Hurwitz–Ran spaces

Given a couple of subspaces
of the complex plane
satisfying some mild conditions (a “nice couple”), and given a PMQ-pair
, consisting of a partially multiplicative quandle (PMQ)
and a group G, we introduce a “Hurwitz–Ran” space
, containing configurations of points in
and in
with monodromies in
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
we prove a homeomorphism between
and the simplicial Hurwitz space
, introduced in previous work of the author: this provides in particular
with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.