(Grothendieck) topoi are leftexact reflective subcategories of a category of presheaves. An important class of quasitopoi (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 quasitopos as the resulting quasitopos is both complete and cocomplete. My question is, is there a way to represent quasitopoi (or nice ones) as reflective subcategories of a Grothendieck topos (with some condition on the reflector)? (Of course, for this, you'd need the quasitopos 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 noncomplete) quasitopos if and only if it can be embedded into a topos such that the embedding has suchandsuch property?

$\begingroup$ a very nice question! $\endgroup$– Grant Olney PassmoreOct 6, 2010 at 20:11
2 Answers
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:
 C is the separated objects for a LawvereTierney topology on a Grothendieck topos (hence reflective in that topos)
 C is the category of sheaves for one Grothendieck topology on a small category which are also separated for a second Grothendieck topology
 C is a locally small, cocomplete quasitopos with a stronggenerating set
 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 generatingset 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 leftexactness 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.

$\begingroup$ Thanks Mike! I did indeed read this on nforum since Urs filled me in on your discussion. Thanks for posting this! $\endgroup$ Oct 17, 2010 at 17:29
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:
 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
 the reflection preserves finite products and monomorphisms and is semileftexact
 the reflection preserves monomorphisms and has stable units
Here the notions of semileftexactness and stable units come from
Cassidy, Hebert, Kelly, Reflective subcategories, localizations, and factorization systems, J. Austral. Math. Soc. Ser. A, 38:287329, 1985.
Let R be the reflection and r the unit of the reflection. Semileftexactness 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.

$\begingroup$ Nice!! Can you extend it to reflective subcategories of nonpresheaf topoi? $\endgroup$ Jun 10, 2011 at 22:46

$\begingroup$ it extends to Grothendieck toposes; not sure about more generally than that. $\endgroup$ Jun 11, 2011 at 5:48