Definition 59.20.2. Let $S$ be a scheme. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, \linebreak[0] Zariski\}$.

1. A big $\tau$-site of $S$ is any of the sites $(\mathit{Sch}/S)_\tau$ constructed as explained above and in more detail in Topologies, Definitions 34.7.8, 34.6.8, 34.5.8, 34.4.8, and 34.3.7.

2. If $\tau \in \{ {\acute{e}tale}, Zariski\}$, then the small $\tau$-site of $S$ is the full subcategory $S_\tau$ of $(\mathit{Sch}/S)_\tau$ whose objects are schemes $T$ over $S$ whose structure morphism $T \to S$ is étale, resp. an open immersion. A covering in $S_\tau$ is a covering $\{ U_ i \to U\}$ in $(\mathit{Sch}/S)_\tau$ such that $U$ is an object of $S_\tau$.

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