The Stacks project

Lemma 83.5.19. Let $B \to S$ as in Section 83.2. Let $j : R \to U \times _ B U$ be a set-theoretic pre-equivalence relation. A morphism $\phi : U \to X$ is an orbit space for $R$ if and only if

1. $\phi \circ s = \phi \circ t$, i.e., $\phi$ is invariant,

2. the induced morphism $(t, s) : R \to U \times _ X U$ is surjective, and

3. the morphism $\phi : U \to X$ is surjective.

This characterization applies for example if $j$ is a pre-equivalence relation, or comes from a groupoid in algebraic spaces over $B$, or comes from the action of a group algebraic space over $B$ on $U$.