How to resolve size issues with the regular epimorphism topology - MathOverflow

How to resolve size issues with the regular epimorphism topology

2010-10-22T20:01:40Z

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.

Answer by David Roberts for How to resolve size issues with the regular epimorphism topology

2010-10-23T00:29:55Z

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.

Answer by Denis-Charles Cisinski for How to resolve size issues with the regular epimorphism topology

2010-10-23T00:57:44Z

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.