The Stacks project

Definition 100.13.1. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.

1. The number of geometric branches of $\mathcal{X}$ at $x$ is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma 100.7.4 hold for $\mathcal{P}_ n$ defined above, or else $\infty$.

2. We say $\mathcal{X}$ is geometrically unibranch at $x$ if the number of geometric branches of $\mathcal{X}$ at $x$ is $1$.