For a Lie group G, we consider the space of smooth singular chains C(G), which is a differential graded Hopf algebra. We show that the category of sufficiently local modules over C(G) can be described infinitesimally, as the category of representations of a dg-Lie algebra which is universal for the Cartan relations. If G is compact and simply connected, the equivalence of categories can be promoted to an A-infinity equivalence of dg-categories.

This result allows for a categorification of the Chern-Weil construction of characteristic classes. The categories mentioned above are quasi-equivalent to the category of infinity-local systems on the classifying space of G. The Chern-Weil homomorphism can be promoted to a Chern-Weil functor taking values in the dg-category of infinity-local systems. The Chern-Weil homomorphism is then recovered by applying the functor to the endomorphisms of the unit object. If time permits, I will discuss further questions which are part of ongoing projects.

The talk is based on joint works with A. Quintero and S. Pineda, and work in progress with M. Rivera.