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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