Qf Functors and (co)monads

One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K : A → B between categories is Frobenius if there exists a functor G : B → A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF → A is a Frobenius functor, where AF denotes the category of F -modules. With these notions, an algebra A over a field k is a Frobenius algebra if and only if A ⊗k − is a Frobenius monad on the category of k-vector spaces. The purpose of this paper is to find characterisations of quasi-Frobenius algebras by just referring to constructions available in any categories. To achieve this we define QF functors between two categories by requiring conditions on pairings of functors which weaken the axioms for adjoint pairs of functors. QF monads on a category A are those monads F for which the forgetful functor UF : AF → A is a QF functor. Applied to module categories (or Grothendieck categories), our notions coincide with definitions first given K. Morita (and others). Further applications show the relations of QF functors and QF monads with Frobenius (exact) categories.