Weak bimonoids in duoidal categories

TL;DR

This paper introduces weak bimonoids in duoidal categories, generalizing existing structures, and proves a fundamental theorem relating modules and Hopf modules under certain conditions.

Contribution

It defines weak bimonoids in duoidal categories and establishes their connection to weak bimonads and Hopf modules, extending previous theories.

Findings

Weak bimonoids induce four weak bimonads with separable Frobenius base (co)monoids.

Under certain assumptions, the fundamental theorem of Hopf modules is proven.

Category equivalences are established under Galois-type conditions.

Abstract

Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg-Moore category of the induced monad. Making appropriate assumptions on the duoidal category in question, the fundamental theorem of Hopf modules is proven which says that the category of modules over one of the base monoids is equivalent to the category of Hopf modules if and only if a Galois-type comonad morphism is an isomorphism.