The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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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