Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.