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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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