FACTORIZATIONS OF CONES ALONG A FUNCTOR

Abstract

First a general Galois correspondence is established, which generalizes at the same time the correspondence between classes of monomorphisms and injective objects and the correspondence between classes of epimorphisms and monomorphisms in a category. This correspondence arises naturally if one tries to generalize some concepts of “topological” or also of “algebraic” functors. Both kinds of functors admit certain factorizations of cones, and just this fact implies some of their common nice properties: lifting limits, continuity and faithfulness, for instance. These properties can be shown without having a left adjoint. Therefore the theory yields also applications to functors which are neither “topological” nor “algebraic”.