Let $\mathcal{C}$ be a site and $X$ a sheaf of sets on $\mathcal{C}$.

Then there exists a hypercover $K_{\bullet}$ of $X$ such that $K_n$ is a coproduct of representable presheaves on $\mathcal{C}$ for all $n$.

Suppose $X$ is a quasi-compact object, i.e. every effective epimorphism $\coprod_{j\in J}\mathcal{F}_j\to X$, with $J$ an index set and $\mathcal{F}_j$ sheaves of sets, can be refined to a finite one by extracting a finite subset of $J$.

(Question 1) Can one arrange, for every $n$, $K_n$ to be a finite coproduct of representable presheaves?

Suppose now that the topos $Sh(\mathcal{C})$ of sheaves of sets on $\mathcal{C}$ is coherent (so every sheaf of sets is quasi-compact).

(Question 2) Can, under this stronger assumption, one arrange that for every $n$, $K_n$ is a finite coproduct of representable presheaves?


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