Topological functors as familiarly-fibrations

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.