For a partially multiplicative quandle (PMQ) $\mathcal{ Q}$ we consider the topological monoid $\mathring{ \mathrm{ HM}}(\mathcal{ Q})$ of Hurwitz spaces of configurations in the plane with local monodromies in $\mathcal{ Q}$. We compute the group completion of $\mathring{ \mathrm{ HM}}(\mathcal{ Q})$: it is the product of the (discrete) enveloping group $\mathcal{ G}(\mathcal{ Q})$ with a component of the double loop space of the relative Hurwitz space $\mathrm{ Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{ Q},G)_{ 1\!\!\,1}$; here $G$ is any group giving rise, together with $\mathcal{ Q}$, to a PMQ-group pair. Assuming further that $\mathcal{ Q}$ is finite and rationally Poincare and that $G$ is finite, we compute the rational cohomology ring of $\mathrm{ Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{ Q},G)_{ 1\!\!\,1}$.
Categories: math.AT