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 Q with completion Q, we introduce the category of Qcrossed topological spaces, and define the Hurwitz space HurΔ (Q): it is a Q-crossed space, and it parametrises Q-branched coverings of the plane. The definition recovers classical Hurwitz spaces when Q is a discrete group G. Finally, we analyse the class of PMQs Sgeo arising from the symmetric groups Sd, and we compute their enveloping groups and their PMQ d completions.