The Stacks project

Definition 83.5.8. 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 $\phi : U \to X$ is set-theoretically $R$-invariant if and only if the map $U(k) \to X(k)$ equalizes the two maps $s, t : R(k) \to U(k)$ for every algebraically closed field $k$ over $B$.

  2. We say $\phi : U \to X$ separates orbits, or separates $R$-orbits if it is set-theoretically $R$-invariant and $\phi (\overline{u}) = \phi (\overline{u}')$ in $X(k)$ implies that $\overline{u}, \overline{u}' \in U(k)$ are in the same orbit for every algebraically closed field $k$ over $B$.

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