Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category A and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on A: we define a bimonad on A as an endofunctor B which is a monad and a comonad with an entwining λ : BB → BB satisfying certain conditions. This λ is also employed to define the category A B B of (mixed) B-bimodules. In the classical situation, an entwining λ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws τ : BB → BB satisfying the Yang-Baxter equation (local prebraidings) which induce an entwining λ and lead to an extension of the theory of braided Hopf algebras.An antipode is defined as a natural transformation S : B → B with special properties and for categories A with limits or colimits and bimonads B preserving them, the existence of an antipode is equivalent to B inducing an equivalence between A and the category A

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.

Interpreting entwining structures as special instances of J. Beck's distributive law, the concept of entwining module can be generalized for the setting of arbitrary monoidal category. In this paper, we use the distributive law formalism to extend in this setting basic properties of entwining modules.

No 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 Bruguières 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, Böhm, 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.