The Stacks project

Lemma 85.2.4. Let $f : Y \to X$ be a morphism of simplicial spaces. In terms of the description of sheaves in Lemma 85.2.2 the morphism $f_{Zar}$ of Lemma 85.2.3 can be described as follows.

  1. If $\mathcal{G}$ is a sheaf on $Y$, then $(f_{Zar, *}\mathcal{G})_ n = f_{n, *}\mathcal{G}_ n$.

  2. If $\mathcal{F}$ is a sheaf on $X$, then $(f_{Zar}^{-1}\mathcal{F})_ n = f_ n^{-1}\mathcal{F}_ n$.

Proof. The first part is immediate from the definitions. For the second part, note that in the proof of Lemma 85.2.3 we have shown that for a $V \subset Y_ n$ open the category $(\mathcal{I}_ V^ u)^{opp}$ contains as a cofinal subcategory the category of opens $U \subset X_ n$ with $f_ n^{-1}(U) \supset V$ and morphisms given by inclusions. Hence we see that the restriction of $u_ p\mathcal{F}$ to opens of $Y_ n$ is the presheaf $f_{n, p}\mathcal{F}_ n$ as defined in Sheaves, Lemma 6.21.3. Since $f_{Zar}^{-1}\mathcal{F} = u_ s\mathcal{F}$ is the sheafification of $u_ p\mathcal{F}$ and since sheafification uses only coverings and since coverings in $Y_{Zar}$ use only inclusions between opens on the same $Y_ n$, the result follows from the fact that $f_ n^{-1}\mathcal{F}_ n$ is (correspondingly) the sheafification of $f_{n, p}\mathcal{F}_ n$, see Sheaves, Section 6.21. $\square$

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