A duoidal category \(M\) is a category with two monoidal products, with certain compatibilities. Aguiar and Mahajan defined the notion of bimonoids in such a category. As there is in general no convolution product in \(M\), there is a problem to define the antipode, and hence the notion of Hopf monoid in \(M\).
An answer is given here by the fundamental theorem of Hopf modules: if \(A\) is a bialgebra over a commutative field \(k\), the following assertions are equivalent: 1. \(A\) has an antipode. 2. \(A\) is an \(A\)-Galois extension of \(k\). 3. (Fundamental theorem of Hopf modules). The category of \(A\)-Hopf modules is equivalent to the category of \(k\)-modules.
A similar theorem is proved under certain conditions on \(M\), namely the Fundamental Theorem of Hopf modules holds for a bimonoid \(A\) if, and only if, the unit of \(A\) determines an \(A\)-Galois extension. These results are applied to small groupoids and to Hopf algebroids over a commutative base algebra.