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.