The Stacks project

Definition 107.3.4. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $T \subset |\mathcal{X}|$ be an irreducible component. The multiplicity of $T$ in $\mathcal{X}$ is defined as $m_{T, \mathcal{X}} = m_{T', U}$ where $f : U \to \mathcal{X}$ is a smooth morphism from a scheme and $T' \subset |U|$ is an irreducible component with $f(T') \subset T$.