Mathematics – Category Theory
Scientific paper
2011-06-27
Journal of Algebra, 355(1):111-126, 2012
Mathematics
Category Theory
v2: 24 pages, several revisions based on suggestions of referee, especially the new theorem 5.2; to appear in the Journal of A
Scientific paper
10.1016/j.jalgebra.2011.12.016
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.
Garner Richard
Lack Stephen
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