The Stacks project

Lemma 18.36.1. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$.

  1. We have $(\mathcal{F}^\# )_ p = \mathcal{F}_ p$ for any presheaf of sets on $\mathcal{C}$.

  2. The stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact (see Categories, Definition 4.23.1) and commutes with arbitrary colimits.

  3. The stalk functor $\textit{PSh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact (see Categories, Definition 4.23.1) and commutes with arbitrary colimits.

Proof. By Sites, Lemma 7.32.5 we have (1). By Sites, Lemmas 7.32.4 we see that $\textit{PSh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is a left adjoint, and by Sites, Lemma 7.32.5 we see the same thing for $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$. Hence the stalk functor commutes with arbitrary colimits (see Categories, Lemma 4.24.5). It follows from the definition of a point of a site, see Sites, Definition 7.32.2 that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact. Since sheafification is exact (Sites, Lemma 7.10.14) it follows that $\textit{PSh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact. $\square$

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