Lemma 7.10.8 (00W9)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.10: Sheafification
Lemma 7.10.8 (cite)

Lemma 7.10.8. The map $\theta : \mathcal{F} \to \mathcal{F}^+$ has the following property: For every object $U$ of $\mathcal{C}$ and every section $s \in \mathcal{F}^+(U)$ there exists a covering $\{ U_ i \to U\} $ such that $s|_{U_ i}$ is in the image of $\theta : \mathcal{F}(U_ i) \to \mathcal{F}^{+}(U_ i)$.

Proof. Namely, let $\{ U_ i \to U\} $ be a covering such that $s$ arises from the element $(s_ i) \in H^0(\{ U_ i \to U\} , \mathcal{F})$. According to Lemma 7.10.6 we may consider the covering $\{ U_ i \to U_ i\} $ and the (obvious) morphism of coverings $\{ U_ i \to U_ i\} \to \{ U_ i \to U\} $ to compute the pullback of $s$ to an element of $\mathcal{F}^+(U_ i)$. And indeed, using this covering we get exactly $\theta (s_ i)$ for the restriction of $s$ to $U_ i$. $\square$

Comments (1)

Comment #8573 by Alejandro González Nevado on July 16, 2023 at 21:57

SS: The canonical map between a presheaf of sets on a site and its corresponding zeroth Čech cohomology admits, for every object and every section of the zeroth Čech cohomology over that object, a covering such that each restriction of the section corresponding to that covering is in the image of this canonical map when applied over the object generating such restriction.

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