Weak bimonads and weak Hopf monads

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.

29 pages; version 2 minor corrections and added references, also added remark 4.3; title changed from "Weak bimonads" to "Weak bimonads and weak Hopf monads"; to appear in Journal of Algebra

References in corpus (2)

• Hopf monads on monoidal categories
• Hopf monoidal comonads

Cited by in corpus (6)

• Idempotent splittings, colimit completion, and weak aspects of the theory of monads
• Weak multiplier bimonoids
• On Generalized Symmetries and Structure of Modular Categories
• Smash coproducts of bicomonads and Hom-entwining structures
• Weak bimonoids in duoidal categories
• Weak bialgebras and monoidal categories