Remark 66.19.10. This remark is the analogue of Étale Cohomology, Remark 59.56.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ be a geometric point of $X$. By Étale Cohomology, Theorem 59.56.3 the category of sheaves on $\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}$ is equivalent to the category of sets (by taking a sheaf to its global sections). Hence it follows from Lemma 66.19.9 part (4) applied to the morphism $\overline{x}$ that the functor

$\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{x}}$

is isomorphic to the functor

$\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}) = \textit{Sets}, \quad \mathcal{F} \longmapsto \overline{x}^*\mathcal{F}$

Hence we may view the stalk functors as pullback functors along geometric morphisms (and not just some abstract morphisms of topoi as in the result of Lemma 66.19.7).

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