The Stacks project

Remark 25.12.8. Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has fibre products and equalizers and let $K$ be a hypercovering. Write $K_ n = \{ U_{n, i}\} _{i \in I_ n}$. Suppose that

  1. $U_ n = \coprod _{i \in I_ n} U_{n, i}$ exists, and

  2. $\coprod _{i \in I_ n} h_{U_{n, i}} \to h_{U_ n}$ induces an isomorphism on sheafifications.

Then we get another simplicial object $L$ of $\text{SR}(\mathcal{C})$ with $L_ n = \{ U_ n\} $, see Remark 25.12.7. Now we claim that $L$ is a hypercovering. To see this we check conditions (1), (2), (3) of Definition 25.6.1. Condition (1) follows from (b) and (1) for $K$. Condition (2) follows in exactly the same way. Condition (3) follows because

\begin{align*} F((\text{cosk}_ n \text{sk}_ n L)_{n + 1})^\# & = ((\text{cosk}_ n \text{sk}_ n F(L)^\# )_{n + 1}) \\ & = ((\text{cosk}_ n \text{sk}_ n F(K)^\# )_{n + 1}) \\ & = F((\text{cosk}_ n \text{sk}_ n K)_{n + 1})^\# \end{align*}

for $n \geq 1$ and hence the condition for $K$ implies the condition for $L$ exactly as in (1) and (2). Note that $F$ commutes with connected limits and sheafification is exact proving the first and last equality; the middle equality follows as $F(K)^\# = F(L)^\# $ by (b).

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