Weak Frobenius monads and Frobenius bimodules

As shown by S. Eilenberg and J.C. Moore (1965), for a monad $F$ with right adjoint comonad $G$ on any catgeory $\mathbb{A}$, the category of unital $F$-modules $\mathbb{A}_F$ is isomorphic to the category of counital $G$-comodules $\mathbb{A}^G$. The monad $F$ is Frobenius provided we have $F=G$ and then $\mathbb{A}_F\simeq \mathbb{A}^F$. Here we investigate which kind of equivalences can be obtained for non-unital monads (and non-counital comonads).Comment: 21 pages, the material is rearranged and the presentation is improve

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

On Rational Pairings of Functors

In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$ with the coalgebra $C$ provided by the evaluation map \ev:C^*\ot_R C\to R. We generalise this situation by defining a {\em pairing} between endofunctors $T$ and $G$ on any category \A as a map, natural in a,b\in \A, \beta_{a,b}:\A(a, G(b)) \to \A(T(a),b), and we call it {\em rational} if these all are injective. In case \bT=(T,m_T,e_T) is a monad and \bG=(G,\delta_G,\ve_G) is a comonad on \A, additional compatibility conditions are imposed on a pairing between \bT and \bG. If such a pairing is given and is rational, and \bT has a right adjoint monad \bT^\di, we construct a {\em rational functor} as the functor-part of an idempotent comonad on the \bT-modules \A_{\rT} which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories