We will describe how modules over the algebra of singular chains on a Lie group G can be described infinitesimally, in terms of the Lie algebra.

The Cartan relations from differential geometry are the fundamental equations in this correspondence.

I will explain this construction, and then describe how it relates to the Chern-Weiltheory for characteristic classesand to the category of

infinity local systems  on the classifying space BG.

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