On weak shape equivalences

Abstract

We prove that weak shape equivalences are monomorphisms in the shape category of uniformly pointed movable continua Sh(M). We use an example of Draper and Keesling to show that weak shape equivalences need not be monomorphisms in the shape category. We deduce that Sh(M) is not balanced. We give a characterization of weak dominations in the shape category of pointed continua, in the sense of Dydak (1979). We introduce the class of pointed movable triples (X,F,Y), for a shape morphism F:X --> Y, and we establish an infinite-dimensional Whitehead theorem in shape theory from which we obtain, as a corollary, that for every pointed movable pair of continua (Y,X) the embedding j: X --> Y is a shape equivalence iff it is a weak shape equivalence.