Hurwitz–Ran spaces

Given a couple of subspaces Y ⊂ X of the complex plane C satisfying some mild conditions (a “nice couple”), and given a PMQ-pair ( Q , G ) , consisting of a partially multiplicative quandle (PMQ) Q and a group G , we introduce a “Hurwitz–Ran” space Hur ( X , Y ; Q , G ) , containing configurations of points in X \ Y and in Y with monodromies in 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 Q we prove a homeomorphism between Hur ( ( 0 , 1 ) 2 ; Q + ) and the simplicial Hurwitz space Hur Δ ( Q ) , introduced in previous work of the author: this provides in particular Hur ( ( 0 , 1 ) 2 ; Q + ) with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.