The Stacks project

Lemma 61.12.19. Let $S$ be a scheme contained in a big pro-étale site $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. A sheaf $\mathcal{F}$ on the big pro-étale site $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is given by the following data:

  1. for every $T/S \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ a sheaf $\mathcal{F}_ T$ on $T_{pro\text{-}\acute{e}tale}$,

  2. for every $f : T' \to T$ in $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ a map $c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$.

These data are subject to the following conditions:

  1. given any $f : T' \to T$ and $g : T'' \to T'$ in $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ the composition $c_ g \circ g_{small}^{-1}c_ f$ is equal to $c_{f \circ g}$, and

  2. if $f : T' \to T$ in $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is weakly étale then $c_ f$ is an isomorphism.

Proof. Identical to the proof of Topologies, Lemma 34.4.20. $\square$

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