  We consider the moduli space $\mathfrak{M}_{g,n}$ of Riemann surfaces of
genus $g\ge0$ with $n\ge1$ ordered and directed marked points. For $d\ge
2g+n-1$ we show that $\mathfrak{M}_{g,n}$ is homotopy equivalent to a component
of the simplicial Hurwitz space
$\mathrm{Hur}^{\Delta}(\mathfrak{S}_d^{\mathrm{geo}})$ associated with the
partially multiplicative quandle $\mathfrak{S}_d^{\mathrm{geo}}$. As an
application, we give a new proof of the Mumford conjecture on the stable
rational cohomology of moduli spaces of Riemann surfaces. We also provide a
combinatorial model for the infinite loop space
$\Omega^{\infty-2}\mathrm{MTSO}(2)$ of Hurwitz flavour.
