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Tag 00YQ

Chapter 7: Sites and Sheaves > Section 7.38: Criterion for existence of points

Proposition 7.38.3. Let $\mathcal{C}$ be a site. Assume that

  1. finite limits exist in $\mathcal{C}$, and
  2. every covering $\{U_i \to U\}_{i \in I}$ has a refinement by a finite covering of $\mathcal{C}$.

Then $\mathcal{C}$ has enough points.

Proof. We have to show that given any sheaf $\mathcal{F}$ on $\mathcal{C}$, any $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$, and any distinct sections $s, s' \in \mathcal{F}(U)$, there exists a point $p$ such that $s, s'$ have distinct image in $\mathcal{F}_p$. See Lemma 7.37.3. Consider the system $(J, \geq, V_j, g_{jj'})$ with $J = \{1\}$, $V_1 = U$, $g_{11} = \text{id}_U$. Apply Lemma 7.38.2. By the result of that lemma we get a system $(I, \geq, U_i, f_{ii'})$ refining our system such that $s_p \not = s'_p$ and such that moreover for every finite covering $\{W_k \to W\}$ of the site $\mathcal{C}$ the map $\coprod_k u(W_k) \to u(W)$ is surjective. Since every covering of $\mathcal{C}$ can be refined by a finite covering we conclude that $\coprod_k u(W_k) \to u(W)$ is surjective for any covering $\{W_k \to W\}$ of the site $\mathcal{C}$. This implies that $u = p$ is a point, see Proposition 7.32.2 (and the discussion at the beginning of this section which guarantees that $u$ commutes with finite limits). $\square$

    The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 8439–8448 (see updates for more information).

    \begin{proposition}
    \label{proposition-criterion-points}
    Let $\mathcal{C}$ be a site. Assume that
    \begin{enumerate}
    \item finite limits exist in $\mathcal{C}$, and
    \item every covering $\{U_i \to U\}_{i \in I}$
    has a refinement by a finite covering of $\mathcal{C}$.
    \end{enumerate}
    Then $\mathcal{C}$ has enough points.
    \end{proposition}
    
    \begin{proof}
    We have to show that given any sheaf
    $\mathcal{F}$ on $\mathcal{C}$, any $U \in \Ob(\mathcal{C})$,
    and any distinct sections $s, s' \in \mathcal{F}(U)$, there exists
    a point $p$ such that $s, s'$ have distinct image in
    $\mathcal{F}_p$. See Lemma \ref{lemma-enough}.
    Consider the system $(J, \geq, V_j, g_{jj'})$
    with $J = \{1\}$, $V_1 = U$, $g_{11} = \text{id}_U$.
    Apply Lemma \ref{lemma-refine-all-at-once}.
    By the result of that lemma we get a system
    $(I, \geq, U_i, f_{ii'})$ refining our system such
    that $s_p \not = s'_p$ and such that moreover for every
    finite covering $\{W_k \to W\}$ of the site $\mathcal{C}$ the map
    $\coprod_k u(W_k) \to u(W)$ is surjective.
    Since every covering of $\mathcal{C}$ can be refined by
    a finite covering we conclude that
    $\coprod_k u(W_k) \to u(W)$ is surjective for {\it any}
    covering $\{W_k \to W\}$ of the site $\mathcal{C}$.
    This implies that $u = p$ is a point, see
    Proposition \ref{proposition-point-limits} (and the discussion
    at the beginning of this section which guarantees that $u$
    commutes with finite limits).
    \end{proof}

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