On Hopf monoids in duoidal categories

TL;DR

This paper explores Hopf monoids within duoidal categories, establishing conditions under which the Fundamental Theorem of Hopf Modules applies, and illustrates these concepts through examples like groupoids and Hopf algebroids.

Contribution

It extends the theory of bimonoids in duoidal categories by linking the Fundamental Theorem of Hopf Modules to A-Galois extensions under specific assumptions.

Findings

The Fundamental Theorem of Hopf Modules holds under certain conditions.

A-Galois extensions are characterized by the unit of A.

Applications to groupoids and Hopf algebroids demonstrate the theory's relevance.

Abstract

Aguiar and Mahajan's bimonoids A in a duoidal category M are studied. Under certain assumptions on M, the Fundamental Theorem of Hopf Modules is shown to hold for A if and only if the unit of A determines an A-Galois extension. Our findings are applied to the particular examples of small groupoids and of Hopf algebroids over a commutative base algebra.