Definition 99.13.1. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.

The

*number of geometric branches of $\mathcal{X}$ at $x$*is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma 99.7.4 hold for $\mathcal{P}_ n$ defined above, or else $\infty $.We say $\mathcal{X}$ is

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

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