## 59.21 The étale topos

A topos is the category of sheaves of sets on a site, see Sites, Definition 7.15.1. Hence it is customary to refer to the use the phrase “étale topos of a scheme” to refer to the category of sheaves on the small étale site of a scheme. Here is the formal definition.

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 $S$.

Note that the small Zariski topos of $S$ is simply the category of sheaves of sets on the underlying topological space of $S$, see Topologies, Lemma 34.3.12. Whereas the small étale topos does not depend on the choices made in the construction of the small étale site, in general the big topoi do depend on those choices.

It turns out that the big or small étale topos only depends on the full subcategory of $(\mathit{Sch}/S)_{\acute{e}tale}$ or $S_{\acute{e}tale}$ consisting of affines, see Topologies, Lemmas 34.4.11 and 34.4.12. We will use this for example in the proof of the following lemma.

Lemma 59.21.2. Let $S$ be a scheme. The étale topos of $S$ is independent (up to canonical equivalence) of the construction of the small étale site in Definition 59.20.2.

Proof. We have to show, given two big étale sites $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$ containing $S$, then $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}')$ with obvious notation. By Topologies, Lemma 34.12.1 we may assume $\mathit{Sch}_{\acute{e}tale}\subset \mathit{Sch}_{\acute{e}tale}'$. By Sets, Lemma 3.9.9 any affine scheme étale over $S$ is isomorphic to an object of both $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$. Thus the induced functor $S_{affine, {\acute{e}tale}} \to S_{affine, {\acute{e}tale}}'$ is an equivalence. Moreover, it is clear that both this functor and a quasi-inverse map transform standard étale coverings into standard étale coverings. Hence the result follows from Topologies, Lemma 34.4.12. $\square$

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