Galois functors and entwining structures

{\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.

💡 Research Summary

The paper is organized around three central themes: (i) a categorical reinterpretation of Galois comodules over corings, (ii) the definition of Galois functors for arbitrary monads or comonads, and (iii) the development of an entwining theory for a monad and a comonad, culminating in applications to Hopf monads, comonoidal functors, and the notion of Galois objects in categories with finite products.

In the first section the authors recall that a coring C over a ring A gives rise to the category of right C‑comodules. A right C‑comodule M is called Galois if the canonical map
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