Journal of Pure and Applied Algebra

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1 Journal o Pure and Applied Algebra 215 (2011) Contents lists available at ScienceDirect Journal o Pure and Applied Algebra journal homepage: Internal Kleisli categories Tomasz Brzezińsi, Adrian Vazquez-Marquez Department o Mathematics, Swansea University, Singleton Par, Swansea SA2 8PP, UK a r t i c l e i n o a b s t r a c t Article history: Received 16 October 2009 Received in revised orm 14 October 2010 Available online 21 January 2011 Communicated by J. Adáme A construction o Kleisli objects in 2-categories o noncartesian internal categories or categories internal to monoidal categories is presented Elsevier B.V. All rights reserved. MSC: 18D05; 18D10; 16T15; 18C20 1. Introduction Internal categories within monoidal categories have been introduced and studied by Aguiar in his Ph.D. thesis [1] as a ramewor or analysing properties o quantum groups. By choosing the monoidal category M appropriately, algebraic structures o recent interest in Hop algebra (or quantum group) theory, such as corings and C-rings, can be interpreted as internal categories. Internal categories can be organised into two dierent 2-categories. The irst one, denoted by Cat(M), has internal unctors as 1-cells, and internal natural transormations as 2-cells. The second one, denoted by Cat c (M), has internal counctors as 1-cells, and internal natural cotransormations as 2-cells. While a unctor can be interpreted as a push-orward o morphisms, a counctor can be interpreted as a liting o morphisms rom one internal category to the other. The aim o this paper is to study adjunctions and monads in both Cat(M) and Cat c (M) and to show that every such monad has a Klesili object. A monad (comonad) in the 2-category Cat(M), i.e., an internal unctor with two natural transormations which satisy the usual associativity and unitality conditions, is simply called an internal monad (respectively, an internal comonad). We show that every internal monad (comonad) arises rom and gives rise to a pair o adjoint unctors, by explicitly constructing the Kleisli internal category. A monad in the 2-category Cat c (M), i.e., an internal counctor with two natural cotransormations which satisy the usual associativity and unitality conditions, is called an internal opmonad. Similarly as or internal monads, we show that every internal opmonad arises rom and gives rise to a pair o adjoint counctors, by explicitly constructing Kleisli objects in Cat c (M). The paper is organised as ollows. In Section 2 we review internal categories and describe 2-categories Cat(M), Cat c (M), ollowing [1]. In Section 3 we irst review elements o the ormal theory o monads o Street and Lac [15,9], which is a ormal ramewor or studying monads in any bicategory. We then prove that Cat(M), Cat c (M) and the vertical dual o Cat(M), embed into the bicategory o Kleisli objects associated to particular bicategories. The existence o these embeddings is then used in Section 4 to deduce the existence o Kleisli internal categories (or internal monads, opmonads and comonads). The paper is completed with explicit examples which interpret internal adjunctions and Klesli objects as twisting o (co)rings, and mae a connection between the internal Kleisli categories and constructions amiliar rom Hop algebra and Hop-Galois theories. 2. Noncartesian internal categories 2.1. The deinition o an internal category Extending the classical approach to internal category theory, M. Aguiar introduced the ollowing notion o an internal category in a monoidal category in [1]. Let M = (M,, 1) be a monoidal category (with tensor product and unit object 1), Corresponding author. addresses: T.Brzezinsi@swansea.ac.u (T. Brzezińsi), @swansea.ac.u (A. Vazquez-Marquez) /$ see ront matter 2011 Elsevier B.V. All rights reserved. doi: /j.jpaa

2 2136 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) regular in the sense that it has equalisers and preserves all equalisers. Typical examples we are interested in are the category Vect o vector spaces over a ield and the opposite o the category Mod o modules over a commutative, associative, unital ring, with the standard tensor products. As a rule, when dealing with monoidal categories, we do not write canonical associativity and let and right unitality isomorphisms nor the canonical morphisms embedded in deinitions o equalisers, etc. The regularity condition ensures that a bicategory Comod(M) o comonoids can be associated to M = (M,, 1). The objects (0-cells) in Comod(M) are coassociative and counital comonoids C = (C, C, e C ) ( C is the comultiplication, e C denotes the counit). The 1-cells C D are C-D-bicomodules M = (M, λ, ϱ). Here λ : M C M denotes the let C-coaction and ϱ : M M D is the right D-coaction that are counital, coassociative and commute with each other in the standard way. Bicolinear maps are the 2-cells o Comod(M). The vertical composition (o 2-cells) is the same as the composition in M. The horizontal composition (o 1-cells and 2-cells) is given by the cotensor product: Start with comonoids C, D, E, a C-D-bicomodule M = (M, λ M, ϱ M ) and a D-E-bicomodule N = (N, λ N, ϱ N ). Deine M D N as the equaliser: M D N M N ϱ M N M λ N M D N. The C-E-bicomodule M D N = (M D N, λ M D N, M D ϱ N ) is then the composite 1-cell C M D N E. Given a map o comonoids : C D, any let C comodule A = (A, λ) can be made into a let D-comodule with the induced coaction ( A) λ. We denote the object part o the resulting D-comodule by A (obviously, A is isomorphic to A in M), and the comodule itsel by A. Similarly, any right C-comodule A = (A, ϱ) is a right D-comodule with coaction (A ) ϱ; we denote it by A. Let C = (C, C, e C ) be a comonoid. Then the category o endo-1-cells on C (in Comod(M)) or the category o C- bicomodules C M C is a monoidal category with the monoidal product given by the cotensor product C and with the unit object C (understood as a C-bicomodule with both coactions provided by C ). Following [1], by an internal category in M (with object o objects C) we mean a monoid in C M C, that is a C-bicomodule with an associative and unital composition. This means, an internal category is a pair: a comonoid C = (C, C, e C ) in M and a monoid A = (A, m A, u A ) in a monoidal category o C-bicomodules C M C (with cotensor product). Here (A, λ, ϱ) is the underlying object, the multiplication in A is denoted by m A and the unit by u A. A is thought o as the object o morphisms (with composition m A and identity morphism u A ), while C is understood as an object o objects. For an internal category we write (C, A) to indicate both: objects and morphisms. The let coaction λ : A C A should be understood as determining the codomain and the right coaction ϱ : A A C as determining the domain. The notation m n A is used to record the n 1-old composite o m A, e.g. m 2 = A m A (m A C A) etc. In case M = Set and = (the cartesian monoidal category) the above deines the usual internal category in Set. I M is the category o vector spaces over a ield, then internal category in M coincides with the notion o a C-ring. O special algebraic interest is the case M = Mod op comonoids in Mod op (the opposite o the category o modules over a commutative ring ). The become A-bimodules. The are the same as the monoids in Mod, i.e. -algebras A; bicomodules in Mod op equalisers in Mod op are the same as coequalisers in Mod ; consequently the cotensor products in Mod op coincide with the usual tensor products o bimodules. Multiplication and unit become comultiplication and a counit respectively. In a word: an internal category in Mod op is the same as an A-coring; see [4] or more details about corings. We return to examples o this in Section 5. While the deinition o a non-cartesian internal category ollows quite naturally the cartesian case, the deinition o internal unctors and natural transormations leaves more scope or reedom. Two possible deinitions are proposed in [1] and we describe them both presently Internal unctors and the 2-category Cat(M) An internal unctor or simply a unctor : (C, A) (D, B) is a pair = ( 0, 1 ), where 0 : C D is a morphism o comonoids, and 1 : 0 A 0 B is a D-bicomodule map that is multiplicative and unital in the sense that the ollowing diagrams commute: A C A A 0 D 0 A 1 D 1 B D B C 0 m A A 1 B, m B u A Du B A 1 B.

3 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) Internal unctors are composed component-wise. In order to relieve the notation, in what ollows we write V or V 0 etc. Typically, we also denote the composition o morphisms in M by juxtaposition. The composition in M taes precedence over all other operations on morphisms in M. An internal natural transormation α : g is a morphism o D-bicomodules (i.e. a 2-cell in Comod(M)) α : g C B such that the ollowing diagram commutes A C C A g g D C g 1 D α B D B ϱ m B A B λ m B C C A C C A B D B. α D 1 Natural transormations can be composed in two dierent ways. The vertical composition is given by the convolution product. That is, the composite o α : g and β : g h is β α := m B (β D α) C. The horizontal composition, also nown as the Godement product, is deined as ollows. In the situation: (C, A) α (D, B) β (D, B ) g the Godement product β α : h g is deined as β α := βg 0 h 1 α = 1 α β 0. h With any internal unctor : (C, A) (D, B) one associates the identity natural transormation on. This is given as := 1 u A = u B 0 : C B. The second equality ollows by the unitality o unctors. In view o the unitality o multiplications, α = α and β = β, or any natural transormations α, β with domain and codomain, respectively. A collection o internal categories in M = (M,, 1) together with internal unctors (with composition) and internal natural transormations (with vertical and horizontal compositions, and the identity natural transormations as units or the vertical composition) orms a 2-category Cat(M). In case M = Mod op coring morphisms and the opposite o 2-cells are representations o corings; see [4, Section 24]. 0-cells o Cat(Mod op ) are corings, 1-cells are 2.3. Internal counctors and the 2-category Cat c (M) An internal counctor or simply a counctor : (C, A) (D, B) is a pair = ( 0, 1 ), where 0 : D C is a morphism o comonoids, and 1 : A C 0 D 0 B is a C-D-bicomodule map that respects multiplications and units in the sense that the ollowing diagrams commute: A C A C D A C 1 A C B A C D D B A 1 C D B m A C D A C D 1 D B B B D B, 1 m B u A C D D, ub where we write B or 0 B etc., as in Section 2.2. The composite h o two counctors : (C, A) (D, B), h : (D, B) (D, B ) has the object component (h ) 0 = 0 h 0, and its morphism component is the composite (in M): A C 0 h 0 D A C D D h D 1 D D B D h D h 1 B. Let, g : (C, A) (D, B) be counctors. A natural cotransormation α : g is a morphism o D-bicomodules (i.e. a 2-cell in Comod(M)) α : D g B such that the ollowing diagram commutes

4 2138 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) A g C B A C λ A g C D D B g 1 C B B D B A C α m B A C D B 1 m B B D D B B D B. λ α D B The vertical composition o natural cotransormations α, β between counctors rom (C, A) to (D, B) is the composite β α : D α B λ D D B β DB B D B m B B. The identity natural cotransormation or a counctor rom (C, A) to (D, B) is equal to u B. In the situation: (C, A) α (D, B) β (D, B ) g the horizontal composition β α : h g is the ollowing composite: h h D D D h D α D β g B D B g B D D D B 1 D B g B D B m B g B. As explained in [1, Section 4.3], while an internal unctor can be understood as a push-orward o arrows rom one category to the other, an internal counctor can be interpreted as a liting o arrows. In the case o M = Mod op, a counctor rom an A-coring C to a B-coring D is equivalent to the ollowing commutative diagram o unctors: D Comod B Mod F 1 F 0 Mod. C Comod A Mod Here C Comod, A Mod etc. denote categories o let comodules respectively modules. The unmared arrows are the orgetul unctors and F 0 is the restriction o scalars unctor corresponding to the -algebra map 0 : A B. Thus a counctor in this case can be interpreted as a let extension o corings [3]. 3. Internal categories and the ormal theory o monads Following [15,8], a Kleisli object o a monad t in a 2-category is an object representing the covariant t-algebra unctor to the 2-category o categories (the reader not amiliar with this concept might lie to consult Section 5 o [11]). A ormal approach to the theory o monads was developed in [15] and [9]. Although not every monad in a given bicategory has a Kleisli object, to any bicategory, say B, a new bicategory KL(B) can be associated. Any monad in KL(B) has a Klesli object (and it is given in a orm o a wreath product). The detailed description o KL(B) is given in [9, Section 1]; here we give the relevant ormulae in the case B = Comod(M). The 0-cells o KL(Comod(M)) are monads in Comod(M), i.e. monoids in the categories o C-bicomodules, i.e. internal categories (C, A) in M. A 1-cell (C, A) (D, B) in KL(Comod(M)) is a pair (M, φ), where M = (M, λ M, ϱ M ) is a C-Dbicomodule and φ : A C M M D B is a C-D-bicomodule map rendering commutative the ollowing diagrams A C A C M A C φ A C M D B φ DB M D B D B A C M φ M D B m A C M A C M φ M D B, M D m B u A C M M. M D u B

5 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) A 2-cell (M,φ) (C, A) χ (D, B) (N,ψ) is a C-D-bicomodule map χ : M N D B rendering commutative the ollowing diagram A C M φ M D B χ D B N D B D B A C χ A C N D B ψ D B N D B D B N D m B N D B. N D m B The vertical composition o 2-cells χ : (M, φ) (N, ψ), χ : (N, ψ) (Q, θ) is the composite morphism M χ N D B χ D B Q D B D B Q Dm B Q D B. The horizontal composition o (M,φ) (M,φ ) (C, A) χ (D, B) χ (D, B ) (N,ψ) (N,ψ ) is the composite M D M χ D M N D B D M N D φ N D M D B N Dχ D B N D N D B D B N D N D m B N D N D B. As all Cat(M), Cat c (M) and KL(Comod(M)) have the same 0-cells it is natural to as, what is the relationship o the ormer two 2-categories to the latter bicategory. This question is addressed presently. For any bicategory B, B denotes the bicategory obtained by reversing 1-cells in B, while B is the bicategory obtained by reversing 2-cells in B. Theorem 3.1. Cat(M) is locally ully embedded in KL(Comod(M)) and Cat(M) is locally ully embedded in KL(Comod(M) ). Proo. The embedding unctor Φ : Cat(M) KL(Comod(M)), is deined as ollows. On 0-cells Φ is the identity. On 1-cells Φ : (C, A) (D, B) C, φ : A C C = A C D B, where φ = (C C 1 ) λ A. On 2-cells (C,φ ) Φ : (C, A) α (D, B) (C, A) χ α (D, B), g where χ α = (C g D α) C. The proo that Φ is a unctor is a standard exercise in the diagram chasing. Since Φ is the identity on 0-cells it is an embedding. To see that it is locally ull, tae internal unctors, g : (C, A) (D, B) and any 2-cell (C,φ ) (C, A) χ (D, B) (C g,φ g ) in KL(Comod(M)). Then α = m B (g 0 D B) χ is an internal natural transormation α : g. Furthermore, Φ(α) = χ. Thereore, Φ is a locally ull embedding as stated. The embedding Φ : Cat(M) KL(Comod(M) ), is obtained rom Φ by the let-right symmetry. That is, on 0-cells Φ is the identity. On 1-cells Φ : (C, A) (D, B) C,φ : C C A = A B D C, (C g,φ g )

6 2140 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) where φ = ( 1 C C) ϱ A. On 2-cells (in Cat(M)) g ( C,φ ) Φ : (C, A) α (D, B) (C, A) χ α (D, B), where χ α = (α D g C) C. ( g C,φ g ) Theorem 3.2. Cat c (M) is locally ully embedded in KL(Comod(M)). Proo. This is even more straightorward than Theorem 3.1. A unctor Ψ : Cat c (M) KL(Comod(M)), is deined as ollows. On 0-cells Ψ is the identity. On 1-cells Ψ : (C, A) (D, B) D, 1, On 2-cells Ψ is the identity, where, by the standard identiication g B = g D D B, α : D g B is now understood as a morphism D g D D B. The aoresaid identiication allows one to see immediately that Ψ is well-deined and compatible with all compositions. 4. Adjunctions and monads on noncartesian internal categories 4.1. Adjunctions and (co)kleisli objects in Cat(M) Let (C, A), (D, B) be internal categories in M = (M,, 1) and consider a pair o internal unctors l : (C, A) (D, B) and r : (D, B) (C, A). The unctor l is said to be let adjoint to r, provided l r is an adjunction in the 2-category Cat(M). That is there are internal natural transormations ε : lr 11, η : 11 rl such that εl 0 l 1 η = l, r 1 ε ηr 0 = r. The conditions (1) are reerred to as triangular identities, ε is called a counit and η is called a unit o the adjunction l r. The aim o this section is to show that, similarly to the standard external category theory, the adjointness can be characterised by an isomorphism o morphism objects. Start with a pair o internal unctors l : (C, A) (D, B) and r : (D, B) (C, A). The object D r C A can be interpreted as an object o all morphisms in (C, A) with codomain r, i.e. o morphisms r( ). Thus, intuitively, D r C A is an internal version o the biunctor Mor A (, r ). Similarly B D l C can be interpreted as a collection o arrows l( ), i.e. as an internalisation o Mor B (l, ). With this interpretation in mind we propose the ollowing Deinition 4.1. Let l : (C, A) (D, B) and r : (D, B) (C, A) be a pair o internal unctors. A D-C-bicomodule map θ : D r C A B D l C, is said to be bi-natural provided it renders commutative the ollowing diagrams: (1) B r C A ϱ C A B D D r C A B D θ B D B D l C λ C A D D B r C A D D r 1 C A D r C A C A D r C m A B l D C θ D r C A m B D l C and D r C A C A θ C ϱ B l D C C A C C D r C m A D r C A θ B D B D B l D C m 2 B D l C B l D C. B D l C l 1 D l C

7 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) The readers can easily convince themselves that the irst o the above diagrams corresponds to the statement that θ is a natural transormation between covariant morphism unctors. Since the category o C-bicomodules is not symmetric (nor even braided), the deinition o an opposite internal category hence contravariant internal unctor does not seem to be possible. On the other hand a condition relecting naturality o θ as a transormation between contravariant hom-unctors can be stated provided all the morphisms are evaluated (or composed) at the end. This is the essence o the second diagram. Theorem 4.2. Let l : (C, A) (D, B) and r : (D, B) (C, A) be a pair o internal unctors. Then l r is an adjunction i and only i the objects D r C A and B D l C are isomorphic by a bi-natural map o D-C-bicomodules. Proo. We give two proos o this theorem. The irst one is based on straightorward albeit lengthy calculations that use generalised elements. The second one, suggested to us by G. Böhm, is more conceptual and based on the interplay between Cat(M) and KL(Comod(M)) (see Theorem 3.1) and description o adjunctions in the latter. Computational proo. Suppose that l r is an adjunction with counit ε and unit η. Then the isomorphism is θ : D r C A B D l C, with the inverse θ 1 : B D l C D r C A, θ = (m B D l C) (ε D l 1 C C) (D r C ϱ), θ 1 = (D r C m A ) (D D r 1 C η) (λ D l C). These maps are well-deined by the colinearity o ε, η and the multiplication maps m A and m B. To prove that θ 1 θ is an identity apart rom the colinearity one uses the naturality o η, multiplicativity o r 1 and the second triangular identity (1). Dually, the proo that θ θ 1 is an identity uses the naturality o ε, multiplicativity o l 1 and the irst triangular identity in (1). The naturality o ε combined with colinearity o m A, multiplicativity o l 1 and unitality o m A yield the bi-naturality o θ. Dually, the bi-naturality o θ 1 ollows by the naturality o η, colinearity o m B, multiplicativity o r 1 and the unitality o m B. Conversely, assume that there is a bi-natural map θ : D r C A B D l C with the inverse θ 1 : B D l C D r C A. Deine the bicomodule maps ε : D lr B, η : rl C A, as composites ε = m B (B D l) θ (D D r) D, η = m A (r C A) θ 1 (l C C) C. The bi-naturality o θ together with the unitality o m A imply that ε is a natural transormation ε : lr 11. Similarly, η is a natural transormation η : 11 rl by bi-naturality o θ 1 and unitality o m B. This is conirmed by straightorward calculations. To prove the irst o triangular identities (1) we introduce an explicit notation or the comultiplication and let coactions, i.e. the Sweedler notation, which we adopt in the orm n C (c) = c (1) c (2) c (n+1) and λ n (b) = b ( n) b ( 1) b (0) ; see [13]. Here c should be understood as a generalised element o C (i.e. a morphism rom any object in M to C), and b is a generalised element o B. Next, tae a generalised element d a o D r C A and a generalised element b c o B D l C, and write a θ d θ := θ(d a), c θ b θ := θ 1 (b c), or the generalised elements obtained ater applying θ and θ 1, respectively. The properties o θ and θ 1 give rise to twelve equalities. There are our equalities corresponding to the colinearity (two or each θ and θ 1 ), two equalities encoding the codomains o θ and θ 1 (e.g. a θ d θ is in an appropriate equaliser), two recording that θ and θ 1 are mutually inverse, inally the bi-naturality o θ results in two equations or θ and two equations or θ 1. The interested reader can easily write all these equalities down; we only show them in action. In the orthcoming calculation, the multiplications m A, m B as well as composition in M are denoted by juxtaposition. We also reely use the ollowing equalities: u A r 0 l 0 = rl 0 = rl = r 1 l etc. Tae a generalised element c o C, and, using the deinition o ε and η, and the act that l 0 is a map o comonoids, compute εl 0 l 1 η(c) = rl 0 (c (2) ) θ l l 0 (c (1) ) θ l 1 r(c (4) θ )l 1 l(c (3) ) θ. Next, use (2) and apply the second o the diagrams in Deinition 4.1 to the irst three terms, then the irst o the diagrams in Deinition 4.1 to the irst three terms and then the second o the diagrams in Deinition 4.1 to the last three terms to obtain: εl 0 l 1 η(c) = r 1 l(c (2) )r(c (4) θ ) θ l l 0 (c (1) ) θ l 1 l(c (3) ) θ = l(c (1) ) r(c (4) θ ) θ l l 0 (c (2) ) θ l 1 l(c (3) ) θ = l(c (1) ) r(c (4) θ )l(c (3)) θ l l 0 (c (2) ) θ. θ (2)

8 2142 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) The let colinearity o θ, unitality o multiplication m B and the let colinearity o θ 1 allow one to transorm the above expression urther: εl 0 l 1 η(c) = u B r(c (3) θ )l(c (2)) θ θ ( 1) r(c (3) θ )l(c (2)) θ θ (0) l l 0 (c (1) ) θ = r(c (3) θ )l(c (2)) θ θ l l 0 (c (1) ) θ = r(c (2) θ (2))l(c (1) ) θ θ l c (2) θ (1) θ. Finally we can use the act that the codomain o θ 1 is equal to the equaliser D r C A, then the unitality o m A, the property θ θ 1 = 11, and the unitality o m B (or the idempotent property l l = l) to compute εl 0 l 1 η(c) = u A l(c (1) ) θ ( 1) l(c (1) ) θ (0) θ l θ c (2) θ = l(c (1) ) θ θ l c (2) θ θ = l(c (1) )l(c (2) ) = l(c), as required. In summary, we used precisely hal o the twelve properties characterising θ and θ 1. The other hal is used to veriy the second triangular equality (1) (a tas let to the reader). Conceptual proo. This proo has been suggested to us by Böhm [2]. Start with a 2-category (or a bicategory) B. Denote vertical composition in B by and the horizontal composition by. Consider a pair o 1-cells in KL(B) (L,φ) (A, C) (B, D), (R,ψ) where the notation (A, C) means a monad A on 0-cell C (in B). Assume urther that L has a right adjoint K in B, with unit ι and counit σ. As explained in [2] there is a bijective correspondence between adjunctions (L, φ) (R, ψ) in KL(B) and isomorphism 2-cells θ : A R K B, satisying the ollowing equalities: and θ (m A R) = (K m B ) (K B σ B) (K φ K B) (ι A K B) (A θ) (K m B ) (θ B) = θ (m A R) (A ψ). Now tae B = Comod(M). Since there is a locally ull embedding Φ : Cat(M) KL(Comod(M)) (see Theorem 3.1), description o adjunctions in Cat(M) is the same as description o adjunctions in Φ(Cat(M)) KL(Comod(M)). Properties o the pair o internal unctors l, r can be translated to properties o 1-cells (C l, φ l ) and (D r, φ r ), where φ l, φ r are deined in the proo o Theorem 3.1. The bicomodule C l has a right adjoint l C in Comod(M). The unit o adjunction is C, and the counit is l 0. In the case o B = Comod(M), the above properties o 2-cells θ correspond precisely to the bi-naturality. A monad on an internal category (C, A) is deined as a monad in the 2-category Cat(M) with underlying 0-cell (C, A). That is, a monad is a triple t = (t, µ, η), where t is an internal endounctor on (C, A), and µ : tt t, η : 11 t are natural transormations satisying the standard associativity and unitality conditions: µ t 1 µ = µ µt 0, µ t 1 η = µ ηt 0 = t. Any adjunction l r rom (C, A) to (D, B) with counit ε and unit η deines a monad on (C, A) by (rl, r 1 εl 0, η). The aim o this section is to construct explicitly the Kleisli object (or the Kleisli internal category) or a monad t in Cat(M). We irst recall the deinition o Kleisli objects (in Cat(M)) rom [8]; see also [11, Section 5]. Start with a monad t = (t, µ, η) on an internal category (C, A) in M. Then the t-algebra 2-unctor Alg t : Cat(M) CAT, is deined as ollows. For every internal category (D, B), Alg t (D, B) is a category consisting o pairs (y, σ ), where y : (C, A) (D, B) is an internal unctor and σ : yt y is an internal natural transormation such that σ y 1 η = y, σ y 1 µ = σ σ t 0. A morphism (y, σ ) (y σ ) in Alg t (D, B) is an internal natural transormation α : y y such that α σ = σ αt 0. Given an internal unctor : (D, B) (D, B ), Alg t () = t, where t (y, σ ) = (y, 1 σ ) and t (α) = 1 α. Finally, given an internal natural transormation β : g, Alg t (β) is a natural transormation β t : t g t given on objects (y, σ ) in terms o the internal natural transormation βy 0 : y gy. A Kleisli object or t is then deined as the representing object or Alg t.

9 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) Theorem 4.3. Let t = (t, µ, η) be a monad on an internal category (C, A) in M. Denote by A t a C-bicomodule with underlying object A t = C t C A and coactions C C A, C t C ϱ. Deine bicomodule maps u t : C A t, u t = (C C η) C, m t = (C t C A) C (C t C A) = C t C A t C A A t, m t = (C t C m 2 A ) (C t C µ C t 1 C A) ( C C A t C A). Then (C, A t, m t, u t ) is an internal category in M and it is the Kleisli object or t. Proo. Let Φ : Cat(M) KL(Comod(M)) be the 2-unctor deined in the proo o Theorem 3.1. Since t is a monad on (C, A), Φ(t) = (C t, φ t ) is a monad on Φ(C, A) = (C, A) in the bicategory KL(Comod(M)), i.e. it is the (Kleisli) wreath in terminology o [9]. As explained in [9, Section 3] the Kleisli object or t is the wreath product o the monad (C, A) with Φ(t). This is a monad in Comod(M) on C with the endomorphism part C t C A = A t. The multiplication is the composite C t C A t C A C t φ t A C tt C A C A Ctt m A C tt C A Φ(µ) C A C t C A C A Ct m A A t, where the standard isomorphisms are already taen into account. An easy calculation conirms that this composite is equal to m t. The unit in the wreath product A t is given by Φ(η) = u t. One o the consequences o Theorem 4.3 is the existence o the Kleisli adjunction l r, (C, A) the unctors r = (r 0, r 1 ) and l = (l 0, l 1 ) are r 0 = t 0, r 1 : A t A, r 1 = m A (µ C t 1 ), l 0 = C, l 1 : A A t, l 1 = (C t C m A ) (C C η C A) λ 2. l r (C, A t ). Explicitly, The unit o the adjunction is η (the unit o the monad t), while the counit is ε = (C t C t) C. Note that (t, µ, η) = (rl, r 1 εl 0, η), as required. Note urther that the bi-natural isomorphism θ corresponding to the Kleisli adjunction l r by Theorem 4.2 is the identity. This is in perect accord with the intuition coming rom the standard (external) category theory in which the Kleisli adjunction isomorphism is the identity by the very deinition o morphisms in the Kleisli category. The Kleisli adjunction can be used to describe explicitly the isomorphism o categories Θ : Alg t (D, B) Cat(M) ((C, A t ), (D, B)), that is natural in (D, B) and whose existence is the contents o Theorem 4.3. Here Cat(M) ((C, A t ), (D, B)) denotes the category with objects internal unctors (C, A t ) (D, B) and morphisms internal natural transormations. On objects (y, σ ) o Alg t (D, B), Θ(y, σ ) is an internal unctor with components Θ(y, σ ) 0 = y 0, Θ(y, σ ) 1 = m B (σ D y 1 ) : C t C A B. On morphisms Θ is the identity map. The inverse o Θ is given on objects as ollows. Tae an internal unctor g : (C, A t ) (D, B), and deine Θ 1 (g) = (y, σ ), y = gl, σ = g 1 εl 0 = g 1 ε. A comonad in a 2-category K is the same as a monad in a vertically dual 2-category K, obtained rom K by reversing the 2-cells. Thus, in the case o Cat(M), a comonad on an internal category (C, A) is the same as a monad in Cat(M), i.e. a triple (g, δ, ε), where g is an internal endounctor on (C, A) and δ : g gg, ε : g 11 are internal natural transormations such that g 1 δ δ = δg 0 δ, g 1 ε δ = εg 0 δ = g. Since, by Theorem 3.1, there is a (locally ull) embedding Cat(M) KL(Comod(M) ), internal comonads have co-kleisli objects. As was the case or monads, these are given in terms o a (co)wreath product, and the explicit orm o the embedding Φ in the proo o Theorem 3.1, yields Theorem 4.4. Let g = (g, δ, ε) be a comonad on an internal category (C, A) in M. Then g has a co-kleisli object (C, g A), where ga = A C g C, with the unit u g : C g A, u g = (ε C C) C, and multiplication m g = (A C g C) C (A C g C) = A C g A C g C g A, m g = (m 2 A C g C) (A C g 1 C δ C g C) (A C g A C C ).

10 2144 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) It has been observed already in [6] that a monad structure on a unctor induces a comonad structure on its adjoint. This observation remains true in any 2-category, and the relationship between monad and comonad structures is that o being mates under the adjunction [8]. It is also well nown that the Kleisli category o a monad with a let adjoint is isomorphic to the co-kleisli category o its comonad mate. The same property is enjoyed by monads on an internal category. More explicitly, consider an adjunction l r : (C, A) (C, A) with unit ι and counit σ. I (r, µ, η) is a monad, then l is a comonad with comultiplication δ being a mate o µ, δ = σ l 2 0 l 1µl 2 0 l 1r 1 ιl 0 l 1 ι, and the counit ε being a mate o η, ε = σ l 1 η. By [10, 2.14 Theorem], the Kleisli object A r o (r, µ, η) is isomorphic to the co-kleisli object l A o (l, δ, ε). In particular, the adjunction isomorphism θ : A r = C r C A A l C C = l A, constructed in the proo o Theorem 4.2 is an isomorphism o internal categories Kleisli objects in Cat c (M) An internal counctor l : (C, A) (D, B) is said to be a let adjoint o a counctor r : (D, B) (C, A) in case l r orm an adjunction in Cat c (M). A monad in the 2-category Cat c (M) is called an internal opmonad (or simply an opmonad). Thans to the existence o the locally ull embedding Ψ : Cat c (M) KL(Comod(M)) described in (the proo o) Theorem 3.2, every opmonad has its Kleisli internal category, which is given in terms o a wreath product. Formally, one can ormulate the ollowing Theorem 4.5. Let t = (t, µ, η) be an opmonad on an internal category (C, A) in M. Then the C-bicomodule t A is a monoid in C M C with unit η and multiplication m t = m 2 A (µ C t 1 C A) (λ C λ). The internal category (C, t A, m t, η) is the Kleisli object or t (in Cat c (M)). Proo. This is simply the wreath product o (C, A) with Ψ (t) = ( t C, t 1 ), once the identiication t C C A = t A is taen into account. Since Ψ is the identity on natural cotransormations, the unit or t A is the same as the unit o the opmonad, and the multiplication o the opmonad enters the wreath product unmodiied. 5. Examples The aim o this section is to explore the meaning o internal adjunctions and internal (co)kleisli categories in purely algebraic terms. As the Kleisli objects considered here are wreath products in the opposite o the category o modules (over a commutative ring) the examples described in this section orm a (dual) special case o the situation studied in [7]. Throughout this section, is a commutative ring, the multiplication in -algebras is denoted by juxtaposition and the unit is denoted by 1. The unadorned tensor product is the usual tensor product o -modules Twisting o corings and algebras Let A be a -algebra. An A-coring twisting datum is a triple D l C, θ, r in which C = (C, C, e C ) and D = (D, D, e D ) are A-corings, l, r are maps o A-corings and θ : D l r C is an isomorphism o D-C-bicomodules. Recall that by an A-coring map we mean a morphism o comonoids in the monoidal category o A- bimodules. Thus l and r are A-bimodule maps compatible with comultiplications and counits. Recall urther that D l means a D-C bicomodule with the let coaction given by the comultiplication D and with the right coaction modiied by l, (D A l) D. The meaning o r C is similar (but now let coaction is modiied by r). The readers can easily convince themselves that an A-coring twisting datum is the same as the internal adjunction (A, l) (A, r) rom (A, C) to (A, D) in Cat(Mod op ). The bicomodule map property o θ is precisely the same as its binaturality. With this interpretation in mind, with no eort and calculations one proves Lemma 5.1. In an A-coring twisting datum D l C r, θ, in which e D θ 1 and e C θ are units in convolution algebras Hom A A (C, A) and Hom A A (D, A) respectively, the A-corings D and C are mutually isomorphic.

11 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) Proo. In view o the proo o Theorem 4.2, e D θ 1 is a unit o adjunction (A, l) (A, r) and e C θ is the counit o (A, l) (A, r). To require that they have convolution inverses is the same as to require that they are invertible 2-cells in Cat(Mod op ). Thus the A-coring maps l and r must be isomorphisms. Explicitly, writing u or e D θ 1, ū or its convolution inverse and or the convolution product, the inverse o l is l 1 = ū r u. The internal category interpretation o an A-coring twisting datum D l C, θ, implies immediately the r existence o twisted A-coring structures on C and D. Twisting o (C, C, e C ): The A-bimodule C is an A-coring with comultiplication and counit θ C = (θ r A C) C, e θ C = e D θ 1. C with this coring structure is denoted by C θ. Twisting o (D, D, e D ): The A-bimodule D is an A-coring with comultiplication and counit θ D = (D A θ 1 l) D, e θ D = e C θ. D with this coring structure is denoted by D θ. C θ is simply the Kleisli object o the monad associated to the adjunction (A, l) (A, r), while D θ is the co-kleisli object o the corresponding comonad. The Kleisli adjunction yields a new A-coring twisting datum l C θ C, θ r, in which r = θ r, l = (ed θ 1 A C) C and θ is the identity morphism on C. As explained in [9, Section 1], the process o associating Kleisli objects terminates ater one step, hence this new twisting A-coring datum does not produce any new corings. More speciically, C θ = C θ and (C θ ) θ = C. Since a -coalgebra is an example o a -coring, the above constructions can be speciied to -coalgebras. In a similar way one can introduce an algebra twisting datum as a pair o -algebra morphisms supplemented with a bimodule isomorphism (with bimodule structures suitably induced by algebra maps). To interpret such a datum as an internal adjunction one should (ormally) assume that is a ield, but algebraic arguments wor or any commutative ring. This is in act an external construction since any -algebra is the same as a -linear category with a single object, and hence an algebra twisting datum is simply an adjunction (in the most classical sense) between -linear categories with single objects Sweedler s corings and Hop-Galois extensions The aim o this section is to illustrate the construction o Kleisli objects on a very speciic example and to mae a connection with the notions appearing in Hop-Galois theory. To any morphism o -algebras B A one associates the canonical Sweedler A-coring A B A with coproduct and counit (a B a ) = a B 1 A A 1 A B a, e(a B a ) = aa. This way (A, A B A) becomes an internal category in Mod op. To illustrate the construction o Kleisli objects we will study internal endounctors t on (A, A B A) which operate trivially on the object component o (A, A B A), i.e. t = (A, t 1 ). We reely use the standard isomorphism, or all A-bimodules M, Hom A A (A B A, M) = M B := {m M b B, bm = mb}. With this identiication at hand, a monad (which is an identity on the object o objects A) is a triple t := i s i B t i (A B A) B, m, u A B satisying the ollowing equations: (a) t is a unctor: i s it i = 1, i,j s i B t i s j B t j = i s i B 1 A B t i ; (b) naturality o multiplication: tm = mt 2 ; (c) naturality o unit: tu = u B 1 A ; (d) the associative law: m 2 = i ms imt i ; (e) the unit law: mu = i ms iut i = 1 A. In (b), t 2 = i s itt i = i,j s is j B t j t i, i.e. it is the square o t with respect to the natural multiplication in (A B A) B. For example, any unit u A B deines a monad on (A, A B A) as a triple (u B u 1, u 1, u). That is, or all a, a A, t 0 (a) = a, t 1 (a B a ) = au B u 1 a, µ(a B a ) = au 1 a, η(a B a ) = aua.

12 2146 T. Brzezińsi, A. Vazquez-Marquez / Journal o Pure and Applied Algebra 215 (2011) The Kleisli coring (A B A) t associated to the datum (t, m, u) equals A B A as an A-bimodule, but has modiied comultiplication and counit, t (a B a ) = amt B a, e t (a B a ) = aua. Lemma 5.2. The element mt is a group-lie element in (A B A) t. Proo. The unit law (e) implies immediately that e t (mt) = 1. Furthermore, t (mt) = i ms i mt B t i = i,j ms i mt i s j t B t j = i m 2 s i t B t i = i m 2 s i tt i t = i m 2 t 2 s i B t i = mt A mt, where the second and ourth equalities ollow by (a), the third one by (d) and the inal one by (b). The Kleisli adjunction comes out as l 1 : A B A (A B A) t, a B a amta, r 1 : (A B A) t A B A, a B a au B a. Lemma 5.2 maes is clear that l 1 is an A-coring map as required. That r 1 is an A-coring map ollows by (c) and (e). O course, both statements ollow by general arguments described in Section 4.1. The unit o the Kleisli adjunction is given by e t and the counit by e. This description o Kleisli objects or monads on the Sweedler canonical coring can be transerred to Hop-Galois extensions. Let H be a Hop algebra, and A be a right H-comodule algebra with coaction ϱ A : A A H. Let B = A coh = {a A ϱ A (a) = a 1 H } be the coinvariant subalgebra o A. Then the inclusion B A is called a Hop-Galois extension provided the canonical map can : A B A A H, a B a a ϱ A (a), is an isomorphism; see e.g. [12, Chapter 8]. The restricted inverse o the canonical map τ : H A B A, h can 1 (1 A h), is called the translation map. The translation map has several nice properties (see [14, 3.4 Remar]), which in particular imply that, or any group-lie element x H (i.e. an element such that H (x) = x x and e H (x) = 1), τ(x) (A B A) B satisies conditions (a). Thus any group-lie element x H induces an endounctor on A B A = A H. Using urther properties o the translation map, one inds that an element m A B satisies condition (b) i and only i ϱ A (m) = m x. Similarly, an element u A B satisies condition (c) i and only i ϱ A (u) = u x 1, (note that this equality maes sense, since every group-lie element o a Hop algebra is necessarily a unit). The Galois condition induces an action o H on A B, nown as the Miyashita Ulbrich action [16,5]. For any h H, write τ(h) = i h(1) i B h (2) i. Then, or any a A B, the Miyashita Ulbrich action is given by a h := i h (1) iah (2) i. In the Hop-Galois case, the associative and unital laws (d) and (e) come out as m 2 = m(m h), mu = m(u x) = 1. Acnowledgements Some o the results o this paper were presented at CT2009 in Cape Town. The authors are grateul to many participants o CT2009, in particular to Gabriella Böhm and Ross Street, or comments and suggestions. T. Brzezińsi would lie to acnowledge the International Travel Grant rom the Royal Society (UK). The research o A. Vazquez Marquez was supported by the CONACYT grant no /

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