  In this paper we develop the theory of topological categories over a base
category, that is, a theory of topological functors. Our notion of topological
functor is similar to (but not the same) the existing notions in the literature
(see Borceux "Handbook of categorical algebra 2, 7.3"), and it aims at the same
examples. In our sense, a (pre) topological functor is a functor that creates
cartesian families. A topological functor is, in particular, a fibration, and
our emphasis is put in this fact.
