Lemma 101.46.1. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent

there is a surjective smooth morphism $U \to \mathcal{X}$ where $U$ is a scheme such that every quasi-compact open of $U$ has finitely many irreducible components,

for every scheme $U$ and every smooth morphism $U \to \mathcal{X}$ every quasi-compact open of $U$ has finitely many irreducible components,

for every algebraic space $Y$ and smooth morphism $Y \to \mathcal{X}$ the space $Y$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 67.49.1, and

for every quasi-compact algebraic stack $\mathcal{Y}$ smooth over $\mathcal{X}$ the space $|\mathcal{Y}|$ has finitely many irreducible components.

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