Definition 104.5.1. We say $\mathcal{M}_{1, 1}$ is an algebraic stack if and only if

1. We have descent for objects for the étale topology on $\mathit{Sch}$.

2. The key fact holds.

3. there exists a surjective and smooth morphism $S \to \mathcal{M}_{1, 1}$.

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