Remark 7.17.9. Let \mathcal{C} be a site. There are several ways to ensure that the hypotheses of part (4) of Lemma 7.17.8 are satisfied. Here are a few.
Assume there exists a set \mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) with the following properties:
for every surjection \mathcal{F} \to * there exist m \geq 0 and U_1, \ldots , U_ m \in \mathcal{B} with \mathcal{F}(U_ j) nonempty and \coprod h_{U_ j}^\# \to * surjective,
for U, U' \in \mathcal{B} the sheaf h_ U^\# \times h_{U'}^\# is quasi-compact.
Assume there exists a set \mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) with the following properties:
there exist m \geq 0 and U_1, \ldots , U_ m \in \mathcal{B} with \coprod h_{U_ j}^\# \to * surjective,
for U \in \mathcal{B} any covering of U can be refined by a finite covering \{ U_ j \to U\} _{j = 1, \ldots , m} with U_ j \in \mathcal{B}, and
for U, U' \in \mathcal{B} there exist m \geq 0, U_1, \ldots , U_ m \in \mathcal{B}, and morphisms U_ j \to U and U_ j \to U' such that \coprod h_{U_ j}^\# \to h_ U^\# \times h_{U'}^\# is surjective.
Suppose that
\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) is quasi-compact,
every object of \mathcal{C} has a covering whose members are quasi-compact objects,
if U and U' are quasi-compact, then the sheaf h_ U^\# \times h_{U'}^\# is quasi-compact.
In cases (1) and (2) we set S \subset \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})) equal to the set of finite coproducts of the sheaves h_ U^\# for U \in \mathcal{B}. In case (3) we set S \subset \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})) equal to the set of finite coproducts of the sheaves h_ U^\# for U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) quasi-compact.
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