Foliated categories (French: catégories feuilletées), or simply foliations (not to be confused with the notion of foliations in differential geometry), were introduced by Jean Bénabou in unpublished work dating back to 1984. They are a weaker structure than fibered categories, but still allow one to test for various standard properties of functors fibre-wise. A functor makes its domain category a foliated category (over ) if the following conditions hold: Every morphism in factors as a...