Definition 59.21.1. Let $S$ be a scheme.

The

*étale topos*, or the*small étale topos*of $S$ is the category $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$ of sheaves of sets on the small étale site of $S$.The

*Zariski topos*, or the*small Zariski topos*of $S$ is the category $\mathop{\mathit{Sh}}\nolimits (S_{Zar})$ of sheaves of sets on the small Zariski site of $S$.For $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $ a

*big $\tau $-topos*is the category of sheaves of set on a big $\tau $-topos of $S$.

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