## 7.52 Points and topologies

Recall from Section 7.32 that given a functor $p = u : \mathcal{C} \to \textit{Sets}$ we can define a stalk functor

\[ \textit{PSh}(\mathcal{C}) \longrightarrow \textit{Sets}, \mathcal{F} \longmapsto \mathcal{F}_ p. \]

Definition 7.52.1. Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. A *point $p$* of the topology is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that

For every covering sieve $S$ on $U$ the map $S_ p \to (h_ U)_ p$ is surjective.

The stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \to \mathcal{F}_ p$ is exact.

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