The Stacks project

Definition 59.10.2. A site1 consists of a category $\mathcal{C}$ and a set $\text{Cov}(\mathcal{C})$ consisting of families of morphisms with fixed target called coverings, such that

  1. (isomorphism) if $\varphi : V \to U$ is an isomorphism in $\mathcal{C}$, then $\{ \varphi : V \to U\} $ is a covering,

  2. (locality) if $\{ \varphi _ i : U_ i \to U\} _{i\in I}$ is a covering and for all $i \in I$ we are given a covering $\{ \psi _{ij} : U_{ij} \to U_ i \} _{j\in I_ i}$, then

    \[ \{ \varphi _ i \circ \psi _{ij} : U_{ij} \to U \} _{(i, j)\in \prod _{i\in I} \{ i\} \times I_ i} \]

    is also a covering, and

  3. (base change) if $\{ U_ i \to U\} _{i\in I}$ is a covering and $V \to U$ is a morphism in $\mathcal{C}$, then

    1. for all $i \in I$ the fibre product $U_ i \times _ U V$ exists in $\mathcal{C}$, and

    2. $\{ U_ i \times _ U V \to V\} _{i\in I}$ is a covering.

[1] What we call a site is a called a category endowed with a pretopology in [Exposé II, Définition 1.3, SGA4]. In [ArtinTopologies] it is called a category with a Grothendieck topology.

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