The Stacks project

Definition 83.5.13. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation over $B$.

1. We say $j$ is a set-theoretic pre-equivalence relation if for all algebraically closed fields $k$ over $B$ the relation $\sim _ R$ on $U(k)$ defined by

   \[ \overline{u} \sim _ R \overline{u}' \Leftrightarrow \begin{matrix} \exists \text{ field extension }K/k, \ \exists \ r \in R(K), \\ s(r) = \overline{u}, \ t(r) = \overline{u}' \end{matrix} \]

   is an equivalence relation.

2. We say $j$ is a set-theoretic equivalence relation if $j$ is universally injective and a set-theoretic pre-equivalence relation.