THE EILENBERG-MOORE CATEGORY AND A BECK-TYPE THEOREM FOR A MORITA CONTEXT

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1 THE EILENBERG-MOORE CATEGORY AND A BECK-TYPE THEOREM FOR A MORITA CONTEXT TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE Abstract. The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the moritability of a pair of functors with a common domain) are derived. Contents 1. Introduction 2 2. Double adjunctions and Morita contexts Adjunctions and (co)monads The category of double adjunctions The category of Morita contexts 4 3. A Beck-type theorem for a Morita context From double adjunctions to Morita contexts The Eilenberg-Moore category of a Morita context From Morita contexts to double adjunctions Every Morita context arises from a double adjunction The comparison functor Moritability Morita theory Preservation of coequalisers by algebras Morita contexts and equivalences of categories of algebras Examples and applications Blowing up one adjunction Morita theory for rings Formal duals Herds versus pretorsors Remarks on dualisations and generalisations 31 Date: November Mathematics Subject Classification. 18A40. 1

2 2 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE 6.1. Dualisations Bicategories 32 References Introduction This paper is directly motivated by recent almost simultaneous appearance of two different approaches to the definition of a categorical (functorial) notion of a herd or pre-torsor in [4] and [3]. The definition of a herd functor [4] is based on two bialgebra functors of two monads connected by certain structure maps, reminiscent of a Morita context. The definition of a pre-trosor in [3] takes a pair of adjunctions (with coinciding codomain category for the left adjoints) as a starting point. The aim of this paper is to show that there is a close relationship between pairs of adjunctions and functorial Morita contexts similar to the correspondence between single adjunctions and monads. These are the main results and the organisation of the paper. In Section 2 we recall from [3] the definition of the category of double adjunctions on categories A and B, Adj(A, B), and introduce the category Mor(A, B) of (functorial) Morita contexts. In Section 3 we describe functors connecting categories of double adjunctions and Morita contexts. More precisely we compare categories Adj(A, B) and Mor(A, B). First we define a functor Υ : Adj(A, B) Mor(A, B). To construct a functor in the converse direction, to each Morita context T we associate its Eilenberg-Moore category (A, B) T. This is very reminiscent of the classical Eilenberg-Moore construction of algebras of a monad (recalled in Section 2.1), and, in a way, is based on doubling of the latter. Objects in (A, B) T are two algebras, one for each monad in T, together with two connecting morphisms. Once (A, B) T is defined, two adjunctions, one between (A, B) T and A the other between (A, B) T and B, are constructed. This construction yields a functor Γ : Mor(A, B) Adj(A, B). Next it is shown that the functors (Γ, Υ) form an adjoint pair, and that Γ is a full and faithful functor. The counit of this adjunction is given by a comparison functor K which compares the common category T in a double adjunction T with the Eilenberg-Moore category of the associated Morita context T = Υ(T). A necessary and sufficent condition for the comparison functor to be an equivalence are derived. This is closely related to the existence of colimits of diagrams of certain type in T and is a Morita double adjunction version of the classical Beck theorem (on precise monadicity). In Section 4 we analyse which objects of Mor(A, B) describe equivalences between categories of algebras of monads. It is also proven that large classes of equivalences between categories of algebras are induced by Morita contexts. In Section 5 examples and special cases of the theory developed in preceding sections are given. In particular, it is shown how the main results of Section 3 can be applied to a single adjunction leading to a new point of view on some aspects of descent theory. The Eilenberg-Moore category associated to the module-theoretic Morita context is identified as the category of modules of the associated matrix Morita ring. We also show that the theory developed in Sections 2 4 is applicable to pre-torsors and herd functors, thus bringing forth means for comparing pre-torsors with balanced herds.

3 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 3 The paper is completed with comments on dual versions of constructions presented and with an outlook. Throughout the paper, the composition of functors is denoted by juxtaposition, the symbol is reserved for composition of natural transformations and morphisms. The action of a functor on an object or morphism is usually denoted by juxtaposition of corresponding symbols (no brackets are typically used). Similarly the morphism corresponding to a natural transformation, say α, evaluated at an object, say X, is denoted by a juxtaposition, i.e. by αx. For an object X in a category, we will use the symbol X as well to denote the identity morphism on X. Typically, but not exclusively, objects and functors are denoted by capital Latin letters, morphisms by small Latin letters and natural transformations by Greek letters. 2. Double adjunctions and Morita contexts The aim of this section is to recall the standard correspondence between adjoint functors and (co)monads and to introduce the main categories studied in the paper Adjunctions and (co)monads. It is well-known from [6] that there is a close relationship between pairs of adjoint functors (L : A B, R : B A), monads A on A and comonads C on B. Starting from an adjunction (L, R) with unit η : A RL and counit ε : LR B, the corresponding monad and comonad are A = (RL, RεL, η) and C = (LR, LηR, ε). Starting from a monad A = (A, m, u) (where m is the multiplication and u is the unit), one first defines the Eilenberg-Moore category A A of A-algebras, whose objects are pairs (X, ρ X ), where X is an object in A and ρ X : AX X is a morphism in A such that ρ X Aρ X = mx ρ X and ux ρ X = X. There is a pair of adjoint functors (F A : A A A, U A : A A A), where U A is a forgetful functor and F A is the induction or free algebra functor, for all objects X A defined by F A X = (AX, mx). Similarly, one defines the category B C of coalgebras over C, and obtains an adjoint pair (U C : B C B, F C : B B C ), where F C is the induction or free coalgebra functor and U C is the forgetful functor. The original and constructed adjunctions are related by the comparision or Kleisli functor K : B A A (or K : A B C in the comonad case). For any Y B, the comparision functor is given by KY = (RY, RεY ). The functor R is said to be monadic if K is an equivalence of categories. Similarly L is said to be comonadic if K is an equivalence of categories. Beck s Theorem [2] provides one with necessary and sufficient conditions for the functors R and L to be monadic or comonadic respectively. For more detailed and comprehensive study of matters described in this section we refer to [1] The category of double adjunctions. In this paper, rather than looking at one adjunction, we consider two adjunctions with right adjoints operating on a common category. More precisely, let A and B be two categories. Following [3], the category Adj(A, B) is defined as follows. An object in Adj(A, B) is a pentuple (or a triple) T = (T, (L A, R A ), (L B, R B )), where T is a category and (L A : A T, R A : T A) and (L B : B T, R B : T B) are adjunctions whose units and counits are denoted respectively by η A, η B and ε A, ε B. A morphism F : T = (T, (L A, R A ), (L B, R B )) T = (T, (L A, R A ), (L B, R B )) is a functor F : T T such that R A F = R A and R BF = R B.

4 4 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE To a morphism F in Adj(A, B), one associates two natural transformations (1) a := (ε A F L A) (L A η A ) : L A F L A, b := (ε B F L B) (L B η B ) : L B F L B, which satisfy the following compatibility conditions (2) (3) (R A a) η A = η A, (R B b) η B = η B, (F ε A ) (ar A) = ε A F, (F ε B ) (br B) = ε B F The category of Morita contexts. Consider two categories A and B. Let A = (A, m A, u A ) be a monad on A, B = (B, m B, u B ) be a monad on B. An A-B bialgebra functor T = (T, λ, ρ), is a functor T : B A equipped with two natural transformations ρ : T B T and λ : AT T such that (4) T BB ρb T B T m B ρ T B ρ T, T B ρ T = T u B T, (5) AAT Aλ AT m A T AT λ λ T, λ AT T u A T = T, AT B Aρ AT λb T B ρ λ T. A bialgebra morphism φ : T T between two A-B bialgebra functors is a natural transformation that satisfies the following conditions T B ρ T AT λ T φb T B φ ρ T, Aφ φ AT λ T. An A-B bialgebra functor T induces a functor B A A, which is denoted again by T and is defined by T Y = (T Y, λy ), for all Y B. A Morita context on A and B is a sextuple T = (A, B, T, T, ev, êv), that consists of a monad A = (A, m A, u A ) on A, a monad B = (B, m B, u B ) on B, an A-B bialgebra functor T, a B-A bialgebra functor T and natural transformations ev : T T A and êv : T T B. These are required to satisfy the following conditions: ev is an A-A bialgebra morphism, êv is a B-B bialgebra morphism, and the following diagrams

5 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 5 commute (6) T T T T bev T B T T T bev b T B T evt AT ρ λ T, bt ev T A ˆρ ˆλ T, (7) T AT bt λ T T T B T ρ b T T T ˆρT T T bev B, bev T ˆλ T T ev ev A. Diagrams (7) mean that ev is B-balanced and êv is A-balanced. A pair of bialgebras T, T that satisfy all the conditions in this section except for diagrams (7) is termed a pair of formally dual bialgebras or a pre-morita context. Thus a Morita context is a balanced pair of formally dual bialgebras. Remark Let k A and k B be unital associative rings. Set A to be the category of left k A -modules and B the category of left k B -modules. For a k A -ring A (i.e. a ring map k A A) with multiplication µ A and unit ι A and a k B -ring B with multiplication µ B and unit ι B consider monads A = (A ka, µ A ka, ι A ka ) and B = (B kb, µ B kb, ι B kb ). Then the definition of a Morita context with monads A and B coincides with the classical definition of a (ring-theoretic) Morita context between rings A and B; see Section 2.3 for a more detailed study of this example. Note however that the definition of a categorical Mortia context introduced in this paper differs from the notion of a wide Morita context [5], which also gives a categorical albeit different interpretation of classical Morita contexts. The connection between categorical Morita contexts and wide Morita contexts is discussed in Section 4. A morphism φ : T = (A, B, T, T, ev, êv) T = (A, B, T, T, ev, êv ) between Morita contexts on A and B is a quadruple (φ 1, φ 2, φ 3, φ 4 ) defined as follows. First, φ 1 : A A and φ 2 : B B are monad morphisms, i.e. φ 1 : A A and φ 2 : B B are natural transformation such that (8) AA m A A 11 A u A A φ 1 φ 1 φ 1 A A m A A, u A φ 1 A, (9) BB m B B 11 B u B B φ 2 φ 2 φ 2 B B m B B, u B φ 2 B, where the shorthand notation φ 1 φ 1 := A φ 1 φ 1 A = φ 1 A Aφ 1 etc. for the Godement product of natural transformations is used. These make T an A-B bialgebra functor

6 6 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE and T a B-A bialgebra functor with structures given by T B T φ 2 T B ρ T, AT φ 1 T A T λ T, T A bt φ 1 T A ˆρ T, B T φ 2 T b B T ˆλ T. Second, φ 3 : T T is a morphism of A-B bialgebras and φ 4 : T T is a morphism of B-A bialgebras. These morphisms are required to satify the following compatibility conditions (10) T T φ 3 φ 4 T T ev A φ 1 ev A, φ 4 φ 3 T T bev B φ 2 T T bev B. This completes the definition of the category Mor(A, B) of Morita contexts on A and B. 3. A Beck-type theorem for a Morita context The aim of this section is to analyse the relationship between double adjunctions described in Section 2.2 and Morita contexts defined in Section 2.3. The same notation and conventions as in Section 2 are used From double adjunctions to Morita contexts. Fix categories A and B. The aim of this section is to construct a functor Υ from the category of double adjunctions Adj(A, B) to the category of Morita contexts Mor(A, B). Take any object T = (T, (L A, R A ), (L B, R B )) in Adj(A, B) and define an object Υ(T) = T := (A, B, T, T, ev, êv) in Mor(A, B) as follows. The adjunction (L A, R A ) defines a monad A := (A := R A L A, m A := R A ε A L A, η A ) on A and the adjunction (L B, R B ) defines a monad B := (B := R B L B, m B := R A ε B L B, η B ) on B; see Section 2.1. Set T := R A L B : B A and view it as an A-B bialgebra functor by R A ε A L B : AT = R A L A R A L B R A L B = T and R A ε B L B : T B = R A L B R B L B R A L B = T, i.e. T = (R A L B, R A ε A L B, R A ε B L B ). To check that T is a bialgebra functor one can proceed as follows: the associativity conditions (of left A-action, right B-action and mixed associativity) are a straightforward consequence of naturality of the counits. The unitality of left and right actions follows by the triangular identities for units and counits of adjunctions (L A, R A ) and (L B, R B ). The B-A bialgebra functor T := (R B L A, R B ε B L A, R B ε A L A ) is defined similarly. Finally define natural transformations ev : T T = R A L B R B L A R A ε B L A RA L A = A, êv : T T = R B L A R A L B R B ε A L B R B L B = B. All compatibility diagrams (6)-(7) follow (trivially) by the naturality of the counits ε A and ε B.

7 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 7 For any morphism F : T T in Adj(A, B), the corresponding morphism of Morita contexts, Υ(F ) = (φ 1, φ 2, φ 3, φ 4 ) : Υ(T) Υ(T ), is defined as follows. The monad morphisms are: φ 1 : R A L A R A a RA F L A = R A L A, φ 2 : R B L B R B b R B F L B = R B L B, where a and b are defined by equations (1) in Section 2.2. Equations (2) express exactly that φ 1 and φ 2 satisfy the second diagrams in (8) and (9), respectively. That φ 1 and φ 2 satisfy the first diagrams in (8) and (9) follows by (3) and by the naturality. The morphisms of bialgebras are: φ 3 : T = R A L B R A b R A F L B = R A L B = T, φ 4 : T = R B L A R B a R B F L A = R B L A = T. That φ 3 and φ 4 are bialgebra morphisms satisfying compatibility conditions (10) follows by equations (3) and by the naturality (the argument is very similar to the one used for checking that φ 1 and φ 2 preserve multiplications) The Eilenberg-Moore category of a Morita context. Before we can make a converse construction for the functor Υ introduced in Section 3.1, we need to define a category of representations for a Morita context. This construction is similar to that of the Eilenberg-Moore category of algebras for a monad; see Section 2.1. Let T = (A, B, T, T, ev, êv) be an object in Mor(A, B). The Eilenberg-Moore category associated to T, (A, B) T, is defined as follows. Objects in (A, B) T are sextuples (or quadruples) X = ((X, ρ X ), (Y, ρ Y ), v, w), where (X, ρ X ) A A is an algebra for the monad A, (Y, ρ Y ) B B is an algebra for the monad B, v : T Y X is a morphism in A A and w : T X Y is a morphism in B B (i.e. (11) AT Y λy T Y Av v AX B T X ρ X X, ˆλX satisfying the following compatibility conditions T X Bw w BY ρ Y Y ), (12) T T X T w T Y T T Y bt v T X evx AX v ρ X X, bevy BY w ρ Y Y, (13) T AX bt ρ X T X T BY T ρ Y T Y ˆρX T X w w Y, ρy T Y v v X.

8 8 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE All these diagrams can be understood as generalised mixed associativity conditions. A morphism X X in (A, B) T is a couple (f, g), where f : X X is a morphism in A A and g : Y Y is a morphism in B B (i.e. (14) AX Af AX BY Bg BY ρ X X f ρ X X, ρ Y Y g ρ Y Y ), such that T b f (15) T X T X T Y T g T Y w Y g w Y, v X f v X. This completes the construction of the Eilenberg-Moore category of a Morita context From Morita contexts to double adjunctions. In this section a functor Γ from the category of Morita contexts on A and B, Mor(A, B), to the category of double adjunctions Adj(A, B) is constructed. For a Morita context T = (A, B, T, T, ev, êv) Mor(A, B), the double adjunction Γ(T) = T = ((A, B) T, (G A, U A ), (G B, U B )), is defined as follows. (A, B) T is the Eilenberg-Moore category for the Morita context T as defined in Section 3.2. The functors U A : (A, B) T A and U B : (A, B) T B are the forgetful functors (i.e. U A X = X and U B X = Y for all objects X = ((X, ρ X ), (Y, ρ Y ), v, w) in (A, B) T ). The definition of the functors G A and G B is slightly more involved. For any X A and Y B, define G A X = ((AX, m A X), ( T X, ˆλX), evx, ˆρX), G B Y = ((T Y, λy ), (BY, m B Y ), ρy, êvy ). (AX, m A X) is simply the free A-algebra on X (see Section 2.1), hence it is an object of A A. From the discussion in Section 2.3 we know that ( T X, ˆλX) B B, with a B- algebra structure induced by the B-A bialgebra functor T. To complete the check that G A X is an object of (A, B) T it remains to verify whether the maps evx : T T X AX and ρx : T AX T X satisfy all needed compatibility conditions. The left hand side of (11) expresses that ev is left A-linear, the left hand side of (12) that ev is right A-linear. The right hand side of (11) follows by the mixed associativity of the B-A bialgebra T (see the last diagram of (5)). The right hand side of (12) is the second Morita identity of the maps ev and êv; see the right hand side of (6). The left hand side of (13) follows again from the properties of T as a B-A bialgebra, in particular by the associativity of its right A-action; compare with the first diagram in (4). Finally, the right hand side of (13) is an application of the fact that ev is B-balanced, which is expressed in the right hand side of (7). We conclude that G A (and, by symmetric arguments, also G B ) is well-defined on objects. For a morphism f : X X in

9 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 9 A, define G A f = (Af, T f) and similarly G B g = (T g, Bg) for any morphism g in B. Verification that G A f and G B g are well defined is very simple and left to the reader. Lemma ((A, B) T, (G A, U A ), (G B, U B )) is a double adjunction. Proof. We construct units ν A, ν B and counits ζ A, ζ B of adjunctions. For any objects X A and Y B, ν A, ν B are defined as morphisms in A and B respectively, ν A X = η A X : X AX, ν B Y = η B Y : Y BY. For any X (A, B) T, the counits ζ A, ζ B are given by the following morphisms in (A, B) T, ζ A X = (f A X, g A X) : G A U A X X, ((AX, m A X), ( T X, ˆλX), evx, ˆρX) ((X, ρ X ), (Y, ρ Y ), v, w), f A X = ρ X : AX X, g A X = w : T X Y, ζ B X = (f B X, g B X) : G B U B X X, ((T Y, λy ), (BY, m B Y ), ρy, êvy ) ((X, ρ X ), (Y, ρ Y ), v, w), f B X = v : T Y X, g B X = ρ Y : BY Y. To check that ζ A X is a morphism in (A, B) T, one has to verify that diagrams (14) and (15) commute. The left hand side of (14) holds, since f A X = ρ X is canonically a morphism in A A, the right hand side holds since g A X = w is a morphism in B B by definition. The left hand side of (15) is exactly the left hand side of (13), and the right hand side of (15) is precisely the left hand side of (12). Similarly one checks that ζ B X is a morphism in (A, B) T. Now take any object X A. The first triangular identity translates to the following diagram G A X (Aη A X, T b η A X) G A U A G A X, (m A X,ˆρX) G A X which commutes by the unit properties of the monad A and the bialgebra functor T (i.e. m A Aη A = A and ˆρ T η A = T ). For the second triangular identity, take any X (A, B) T and consider the diagram U A X = X η A X U A G A U A X = AX, ρ X U A X = X which commutes by the unit property of the A-algebra (X, ρ X ). In the same way one verifies that (G B, U B ) is an adjoint pair.

10 10 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE Let φ = (φ 1, φ 2, φ 3, φ 4 ) : T T be a morphism of Morita contexts. We need to construct a morphism Γ(φ) : Γ(T) Γ(T ) in Adj(A, B), i.e. a functor F = Γ(φ) : (A, B) T (A, B) T such that U A F = U A and U BF = U B. It is well-known that a morphism of monads φ 1 : A A induces a functor Φ 1 : A A A A, given by Φ 1 (X, ρ X ) = (X, ρ X φ 1 X), for all objects (X, ρ X ) A A, and Φ 1 f = f for all morphisms f in A A (the naturality of φ 1 implies that f is indeed a morphism in A A ). Similarly, a morphism of monads φ 2 : B B induces a functor Φ 2 : B B B B. Take any X = ((X, ρ X ), (Y, ρ Y ), v, w) (A, B) T and define F X = ((X, ρ X φ 1 X), (Y, ρ Y φ 2 Y ), v φ 3 Y, w φ 4 X). To check diagrams (11), (12), (13), one has to rely on the naturality of φ 1, φ 2, φ 3, φ 4, and their properties as morphisms of monads and bialgebras. For a morphism (f, g) : X X in (A, B) T, define F (f, g) = (f, g). Then, by construction (or by naturality of φ 1 and φ 2 ), f is a morphism in A A and g is a morphism in B B. Diagram (15) for F(f, g) to be a morphism in (A, B) T follows by the naturality of φ 3 and φ 4, combined with the corresponding diagram for (f, g) as a morphism in (A, B) T. Finally, the construction of F immediately implies that, for any X (A, B) T, U A F X = U A X = X and U BF X = U B X = Y. This completes the construction of a functor Γ : Mor(A, B) Adj(A, B) Every Morita context arises from a double adjunction. Here we prove that starting with a Morita context and performing subsequent constructions of a double adjunction and a Morita context gives back the original Morita context, i.e. we prove the following Lemma The composite functor ΥΓ is the identity functor on Mor(A, B). Proof. The computation that, for any Morita context T Mor(A, B), ΥΓT = T is easy and left to the reader. Consider two Morita contexts T = (A, B, T, T, ev, êv), T = (A, B, T, T, ev, êv ) and a morphism φ = (φ 1, φ 2, φ 3, φ 4 ) : T T. Write (φ 1 Γ(φ), φ 2 Γ(φ), φ 3 Γ(φ), φ 3 Γ(φ)) := ΥΓφ. In view of the definition of functor Υ in Section 3.1, to compute the φ i Γ(φ) one first needs to compute natural transformations a, b (see equations (1) in Section 2.2) corresponding to double adjunctions ΓT = ((A, B) T, (G A, U A ), (G B, U B )) and ΓT = ((A, B) T, (G A, U A ), (G B, U B )); see Section 3.3. These are given by a = ζ A Γ(φ)G A G A ν A, b = ζ B Γ(φ)G B G B ν B where ζ A is the counit of adjunction (G A, U A ), ζ B is the counit of adjunction (G B, U B ), ν A is the unit of adjunction (G A, U A ) and ν B is the unit of adjunction (G B, U B ). Since, for all X = ((X, ρ X ), (Y, ρ Y ), v, w) (A, B) T, we obtain Thus Γ(φ)(X) = ((X, ρ X φ 1 X), (Y, ρ Y φ 2 Y ), v φ 3 Y, w φ 4 X), ζ A Γ(φ)G A = (m A φ 1 A, ˆρ φ 4 A ), G A ν A = (Au A, T u A ). φ 1 Γ(φ) = U A a = m A φ 1 A Au A = m A A u A φ 1 = φ 1,

11 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 11 where the second equality follows by the naturality of φ 1, while the third one is a consequence of the unitality of a monad. Similarly, φ 4 Γ(φ) = U B a = ˆρ φ 4 A T u A = ˆρ T u A φ 4 = φ 4. The identities φ 2 Γ(φ) = φ2 and φ 3 Γ(φ) = φ3 are obtained by symmetric calculations. The just computed identification thus defines a natural transformation (the identity transformation) λ : 11 Mor(A,B) ΥΓ The comparison functor. Consider a double adjunction on categories A and B, i.e. an object T = (T, (L A, R A ), (L B, R B )) in Adj(A, B). Let ΥT = T be the associated Morita context on A and B, and consider (A, B) T, the Eilenberg-Moore category of representations of T. In this section we construct a comparison functor For any object Z T, define K : T (A, B) T. K(Z) = ((R A Z, R A ε A Z), (R B Z, R B ε B Z), R A ε B Z, R B ε A Z). The first two components in K(Z), that is, (R A Z, R A ε A Z) A A and (R B Z, R B ε B Z) B B are an application of the Kleisli functors K A : T A A and K B : T B B, corresponding to the adjunctions (L A, R A ) and (L B, R B ) respectively (see Section 2.1). Obviously, R A ε B Z : R A L B R B Z R A Z and R B ε A Z : R B L A L B Z R B Z are well-defined. They satisfy conditions (11), (12) and (13) by the naturality of counits. For a morphism f : Z Z in T, define K(f) = (R A f, R B f). In view of the definition of the comparison functors K A and K B, it is clear that R A f = K A f and R B f = K B f, so R A f and R B f are morphisms in A A and B B respectively. Diagrams (15) follow by the naturality of ε A and ε B, respectively. Definition Let T = (T, (L A, R A ), (L B, R B )) be an object in Adj(A, B). The pair (R A, R B ) is said to be moritable if and only if the comparison functor K is an equivalence of categories. Proposition (Γ, Υ) is an adjoint pair and Γ is a full and faithful functor. Proof. Note that the definition of the comparison functor K immediately implies that K is a morphism in Adj(A, B). Furthermore, K can be defined for any T Adj(A, B). We claim that the assignment T (K : T (A, B) T ) induces a natural transformation κ : ΓΥ 11 Adj(A,B). Take double adjunctions T = (T, (L A, R A ), (L B, R B )), T = (T, (L A, R A ), (L B, R B )) and a functor F : T T such that R A F = R A and R BF = R B (in other words, take a morphism in Adj(A, B)). Let K : T (A, B) ΥT and K : T (A, B) ΥT be the

12 12 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE associated comparison functors. The naturality of κ is equivalent to the commutativity of the following diagram For any object Z of T, and T F T K (A, B) ΥT ΓΥ(F ) K (A, B) ΥT. KF Z = ((R A F Z, R A ε A F Z), (R B Z, R B ε B F Z), R A ε B F Z, R B ε A F Z), (ΓΥ(F )K )(Z) = ((R AZ, R Aε A Z R A ar AZ), (R BZ, R Bε B Z R B br BZ), R Aε B Z R A br BZ, R Bε A Z R B ar AZ), where a and b are natural transformations (1) associated to a morphism of double adjunctions F. The equalities R A F = R A and R BF = R B together with equations (3) yield the required equality KF = ΓΥ(F )K. Let λ : 11 Mor(A,B) ΥΓ be the natural (identity) transformation described in Section 3.4. That the composite κγ Γλ is the identity natural transformation Γ Γ is immediate. To compute the other composite Υκ λυ : Υ Υ, take a double adjunction T = (T, (L A, R A ), (L B, R B )), so that ΥT = (R A L A, R B L B, R A L B, R B L A, R A ε B L A, R B ε A L B ). Then κt = K : T (A, B) ΥT is the comparison functor, and hence where ΥκT = Υ(K) = (φ 1, φ 2, φ 3, φ 4 ), φ 1 = U A ζ A KL A U A G A η A, φ 2 = U B ζ B KL B U B G B η B, φ 3 = U A ζ B KL B U A G B η B, φ 4 = U B ζ A KL A U B G A η A. Here (G A, U A ), (G B, U B ) are adjoint pairs given by ((A, B) ΥT, (G A, U A ), (G B, U B )) := Γ(R A L A, R B L B, R A L B, R B L A, R A ε B L A, R B ε A L B ), so G A η A = (R A L A η A, R B L A η A ), G B η B = (R B L B η B, R A L B η B ). Furthermore, using the definition of the comparison functor (applied to L A X and L B Y, for any objects X A, Y B), we obtain Therefore, ζ A KL A = (R A ε A L A, R B ε B L A ), ζ B KL B = (R B ε B L B, R A ε A L B ). φ 1 = R A ε A L A R A L A η A = R A L A, φ 4 = R B ε A L A R B L A η A = R B L A, since η A is the unit and ε A is the counit of the adjunction (L A, R A ). Similarly, φ 2 = R B L B and φ 3 = R A L B. Thus Υκ is the identity natural transformation, and since also λυ is the identity, so is their composite Υκ λυ. This proves that λ is a unit and κ is a counit of the adjunction (Γ, Υ). Since the unit λ is a natural isomorphism, Γ is a full and faithful functor.

13 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 13 Corollary (Γ, Υ) is a pair of inverse equivalences if and only if, for all double adjunctions T = (T, (L A, R A ), (L B, R B )) Adj(A, B), (R A, R B ) is a moritable pair. Proof. The moritability of each of (R A, R B ) is paramount to the comparison functor K being an equivalence, for all T Adj(A, B), i.e. to the natural transformation κ in the proof of Proposition being an isomorphism. Since the latter is the counit of adjunction (Γ, Υ), the corollary is an immediate consequence of Proposition Moritability. Let T = (T, (L A, R A ), (L B, R B )) be a double adjunction on A and B. This section is devoted to a study when the pair (R A, R B ) is moritable in the sense of Definition We begin with the following simple Lemma Let T be an object in Mor(A, B) and let (f, g) be a morphism in (A, B) T. Then (f, g) is an isomorphism in (A, B) T if and only if f is an isomorphism in A and g is an isomorphism in B. Proof. If (f, g) is an isomorphism in (A, B) T, then clearly f and g are isomorphisms (as all functors, in particular forgetful functors, preserve isomorphisms). Conversely, let f 1 be the inverse (in A) of f and g 1 be the inverse of g (in B). By applying f 1, g 1 to both sides of equalities described by diagrams (14) and (15) one immediately obtains that (f 1, g 1 ) is a morphism in (A, B) T. From now on we fix a double adjunction T = (T, (L A, R A ), (L B, R B )) (with counits ε A, ε B and units η A, η B ) and set T := Υ(T) = (R A L A, R B L B, R A L B, R B L A, R A ε B L A, R B ε A L B ) to be the corresponding Morita context. K : T (A, B) T is the comparison functor. The aim is to determine, when K is an equivalence of categories. Proposition Suppose that (R A, R B ) is a moritable pair and let f be a morphism in T. If both R A f and R B f are isomorphisms, then so is f. Proof. Note that R A = U A K and R B = U B K, where U A : (A, B) T A, U B : (A, B) T B are forgetful functors. If R A f and R B f are isomorphisms, then, by Lemma also Kf is an isomorphism. Since an equivalence of categories reflects isomorphisms, also f is an isomorphism. Definition The pair (R A, R B ) is said to reflect isomorphisms if the fact that both R A f and R B f are isomorphisms for a morphism f T implies that f is an isomorphism. Note that if R A or R B reflects isomorphisms, then the pair (R A, R B ) reflects isomorphisms, but not the other way round. By Proposition 3.6.2, a moritable pair (R A, R B ) reflects isomorphisms. To analyse the comparison functor K further we assume the existence of particular colimits in T. For any object X = ((X, ρ X ), (Y, ρ Y ), v, w) (A, B) T consider the

14 14 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE following diagram (16) L A R A L A X L A ρ X L B R B L A X L A R A L B Y ε B L A X L L B w ε ε A L A X A A v L B Y ε B L B Y L B R B L B Y L A X L B Y. For the rest of this section until Theorem assume that T has colimits of all such diagrams, and let (DX, d A X : L A X DX, d B X : L B Y DX), be the colimit of (16). Any morphism (f, g) : X X in (A, B) T determines a morphism of diagrams (16) (i.e. it induces a natural transformation between functors from a six-object category to T that define diagrams (16)) by applying suitable combinations of the L and R to f and g. Therefore, by the universality of colimits, there is a unique morphism D(f, g) : DX DX in T which satisfies the following identies (17) d A X L A f = D(f, g) d A X, d B X L B f = D(f, g) d B X. This construction yields a functor D : (A, B) T T, X DX, (f, g) D(f, g). Proposition The functor D is the left adjoint of the comparison functor K. Proof. L A R A L A R A Z L A R A ε A Z L A R A Z For any object Z in T there is a cocone L B R B L A R A Z L A R A L B R B Z L B ρ Y ε B L A R A Z L L B R B ε ε A L A R A Z A Z ε A A R A ε L B R Z B Z ε B L B R B Z ε A Z ε B Z Z. L B R B L B R B Z L B R B Z This is a cocone under the diagram of type (16) corresponding to the object KZ. By the universal property of colimits there is a unique morphism εz : DKZ Z, such that (18) εz d A KZ = ε A Z, εz d B KZ = ε B Z. This construction defines a natural transformation ε from the functor DK to the identity functor on T. For all objects X in (A, B) T, define η A X : X R A DX and η B X : Y R B DX as composites η A X = R A d A X η A X, η B X = R B d B X η B Y. Using the naturality of ε A, ε B, η A and η B, the triangular identities for units and counits of adjunctions, and the definition of (DX, d A X, d B X) as a cocone under the diagram (16), one can verify that the pair ηx := (η A X, η B X) is a morphism in (A, B) T. L B R B ε B Z

15 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 15 The assignment X ηx defines a natural transformation η from the identity functor on (A, B) T to KD. We now prove the triangular identities for ε and η. Take any object Z in T and compute R A εz η A KZ = R A εz R A d A KZ η A R A Z = R A ε A Z η A R A Z = R A Z, where the second equality follows by (18) and the third by the triangular identities for ε A and η A. Similarly, R B εz η B KZ = R B Z. This proves that the composite Kε ηk is the identity natural transformation on K. Next take any object X in (A, B) T. Since DηX = D(η A X, η B X), DηX satisfies equalities (17). In particular Therefore, DηX d A X = d A KDX L A R A d A X L A η A X. εdx DηX d A X = εdx d A KDX L A R A d A X L A η A X = ε A DX L A R A d A X L A η A X = d A X ε A L A X L A η A X = d A X, where the second equality follows by (18), the third one by the naturality of ε A, and the final one is one of the triangular identities for ε A and η A. Similarly, εdx DηX d B X = d B X. The universality of colimits now yields εdx DηX = DX, i.e. the second triangular identity for ε and η. Definition The pair (R A, R B ) is said to convert colimits into coequalisers if, for all objects X = ((X, ρ X ), (Y, ρ Y ), v, w) (A, B) T, the diagrams and R A L A R A L A X R B L B R B L B Y R A ε A L A X R A L A ρ X R B ε B L B Y R B L B ρ Y are equalisers in A and B respectively. R A L A X R Ad A X R A DX R B L B Y R B d B X R B DX Lemma The functor D is fully faithful if and only if the pair (R A, R B ) converts colimits into coequalisers. Proof. For all objects X = ((X, ρ X ), (Y, ρ Y ), v, w) (A, B) T, consider the following diagram R A L A R A L A X R A ε A L A X R A L A ρ X ρ X R A L A X R A d A X R A DX. Since the multiplication in monad R A L A is given by R A ε A L A, the top row is a coequaliser; see [1, Proposition 4, Section 3.3]. Thus there exists a morphism X R A DX, which, by its uniqueness, must coincide with η A X. By considering a similar diagram for the other adjoint pair, one fits into it the other component of the unit of X

16 16 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE adjunction, η B X. If (R A, R B ) converts colimits into coequalisers, then (by the uniqueness of coequalisers) both η A X and η B X are isomorphisms. Hence, by Lemma 3.6.1, η is an isomorphism, i.e. D is fully faithful. Conversely, if η is an isomorphism, then both (R A DX, R A d A X) and (R B DX, R B d B X) are (isomorphic to) coequalisers, i.e. the pair (R A, R B ) converts colimits into coequalisers. The main result of this section is contained in the following precise moritability theorem (Beck s theorem for double adjunctions). For this theorem the existence of colimits of diagrams (16) needs not to be assumed a priori. Theorem Let T = (T, (L A, R A ), (L B, R B )) be a double adjunction. Then the pair (R A, R B ) is moritable if and only if T has colimits of all the diagrams (16) and the pair (R A, R B ) reflects isomorphisms and converts colimits into coequalisers. Proof. Assume that K is an equivalence of categories. Consider a diagram of the form (16) in T. We can choose X = KZ for some Z T and we know that (Z, ε A, ε B ) is a cocone for this diagram. Now apply the functor K to diagram (16), then we claim that (KZ, Kε A, Kε B ) is a colimit for the new diagram in (A, B) T. To check this, consider any cocone (H, (f A, g A ), (f B, g B )) on the diagram. Define (h, k) : KZ H by putting h = f A η A R A Z and k = g B η B R B Z. One can verify that (h, k) is indeed a morphism in (A, B) T (apply the fact that H is a cocone and that (f A, g A ) and (f B, g B ) are morphisms in (A, B) T, together with the adjunction properties of (L A, R A ) and (L B, R B )). Since K is an equivalence of categories, it reflects colimits and therefore (Z, ε A, ε B ) is the colimit of the original diagram (16) in T. Furthermore, (R A, R B ) reflects isomorphisms by Proposition and it converts colimits into coequalisers by Lemma In the converse direction, the counit η is an isomorphism by Lemma Applying R A to the first column of the cocone defining εz we obtain the following diagram R A L A R A L A R A Z R A L A R A L A R A Z R A L A R A ε A Z R A L A R A Z R A ε A L A R A Z R A L A R A ε A Z R A L A R A Z R A ε A L A R A Z R A ε A Z R A Z R A εz R A d A KZ R A DKZ. The first column is a (contractible) coequaliser, the second is a coequaliser by the assumption that (R A, R B ) converts colimits into coequalisers. Thus R A εz is an isomorphism. Similarly, R B εz is an isomorphism. Since the pair (R A, R B ) reflects isomorphisms, also εz is an isomorphism in T. 4. Morita theory In this section we study when, given a Morita context T = (A, B, T, T, ev, êv) on A and B, the bialgebra functors T and T induce an equivalence of categories between the categories A A and B B. In particular, we prove that if A and B have coequalisers and U A and U B preserve coequalisers, then any equivalence between two categories A A and B B of algebras of monads A and B is induced by a Morita context in Mor(A, B).

17 A BECK-TYPE THEOREM FOR A MORITA CONTEXT Preservation of coequalisers by algebras. Consider a monad A = (A, m A, u A ) on a category A and a functor S : A B. A pair (S, σ) is called a right A-algebra functor if (S, S, σ) is a B-A bialgebra functor, where B is the trivial monad on B. Thus (S, σ) is a right A-algebra functor if and only if σ : SA S is a natural transformation such that σ Sm A = σ σa and S = σ Su A. Similarly, one introduces the notion of a left A-algebra functor. Note that if T = (T, λ, ρ) is an A-B bialgebra functor, then (T, λ) is a left A-algebra functor and (T, ρ) is a right B-algebra functor. Lemma Let A = (A, m A, u A ) be a monad on A and S = (S, σ) a right A- algebra functor. Any coequaliser preserved by SA is also preserved by S. If (S, σ ) is a left A-algebra functor, then any coequaliser preserved by AS is also preserved by S. Proof. Consider a coequaliser X f g Y in A and assume that it is preserved by SA. Applying the functors S and SA to this coequaliser, one obtains the following diagram in B Su A X SX σx SAX Sf Sg SAf SAg SY Su A Y SAY σy z Sz Z Su A Z SZ σz SAz SAZ. By assumption the lower row is a coequaliser. Suppose that there exists a pair (h, H), where H is an object in B and h : SY H is a morphism in B such that h Sf = h Sg. Since σ is a natural transformation, h σy SAf = h σy SAg. By the univeral property of the coequaliser (SAZ, SAz), there is a unique morphism k : SAZ H such that k SAz = h σy. The naturality of u A and the unitality of the right A-algebra S imply that k Su A Z Sz = h. Then k = k Su A Z : SZ H is a unique morphsim such that h = k Sz. The second statement is verified by a similar computation. Lemma Let f, g : X Y be morphisms in A A. Suppose that the coequaliser (E, e) of (U A (f), U A (g)) exists in A. If AA preserves this coequaliser, then the coequaliser (E, ɛ) of the pair (f, g) exists in A A and U A (E, ɛ) = (E, e). Proof. By Lemma 4.1.1, A preserves the coequaliser (E, e). The universal property of coequalisers implies the existence of a unique map ρ E : AE E such that ρ E Ae = e ρ Y and ρ E u A E = E. Using the fact that the coequaliser (E, e) is preserved by AA, one checks that ρ E defines a (associative) left action of A on E, i.e. (E, ρ E ) is an object of A A. Lemma If A has (all) coequalisers, then the following statements are equivalent: (i) A : A A preserves coequalisers; (ii) AA : A A preserves coequalisers; (iii) A A has (all) coequalisers and they are preserved by U A : A A A;

18 18 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE (iv) U A : A A A preserves coequalisers. Proof. Implications (i) (ii) and (iii) (iv) are obvious. (ii) (iii). Follows by Lemma (iv) (i). The free algebra functor F A : A A A preserves coequalisers, since it is a left adjoint functor. By assumption U A preserves coequalisers as well and A = U A F A, so the statement (i) follows Morita contexts and equivalences of categories of algebras. Let B = (B, m B, u B ) be a monad on B and let (T : B A, ρ) be a right B-algebra functor. For any (X, ρ X ) in B B, let (T B X, τx) be the following coequaliser in A (if it exists) (19) T BX ρx T ρ X T X τx T B X. Similarly, given a monad A = (A, m A, u A ) on A and a right A-algebra functor ( T : A B, ˆρ), consider an object (Y, ρ Y ) in A A, and set ( T A Y, ˆτY ) to be the following coequaliser in B (if it exists) (20) T AY ˆρY bt ρ Y T Y ˆτY TA Y. Finally, recall that for any B-algebra (X, ρ X ), the following diagram is contractible coequaliser in B and a (usual) coequaliser in B B, (21) BBX m B X Bρ X BX ρ X X. As in Section 2.1, the free forgetful adjunctions for A and B are denoted by (F A, U A ), (F B, U B ), respectively. The counits are denoted by ε A, ε B. We are now ready to state the following lifting theorem. Proposition Let A be a monad on A and B a monad on B. There is a bijective correspondence between the following data: (i) an A-B bialgebra functors T, such that coequalisers of the form (19) exist in A for any X B, and they are preserved by AA; (ii) functors T B : B B A A such that AAU A T B : B B A preserves coequalisers of the form (21) for all (X, ρ X ) B B. Let T be a bialgebra functor as in (i) and T B the corresponding functor of (ii), then given a functor P : A X, the functor P preserves coequalisers of the form (19) if and only if the functor P U A T B : B B X preserves coequalisers of the form (21). Proof. (i) (ii). Given a B-algebra (X, ρ A ), define T B X by (19). Then it follows by Lemma that there is a morphism ρ TBX : AT B X T B X such that (T B X, ρ TBX ) is an object in A A. For a morphism f : X Y in B B, the universality of the coequaliser induces a morphism T B f : T B X T B Y such that T B f τx = τy T f. To check that T B f is a morphism in A A, one can proceed as follows. By Lemma 4.1.1, A preserves coequalisers of the form (19), so in particular, AτX is an epimorphism. Therefore, it is enough to verify that T B f ρ TBX AτX = ρ TBY AT B f AτX, which follows from the defining properties of ρ TBX, ρ TBY and T B f, combined with the

19 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 19 naturality of the right action ρ of T. Thus there is a well-defined functor T B : B B A A. Consider now a functor P : A X which preserves all the coequalisers of the form (19). For any (X, ρ X ) B B, construct the following diagram in A P T BBBX P T BmBX P T BBX P T m B BX P ρbbx P T BBX P τbbx P T BBρ X P T m B X P T Bρ X P T m B X P T BX P τbx P ρbx P T Bρ X P T ρ X P U A T B BBX P T Bm B X P T B Bρ X P U A T B BX P TBρX P T ρ X T BX P T X P ρx P τx P U A T B X. The first and second rows of this diagram are coequalisers, as they are obtained by applying functors to the contractible coequaliser (21). All three columns are coequalisers, since they are coequalisers of type (19) to which the functor P is applied. The diagram chasing arguments then yield that the lower row is a coequaliser too. The first part of the proof is then completed by setting P = AA. (ii) (i). Define T = U A T B F B, and natural transformations ρ = U A T B ε B F B = U A T B m B : T B T and λ = U A ε A T B F B : AT T. Then (T, λ, ρ) is an A- B bialgebra functor. Note that U A T B is also a left A-algebra functor with action U A ε A T B. Lemma implies that U A T B preserves the coequaliser (21). This means that, for any X = (X, ρ X ) B B, the following diagram is a coequaliser in A (22) U A T B F B BX U AT B m B X U A T B F B ρ X U A T B F B X UATBρX U A T B X. This diagram is exactly the coequaliser (19) in this situation. Applying the functor AA (resp. any functor P : A X ) to the coequaliser (22) yields the same result as applying the functor AAU A T B (resp. P U A T B ) to (21). Therefore, the functor AA (resp. P ) preserves coequalisers (19). In the following the notation introduced in Proposition is used. For an A- B bialgebra functor T, T B denotes the functor defined by coequalisers (19), and T = U A T B F B. Lemma Suppose that the coequalisers of the form (19) and (20) exist and they are preserved by AA, T A and by BB respectively, then the functor T A : A A B B preserves coequalisers of the form (19).

20 20 TOMASZ BRZEZIŃSKI, ADRIAN VAZQUEZ MARQUEZ, AND JOOST VERCRUYSSE Proof. Note that the functor T A is well-defined by (the dual version of) Proposition We can now consider the following diagram in B: (23) T AT BX b T AρX ˆρT BX T T BX bt λbx ˆτT BX T A T BX bt AT ρ X bt ρx bt T ρ X bt A ρx bt A ρ X T AT X ˆρT X T T X bt λx ˆτT X TA T X bt AτX bt τx bt A τx T AT B X ˆρT B X T T B X bt ρ T B X ˆτT B X TA T B X. By assumption, the first row is a coequaliser; by Lemma 4.1.1, the second row is a coequaliser too. All three columns are coequalisers by definition. Therefore, the lower row is an equaliser as well. If we apply the functor BB to this diagram, the same reasoning yields that BB preserves the equaliser in the lower row of (23), therefore it is an equaliser in B B by Lemma Recall that a wide Morita context (F, G, µ, τ) between categories C and D, consists of two functors F : C D and G : D C, and two natural transformations µ : GF C and τ : F G D, satisfying F µ = τf and µg = Gτ. In [5], left (resp. right) wide Morita contexts are studied. In this setting, C and D are abelian (or Grothendieck) categories and F and G are left (resp. right) exact functors. Left and right wide Morita contexts are used to characterise equivalences between the categories C and D. In the remainder of this section we study wide Morita contexts between categories of algebras over monads. This allows us to weaken the assumptions made in [5]. Moreover, we study the relationship between wide Morita contexts and Morita contexts. Consider monads A on A and B on B. A pair of functors T A : A A B B and T B : B B A A is said to be algebraic if the functors AAU A T B and U B TA AAU A T B preserve coequalisers of the form (21) for all (X, ρ X ) B B and the functors BBU B TA and U A T B BBU B TA preserve coequalisers of the form (24) AAX m A X Aρ X AX ρ X X, for all (X, ρ X ) A A. By an algebraic wide Morita context we mean a wide Morita context (T B, T A, ˆω, ω) between categories of algebras B B and A A, such that the functors T B and T A form an algebraic pair of functors. Remark If the forgetful functors U B and U A preserve coequalisers and the categories B B and A A have all coequalisers (e.g. B B and A A are abelian categories, as in [5]) or equivalently the categories B and A have all coequalisers, see Lemma 4.1.3, then the situation simplifies significantly. In this case, for any bialgebra functors T and T, all coequalisers of the form (19) and (20) exist and, by Lemma 4.1.3, are preserved by AA and BB respectively. Furthermore, in this situation, (T B, T A ) is an algebraic pair if and only if the functors T B and T A preserve coequalisers (of the form (21) and (24) respectively). In particular, it is an adjoint algebraic pair if T A preserves

21 A BECK-TYPE THEOREM FOR A MORITA CONTEXT 21 coequalisers (as any left adjoint preserves all coequalisers). Also, if B B and A A are abelian, then a right wide Morita context between B B and A A is always algebraic. Theorem Let A be a monad on A and B a monad on B. There is a bijective correspondence between the following data: (i) Morita contexts T = (A, B, T, T, ev, êv) on A and B, such that all coequalisers of the form (19) and (20) exist, and they are preserved by AA, T A and BB, T B respectively; (ii) functors T B : B B A A and T A : A A B B, and natural transformations ˆω : TA T B B B and ω : T B TA A A such that (T B, T A, ˆω, ω) is an algebraic wide Morita context between B B and A A. Proof. (i) (ii). The existence of the functors T A and T B, forming an algebraic pair, follows from Proposition Moreover, in light of Lemma 4.2.2, T A preserves coequalisers of the form (19) and T B preserves coequalisers of the form (20). Consider again diagram (23) in which now all the rows and columns are coequalisers. The natural transformation êv induces a map êvx : T T X BX that equalises both the horizontal and vertical arrows. A diagram chasing argument affirms the existence of a map ˆω X : TA T B X BX. For all (X, ρ X ) B B, define ˆω : TA T B B B by ˆωX = ρ X ˆω X. The naturality of êv and the construction of ˆω X imply that ˆω is a natural transformation. Similarly, one defines ω : T B TA A A. To check that the compatibility conditions between ˆω and ω hold, one starts with objects T B T AT BX and T AT B T AY and constructs coequaliser diagrams resulting in T B TA T B X and T A T B TA Y, respectively. The compatibility conditions between ev and êv (diagrams (6)-(7)), as well as the facts that they are bialgebra morphisms and that T and T are bialgebra functors, induce the needed compatibility conditions for ˆω and ω. (ii) (i). By Proposition there exist bialgebra functors T and T, where T = U A T B F B and T = U B TA F A, such that all coequalisers of the form (19) and (20) exist, and they are preserved by AA, T A and BB, T B respectively. Define ev : T T = U A T B F B U B TA F A U A T B ε B b T A F A U A T B TA F A U A ωf A UA F A = A, êv : T T = U B TA F A U A T B F B U B b TA ε A T B F B U B TA T B F B U B ˆωF B UB F B = B, where ε B and ε A are the counits of the adjunctions (F B, U B ) and (F A, U A ). Then (A, B, T, T, ev, êv) is a Morita context. Consider a Morita context (A, B, T, T, ev, êv) on A and B. Suppose that the coequalisers of the form (19) and (20) exist and are preserved by AA, T A and BB, T B respectively. Then there exist natural transformations π : T B T A and ˆπ : TA T B such that, for all objects Y A and X B, evy = πy τ T Y and êvx = ˆπX ˆτT X.