The Stacks project

Definition 96.24.1. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. The associated affine site is the full subcategory $\mathcal{X}_{affine}$ of $\mathcal{X}$ whose objects are those $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over a scheme $U$ such that $U$ is affine. The topology on $\mathcal{X}_{affine}$ will be the chaotic one, i.e., such that sheaves on $\mathcal{X}_{affine}$ are the same as presheaves.

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