The Stacks project

Lemma 7.14.10. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the functor $u : \mathcal{C} \to \mathcal{D}$. Given any object $V$ of $\mathcal{D}$ there exists a covering $\{ V_ j \to V\} $ such that for every $j$ there exists a morphism $V_ j \to u(U_ j)$ for some object $U_ j$ of $\mathcal{C}$.

Proof. Since $f^{-1} = u_ s$ is exact we have $f^{-1}* = *$ where $*$ denotes the final object of the category of sheaves (Example 7.10.2). Since $f^{-1}* = u_ s*$ is the sheafification of $u_ p*$ we see there exists a covering $\{ V_ j \to V\} $ such that $(u_ p*)(V_ j)$ is nonempty. Since $(u_ p*)(V_ j)$ is a colimit over the category $\mathcal{I}^ u_{V_ j}$ whose objects are morphisms $V_ j \to u(U)$ the lemma follows. $\square$

Comments (3)

Comment #3433 by Remy on

The categories may be empty. This happens for example if is the empty site, but it could also just happen that none of the admit a map to any .

Comment #3434 by on

Dear Remy, I think the lemma (and the proof) is OK. The key is that the definition of a "morphism of sites" forces exactness of the functor . For example, if is the empty site, then has to be the empty topos (because the final and initial object of it have to be the same as the pullback of such on by an exact functor). And this in turn means that every object of has an EMPTY covering! Thus the lemma holds in this case. OK?

Comment #3437 by Remy on

You're right; I just realised the same thing.

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  • 1 comment(s) on Section 7.14: Morphisms of sites

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