The Stacks project

Remark 21.16.3. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Let $S \subset \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}))$ be the set of sheaves $K$ which have the form

with $U_1, \ldots , U_ n \in \mathcal{B}$. Then we can ask: when does this set satisfy the assumptions of Lemma 21.16.2? One answer is that it suffices if

1. for some $n \geq 0$, $U_1, \ldots , U_ n \in \mathcal{B}$ the map $\coprod _{i = 1, \ldots , n} h_{U_ i}^\# \to *$ is surjective,

2. every covering of $U \in \mathcal{B}$ can be refined by a covering of the form $\{ U_ i \to U\} _{i = 1, \ldots , n}$ with $U_ i \in \mathcal{B}$,

3. given $U, U' \in \mathcal{B}$ there exist $n \geq 0$, $U_1, \ldots , U_ n \in \mathcal{B}$, maps $U_ i \to U$ and $U_ i \to U'$ such that $\coprod _{i = 1, \ldots , n} h_{U_ i}^\# \to h_ U^\# \times h_{U'}^\# $ is surjective,

4. given morphisms $a, b : U \to U'$ in $\mathcal{C}$ with $U, U' \in \mathcal{B}$, there exist $U_1, \ldots , U_ n \in \mathcal{B}$, maps $U_ i \to U$ equalizing $a, b$ such that $\coprod _{i = 1, \ldots , n} h_{U_ i}^\# \to \text{Equalizer}(h_ a^\# , h_ b^\# : h_ U^\# \to h_{U'}^\# )$ is surjective.

We omit the detailed verification, except to mention that part (2) above insures that every element of $\mathcal{B}$ is quasi-compact and hence every $K \in S$ is quasi-compact as well by Sites, Lemma 7.17.6.