Galois functors and entwining structures

TL;DR

This paper develops a generalized theory of Galois comodules, functors, and entwining structures involving monads and comonads, extending classical concepts to broader categorical contexts including Hopf monads and Galois objects.

Contribution

It introduces the notion of Galois functors over comonads and monads, generalizes corings to entwining of monads and comonads, and applies this to Hopf monads and Galois objects in arbitrary categories.

Findings

Characterization of Galois comodules via relative injectives.

Generalization of corings to entwining of monads and comonads.

Extension of Galois objects concept to categories with finite products.

Abstract

{\em Galois comodules} over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of {\em Galois functors} over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a {\em grouplike natural transformation} $g:I\to G$ generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to comonoidal functors on monoidal categories in the sense of A. Brugui\`{e}res and A. Virelizier. As well-know, for any set $G$ the product $G\times-$ defines an endofunctor on the category of sets and this is a Hopf monad if and only if $G$ allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to {\em Galois objects} in the sense of Chase and Sweedler.