The notion of a via-topological space is introduced on the basis of directed paths connecting its points and different various examples of such spaces are presented. Further the corresponding to their structure definitions of continuity, homeomorphism, and homogeneity are given, which are interpreted in the language of category theory. In terms of these new notions a theorem about the existence on the homogeneous via-topological spaces a structure of a smooth manifold diffeomorphic to the three-dimensional sphere is proved.