Lemma 106.12.2. 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 97.17.2 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 78.22.2) 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 78.23.2. You can also construct $f$ from $\phi $ using the explicit description of the quotient stack in Groupoids in Spaces, Lemma 78.24.1.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)