Deloopings of Hurwitz spaces

For a partially multiplicative quandle (PMQ) Q we consider the topological monoid HM(Q) of Hurwitz spaces of configurations in the plane with local monodromies in Q. We compute the group completion of HM(Q): it is the product of the (discrete) enveloping group G(Q) with a component of the double loop space of the relative Hurwitz space Hur+([0, 1]2, [0, 1]2;Q,G)1; here G is any group giving rise, together with Q, to a PMQ.group pair. Under the additional assumption that Q is finite and rationally Poincaré and that G is finite, we compute the rational cohomology ring of Hur+([0, 1]2, [0, 1]2;Q,G)1.