Higher Riemann-Hilbert Correspondence and Descent for Regular Foliations

Abstract

Let ${\mathcal F}\subset TM$ be a regular foliation. We construct an $A_\infty$ integration functor from globally bounded finite-rank leafwise cohesive modules with locally constant fiber-cohomology rank to global plot-smooth infinity-local systems with the same regularity. The target retains the global higher-transport objects and sheafifies their raw Block--Smith Hom presheaves. The construction combines the Gugenheim--Arias Abad--Sch\"atz iterated-integral map with the Lie-algebroid higher-holonomy formulas. For every $M$, integration is quasi-fully faithful. On each star-shaped foliated box, a strictly unital simplicial prism gives an explicit raw local inverse. If $M$ is compact, effective descent for cohesive modules and \v{C}ech descent for the target Hom sheaves promote the local comparison to an $A_\infty$ quasi-equivalence on the regular full subcategories; compact proper submersions give a distinguished class of examples. At every fixed finite amplitude, the local comparison also induces an equivalence of $\operatorname{Cat}_\infty$-valued hypersheafifications. No global raw-Hom comparison or analogous correspondence for singular foliations or general $L_\infty$-algebroids is asserted.