The Stacks project

Lemma 25.3.7. Let $\mathcal{C}$ be a site with fibre products. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object of $\mathcal{C}$. The collection of all hypercoverings of $X$ forms a set.

Proof. Since $\mathcal{C}$ is a site, the set of all coverings of $X$ forms a set. Thus we see that the collection of possible $K_0$ forms a set. Suppose we have shown that the collection of all possible $K_0, \ldots , K_ n$ form a set. Then it is enough to show that given $K_0, \ldots , K_ n$ the collection of all possible $K_{n + 1}$ forms a set. And this is clearly true since we have to choose $K_{n + 1}$ among all possible coverings of $(\text{cosk}_ n \text{sk}_ n K)_{n + 1}$. $\square$

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