Shape via multi-nets

Abstract

We give in this paper a description of a new category related to shape category. We consider families of multi-valued functions between topological spaces which we call multi-nets. In a well-controlled way functions of a multi-net more and more resemble single-valued functions. We introduce a notion of homotopy for multi-nets and a composition of homotopy classes. The resultant homotopy category of multi-nets $\mathcal{H}M$ is naturally equivalent to the shape category provided we restrict to spaces which have ANRresolutions with onto projections. However, the homotopy category of multi-nets is interesting because it provides an intrinsic method of studying global properties of spaces. Our idea is to extend Borsuk's approach based on fundamental sequences to arbitrary topological spaces in analogy with Sanjurjo's description of shape category of compact metric spaces in terms of upper semi-continuous multi-valued functions.