃ : G δ −→ GG Gε −→ G ε −→ IA, (G, δ, ε̃) is an r-counital comonad. (3) If (G, δ, ε) is a regular quasi-comonad, then, for δ̂ : G δ −→ GG Gδ −→ GGG GGδ −→ GGGG εGGε −→ GG, (G, δ̂, ε) is a weak comonad. As a special case, consider non-counital comonads on the category RM of modules over a commutative ring R with unit. In our terminology this comes out as follows. 4.7. Non-counital coalgebras. A q-counital coalgebra (C,∆, ε) is a non-counital R-coalgebra (C,∆) with some R-linear map ε : C → R. Writing ∆(c) = ∑ c1 ⊗ c2 for c ∈ C, we have: (1) ε is regular if and only if for any c ∈ C, ε(c) = ∑ ε(c1)ε(c2). (2) ε is symmetric if and only if ∑ c1ε(c2) = ∑ ε(c1)c2. (3) ∆ is compatible if and only if ∆(c) = ∑ c1 ⊗ c2ε(c3). (4) If ε is regular, then ∆̃(c) := ∑ c1 ⊗ ε(c2)c3 defines an r-counital coalgebra (C, ∆̃, ε). (5) If (C,∆, ε) is an r-counital comonad, then ∆̂(c) := ∑ ε(c1)c2 ⊗ c3ε(c4) defines an r-counital coalgebra (C, ∆̂, ε) with ε symmetric. Clearly, the q-counital coalgebras (C,∆, ε) over R correspond to the q-counital comonads given by (C ⊗R −,∆ ⊗ −, ε⊗ −) on RM. From this the compatibility conditions for C-comodules are derived (see 4.2). 4.8. Weak corings and pre-A-corings. Let A be a ring with unit 1A and C a non-unital (A,A)-bimodule which is unital as right A-module. Assume there are (A,A)-bilinear maps ∆ : C → C ⊗A C, ε : C → A, where ∆ is coassociative. (C,∆, ε) is called a right unital weak A-coring in [21], provided for all c ∈ C, (ε⊗ IC) · ∆(c) = 1A · c = (IC ⊗ ε) · ∆(c), R. Wisbauer 141 which reads in (obvious) Sweedler notation as ∑ ε(c1)c2 = 1A · c =∑ c1ε(c2). From the equations (IC ⊗ ε⊗ IC) · (IC ⊗ ∆) · ∆(c) = ∑ c1 ⊗ 1A · c2 = ∑ c1 ⊗ c2 = ∆(c), (IC ⊗ ε⊗ IC) · (∆ ⊗ IC) · ∆(c) = ∑ 1A · c1 ⊗ c2 = 1A · ∆(c), it follows by coassociativity that 1A · ∆(c) = ∆(c). Summarising we see that, in this case, (C,∆, ε) induces a weak comonad on the category AM−→ of left non-unital A-modules (=AM since A has a unit). (C,∆, ε) is called an A-pre-coring in [7, Section 6], if (ε⊗ IC) · ∆(c) = c, (IC ⊗ ε) · ∆(c) = 1A · c, which reads (in Sweedler notation) as c = ∑ ε(c1)c2, 1A ·c = ∑ c1ε(c2). Similar to the computation above we obtain that 1A·∆(c) = ∆(c). Now (C,∆, ε) induces an r-counital comonad on AM−→ but ε is not symmetric. Notice that in both cases considered above, restriction and corestriction of ∆ and ε yield an A-coring (AC,∆, ε) (e.g. [21, Proposition 1.3]). 4.9. Comonads acting on functors. Let T : A → B be a functor and (G, δ, ε) a weak comonad on B. We call T a left (non-counital) G-comodule if there exists a natural transformation υ : T → GT such that T υ −→ GT υG −→ GGT = T υT −→ GT δ −→ GGT, and we call it a compatible G-comodule if, in addition, T υ −→ GT = T υ −→ GT δ −→ GGT Gε −→ GT. Dual to Proposition 3.9, given a weak comonad (G, δ, ε) on B, a functor T : A → B is a compatible G-comodule if and only if there is a functor T : A → B G with T = UGT . The motivation for considering generalised monads and comonads came from structures observed while handling full pairings of functors (see end of Section 2). Now we want to reconsider the pairings in view of these constructions. For any pairing (L,R, α, β) between categories A and B, (RL,RεL, η) is a q-unital monad and (LR,LηR, ε) is a q-counital comonad. It is easy to see that (i) if β is regular, then for any B ∈ B, Rε : RLR(B) → R(B) is a compatible RL-module. 142 Regular pairings (ii) if α is regular, then for any A ∈ A, L(A), Lη : L(A) → LRL(A) is a compatible LR-comodules. 4.10. Comparison functors. For a regular pairing (L,R, α, β) between A and B, (RL,RεL, η) is an r-unital monad on A with a (comparison) functor R̂ : B → ARL, B 7→ (R(B), Rε : RLR(B) → R(B)), (LR,LηR, ε) is an r-counital comonad on B with a (comparison) functor L̃ : A → B −→ LR, A 7→ (L(A), Lη : L(A) → LRL(A)), inducing commutativity of the diagrams A L // φRL !! B R̂ �� R // A ARL URL == , B R // φLR A L̃ �� L // B B LR ULR >> . It follows from 3.2 that for the r-unital monad (RL,RεL, η), we have a regular pairing (φRL, URL, αRL, βRL) between A and ARL. Similarly, by 4.2, for the R-counital comonad (LR,LηR, ε), (ULR, φLR, αLR, βLR) is a regular pairing between B and B LR. 4.11. Relating (L,R) with (φRL, URL) and (ULR, φLR). With the above notions we form the diagram MorB(L(A), B) R̂ −,− �� α // MorA(A,R(B)) β // MorB(L(A), B) R̂ −,− �� MorRL(φRL(A), R(B)) αRL // MorA(A,URLR(B)) βRL // MorRL(φRL(A), R(B)). This diagram is commutative if and only if α is symmetric (see Defi- nitions 2.5). Similar constructions apply for (L,R), (ULR, φLR) and L̃−,−, and β is symmetric if and only if K̃−,− · α · β = αLR · βLR · L̃−,−. R. Wisbauer 143 4.12. Corollary. Consider a pairing (L,R, α, β) (see 2.2). (1) The following are equivalent: (a) (L,R, α, β) is a regular pairing; (b) (RL,RεL, η) is an r-unital monad on A and (LR,LηR, ε) is an r-counital comonad on B. (2) The following are equivalent: (a) (L,R, α, β) is a regular pairing with α and β symmetric; (b) (RL,RεL, η) is a weak monad on A and (LR,LηR, ε) is a weak comonad on B. 5. Entwining monads and comonads 5.1. Lifting of functors to module categories. Let (F, µ, η) and (L, µ′, η′) be r-unital monads on the categories A and B, respectively, and AF , BL the categories of the corresponding compatible modules (see 3.2). Given functors T : A → B and T : AF → BL, we say that T is a lifting of T provided the diagram AF T // UF �� BL UL �� A T // B (5.1) is commutative, where the U ’s denote the forgetful functors. 5.2. Proposition. With the data given in 5.1, consider the functors TF, LT : A → B and a natural transformation λ : LT → TF . The non-unital F -module (F, µ) induces an L-action on TF , χ : LTF λF −→ TFF Tµ −→ TF. (1) If (TF, χ) is a (non-unital) L-module, then we get the commutative diagram LLT Lλ // µ′T �� LTF LTϑ // LTF λF // TFF Tµ �� LT λ // TF Tϑ // TF. (5.2) 144 Regular pairings (2) If (TF, χ) is a compatible L-module, then (with ϑ′ = µ′ · Fη′) LT ϑ′T // LT λ // TF Tϑ // TF = LT λ // TF Tϑ // TF. (5.3) (3) If η is symmetric in (F, µ, η) and (A,ϕ) is a compatible F -module, then Tϕ · λA = Tϕ · λA · LTϕ · LTηA. (5.4) Proof. The proof follows essentially as in the monad case replacing the identity on F at some places by ϑ = µ · Fη (see 3.1). To show (3), Proposition 3.5 is needed. ⊔⊓ 5.3. Proposition. Let (F, µ, η) and (L, µ′, η′) be r-unital monads on A and B, respectively, and T : A → B any functor. Then a natural transformation λ : LT → TF induces a lifting to the compatible modules, T : AF → BL, (A,ϕ) 7→ (T (A), Tϕ · λA : LT (A) → T (A)), if and only if the diagram (5.2) is commutative and equation (5.3) holds. Proof. One direction follows from Proposition 5.2, the other one by a slight modification of the proof in the monad case. ⊔⊓ To show that the lifting property implies the existence of a natural transformation λ : LT → TF we need the symmetry of the units, that is, we require the r-unital monads to be weak monads. Then we can extend Applegate’s lifting theorem for monads (and unital modules) (e.g. [13, Lemma 1], [22, 3.3]) to weak monads (and compatible modules). 5.4. Theorem. Let (F, µ, η) and (L, µ′, η′) be weak monads on A and B, respectively. For any functor T : A → B, there are bijective correspon- dences between (i) liftings of T to T : AF → BL; (ii) compatible L-module structures ̺ on TUF : AF → B; (iii) natural transformations λ : LT → TF with commuting diagrams LLT Lλ // µ′T �� LTF λF // TFF Tµ �� LT λ // TF, LT ϑ′T // λ �� λ "" LT λ �� TF Tϑ // TF. (5.5) R. Wisbauer 145 Proof. (i)⇔(ii) follows by Proposition 3.9. (ii)⇒(iii) Given the compatible L-module structure map ̺, put λ := ̺F · LTη : LT LTη −→ LTF ̺F −→ TF. Notice that for λ we can take Tϑ · λ from Proposition 5.2. (iii)⇒(i) Given λ with the commutative diagram in (iii), it follows by Propositions 5.3 that ̺A := Tϕ · λA induces a lifting. ⊔⊓ 5.5. Lifting of functors to comodules. Let (G, δ, ε) and (H, δ′, ε′) be r-unital comonads on the categories A and B, respectively, and A G, BH the corresponding categories of the compatible comodules (see 4.2). Given a functor T : A → B, a functor T̂ : AG → B H , is said to be a lifting of T if the diagram A G T̂ // UG �� B H UH �� A T // B (5.6) is commutative where the U ’s denote the forgetful functors. 5.6. Proposition. With the data given in 5.5, consider the functors TG, HT : A → B and a natural transformation ψ : TG → HT . The (non-counital) G-comodule (G, δ) induces an H-coaction on TG, ζ : TG Tδ −→ TGG ψG −→ HTG. (1) If (TG, ζ) is a (non-counital) H-comodule, we get the commutative diagram TG Tγ // Tδ �� TG ψ // HT δ′T �� TGG ψG // HTG HTγ // HTG Hψ // HHT. (5.7) (2) If H (TG, ζ) is a compatible H-module, then TG Tγ // TG ψ // HT γ′T // HT = TG Tγ // TG ψ // HT. (5.8) (3) If ε is symmetric and (A, υ) is a compatible G-comodule, then ψ · Tυ = HTε ·HTυ · ψ · Tυ. 146 Regular pairings Proof. The situation is dual to that of Proposition 5.2. ⊔⊓ 5.7. Proposition. Let (G, δ, ε) and (H, δ′, ε′) be r-counital comonads on the categories A and B, respectively, and T : A → B any functor. A natural transformation ψ : TG → HT induces a lifting T̂ : AG → B H , (A, υ) 7→ (T (A), ψ · Tυ : T (A) → HT (A)), if and only if the diagram (5.7) is commutative and equation (5.8) holds. Proof. The proof is dual to that of Proposition 5.3. ⊔⊓ Dualising Theorem 5.4, we obtain an extension of Applegate’s lifting theorem for comonads (and comodules) (e.g. [22, 3.5]) to weak comonads (and compatible comodules). 5.8. Theorem. Let (G, δ, ε) and (H, δ′, ε′) be weak comonads on A and B, respectively. For any functor T : A → B, there are bijective correspon- dences between (i) liftings of T to T̂ : AG → B H ; (ii) compatible H-comodule structures υ : TUG → HTUG; (iii) natural transformations ψ : TG → HT with commutative diagrams TG Tδ �� ψ // HT δ′T �� TGG ψG // HTG Hψ // HHT, TG Tγ // ψ �� ψ ## TG ψ �� HT γ′T // HT. Proof. In view of 5.6 and 5.7, the proof is dual to that of Theorem 5.4. Here we take ψ as the composition ψ · Tγ (with ψ from 5.6). ⊔⊓ 6. Lifting of endofunctors to modules and comodules Given a weak monad (F, µ, η), or a weak comonad (Gδ, ε), and any endofunctor T on the category A, we have learned in the preceding sections when T can be lifted to an endofunctor of the compatible modules or comodules, respectively. Now, one may also ask if the lifting is again a weak monad or a weak comonad, respectively. 6.1. Entwining r-unital monads. For weak monads (F, µ, η) and (T, µ̌, η̌) on A and a natural transformation λ : FT → TF , the following are equiv- alent: R. Wisbauer 147 (a) defining product and quasi-unit on TF by µ : TFTF TλF −→ TTFF TTµ −→ TTF µ̌F −→ TF, η : IA η −→ F F η̌ −→ FT λ −→ TF, yields a weak monad (TF, µ, η) on A; (b) λ induces commutativity of the diagrams FFT Fλ // µT �� FTF λF // TFF Tµ �� FT λ // TF, FT ϑT // λ �� λ "" FT λ �� TF Tϑ // TF, (6.1) FTT Fµ̌ �� λT // TFT Tλ // TTF µ̌F �� FT λ // TF, FT Fϑ̌ // λ �� λ "" FT λ �� TF ϑ̌F // TF ; (6.2) (c) λ induces commutativity of the diagrams in (6.1) and the square in (6.2), and there are natural transformations µ̌F : TTF → TF and λ · F η̌ : F → TF where µ̌F is a left and right F -module morphism and λ · F η̌ is an F -module morphism. If these conditions hold, we obtain morphisms of q-unital monads, λ · F η̌ : F → TF and λ · ηT : T → TF . Proof. The assertions follow from the general results in Section 5 and some routine computations. ⊔⊓ 6.2. Weak crossed products. Given (F, µ, η) and T : A → A, the composition TF may have a weak monad structure without requiring such a structure on T . For example, replacing the natural transformations µ̌F and λ · F η̌ in 6.1(c) by some natural transformations ν : TTF → TF, ξ : F → TF, similar to 6.1(a), a multiplication and a quasi-unit can be defined on TF . To make this a weak monad on A, special conditions are to be imposed on ν and ξ which can be obtained by routine computations. 148 Regular pairings Having ν and ξ, one also has natural transformations ν̄ : TT TTη // TTF ν // TF, η : IA η // F ξ // TF, and it is easy to see that ν̄ leads to the same product on TF as ν does. Thus ν̄ and η may be used to define a weak monad structure on TF and the conditions required come out as cocycle and twisted conditions. For more details we refer, e.g., to [1], [11, Section 3]. For a weak comonad (G, δ, ε) and an endofunctor T : A → A, we now consider liftings to the category of compatible G-comodules, T̂ : AG → A G. The case when T has a weak comonad structure is dual to 6.1: 6.3. Entwining weak comonads. For weak comonads (F, δ, ε), (T, δ̌, ε̌), and a natural transformation ψ : TG → GT , the following are equivalent: (a) defining coproduct and quasi-counit on TG by δ̂ : TG δ̌G −→ TTG TTδ −→ TTGG TψG −→ TGTG, ε̂ : TG ψ −→ GT Gε̌ −→ G ε −→ IA, yields a weak comonad (TG, δ̂, ε̂) on A; (b) ψ induces commutativity of the diagrams, where γ = Tε·δ, γ̌ = T ε̌·δ̌, TG Tδ �� ψ // GT δT �� TGG ψG // GTG Gψ // GGT, TG Tγ // ψ �� ψ "" TG ψ �� GT γT // GT, (6.3) TG ψ // δ̌G �� GT Gδ̌ �� TTG Tψ // TGT ψT // GTT, TG γ̌G // ψ �� ψ "" TG ψ �� GT Gγ̌ // GT, (6.4) (c) ψ induces commutativity of the diagrams (6.3) and the square in (6.4) and we have natural transformations δ̌G : TG → TTG, Gε̌ · ψ : TG → G, where δ̌G is a left and right G-comodule morphism and Gε̌ · ψ is a left G-comodule morphism. R. Wisbauer 149 If these conditions hold, we obtain morphisms of q-unital comonads, Gε̌ · ψ : TG → G and εT · ψ : TG → T . 6.4. Weak crossed coproducts. In the situation of 6.3, the coproduct on TG can also be expressed by replacing the natural transformations δ̌G and Gε̌ · ψ by any natural transformations ν : TG → TTG and ζ : TG → G, subject to certain conditions to obtain a weak comonad structure on TG. Given ν and ζ as above, one may form ν̂ : TG ν // TTG TTε // TT , ζ̂ : TG ζ // G ε // IA , and it is easy to see that these induce a weak comonad structure on TG. This leads to the weak crossed coproduct as considered (for coalgebras) in [11] and [12], for example. 7. Mixed entwinings and liftings Throughout this section let (F, µ, η) denote a weak monad and (G, δ, ε) a weak comonad on any category A. In this section we investigate the lifting properties to compatible F -modules and compatible G-comodules, respectively. 7.1. Liftings of monads and comonads. Consider the diagrams AF G // UF �� AF UF �� A G // A, A G F̂ // UG �� A G UG �� A F // A. In both cases the lifting properties are related to a natural transformation ω : FG → GF. The lifting in the left hand case requires commutativity of the diagrams (Proposition 5.3) FFG Fω // µG �� FGF ωF // GFF Gµ �� FG ω // GF, FG ω // ϑG �� ω "" GF Gϑ �� FG ω // GF, (7.1) 150 Regular pairings whereas the lifting to A G needs commutativity of the diagrams (Proposi- tion 5.7) FG Fδ �� ω // GF δF �� FGG ωG // GFG Gω // GGF, FG ω // Fγ �� ω ## GF γF �� FG ω // GF. (7.2) To make G a non-counital comonad with coproduct δ, the latter has to be an F -module morphism, in particular, δF : GF → GGF has to be an F -morphism and this follows by commutativity of the rectangle in (7.2) provided the square in (7.1) is commutative. To make the lifting F̂ a non-unital monad with multiplication µ, the latter has to be a G-comodule morphism, in particular, µG : FFG → FG has to be a G-module morphism and this follows by commutativity of the rectangle in (7.1) provided the square in (7.2) is commutative. 7.2. Natural transformations. The data given in 7.1 allow for natural transformations ξ : G ηG // FG ω // GF εF // F , κ̂ : GF ηGF // FGF ωF // GFF Gµ // GF , τ̂ : FG Fδ // FGG ωG // GFG εFG // FG, with the properties Gµ · κ̂F = κ̂ ·Gµ, τ̂G · Fδ = Fδ · τ̂ , µ · ξF = εF · κ̂, ξG · δ = τ̂ · ηG. (i) If the rectangle in (7.1) is commutative, then κ̂ is idempotent. (ii) If the rectangle in (7.2) is commutative, then τ̂ is idempotent. To make the liftings weak comonads or weak monads, respectively, we have to find pre-units or pre-counits, respectively. In what follows we consider these questions. 7.3. Lemma. (Pre-counits for G) Assume the diagrams in (7.1) to be commutative. Then the following are equivalent: (a) for any (A,ϕ) ∈ AF , εA : G(A) → A is an F -module morphism; (b) εF : GF → F is an F -morphism; R. Wisbauer 151 (c) ϑ = µ · Fη induces commutativity of the diagram FG Fε // ω �� F ϑ �� GF εF // F. (7.3) If these conditions are satisfied, then (with γ = Gε · ϑ) µG · F τ̂ = τ̂ · µG and τ̂ = ϑγ. Proof. This is shown by straightforward verification. ⊔⊓ 7.4. Proposition. Assume the diagrams in (7.1), (7.2) and (7.3) to be commutative. Then (G, δ, ε) is a weak comonad on AF . Proof. This follows from the preceding observations. ⊔⊓ Dual to Lemma 7.3 and 7.4 we obtain for the quasi-units for F̂ : 7.5. Lemma. (Pre-units for F̂ ) Assume the diagrams in (7.2) to be commutative. Then the following are equivalent: (a) for any (A, υ) ∈ A G, ηA : A → F (A) is a G-comodule morphism; (b) ηG : G → FG is G-colinear; (c) γ = Gε · δ induces commutativity of the diagram G γ �� ηG // FG ω �� G Gη // GF. (7.4) If these conditions are satisfied, then Gκ̂ · δF = δF · κ̂ and κ̂ = γϑ. Summing up the above observations yields the 7.6. Proposition. Assume the diagrams in (7.1), (7.2) and (7.4) to be commutative. Then (F̂ , µ, η) is a weak monad on A G. One may consider alternative choices for a pre-counit for G or a pre-unit for F̂ . 152 Regular pairings 7.7. Lemma. Assume the diagrams in (7.1) to be commutative. With the notations from 7.2, the following are equivalent: (a) for any (A,ϕ) ∈ AF , εA : G(A) ξA −→ F (A) ϕ −→ A is an F -module morphism; (b) εF : GF ξF −→ FF µ −→ F (= GF κ̂ −→ GF εF −→ F ) is an F -morphism; (c) commutativity of the diagram FFG Fω // FGF FεF // FF µ �� FG FηG OO ω // GF εF // F. (7.5) If these conditions are satisfied, then τ̂ = µG · F τ̂ · FηG. Proof. The proof is obtained by some diagram constructions. ⊔⊓ Notice that commutativity of (7.3) implies commutativity of (7.5). 7.8. Lemma. Assume the diagrams in (7.2) to be commutative. Then the following are equivalent: (a) for any (A, υ) ∈ A G, η̂ : A υ −→ G(A) ξA −→ F (A) is a G-comodule morphism; (b) η̂G : G ηG −→ FG τ̂ −→ FG (= G δ −→ GG ξG −→ FG) is G-colinear; (c) commutativity of the diagram G δ �� ηG // FG ω // GF GG GηG // GFG Gω // GGF. GεF OO (7.6) If these conditions are satisfied, then κ̂ = GεF ·Gκ̂ · δF. Proof. The situation is dual to Lemma 7.7. ⊔⊓ Notice that commutativity of (7.4) implies commutativity of (7.6). 7.9. Proposition. With the data given in 7.1, assume the diagrams in (7.1), (7.2) and (7.5) to be commutative. R. Wisbauer 153 (1) If (7.6) is commutative, then ε from 7.7 is regular for δ, and for δ : G → GG with δF : GF δF // GGF Gκ̂ // GGF, (G, δ, ε) is an r-counital comonad on AF . (2) If (7.4) is commutative, then δF = δF · κ̂ and (G, δ, ε) is a weak comonad on AF . Proof. This can be shown by suitable diagram constructions. ⊔⊓ 7.10. Proposition. With the data given in 7.1, assume the diagrams in (7.1), (7.2), and (7.6) to be commutative. (1) If (7.5) is commutative, then η̂ in 7.8 is regular for µ, and for µ̂ : FF → F with µ̂G : FFG F τ̂ // FFG µG // FG, (F̂ , µ̂, η̂) is an r-unital monad on A G. (2) If (7.3) is commutative, then µ̂G = τ̂ · µG and (F̂ , µ̂, η̂) is a weak monad on A G. Proof. This is dual to Proposition 7.9. ⊔⊓ Acknowledgments. The author wants to thank Gabriella Böhm, Tomasz Brzeziński and Bachuki Mesablishvili for their interest in this work and for helpful comments on a previous version of this paper. References [1] Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R., and Rodríguez Raposo, A.B., Crossed products in weak contexts, Appl. Categ. Struct. 18(3) (2010), 231-258. [2] Beck, J., Distributive laws, [in:] Seminar on Triples and Categorical Homology Theory, B. Eckmann (ed.), Springer LNM 80 (1969), 119-140. [3] Böhm, G., The weak theory of monads, Adv. Math. 225(1) (2010), 1-32. 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Contact information Robert Wisbauer Mathematisches Institut, Heinrich Heine University, 40225 Düsseldorf, Germany E-Mail: wisbauer@math.uni-duesseldorf.de URL: www.math.uni-duesseldorf.de/∼wisbauer Received by the editors: 24.08.2012 and in final form 12.09.2012.

Regular pairings of functors and weak (co)monads

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