The Stacks project

Definition 61.12.1. Let $T$ be a scheme. A pro-étale covering of $T$ is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ of schemes such that each $f_ i$ is weakly-étale and such that for every affine open $U \subset T$ there exists $n \geq 0$, a map $a : \{ 1, \ldots , n\} \to I$ and affine opens $V_ j \subset T_{a(j)}$, $j = 1, \ldots , n$ with $\bigcup _{j = 1}^ n f_{a(j)}(V_ j) = U$.