The Stacks project

Lemma 85.3.2. Let $\mathcal{C}$ be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. With notation as above, assume the functors $u_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$ have property $P$ of Sites, Remark 7.20.5. Then we can 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 $f : u_\varphi (U) \to V$ of $\mathcal{C}_ m$, 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 $(\varphi \circ \psi , g \circ u_\psi (f))$. This uses that $u_\psi \circ u_\varphi = u_{\varphi \circ \psi }$.

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

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

in the category $\mathcal{C}_{total}$ where the right hand side is the object of $\mathcal{C}_ m$ defined in Sites, Remark 7.20.5 which exists by property $P$. Compatibility of this type of fibre product with compositions of functors implies the claim (details omitted). Since the family $\{ W \times _{g, U, f_ i} U_ i \to W\} $ is a covering of $\mathcal{C}_ m$ by property $P$ 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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