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For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguièeres 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 pre-Hopf 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 pre-Hopf 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.
@article{TAC_2012_26_a9,
author = {Bachuki Mesablishvili and Robert Wisbauer},
title = {Notes on {Bimonads} and {Hopf} {Monads}},
journal = {Theory and applications of categories},
pages = {281--303},
publisher = {mathdoc},
volume = {26},
year = {2012},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2012_26_a9/}
}
Bachuki Mesablishvili; Robert Wisbauer. Notes on Bimonads and Hopf Monads. Theory and applications of categories, Tome 26 (2012), pp. 281-303. http://geodesic.mathdoc.fr/item/TAC_2012_26_a9/