The higher Riemann–Hilbert correspondence

We construct an A∞A∞-quasi-equivalence of dg-categories between PAPA — the category of prefect A0A0-modules with flat ZZ-connection, associated to the de Rham dga A•A• of a compact manifold M — and the dg-category of infinity-local systems on M — homotopy-coherent representations of the smooth singular simplicial set of M  . We understand this as a generalization of the classical Riemann–Hilbert correspondence to ZZ-connections (ZZ-graded superconnections in some circles). This theory makes crucial use of Igusaʼs notion of higher holonomy transport for ZZ-connections which is a derivative of Chenʼs idea of generalized holonomy.