The Stacks project

Definition 25.3.1. Let $\mathcal{C}$ be a site. Let $f = (\alpha , f_ i) : \{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ be a morphism in the category $\text{SR}(\mathcal{C})$. We say that $f$ is a covering if for every $j \in J$ the family of morphisms $\{ U_ i \to V_ j\} _{i \in I, \alpha (i) = j}$ is a covering for the site $\mathcal{C}$. Let $X$ be an object of $\mathcal{C}$. A morphism $K \to L$ in $\text{SR}(\mathcal{C}, X)$ is a covering if its image in $\text{SR}(\mathcal{C})$ is a covering.

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