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.

1. Assume there exists a set $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ with the following properties:

1. 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,

2. for $U, U' \in \mathcal{B}$ the sheaf $h_ U^\# \times h_{U'}^\#$ is quasi-compact.

2. Assume there exists a set $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ with the following properties:

1. there exist $m \geq 0$ and $U_1, \ldots , U_ m \in \mathcal{B}$ with $\coprod h_{U_ j}^\# \to *$ surjective,

2. 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

3. 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.

3. Suppose that

1. $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is quasi-compact,

2. every object of $\mathcal{C}$ has a covering whose members are quasi-compact objects,

3. 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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