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 $S$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).