  A full reflective subcategory E of a presheaf category [C*,Set] is the
category of sheaves for a topology j on C if and only if the reflection
preserves finite limits. Such an E is called a Grothendieck topos. More
generally, one can consider two topologies, j contained in k, and the category
of sheaves for j which are 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*,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.
