Lemma 100.6.2. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent:
$\mathcal{X}$ is quasi-compact,
there exists a surjective smooth morphism $U \to \mathcal{X}$ with $U$ an affine scheme,
there exists a surjective smooth morphism $U \to \mathcal{X}$ with $U$ a quasi-compact scheme,
there exists a surjective smooth morphism $U \to \mathcal{X}$ with $U$ a quasi-compact algebraic space, and
there exists a surjective morphism $\mathcal{U} \to \mathcal{X}$ of algebraic stacks such that $\mathcal{U}$ is quasi-compact.
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