# Bimonads and Hopf monads on categories

@article{Mesablishvili2007BimonadsAH, title={Bimonads and Hopf monads on categories}, author={Bachuki Mesablishvili and Robert Wisbauer}, journal={Journal of K-theory}, year={2007}, volume={7}, pages={349-388} }

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 structure of the category of endofunctors on any category…

## 49 Citations

ON BIMONADS AND HOPF MONADS

- Mathematics
- 2012

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…

NOTES ON BIMONADS AND HOPF MONADS

- Mathematics
- 2010

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…

GENERALIZED HOPF MODULES FOR BIMONADS

- Mathematics
- 2013

Brugui eres, Lack and Virelizier have recently obtained a vast generaliza- tion of Sweedler's Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We…

CYCLIC HOMOLOGY ARISING FROM ADJUNCTIONS

- Mathematics
- 2015

Given a monad and a comonad, one obtains a distributive law between them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfel'd…

Galois functors and generalised Hopf modules

- Mathematics
- 2013

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…

Monads on Higher Monoidal Categories

- MathematicsAppl. Categorical Struct.
- 2018

The structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad are identified.

Semicorings and Semicomodules

- Mathematics
- 2013

In this paper, we introduce and investigate semicorings over associative semirings and their categories of semicomodules. Our results generalize old and recent ones on corings over rings and their…

The Tangent Functor Monad and Foliations

- Mathematics
- 2012

In category theory, monads, which are monoid objects on endofunctors, play a central role closely related to adjunctions. Monads have been studied mostly in algebraic situations. In this…

## References

SHOWING 1-10 OF 62 REFERENCES

MONADS OF EFFECTIVE DESCENT TYPE AND COMONADICITY

- Mathematics
- 2006

We show, for an arbitrary adjunction FU : B→ Awith B Cauchy complete, that the functor F is comonadic if and only if the monad T on A induced by the adjunction is of effective descent type, meaning…

Dualizations and Antipodes

- MathematicsAppl. Categorical Struct.
- 2003

This work defines right autonomous monoidal functors and their higher-dimensional analogue and explains why the category Comodf(H) of finite-dimensional representations of H is autonomous is that the ComodF operation is autonomous and so preserves dualization.

Frobenius monads and pseudomonoids

- Mathematics
- 2004

Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only if it is star autonomous. Autonomous…

Algebras Versus Coalgebras

- MathematicsAppl. Categorical Struct.
- 2008

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.