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 text {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 text {Hur} ((0,1)^2;{mathcal {Q}}_+) and the simplicial Hurwitz space text {Hur} ^{Delta }({mathcal {Q}}), introduced in previous work of the author: this provides in particular text {Hur} ((0,1)^2;{mathcal {Q}}_+) with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.
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