Notes on bimonads and Hopf monads

TL;DR

This paper explores the relationship between pre-Hopf monads, bimonads, and Galois entwinings, providing new insights into their properties and applications within monoidal categories, especially in Cauchy complete and cartesian contexts.

Contribution

It demonstrates that pre-Hopf monads are a special case of Galois entwinings and identifies conditions under which bimonads become Hopf monads, extending the theoretical framework.

Findings

Pre-Hopf monads are a special case of Galois entwinings.

Certain properties make a bimonad into a Hopf monad on Cauchy complete categories.

Applications to cartesian monoidal categories are discussed.

Abstract

For a generalisation of the classical theory of Hopf algebra over fields, A. Brugui\`eres and A. Virelizier study opmonoidal monads on monoidal categories (which they called {\em bimonads}). In a recent joint paper with S. Lack the same authors define the notion of a {\em 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 %Hopf monads may be considered as 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.