We shall describe a modification of homotopy theory of maps which we call shape theory of maps. This is accomplished by constructing the shape category of maps . The category is built using multi-valued functions. Its objects are maps of topological spaces while its morphisms are homotopy classes of collections of pairs of multi-valued functions which we call multi-binets. Various authors have previously given other descriptions of shape categories of maps. Our description is intrinsic in the sense that we do not use any outside objects. It is a version of the author’s extension to arbitrary topological spaces of Sanjurjo’s approach to shape theory via small multi-valued functions adapted to maps.

1991 Mathematics Subject Classification: 54B25, 54F45, 54C56.