Lemma 7.8.8 (00VY)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.8: Families of morphisms with fixed target
Lemma 7.8.8 (cite)

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Lemma 7.8.8. Let $\mathcal{C}$ be a category. Let $\text{Cov}(\mathcal{C})$ be a proper class of coverings satisfying conditions (1), (2) and (3) of Definition 7.6.2. Let $\text{Cov}_1, \text{Cov}_2 \subset \text{Cov}(\mathcal{C})$ be two subsets of $\text{Cov}(\mathcal{C})$ which endow $\mathcal{C}$ with the structure of a site. If every covering $\mathcal{U} \in \text{Cov}(\mathcal{C})$ is combinatorially equivalent to a covering in $\text{Cov}_1$ and combinatorially equivalent to a covering in $\text{Cov}_2$ , then $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_1) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_2)$ .

Proof. This is clear from Lemmas 7.8.7 and 7.8.3 above as the hypothesis implies that every covering $\mathcal{U} \in \text{Cov}_1 \subset \text{Cov}(\mathcal{C})$ is combinatorially equivalent to an element of $\text{Cov}_2$ , and similarly with the roles of $\text{Cov}_1$ and $\text{Cov}_2$ reversed. $\square$

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