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) func-tors 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 in [9], but we prove them here with different assumptions, and under a completely different light.