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The Stacks project

Lemma 85.16.1. Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\text{SR}(\mathcal{C})$. The localization functor $j_0 : \mathcal{C}/K_0 \to \mathcal{C}$ defines an augmentation $a_0 : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_0) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, as in case (B) of Remark 85.4.1. The corresponding morphisms of topoi

\[ a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}),\quad a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]

of Lemma 85.4.2 are equal to the morphisms of topoi associated to the continuous and cocontinuous localization functors $j_ n : \mathcal{C}/K_ n \to \mathcal{C}$ and $j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}$.

Proof. This is immediate from working through the definitions. See in particular the footnote in the proof of Lemma 85.4.2 for the relationship between $a$ and $j_{total}$. $\square$

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