Quasitopoi over a base category

In this paper we develop the theory of quasispaces (for a Grothendieck topology) and of concrete quasitopoi, over a suitable base category. We introduce the notion of f-regular category and of f-regular functor. The f-regular categories are regular categories in which every family with a common codomain can be factorized into a strict epimorphic family followed by a (single) monomorphism. The f-regular functors are (essentially) functors that preserve finite strict monomorphic and arbitrary strict epimorphic families. These two concepts furnish the context to develop the constructions of the theory of concrete quasitopoi over a suitable base category, which is a theory of pointed quasitopoi. Our results on quasispaces and quasitopoi, or closely related ones, were already established by Penon, but we prove them here with different assumptions and generality, and under a completely different light.