The Stacks project

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.

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

  2. 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

  3. 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 $\mathcal{C}_ n$.

Proof. Composition of $(\varphi , f) : U \to V$ with $(\psi , g) : V \to W$ is given by $(\varphi \circ \psi , u_\varphi (g) \circ f)$. This uses that $u_\varphi \circ u_\psi = u_{\varphi \circ \psi }$.

Let $\{ (\text{id}, f_ i) : U_ i \to U\} $ be a covering as in (3) and let $(\varphi , g) : W \to U$ be a morphism with $W \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ m)$. We claim that

\[ W \times _{(\varphi , g), U, (\text{id}, f_ i)} U_ i = W \times _{g, u_\varphi (U), u_\varphi (f_ i)} u_\varphi (U_ i) \]

in the category $\mathcal{C}_{total}$. This makes sense as by our definition of morphisms of sites, the required fibre products in $\mathcal{C}_ m$ exist since $u_\varphi $ transforms coverings into coverings. The same reasoning implies the claim (details omitted). Thus we see that the collection of coverings is stable under base change. The other axioms of a site are immediate. $\square$

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