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 {Q} -crossed 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 Sdᵍᵉᵒ arising from the symmetric groups Sd , and we compute their enveloping groups and their PMQ completions.