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Tag 00Y5

Chapter 7: Sites and Sheaves > Section 7.31: Points

Definition 7.31.2. Let $\mathcal{C}$ be a site. A point $p$ of the site $\mathcal{C}$ is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that

  1. For every covering $\{U_i \to U\}$ of $\mathcal{C}$ the map $\coprod u(U_i) \to u(U)$ is surjective.
  2. For every covering $\{U_i \to U\}$ of $\mathcal{C}$ and every morphism $V \to U$ the maps $u(U_i \times_U V) \to u(U_i) \times_{u(U)} u(V)$ are bijective.
  3. The stalk functor $\mathop{\textit{Sh}}\nolimits(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_p$ is left exact.

    The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 7010–7024 (see updates for more information).

    \begin{definition}
    \label{definition-point}
    Let $\mathcal{C}$ be a site. A {\it point $p$ of the site
    $\mathcal{C}$} is given by a functor $u : \mathcal{C}
    \to \textit{Sets}$ such that
    \begin{enumerate}
    \item For every covering $\{U_i \to U\}$ of $\mathcal{C}$ the map
    $\coprod u(U_i) \to u(U)$ is surjective.
    \item For every covering $\{U_i \to U\}$ of $\mathcal{C}$ and
    every morphism $V \to U$ the maps
    $u(U_i \times_U V) \to u(U_i) \times_{u(U)} u(V)$ are bijective.
    \item The stalk functor $\Sh(\mathcal{C}) \to \textit{Sets}$,
    $\mathcal{F} \mapsto \mathcal{F}_p$ is left exact.
    \end{enumerate}
    \end{definition}

    Comments (2)

    Comment #2344 by Katharina on January 6, 2017 a 7:07 am UTC

    Why does the stalk functor only have to be left exact and not exact?

    Comment #2413 by Johan (site) on February 17, 2017 a 1:35 pm UTC

    OK, yes we could change the definition and say this, but I wanted to write down minimal conditions. The discussion following the definition shows that indeed the stalk functor is exact (as it defines a point of the topos).

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