The Stacks project

Lemma 78.16.1. Let $B \to S$ as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The algebraic space $G$ defined by the cartesian square

\[ \xymatrix{ G \ar[r] \ar[d] & R \ar[d]^{j = (t, s)} \\ U \ar[r]^-{\Delta } & U \times _ B U } \]

is a group algebraic space over $U$ with composition law $m$ induced by the composition law $c$.

Proof. This is true because in a groupoid category the set of self maps of any object forms a group. $\square$

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