The Stacks project

Lemma 79.14.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. Assume the morphisms $s, t$ are separated and locally of finite type. There exists a canonical morphism

\[ (U', Z_{univ}, s', t', c', e', i') \longrightarrow (U, R, s, t, c, e, i) \]

of groupoids in algebraic spaces over $B$ where

  1. $g : U' \to U$ is identified with $(R_ s/U, e)_{fin} \to U$, and

  2. $Z_{univ} \subset R \times _{s, U, g} U'$ is the universal open (and closed) subspace finite over $U'$ which contains the base change of the unit $e$.

Proof. See discussion above. $\square$

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