Hurwitz-Ran spaces

Abstract

Given a couple of subspaces $\mathcal{Y}\subset\mathcal{X}$ of the complex plane $\mathbb{C}$ satisfying some mild conditions (a ``nice couple''), and given a PMQ-pair $(\mathcal{Q},G)$, consisting of a partially multiplicative quandle (PMQ) $\mathcal{Q}$ and a group $G$, we introduce a ``Hurwitz-Ran'' space $\mathrm{Hur}(\mathcal{X},\mathcal{Y};\mathcal{Q},G)$, containing configurations of points in $\mathcal{X}\setminus\mathcal{Y}$ and in $\mathcal{Y}$ with monodromies in $\mathcal{Q}$ and in $G$, respectively. We further introduce a notion of morphisms between nice couples, and prove that Hurwitz-Ran spaces are functorial both in the nice couple and in the PMQ-group pair. For a locally finite PMQ $\mathcal{Q}$ we prove a homeomorphism between $\mathrm{Hur}((0,1)^2;\mathcal{Q}_+)$ and the simplicial Hurwitz space $\mathrm{Hur}^{\Delta}(\mathcal{Q})$, introduced in previous work of the author: this provides in particular $\mathrm{Hur}((0,1)^2;\mathcal{Q}_+)$ with a cell stratification in the spirit of Fox-Neuwirth and Fuchs.