The Stacks project

Definition 61.12.7. A big pro-étale site is any site $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ as in Sites, Definition 7.6.2 constructed as follows:

1. Choose any set of schemes $S_0$, and any set of pro-étale coverings $\text{Cov}_0$ among these schemes.

2. Change the function $Bound$ of Sets, Equation (3.9.1.1) into

   $$Bound(\kappa ) = \max \{ \kappa ^{2^{2^{2^\kappa }}}, \kappa ^{\aleph _0}, \kappa ^+\} .$$

3. As underlying category take any category $\mathit{Sch}_\alpha$ constructed as in Sets, Lemma 3.9.2 starting with the set $S_0$ and the function $Bound$.

4. Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha$ and the class of pro-étale coverings, and the set $\text{Cov}_0$ chosen above.