Partially multiplicative quandles and simplicial Hurwitz spaces

We introduce partially multiplicative quandles (PMQ), a generalisation of both partial monoids and quandles. We set up the basic theory of PMQs, focusing on the properties of free PMQs and complete PMQs. For a PMQ $\mathcal{Q}$ with completion $\hat{\mathcal{Q}}$, we introduce the category of $\hat{\mathcal{Q}}$-crossed topological spaces, and define the Hurwitz space $\mathrm{Hur}^{\Delta}(\mathcal{Q})$: it is a $\hat{\mathcal{Q}}$-crossed space, and it parametrises $\mathcal{Q}$-branched coverings of the plane. The definition recovers classical Hurwitz spaces when $\mathcal{Q}$ is a discrete group $G$. Finally, we analyse the class of PMQs $\mathfrak{S}_d^{\mathrm{geo}}$ arising from the symmetric groups $\mathfrak{S}_d$, and we compute their enveloping groups and their PMQ completions.