The Stacks project

Definition 96.4.3. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.

1. We say $\mathcal{F}$ is a Zariski sheaf, or a sheaf for the Zariski topology if $\mathcal{F}$ is a sheaf on the associated Zariski site $\mathcal{X}_{Zar}$.

2. We say $\mathcal{F}$ is an étale sheaf, or a sheaf for the étale topology if $\mathcal{F}$ is a sheaf on the associated étale site $\mathcal{X}_{\acute{e}tale}$.

3. We say $\mathcal{F}$ is a smooth sheaf, or a sheaf for the smooth topology if $\mathcal{F}$ is a sheaf on the associated smooth site $\mathcal{X}_{smooth}$.

4. We say $\mathcal{F}$ is a syntomic sheaf, or a sheaf for the syntomic topology if $\mathcal{F}$ is a sheaf on the associated syntomic site $\mathcal{X}_{syntomic}$.

5. We say $\mathcal{F}$ is an fppf sheaf, or a sheaf, or a sheaf for the fppf topology if $\mathcal{F}$ is a sheaf on the associated fppf site $\mathcal{X}_{fppf}$.

A morphism of sheaves is just a morphism of presheaves. We denote these categories of sheaves $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{Zar})$, $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$, $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{smooth})$, $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{syntomic})$, and $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$.