Idempotent monads and ⋆-functors

Abstract

a b s t r a c tFor an associative ring R, let P be an R-module with S = End R (P). C. Menini and A. Orsatti posed the question of when the related functor Hom R (P, −) (with left adjoint P ⊗ S −) induces an equivalence between a subcategory of R M closed under factor modules and a subcategory of S M closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphism and the counit is a monomorphism. A module P inducing these properties is called a -module.The purpose of this paper is to consider the corresponding question for a functor G : B → A between arbitrary categories. We call G a -functor if it has a left adjoint F : A → B such that the unit of the adjunction is an extremal epimorphism and the counit is an extremal monomorphism. In this case (F , G) is an idempotent pair of functors and induces an equivalence between the category A GF of modules for the monad GF and the category B FG of comodules for the comonad FG. Moreover, B FG = Fix(FG) is closed under factor objects in B, A GF = Fix(GF ) is closed under subobjects in A.