Definition 7.30.4. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$.

1. The topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is called the localization of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ at $\mathcal{F}$.

2. The morphism of topoi $j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of Lemma 7.30.1 is called the localization morphism.

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