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    $\begingroup$ Very nice example. I was initially taken aback, thinking: But those are sheaves of posets, and we know that for sheaves of posets, isomorphism can be detected by stalks! For others with the same confusion, the resolution is that the inclusion from (countably) sup-complete posets into posets doesn’t preserve colimits, so the stalks are different when computed in the sup-complete category — in particular, the stalk of $G$ collapses. Giving that stalk computation more directly, we can see exactly how countable joins cause the collapse: [cont’d] $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Nov 23 at 20:16
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    $\begingroup$ [cont’d] $\newcommand{\R}{\mathbb{R}}$Writing the stalk inclusions as $\alpha : G(U) \to G_0$, we have for any $V \subseteq U$ that $\alpha(V) = \bigvee_{n} \alpha(V) = \bigvee_{n} \alpha(V \cup (-\infty,-1/n) \cup (1/n,\infty)) = \alpha(\bigvee_n (V \cup (-\infty,-1/n) \cup (1/n,\infty))) = \alpha (V \cup (\R \setminus \{0\}))$, and hence no information can be retained in the stalk beyond whether $0$ is in each “germ”. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Nov 23 at 20:16
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    $\begingroup$ Interesting! So perhaps the lesson here is that stalks should not be computed using the filtered colimits that already exist if they are not known to be well behaved. That it happens even for locally presentable categories means that failure of sheaf-locality does not prevent sheafification – we just cannot use the Godement comonad to calculate it (because we do not have comonadicity of sheaves over stalks in this case). $\endgroup$
    – Zhen Lin
    Commented Nov 23 at 23:08