  We extend calculus from smooth manifolds to topological manifolds making use
of a theory of generalized functions developed for this aim. Actually such
extension fits into a boarder context: the universal construction of a site
containing all continuous maps between topological spaces and whose arrows are
smooth in the generalized sense. As an application we prove the existence of
non singular generalized tangent vector fields on spheres of any dimension,
showing how this result is coherent with the non existence of smooth vector
fields on spheres of even dimension. Many other applications are outlined or
suggested, some of which are under development by the author.
