Question

(Grothendieck) topoi are left-exact reflective subcategories of a category of presheaves. An important class of quasi-topoi (see: http://ncatlab.org/nlab/show/quasitopos) arise as the category of concrete sheaves on a concrete site. Concrete sheaves are those sheaves $X$ such that the induced map $Hom(C,X) \to Hom(\underline{C},\underline{X})$ is injective for all objects $C$, where $\underline{C}$ is the underlying set of $C$ and $\underline{X}$ is the value of $X$ on the terminal object. Concrete sheaves are a reflective subcategory of all sheaves. Concrete sheaves are a particularly nice example of a quasi-topos as the resulting quasi-topos is both complete and cocomplete. My question is, is there a way to represent quasi-topoi (or nice ones) as reflective subcategories of a Grothendieck topos (with some condition on the reflector)? (Of course, for this, you'd need the quasi-topos to be complete, since reflective subcategories of complete categories are again complete). More generally, is there some theorem saying that a category is a (possibly non-complete) quasi-topos if and only if it can be embedded into a topos such that the embedding has such-and-such property?

Answer 1

This is only a partial answer, and you may know it already since I mentioned it recently at the nForum, but for completeness, here it is. Theorem C2.2.13 in Sketches of an Elephant shows that the following are equivalent for a category C:

1. C is the separated objects for a Lawvere-Tierney topology on a Grothendieck topos (hence reflective in that topos)
2. C is the category of sheaves for one Grothendieck topology on a small category which are also separated for a second Grothendieck topology
3. C is a locally small, cocomplete quasitopos with a strong-generating set
4. C is locally presentable, locally cartesian closed, and every strong equivalence relation is effective

A category of this sort is called a Grothendieck quasitopos. The third characterization seems most similar to what you're looking for. I doubt you can get away without some generating-set condition, since it seems very unlikely that the (complete, cocomplete, locally small) quasitopos of pseudotopological spaces (for example) can be reflectively embedded in a topos.

What I don't know is whether one can put conditions directly on a reflective subcategory of a topos, analogous to left-exactness of the reflector, to guarantee that it is of this form. The reflector for separated objects preserves finite products and monics, but I have no idea whether that would be sufficient as a characterization.

Answer 2

This is really an answer to the question raised in Mike's reformulation of the question, but is too long for a comment and may be of interest.

Richard Garner and I have considered when a reflective subcategory of a presheaf category has the form considered in condition 2 of his answer. It turns out that preservation of finite products and monomorphisms is not enough: to see this, consider the reflection of directed graphs into preorders, which preserves finite products and monomorphisms but is not of this form.

In fact for a full reflective subcategory E of a presheaf category [C^{op},Set], the following conditions are equivalent:

1. there is a topology j and a larger topology k for which E consists of the objects which are sheaves for j and separated for k
2. the reflection preserves finite products and monomorphisms and is semi-left-exact
3. the reflection preserves monomorphisms and has stable units

Here the notions of semi-left-exactness and stable units come from

Cassidy, Hebert, Kelly, Reflective subcategories, localizations, and factorization systems, J. Austral. Math. Soc. Ser. A, 38:287-329, 1985.

Let R be the reflection and r the unit of the reflection. Semi-left-exactness says that R preserves each pullback of a component rX:X->RX of the unit along a map A->RX with A in the subcategory.

Stable units says the same thing, but without the requirement that A be in the subcategory. This turns out to be equivalent to R preserving all pullbacks over an object of the subcategory.