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*}

Proposition 7.33.2. Let $\mathcal{C}$ be a site. Assume that finite limits exist in $\mathcal{C}$. (I.e., $\mathcal{C}$ has fibre products, and a final object.) A point $p$ of such a site $\mathcal{C}$ is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that

  1. $u$ commutes with finite limits, and

  2. if $\{ U_ i \to U\} $ is a covering, then $\coprod _ i u(U_ i) \to u(U)$ is surjective.

Proof. Suppose first that $p$ is a point (Definition 7.32.2) given by a functor $u$. Condition (2) is satisfied directly from the definition of a point. By Lemma 7.32.3 we have $(h_ U)_ p = u(U)$. By Lemma 7.32.5 we have $(h_ U^\# )_ p = (h_ U)_ p$. Thus we see that $u$ is equal to the composition of functors

\[ \mathcal{C} \xrightarrow {h} \textit{PSh}(\mathcal{C}) \xrightarrow {{}^\# } \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \xrightarrow {()_ p} \textit{Sets} \]

Each of these functors is left exact, and hence we see $u$ satisfies (1).

Conversely, suppose that $u$ satisfies (1) and (2). In this case we immediately see that $u$ satisfies the first two conditions of Definition 7.32.2. And its stalk functor is exact, because it is a left adjoint by Lemma 7.32.5 and it commutes with finite limits by Lemma 7.33.1. $\square$


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