MONADS AND COMONADS ON MODULE CATEGORIES

@article{Bhm2009MONADSAC,
  title={MONADS AND COMONADS ON MODULE CATEGORIES},
  author={Gabriella B{\`o}hm and Tomasz Brzezinski and Robert Wisbauer},
  journal={Journal of Algebra},
  year={2009},
  volume={322},
  pages={1719-1747}
}
COMODULES AND CONTRAMODULES
Abstract Algebras A and coalgebras C over a commutative ring R are defined by properties of the (endo)functors A ⊗R – and C ⊗R – on the category of R-modules R. Generalising these notions, monads and
Galois functors and entwining structures
Galois functors and generalised Hopf modules
As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the
Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras
For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors − ⊗ k V and V ⊗ k − are known to be left adjoint to some kind of Hom-functors as endofunctors of H 𝕄. The
Hom-Tensor Relations for Two-Sided Hopf Modules Over Quasi-Hopf Algebras
For a Hopf algebra H over a commutative ring k, the category of right Hopf modules is equivalent to the category 𝕄 k of k-modules, that is, the comparison functor is an equivalence (Fundamental
Categories of modules, comodules and contramodules over representations
We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As
Cofree objects in the categories of comonoids in certain abelian monoidal categories
We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or
NOTES ON BIMONADS AND HOPF MONADS
For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguieres and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent
ON BIMONADS AND HOPF MONADS
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
Bimonads and Hopf monads on categories
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
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References

SHOWING 1-10 OF 63 REFERENCES
The Structure of Corings: Induction Functors, Maschke-Type Theorem, and Frobenius and Galois-Type Properties
Given a ring A and an A-coring C, we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint −⊗AC are separable. We then
Bimonads and Hopf monads on categories
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
Algebras Versus Coalgebras
TLDR
This survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s, to look at the interplay between algebraic and coalgebraic notions.
TRIPLES, ALGEBRAS AND COHOMOLOGY
It is with great pleasure that the editors of Theory and Applications of Categories make this dissertation generally available. Although the date on the thesis is 1967, there was a nearly complete
Homological Algebra of Semimodules and Semicontramodules
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call
The formal theory of monads II
Compatibility Conditions Between Rings and Corings
We introduce the notion of “bi-monoid” in general monoidal category, generalizing by this the notion of “bialgebra”. In the case of bimodules over a noncommutative algebra, we obtain compatibility
Equivalences between categories of modules and categories of comodules
We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gómez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm
Homological algebra of semimodules and semicontramodules. Semi-infinite homological algebra of assoc
Preface.- Introduction.- 0 Preliminaries and Summary.- 1 Semialgebras and Semitensor Product.- 2 Derived Functor SemiTor.- 3 Semicontramodules and Semihomomorphisms.- 4 Derived Functor SemiExt.- 5
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