The Stacks project

Lemma 7.33.1. Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. If the category of neighbourhoods of $p$ is cofiltered, then the stalk functor ( is left exact.

Proof. Let $\mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a finite diagram of sheaves. We have to show that the stalk of the limit of this system agrees with the limit of the stalks. Let $\mathcal{F}$ be the limit of the system as a presheaf. According to Lemma 7.10.1 this is a sheaf and it is the limit in the category of sheaves. Hence we have to show that $\mathcal{F}_ p = \mathop{\mathrm{lim}}\nolimits _\mathcal {I} \mathcal{F}_{i, p}$. Recall also that $\mathcal{F}$ has a simple description, see Section 7.4. Thus we have to show that

\[ \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _{\{ (U, x)\} ^{opp}} \mathcal{F}_ i(U) = \mathop{\mathrm{colim}}\nolimits _{\{ (U, x)\} ^{opp}} \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i(U). \]

This holds, by Categories, Lemma 4.19.2, because the opposite of the category of neighbourhoods is filtered by assumption. $\square$

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