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
u(X) is a singleton set, and
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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