Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A well-known theorem states that a category E is a Grothendieck topos if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite limits.

In their paper Grothendieck quasitoposes, Garner and Lack proved that a category E is a Grothendieck quasitopos (the category of sheaves for one topology that are separated for another topology) if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector has stable units (i.e. preserves all pullbacks over objects in the subcategory) and preserves monomorphisms.

Is there a characterization of when a category E can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite products? (The reflector preserving finite products is equivalent to the reflective subcategory being an exponential ideal, i.e. $Y^X$ lies in the subcategory as soon as $Y$ does.) Of course, any such category must be locally presentable and cartesian closed. Can this be extended to a set of conditions which is both necessary and sufficient?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.