The Stacks project

Definition 66.19.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$.

1. An étale neighborhood of $\overline{x}$ of $X$ is a commutative diagram

   $$\xymatrix{ & U \ar[d]^\varphi \\ {\bar x} \ar[r]^{\bar x} \ar[ur]^{\bar u} & X }$$

   where $\varphi$ is an étale morphism of algebraic spaces over $S$. We will use the notation $\varphi : (U, \overline{u}) \to (X, \overline{x})$ to indicate this situation.

2. A morphism of étale neighborhoods $(U, \overline{u}) \to (U', \overline{u}')$ is an $X$-morphism $h : U \to U'$ such that $\overline{u}' = h \circ \overline{u}$.