Lemma 99.6.2. 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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