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Given a Grothendieck topology $T$, say subcanonical, on a category $C$, we are able to talk about sheaves in $(C, T)$. Since pre/sheaves can be viewed as generalized objects of $C$ (via Yoneda functor), one natural question arises, is $Shv(C, T)$ or $Psh(C)$ a site extending the site $(C, T)$? If it is, then we take pre/sheaves on that new site, can we get more pre/sheaves?

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up vote 5 down vote accepted

Yes. Every topos has the so-called canonical topology, in which a sieve is covering if and only if it is generated by a small jointly epimorphic family. (In the Grothendieck topos case, this is the same as being jointly epimorphic, because the small-generation condition is automatic.) Moreover, the category of sheaves in the canonical topology of a Grothendieck topos is equivalent to the original topos, so there are no new sheaves. See Proposition 2.2.7 in [Johnstone, Sketches of an elephant, Part C].

That said, it is not legitimate to form presheaves on a large site: for one thing, the resulting category is not a topos.

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I saw the following way to extend the covering, although it doesn't spell out as a topology, on the presheaves. Declare a family $\{F_i\to F\}$ of presheaves as a cover if for every representable $T$ and $T\to F$ the pullback $T\times_F F_i$ is representable and $\{T\times_F F_i\to T\}$ is a cover. I didn't think this is the same as the canonical topology you described. Maybe there is some relation? – Ma Ming May 23 '13 at 14:47
That defines, at best, a pretopology. However the sieve generated by any such family is a covering sieve in the canonical topology. – Zhen Lin May 23 '13 at 16:35
Yes, I always work with pretopology. – Ma Ming May 23 '13 at 18:04
So if $(\mathcal C,\mathcal T)$ is a subcanonical site, is the restriction of the canonical topology on $Shv(\mathcal C,\mathcal T)$ to the subcategory of representable sheaves equivalent to the original topology $\mathcal T$ on $\mathcal C$? – Donkey_2009 Apr 7 '15 at 15:31
That is correct. – Zhen Lin Apr 7 '15 at 17:00

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