The Stacks project

Definition 7.6.2. A site1 is given by a category $\mathcal{C}$ and a set $\text{Cov}(\mathcal{C})$ of families of morphisms with fixed target $\{ U_ i \to U\} _{i \in I}$, called coverings of $\mathcal{C}$, satisfying the following axioms

  1. If $V \to U$ is an isomorphism then $\{ V \to U\} \in \text{Cov}(\mathcal{C})$.

  2. If $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})$ and for each $i$ we have $\{ V_{ij} \to U_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{C})$, then $\{ V_{ij} \to U\} _{i \in I, j\in J_ i} \in \text{Cov}(\mathcal{C})$.

  3. If $\{ U_ i \to U\} _{i\in I}\in \text{Cov}(\mathcal{C})$ and $V \to U$ is a morphism of $\mathcal{C}$ then $U_ i \times _ U V$ exists for all $i$ and $\{ U_ i \times _ U V \to V \} _{i\in I} \in \text{Cov}(\mathcal{C})$.

[1] This notation differs from that of [SGA4], as explained in the introduction.

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