Galois functors and entwining structures

Abstract

Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of 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 grouplike natural transformation g : I → G generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to opmonoidal monads with antipode on autonomous monoidal categories (named Hopf monads by Bruguières and Virelizier) which can be understood as an entwining of two related functors.As well known, for any set G the product G × − 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 Galois objects in the sense of Chase and Sweedler.

The Fundamental Theorem for weak braided bimonads

“…As easily seen, condition (5) in 6.2 yield the equalities (5.3) and (5.11) and also implies (5.2(i)) and (5.10(i)) for ω and ω, respectively (e.g. [8], [23]). Now Propositions 2.3 and 2.5 in [2] show the equations in (5.2)(ii) and (5.10)(ii).…”

“…[1,2] and we can -and will -freely use essential parts of their results in our situation. Note that if ∇ is the identity of H, the conditions (1)-(4) in the definition describe the invertible double entwinings considered in [23].…”

“…Opmonoidal monads T = (T, m, e) on strict monoidal categories (V, ⊗, I) were also called bimonads by Bruguières et al in [13] and these were generalised to weak bimonads in monoidal categories by Böhm et al in [10]. As pointed out in [23,Section 5], the bimonads from [13] yield a special case of an entwining of the monad T with the comonad − ⊗ T (I), where T (I) has a coalgebra structure derived from the opmonoidality of T. To transfer the structures from [10] to arbitrary categories one has to consider weak entwinings between monads and comonads. It is planned to elaborate details for this in a subsequent article.…”

Azumaya Monads and Comonads

“…The work is part of a joint project of the two authors (see references [5,6,21,33]) in which algebraic and coalgebraic structures are to be formulated and studied in general categories.…”

Generalised bialgebras and entwined monads and comonads

“…These observations led to a revival of the investigation of various forms of distributive laws. In a series of papers [12,13,14] it was shown how they allow for formulating the theory of Hopf algebras and Galois extensions in a general categorical setting.…”