A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hompresheaves are sheaves). Is there a common name for a site whose codomain fibration is a stack? The canonical topology on a Grothendieck topos has this property, as does the coherent topology on a pretopos, the regular topology on a Barrexact category, the extensive topology on a lextensive category, etc.
I don't have an answer to your question, but I'm going to post whatever thoughts I had about it. Maybe something here will help someone answer the question, or at least help more people understand what's involved. I'm sorry that it's come out so long. Definitions


Wait. Aren't algebraic spaces usually defined to be quasiseperated? Doesn't this mean that you also need a condition on the diagonal map $U \to U \times_{U/R} U$? or on the map $U \times_{U/R} U \to U \times U$? This seems to suggest that algebraic spaces (viewed as sheaves) are not closed in the way you describe. See also this MO question: mathoverflow.net/questions/9043/…
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Chris SchommerPries
Dec 24 '09 at 13:14



Though quasiseparatedness is necessary for many results, it just doesn't make sense to build it into the definition. After all, not all schemes are quasiseparated, and all schemes should be algebraic spaces. But there is a separation hypothesis that I swept under the rug: when I take the "gluing closure" of a site, I only want to throw in sheaves with representable diagonal (otherwise I wouldn't be able to extend the site structure to the enlarged category). I seem to recall that this isn't completely automatic for algebraic spaces, but very close to free.
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Anton Geraschenko
Dec 24 '09 at 18:03



Incidently, you may need to apply the "gluing closure construction" several times (possibly countably many?) to get a site closed under gluing. For example, the applying it to affine schemes with the Zariski topology gives you the category of separated schemes (because of the representability condition on the diagonal). You have to apply it again to get the category of all schemes.
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Anton Geraschenko
Dec 24 '09 at 18:05



Thanks! My definition of "prestack" is the same as yours, except that I would consider arbitrary covering families, not just single covers (see mathoverflow.net/questions/9705/… ). Then regarding prestack ⇒ subcanonical, $C$ is subcanonical if for any covering family $(U_i\to X)$, maps $X\to Y$ are the same as compatible families of maps $(U_i\to Y)$. But maps $X\to Y$ are the same as maps $X\to Y\times X$ in the fiber $C/X$ of the codomain fibration, and similarly for maps $U_i\to Y$, which the prestack condition lets you glue.
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Mike Shulman
Dec 24 '09 at 22:01



Also, I wonder if our definitions of "covering" are different? The projections $R\to U$ of any equivalence relation are split epic, and hence coverings in any Grothendieck topology, so it seems to me like any equivalence relation is a "covering relation" in your sense. I think your notion of "closed under gluing" would then just reduce to saying that the category is Barrexact and its topology includes the regular topology (a "superexact site").
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Mike Shulman
Dec 24 '09 at 22:07
