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Notes on Bimonads and Hopf Monads

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Bachuki Mesablishvili and Robert Wisbauer

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.

Keywords:
Opmonoidal functors, bimonads, Hopf monads, Galois entwinings

2000 MSC:
18A40, 16T15, 18C20

*Theory and Applications of Categories,*
Vol. 26, 2012,
No. 10, pp 281-303.

Published 2012-06-05.

http://www.tac.mta.ca/tac/volumes/26/10/26-10.dvi

http://www.tac.mta.ca/tac/volumes/26/10/26-10.ps

http://www.tac.mta.ca/tac/volumes/26/10/26-10.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/10/26-10.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/10/26-.10ps

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