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GALOIS FUNCTORS AND ENTWINING STRUCTURES
, 909
"... Abstract. Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injecti ..."
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Abstract. Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a grouplike natural transformation g: I → G generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to comonoidal functors on monoidal categories in the sense of A. Bruguières and A. Virelizier. As wellknow, for any set G the product G × − defines an endofunctor on the category of sets and this is a Hopf monad if and only if G allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to Galois objects in the sense of Chase and Sweedler.
Lifting theorems for tensor functors on module categories
 Department of Mathematics, Heinrich Heine University Düsseldorf, Germany
"... groups. These notions were generalised to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate’s lifting theorem of functors ..."
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groups. These notions were generalised to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate’s lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory. The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focussing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (Crings). We also highlight the relevance of the YangBaxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).
BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
BIMONADS AND HOPF MONADS ON CATEGORIES BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
SEMIUNITAL SEMIMONOIDAL CATEGORIES (APPLICATIONS TO SEMIRINGS AND SEMICORINGS)
"... Abstract. The category ASA of bisemimodules over a semialgebra A, with the so called Takahashi’s tensorlike product − ⊠A −, is semimonoidal but not monoidal. Although not a unit in ASA, the base semialgebra A has properties of a semiunit (in a sense which we clarify in this note). Motivated by this ..."
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Abstract. The category ASA of bisemimodules over a semialgebra A, with the so called Takahashi’s tensorlike product − ⊠A −, is semimonoidal but not monoidal. Although not a unit in ASA, the base semialgebra A has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories (V, •, I) as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call Jmonads (Jcomonads) with respect to the endofunctor J: = I• − ≃ −•I: V − → V. This motivated also introducing a more generalized notion of monads (comonads) in arbitrary categories with respect to arbitrary endofunctors. Applications to the semiunital semimonoidal variety (ASA, ⊠A, A) provide us with examples of semiunital Asemirings (semicounital Asemicorings) and semiunitary semimodules (semicounitary semicomodules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules). 1.
NOTES ON BIMONADS AND HOPF MONADS
"... Abstract. For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a preHopf monad by requiring on ..."
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Abstract. For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a preHopf monad by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the preHopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered. 1.
COALGEBRAIC STRUCTURES IN MODULE THEORY
"... Abstract. Although coalgebras and coalgebraic structures are wellknown for a long time it is only in recent years that they are getting new attention from people working in algebra and module theory. The purpose of this survey is to explain the basic notions of the coalgebraic world and to show the ..."
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Abstract. Although coalgebras and coalgebraic structures are wellknown for a long time it is only in recent years that they are getting new attention from people working in algebra and module theory. The purpose of this survey is to explain the basic notions of the coalgebraic world and to show their ubiquity in classical algebra. For this we recall the basic categorical notions and then apply them to linear algebra and module theory. It turns out that a number of results proven there were already contained in categorical papers from decades ago. Key Words: monads and comonads, algebras and coalgebras, module and
HOMTENSOR RELATIONS FOR TWOSIDED HOPF MODULES OVER QUASIHOPF ALGEBRAS
"... the category MH H of right ..."
3. Actions on functors and Galois fun...
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
QF FUNCTORS AND (CO)MONADS
"... Abstract. One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K: A → B between categories is Frobenius if there exists a functor G: B → A which is at the same time a right and left adjoint of K; a monad F on ..."
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Abstract. One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K: A → B between categories is Frobenius if there exists a functor G: B → A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF → A is a Frobenius functor, where AF denotes the category of Fmodules. With these notions, an algebra A over a field k is a Frobenius algebra if and only if A ⊗k − is a Frobenius monad on the category of kvector spaces. The purpose of this paper is to find characterisations of quasiFrobenius algebras by just referring to constructions available in any categories. To achieve this we define QF functors between two categories by requiring conditions on pairings of functors which weaken the axioms for adjoint pairs of functors. QF monads on a category A are those monads F for which the forgetful functor UF: AF → A is a QF functor. Applied to module categories (or Grothendieck categories), our notions coincide with definitions first given K. Morita (and others). Further applications