Weak bimonads and weak Hopf monads

TL;DR

This paper introduces the concept of weak bimonads in monoidal categories, characterizes their structure, and explores their relation to weak bimonoids and weak Hopf monads, expanding the theoretical framework of monad theory.

Contribution

It defines weak bimonads with simple axioms, linking them to weak bimonoids and weak Hopf monads, and clarifies their monoidal structure and properties.

Findings

Characterization of weak bimonads via simple axioms

Relation established between weak bimonads and weak bimonoids

Introduction of weak Hopf monads with antipodes

Abstract

We define a weak bimonad as a monad T on a monoidal category M with the property that the Eilenberg-Moore category M^T is monoidal and the forgetful functor from M^T to M is separable Frobenius. Whenever M is also Cauchy complete, a simple set of axioms is provided, that characterizes the monoidal structure of M^T as a weak lifting of the monoidal structure of M . The relation to bimonads, and the relation to weak bimonoids in a braided monoidal category are revealed. We also discuss antipodes, obtaining the notion of weak Hopf monad.