A Riemann--Hilbert correspondence for infinity local systems

2009/08/20 by Jonathan Block, Aaron Smith, Block, Jonathan +1
Mathematics · #35Q15 #53C29 #57R15 #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0908.2843

openalex publication_date 2009/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe an A_∞-quasi-equivalence of dg-categories between the first authors' P_A ---the category of category of prefect A⁰-modules with flat \Z-connection, corresponding to the de Rham dga 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, \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 π_∞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.

Citations

• Homotopical and Higher Categorical Structures in Algebraic Geometry
• Enriched model categories and an application to additive endomorphism spectra

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