Monomorphisms and epimorphisms in pro-categories

摘要

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. We give characterizations of mpnomorphisms (respectively, epimorphisms) in pro-category pro-C, provided C has direct sums (respectively, pushouts). Let E(C) (respectively, M(C)) be the subcategory of C whose morphisms are epimorphisms (respectively, monomorphisms) of C. We give conditions in some categories C for an object X of pro-C to be isomorphic to an object of pro-E(C) (respectively, pro-M(C)). A related class of objects of pro-C consists of X such that there is an epimorphism X → P ∈ Ob(C) (respectively, a monomorphism P ∈ Ob(C) → X). Characterizing those objects involves conditions analogous (respectively, dual) to the Mittag-Leffler property. One should expect that the object belonging to both classes ought to be stable. It is so in the case of pro-groups. The natural environment to discuss those questions are balanced categories with epimorphic images. The last part of the paper deals with that question in pro-homotopy.