Idempotent monads and $\star$-functors

Abstract

For 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 $P\ot_S-$) 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 $\star$-module. The purpose of this paper is to consider the corresponding question for a functor $G:\B\to \A$ between arbitrary categories. We call $G$ a {\em $\star$-functor} if it has a left adjoint $F:\A\to \B$ such that the unit of the adjunction is an {\em extremal epimorphism} and the counit is an {\em 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 $\A$.