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We introduce the notion of partially multiplicative quandle (PMQ),generalising both classical notions of partial monoid and of quandle. We set upthe basic theory of PMQs, in analogy with classical group theory. For acomplete PMQ $\hat{\mathcal{Q}}$, and for a braided closed monoidal category$\mathbf{A}$, we introduce a braided monoidal category of$\hat{\mathcal{Q}}$-crossed object in $\mathbf{A}$, in analogy with theclassical notion of $G$-crossed objects, depending on a discrete group $G$. Weuse the previous to give a definition of Hurwitz spaces$\mathrm{Hur}^{\Delta}(\mathcal{Q})$ of configurations of points in the planewith monodromies in a PMQ: the definition has a simplicial flavour, and givesrise to topological spaces with a natural cell stratification; it recovers theclassical definition of Hurwitz spaces in the case of a discrete group $G$considered as a PMQ with trivial multiplication. We give a criterion to detectwhether a PMQ $\mathcal{Q}$ is Poincare, i.e. whether all connected componentsof $\mathrm{Hur}^{\Delta}(\mathcal{Q})$ are topological manifolds. Finally, weanalyse thoroughly the PMQ $\mathfrak{S}_d^{\mathrm{geo}}$, for $d\ge2$, and wecompute its enveloping group and its completion as PMQ.

Partially multiplicative quandles and simplicial Hurwitz spaces