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Tag 00VH

Chapter 7: Sites and Sheaves > Section 7.6: Sites

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.

The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 677–695 (see updates for more information).

\begin{definition}
\label{definition-site}
A {\it site}\footnote{This notation differs from that of \cite{SGA4}, as
explained in the introduction.} 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 {\it coverings of $\mathcal{C}$},
satisfying the following axioms
\begin{enumerate}
\item If $V \to U$ is an isomorphism then $\{V \to U\} \in
\text{Cov}(\mathcal{C})$.
\item 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
$\{V_{ij} \to U\}_{i \in I, j\in J_i} \in \text{Cov}(\mathcal{C})$.
\item 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
$\{U_i \times_U V \to V \}_{i\in I} \in \text{Cov}(\mathcal{C})$.
\end{enumerate}
\end{definition}

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