Galois functors and generalised Hopf modules

TL;DR

This paper demonstrates that fundamental theorems for Hopf modules in various categorical contexts can be derived from the theory of Galois functors, unifying different approaches under a common framework.

Contribution

It shows that existing fundamental theorems for Hopf modules in monoidal and duoidal categories follow from the Galois functor theory, providing a unifying categorical perspective.

Findings

Fundamental theorems are derivable from Galois functor theory.

Unification of different categorical approaches to Hopf modules.

Extension of Galois functor framework to more general settings.

Abstract

As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the characterisation of opmonoidal monads on monoidal categories as right Hopf monads in the sense of Bruguieres and Virelizier. Hereby the central part is to describe conditions under which a comparison functor between the base category and the category of Hopf modules becomes an equivalence (Fundamental Theorem). For monoidal categories, Aguiar and Chase extended the setting by replacing the base category by a comodule category for some comonoid and considering a comparison functor to generalised Hopf modules. For duoidal categories, Bohm, Chen and Zhang investigated a comparison functor to the Hopf modules over a bimonoid induced by the two monoidal structures given in such categories. In both approaches fundamental theorems are proved and the purpose of this paper is to show that these can be derived from the theory of Galois functors.