I will discuss joint work with Jonathan Block, in which we prove a higher-homotopical version of the smooth Riemann-Hilbert correspondence. Specifically, this is an (A_\infty, quasi-) equivalence between two geometric dg-categories naturally associated to a smooth, compact manifold. The objects of one consist of Z-graded modules over the functions on the manifold with a kind of homotopy-flat Z-graded connection. The other consists of "infinity-local systems" which, from a certain perspective can be regarded as linear representations of the singular simplicial set of the manifold. The functor is constructed explicitly as a holonomy calculation using the theory of iterated integrals.