# Grothendieck topology on pre/sheaves

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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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
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 at 15:31