Definition 94.4.1. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.

The

*associated Zariski site*, denoted $\mathcal{X}_{Zar}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{Zar}$.The

*associated étale site*, denoted $\mathcal{X}_{\acute{e}tale}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{\acute{e}tale}$.The

*associated smooth site*, denoted $\mathcal{X}_{smooth}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{smooth}$.The

*associated syntomic site*, denoted $\mathcal{X}_{syntomic}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{syntomic}$.The

*associated fppf site*, denoted $\mathcal{X}_{fppf}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{fppf}$.

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