Paper by S. Holtrop and J. Pedregal.<\/p>\n
Abstract:\u00a0We prove the existence of a Riemann–Hilbert \\(A_\\infty\\)-functor from the curved dg-category of \\(\\mathscr{A}\\)-connections on graded vector bundles, where \\(\\mathscr{A}\\) is a higher Lie algebroid, to the curved dg-category of \\(\\Pi_\\infty(\\mathscr{A})\\)-connections on grded vector bundles, where \\(\\Pi_\\infty(\\mathscr{A})\\) is the homotopy groupoid of \\(\\mathscr{A}\\). When restricted to the sub dg-categories of flat objects, it induces an \\(A_\\infty\\)-functor from the dg-category of representations up to homotopy of \\(\\mathscr{A}\\) to the dg-category of representations up to homotopy of \\(\\Pi_\\infty(\\mathscr{A})\\), which recovers the one from arXiv:1011.4693 and arXiv:0908.2843 when \\(\\mathscr{A}\\) is a Lie algebroid. The key ingredient for defining such functors is a non-abelian Stokes’s theorem for endomorphism-valued differential forms. Furthermore, for a graded vector bundle \\(E\\), we introduce the notion of its higher general linear groupoid in such a way that \\(G\\)-connections on \\(E\\), for a higher Lie groupoid \\(G\\), become smooth functors from \\(G\\) to the higher general linear groupoid of \\(E\\). We combine this with the Riemann–Hilbert functor to define higher parallel transport functors for \\(\\mathscr{A}\\)-connections. Finally, we use these functors to define the higher holonomy groups of \\(\\mathscr{A}\\)-connections and endow them with the structure of augmented simplicial Lie groups.<\/p>\n