Lemma 7.42.2. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. The following are equivalent:
$U$ is sheaf theoretically empty,
$\mathcal{F}(U)$ is a singleton for each sheaf $\mathcal{F}$,
$\emptyset ^\# (U)$ is a singleton,
$\emptyset ^\# (U)$ is nonempty, and
the empty family is a covering of $U$ in $\mathcal{C}$.
Moreover, if $U$ is sheaf theoretically empty, then for any morphism $U' \to U$ of $\mathcal{C}$ the object $U'$ is sheaf theoretically empty.
Proof. For any sheaf $\mathcal{F}$ we have $\mathcal{F}(U) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F})$. Hence, we see that (1) and (2) are equivalent. It is clear that (2) implies (3) implies (4). If every covering of $U$ is given by a nonempty family, then $\emptyset ^+(U)$ is empty by definition of the plus construction. Note that $\emptyset ^+ = \emptyset ^\# $ as $\emptyset $ is a separated presheaf, see Theorem 7.10.10. Thus we see that (4) implies (5). If (5) holds, then $\mathcal{F}(U)$ is a singleton for every sheaf $\mathcal{F}$ by the sheaf condition for $\mathcal{F}$, see Remark 7.7.2. Thus (5) implies (2) and (1) – (5) are equivalent. The final assertion of the lemma follows from Axiom (3) of Definition 7.6.2 applied the empty covering of $U$. $\square$
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