Definition 7.25.1. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

1. The site $\mathcal{C}/U$ is called the localization of the site $\mathcal{C}$ at the object $U$.

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

3. The functor $j_{U*}$ is called the direct image functor.

4. For a sheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_ U^{-1}\mathcal{F}$ is called the restriction of $\mathcal{F}$ to $\mathcal{C}/U$.

5. For a sheaf $\mathcal{G}$ on $\mathcal{C}/U$ the sheaf $j_{U!}\mathcal{G}$ is called the extension of $\mathcal{G}$ by the empty set.

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