Bimonads and Hopf monads on categories

Abstract

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

Coalgebraic structures in module theory

“…An endofunctor allowing for a monad and a comonad structure may be a Frobenius monad if the compatibility conditions in Proposition 1.17 are satisfied, or one may impose other compatibility requirements leading to the definition of (see [11,Definition 4.1]) 2.4. Bimonads.…”

“…The and it can be shown that this λ is a mixed entwining (see [11,Proposition 6.3]). Of course, not every mixed entwining has to be of this form.…”

Lifting Theorems for Tensor Functors on Module Categories

“…The conditions required in 5.4 make the functor B ⊗ R − a bimonad on R M in the sense of [20,Definition 4.1]. In general they do not imply the same property for the functor − ⊗ R B on M R .…”

“…Among bialgebras, Hopf algebras are characterised by the fact that the functor B ⊗ R − : R M → B B M is an equivalence (e.g. [20,Theorem 6.12]). …”

Comodules and Contramodules

“…Now, considering the monad and comonad [B, −] = Hom(B, −), we can also define bimodules leading to the category M These relationships are explained in more detail in [1]. It is shown in [5] and [7] how they can be transferred to bimonads on arbitrary categories.…”