The Stacks project

Definition 96.7.1. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.

1. A presheaf of modules on $\mathcal{X}$ is a presheaf of $\mathcal{O}_\mathcal {X}$-modules. The category of presheaves of modules is denoted $\textit{PMod}(\mathcal{O}_\mathcal {X})$.

2. We say a presheaf of modules $\mathcal{F}$ is an $\mathcal{O}_\mathcal {X}$-module, or more precisely a sheaf of $\mathcal{O}_\mathcal {X}$-modules if $\mathcal{F}$ is an fppf sheaf. The category of $\mathcal{O}_\mathcal {X}$-modules is denoted $\textit{Mod}(\mathcal{O}_\mathcal {X})$.