Given a couple of subspaces Y⊂X\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Y}}\subset {\mathcal {X}}$$\end{document} of the complex plane C\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}$$\end{document} satisfying some mild conditions (a “nice couple”), and given a PMQ-pair (Q,G)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathcal {Q}},G)$$\end{document}, consisting of a partially multiplicative quandle (PMQ) Q\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Q}}$$\end{document} and a group G, we introduce a “Hurwitz–Ran” space Hur(X,Y;Q,G)\documentclass[12pt]{minimal}
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\begin{document}$$\text {Hur}({\mathcal {X}},{\mathcal {Y}};{\mathcal {Q}},G)$$\end{document}, containing configurations of points in X\Y\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {X}}\setminus {\mathcal {Y}}$$\end{document} and in Y\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Y}}$$\end{document} with monodromies in Q\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Q}}$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Q}}$$\end{document} we prove a homeomorphism between Hur((0,1)2;Q+)\documentclass[12pt]{minimal}
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\begin{document}$$\text {Hur} ((0,1)^2;{\mathcal {Q}}_+)$$\end{document} and the simplicial Hurwitz space HurΔ(Q)\documentclass[12pt]{minimal}
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\begin{document}$$\text {Hur} ^{\Delta }({\mathcal {Q}})$$\end{document}, introduced in previous work of the author: this provides in particular Hur((0,1)2;Q+)\documentclass[12pt]{minimal}
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\begin{document}$$\text {Hur} ((0,1)^2;{\mathcal {Q}}_+)$$\end{document} with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.
