# On the existence of nice hypercovers

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?