The Stacks project

Proposition 7.33.3. 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 7.33.2. $\square$

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