How to resolve size issues with the regular epimorphism topology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:01:17Z http://mathoverflow.net/feeds/question/43228 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43228/how-to-resolve-size-issues-with-the-regular-epimorphism-topology How to resolve size issues with the regular epimorphism topology David Carchedi 2010-10-22T20:01:40Z 2010-10-23T02:07:27Z <p>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.</p> http://mathoverflow.net/questions/43228/how-to-resolve-size-issues-with-the-regular-epimorphism-topology/43256#43256 Answer by David Roberts for How to resolve size issues with the regular epimorphism topology David Roberts 2010-10-23T00:29:55Z 2010-10-23T02:07:27Z <p>In a word <a href="http://www.ncatlab.org/nlab/show/universe" rel="nofollow">universes</a>! 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.</p> <p>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.</p> http://mathoverflow.net/questions/43228/how-to-resolve-size-issues-with-the-regular-epimorphism-topology/43257#43257 Answer by Denis-Charles Cisinski for How to resolve size issues with the regular epimorphism topology Denis-Charles Cisinski 2010-10-23T00:57:44Z 2010-10-23T00:57:44Z <p>In [SGA 4, Exposé II], a site is <em>defined</em> 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 <em>loc. cit.</em>), and this is a topos because its satisfies the Giraud axioms (combine Proposition 4.8 and Corollary 4.11 in <em>loc. cit.</em>).</p> <p>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 <em>canonical topology</em>.</p>