I will describe the structure of the category whose morphisms are riemann surfaces endowed with a principal flat connection for a fixed compact connected structure group. We will relate this space to a suitable stabilization (in genus) of the universal moduli space of riemann surfaces endowed with a holomorphic bundle. The proof involves classical results of Atiyah-Bott along with results of Madsen-Cohen in relation to the parametrized Mumford conjecture. This is joint work with Ralph Cohen and Soren Galatius.