Definition 100.10.4. Let $\mathcal{X}$ be an algebraic stack. Let $Z \subset |\mathcal{X}|$ be a closed subset. An algebraic stack structure on $Z$ is given by a closed substack $\mathcal{Z}$ of $\mathcal{X}$ with $|\mathcal{Z}|$ equal to $Z$. The reduced induced algebraic stack structure on $Z$ is the one constructed in Lemma 100.10.1. The reduction $\mathcal{X}_{red}$ of $\mathcal{X}$ is the reduced induced algebraic stack structure on $|\mathcal{X}|$.
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