The Stacks project

Lemma 83.5.16. 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. If $j$ is a pre-equivalence relation, then $j$ is a set-theoretic pre-equivalence relation. This holds in particular when $j$ comes from a groupoid in algebraic spaces, or from an action of a group algebraic space on $U$.

  2. If $j$ is an equivalence relation, then $j$ is a set-theoretic equivalence relation.

Proof. Omitted. $\square$

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