Grothendieck quasitoposes

TL;DR

This paper characterizes Grothendieck quasitoposes as certain reflective subcategories of presheaf categories, based on the preservation of monomorphisms and pullbacks, extending the classical theory of sheaves and topoi.

Contribution

It provides a new characterization of Grothendieck quasitoposes through the preservation of monomorphisms and pullbacks, generalizing the concept of sheaf categories.

Findings

Characterization of Grothendieck quasitoposes via reflection properties.

Includes many examples of categories of spaces and concrete sheaves.

Extends classical sheaf theory to broader categorical contexts.

Abstract

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.