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 $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$; here $G$ is any group giving rise, together with ${\mathcal {Q}}$, to a PMQ–group pair. Under the additional assumption that ${\mathcal {Q}}$ is finite and rationally Poincaré and that $G$ is finite, we compute the rational cohomology ring of $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$.

We prove the analog for the $K$-theory of forms of the $Q=+$ theorem in algebraic $K$-theory. That is, we show that the $K$-theory of forms defined in terms of an $S_{\bullet }$-construction is a group completion of the category of quadratic spaces for form categories in which all admissible exact sequences split. This applies for instance to quadratic and hermitian forms defined with respect to a form parameter.