The Stacks project

Lemma 7.33.2. Let $\mathcal{C}$ be a site. Assume that $\mathcal{C}$ has a final object $X$ and fibred products. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor such that

  1. $u(X)$ is a singleton set, and

  2. for every pair of morphisms $U \to W$ and $V \to W$ with the same target the map $u(U \times _ W V) \to u(U) \times _{u(W)} u(V)$ is bijective.

Then the the category of neighbourhoods of $p$ is cofiltered and consequently the stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \to \mathcal{F}_ p$ commutes with finite limits.

Proof. Please note the analogy with Lemma 7.5.2. The assumptions on $\mathcal{C}$ imply that $\mathcal{C}$ has finite limits. See Categories, Lemma 4.18.4. Assumption (1) implies that the category of neighbourhoods is nonempty. Suppose $(U, x)$ and $(V, y)$ are neighbourhoods. Then $u(U \times V) = u(U \times _ X V) = u(U) \times _{u(X)} u(V) = u(U) \times u(V)$ by (2). Hence there exists a neighbourhood $(U \times _ X V, z)$ mapping to both $(U, x)$ and $(V, y)$. Let $a, b : (V, y) \to (U, x)$ be two morphisms in the category of neighbourhoods. Let $W$ be the equalizer of $a, b : V \to U$. As in the proof of Categories, Lemma 4.18.4 we may write $W$ in terms of fibre products:

\[ W = (V \times _{a, U, b} V) \times _{(pr_1, pr_2), V \times V, \Delta } V \]

The bijectivity in (2) guarantees there exists an element $z \in u(W)$ which maps to $((y, y), y)$. Then $(W, z) \to (V, y)$ equalizes $a, b$ as desired. The “consequently” clause is Lemma 7.33.1. $\square$

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