Remark 34.11.1. Take any category $\mathit{Sch}_\alpha$ constructed as in Sets, Lemma 3.9.2 starting with the set of schemes $\{ X, Y, S\}$. Choose any set of coverings $\text{Cov}_{fppf}$ on $\mathit{Sch}_\alpha$ as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha$ and the class of fppf coverings. Let $\mathit{Sch}_{fppf}$ denote the big fppf site so obtained. Next, for $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic\}$ let $\mathit{Sch}_\tau$ have the same underlying category as $\mathit{Sch}_{fppf}$ with coverings $\text{Cov}_\tau \subset \text{Cov}_{fppf}$ simply the subset of $\tau$-coverings. It is straightforward to check that this gives rise to a big site $\mathit{Sch}_\tau$.

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