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If $E$ is a Grothendieck topos, one can equip it with the "regular epimorphism topology"- this is the topology generated by covering families consisting of singleton epimorphisms. A presheaf on $E$ is a sheaf for this topology if and only if it is representable, so that the topos of sheaves with respect to this topology is equivalent to $E$ itself. This is primarly done in order to introduce stacks over a topos. My question is, how is this done rigorously since $E$, as a Grothendieck topos, is not a small category, so the "category of presheaves on E" only exists as a metacategory.

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I thought for that to work you had to take the canonical topology, consisting of all jointly-epimorphic families of maps. Why is it enough to take only single epimorphisms? –  Mike Shulman Oct 23 '10 at 0:54
Because toposes have small sums, so that you may always see your covering family as a single map. –  Denis-Charles Cisinski Oct 23 '10 at 1:00
Because a Grothendieck topos is infinitary extensive ;), and any pretopology can be turned into a singleton pretopology as Denis-Charles intimates. –  David Roberts Oct 23 '10 at 1:11
Indeed, that property guarantees that any pretopology is the union of a singleton pretopology with the extensive one. But it seems that a presheaf might be a sheaf for the singleton pretopology without being a sheaf for the extensive topology. In less hifalutin' language, small sums let you see your covering family as a single map, but unless you know that your presheaf preserves the small sums, I don't see why being a sheaf for the single map implies that it is also a sheaf for the original covering family. –  Mike Shulman Oct 23 '10 at 5:39
In particular, I think any singleton pretopology is a locally connected site ncatlab.org/nlab/show/locally+connected+site, which implies that any constant presheaf is a sheaf for such a pretopology. But this is certainly not the case for the extensive or canonical topologies. –  Mike Shulman Oct 23 '10 at 6:14

2 Answers 2

In [SGA 4, Exposé II], a site is defined as a category $C$ that is endowed with a (Grothendieck) topology, and which admits a small topologically generating set. This last property means that there exists a small family $G$ of objects of $C$ such that, for any object $X$ of $C$, any covering of $X$ admits a refinement by a covering family of shape $\{U_i\to X\}_{i\in I}$, such that the $U_i$'s all belong to $G$. The theory of sheaves on such a site is very well defined (see Theorème 3.4 in loc. cit.), and this is a topos because its satisfies the Giraud axioms (combine Proposition 4.8 and Corollary 4.11 in loc. cit.).

Otherwise, what you want is precisely stated as one of the characterizations of toposes in the original Theorem of Giraud (i.e. Théorème 1.2 in [SGA 4, Exposé IV]). And, by the way, what you call the 'regular epimorphism topology' is also known to be the canonical topology.

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A slightly weaker condition than the existence of a small topologically generating set is WISC (ncatlab.org/nlab/show/WISC), which says that each category of covers of each object in C has a weakly initial set. Would this be enough to recover the results you mention? –  David Roberts Oct 23 '10 at 1:11
I am not very optimistic with this one. For instance consider the site $Top$ of all topological spaces, endowed with the topology defined by open coverings (or, equivalently, local homeomorphisms). It satisfies WISC, but small sheaves on Top won't define a topos (because, otherwise, you would get a small generating family of Top itself). –  Denis-Charles Cisinski Oct 23 '10 at 1:37
Thanks! Actually, it seems that 1.1.1 from SGA 4 is really what "saves" everything- because first you need presheaves to EXIST :). –  David Carchedi Oct 23 '10 at 2:06
@Denis-Charles - ah, you are right. Thanks. –  David Roberts Oct 24 '10 at 21:35

In a word universes! Define $U-Set$ to be the small sets for a universe $U$, and pick two universes $U \in V$. Thus let $E \simeq Sh^U(C)$ where $C$ is a category in $U-Set$ (Edit: with a Grothendieck topology), and the sheaves take values in $U-Set$. $E$ is not a $U$-small category, but a $V$-small category. One can then take sheaves on $E$ with values in $V-Set$, and this will still be a category for the ambient metatheory.

There are theorems about properties of these various levels of categories of sheaves, and you may be able to get away with slightly less than what I have assumed, but this is a start.

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So then I'd get that $Sh^V\left(E\right) \cong E$ as $V$-small categories? –  David Carchedi Oct 23 '10 at 1:47
Actually, taking Denis-Charles' answer into account, you probably only need one universe, U; then E=Sh^U(D) is a large category, but with a U-small generating set, which I think would make E with the regular epimorphism pretopology a site as in SGA4, Expose II. Then the sheaves on E would form a large category (of the same 'size'), and this would be equivalent to E. –  David Roberts Oct 24 '10 at 10:43
On the other hand, without referring to conjectural connections between a generating family and a topologically generating set, you should get Sh^V(E) is a V-large category (but still a category in the metalogic), which is essentially V-small: it is equivalent to E. –  David Roberts Oct 24 '10 at 21:30

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