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

The site $\mathcal{C}/U$ is called the

*localization of the site $\mathcal{C}$ at the object $U$*.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*.The functor $j_{U*}$ is called the

*direct image functor*.For a sheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_ U^{-1}\mathcal{F}$ is called the

*restriction of $\mathcal{F}$ to $\mathcal{C}/U$*.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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