Cartesian functors and foliated categories

In 1961 Grothendieck defined cartesian functors between prefibered and fibered categories. Since then they have been studied only in the case
of fibrations. I shall give a general definition covering many other situations, which reduces to the classical one in the previous cases, and give many important examples of the more general situation.

I shall prove for such functors many results which were not known even in the "classical" situation. Some of these results need assumptions, much weaker than fibrations, which I call prefoliations and foliations which I shall introduce and study briefly. Again many significant examples shall be given.

Although distributors are underlying many such situations I shall keep things "elementary" and use this "tool" very little. Whatever is needed shall be briefly recalled.