Isomorphisms in pro-categories

Abstract

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. In \cite{DR2} we gave characterizations of monomorphisms (resp. epimorphisms) in arbitrary pro-categories, pro-(C), where (C) has direct sums (resp. weak push-outs). In this paper we introduce the notions of strong monomorphism and strong epimorphism. Part of their significance is that they are preserved by functors. These notions and their characterizations lead us to important classical properties and problems in shape and pro-homotopy. For instance, strong epimorphisms allow us to give a categorical point of view of uniform movability and to introduce a new kind of movability, the sequential movability. Strong monomorphisms are connected to a problem of K.Borsuk regarding a descending chain of retracts of ANRs. If (f: X \to Y) is a bimorphism in the pointed shape category of topological spaces, we prove that (f) is a weak isomorphism and (f) is an isomorphism provided (Y) is sequentially movable and $X$ or $Y$ is the suspension of a topological space. If (f: X \to Y) is a bimorphism in the pro-category pro-(H_0) (consisting of inverse systems in (H_0), the homotopy category of pointed connected CW complexes) we show that (f) is an isomorphism provided (Y) is sequentially movable.