Lemma 85.3.1. Let $\mathcal{C}$ be a simplicial object in the category of sites. With notation as above we construct a site $\mathcal{C}_{total}$ as follows.

An object of $\mathcal{C}_{total}$ is an object $U$ of $\mathcal{C}_ n$ for some $n$,

a morphism $(\varphi , f) : U \to V$ of $\mathcal{C}_{total}$ is given by a map $\varphi : [m] \to [n]$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$, $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ m)$ and a morphism $f : U \to u_\varphi (V)$ of $\mathcal{C}_ n$, and

a covering $\{ (\text{id}, f_ i) : U_ i \to U\} $ in $\mathcal{C}_{total}$ is given by an $n$ and a covering $\{ f_ i : U_ i \to U\} $ of $\mathcal{C}_ n$.

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