The Stacks project

Definition 59.21.1. Let $S$ be a scheme.

  1. 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 $S$.

  2. 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 $S$.

  3. 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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