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

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

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

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

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

5. 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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