Grothendieck quasitoposes

A full reflective subcategory E of a presheaf category [C ^op,Set] is the category of sheaves for a topology j on C if and only if the reflection from [C ^op,Set] into E preserves finite limits. Such an E is then called a Grothendieck topos. More generally, one can consider two topologies, j⊆ k, and the category of sheaves for j which are also separated for k. The categories E of this form for some C, j, and k are the Grothendieck quasitoposes of the title, previously studied by Borceux and Pedicchio, and include many examples of categories of spaces. They also include the category of concrete sheaves for a concrete site. We show that a full reflective subcategory E of [C ^op,Set] arises in this way for some j and k if and only if the reflection preserves monomorphisms as well as pullbacks over elements of E. More generally, for any quasitopos S, we define a subquasitopos of S to be a full reflective subcategory of S for which the reflection preserves monomorphisms as well as pullbacks over objects in the subcategory, and we characterize such subquasitoposes in terms of universal closure operators.