  We describe an $A_\infty$-quasi-equivalence of dg-categories between the
first authors' $\mathcal{P}_{\mathcal{A}}$ ---the category of category of
prefect $A^0$-modules with flat $\Z$-connection, corresponding to the de Rham
dga $\mathcal{A}$ of a compact manifold $M$--- and the dg-category of
\emph{infinity-local systems} on $M$ ---homotopy coherent representations of
the smooth singular simplicial set of $M$, $\Pinf$. We understand this as a
generalization of the Riemann--Hilbert correspondence to $\Z$-connections
($\Z$-graded superconnections in some circles). In one formulation an
infinity-local system is simplicial map between the simplicial sets
${\pi}_{\infty}M$ and a repackaging of the dg-category of cochain complexes by
virtue of the simplicial nerve and Dold-Kan. This theory makes crucial use of
Igusa's notion of higher holonomy transport for $\Z$-connections which is a
derivative of Chen's main idea of generalized holonomy.
