The Stacks project

Lemma 85.21.5. Let $U$ be a simplicial object of a site $\mathcal{C}$ with fibre products.

  1. $\mathcal{C}/U$ has the structure of a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites,

  2. the construction of Lemma 85.3.1 applied to the structure in (1) reproduces the site $(\mathcal{C}/U)_{total}$ above,

  3. if $a : U \to X$ is an augmentation, then $a_0 : \mathcal{C}/U_0 \to \mathcal{C}/X$ is an augmentation as in Remark 85.4.1 part (A) and gives the same morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/U)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$ as the one above.

Proof. Given a morphism of objects $V \to W$ of $\mathcal{C}$ the localization morphism $j : \mathcal{C}/V \to \mathcal{C}/W$ is a left adjoint to the base change functor $\mathcal{C}/W \to \mathcal{C}/V$. The base change functor is continuous and induces the same morphism of topoi as $j$. See Sites, Lemma 7.27.3. This proves (1).

Part (2) holds because a morphism $V/U_ n \to W/U_ m$ of the category constructed in Lemma 85.3.1 is a morphism $V \to W \times _{U_ m, f_\varphi } U_ n$ over $U_ n$ which is the same thing as a morphism $f : V \to W$ over the morphism $f_\varphi : U_ n \to U_ m$, i.e., the same thing as a morphism in the category $(\mathcal{C}/U)_{total}$ defined above. Equality of sets of coverings is immediate from the definition.

We omit the proof of (3). $\square$

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