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--Schätz 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 Čech 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 ${\mathrm{Cat}}_{\infty }$-valued hypersheafifications. No global raw-Hom comparison or analogous correspondence for singular foliations or general $L_{\infty }$-algebroids is asserted.

52 pages, 3 commutative diagrams. Substantially revised and expanded from v1: strengthened foundations and proofs, fuller treatments of higher integration and descent, additional examples. Comments welcome!