The Stacks project

Lemma 105.12.2. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces with $s, t : R \to U$ flat and locally of finite presentation. Consider the algebraic stack $\mathcal{X} = [U/R]$. Given an algebraic space $Y$ there is a $1$-to-$1$ correspondence between morphisms $f : \mathcal{X} \to Y$ and $R$-invariant morphisms $\phi : U \to Y$.

Proof. Criteria for Representability, Theorem 96.17.2 tells us $\mathcal{X}$ is an algebraic stack. Given a morphism $f : \mathcal{X} \to Y$ we let $\phi : U \to Y$ be the composition $U \to \mathcal{X} \to Y$. Since $R = U \times _\mathcal {X} U$ (Groupoids in Spaces, Lemma 77.22.2) it is immediate that $\phi $ is $R$-invariant. Conversely, if $\phi : U \to Y$ is an $R$-invariant morphism towards an algebraic space, we obtain a morphism $f : \mathcal{X} \to Y$ by Groupoids in Spaces, Lemma 77.23.2. You can also construct $f$ from $\phi $ using the explicit description of the quotient stack in Groupoids in Spaces, Lemma 77.24.1. $\square$


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