Model Categories and the Higher Riemann-Hilbert Correspondence

TL;DR

This work provides a homotopical framework for higher Riemann–Hilbert phenomena by endowing dg module and presheaf categories with model structures of second kind and performing right Bousfield localizations to realize Quillen equivalences. It proves zig-zags of dg Quillen equivalences between $\Omega(X)$-Mod (and $A^{0*}(X)$-Mod) and dg presheaves on $X$, including a Dolbeault analogue, and it identifies the localized homotopy categories with categories of locally constant or hypersheaf objects, thereby generalizing RH-type correspondences to higher, infinity-local systems. Extending further, it develops the singular analogue via contramodules over the singular cochain algebra $C^{*}(X)$, establishing a contraderived- presheaf equivalence and connecting to the co-contra correspondence. Collectively, these results unify de Rham, Dolbeault, and singular perspectives under a coherent homotopical formalism, yielding a robust higher RH correspondence with concrete model-categorical realizations and explicit triangulated equivalences to categories of infinity-local systems.

Abstract

We construct a new model structure on the category of dg presheaves over a topological space $X$, obtained through the right Bousfield localization of the local projective model structure. The motivation for this construction arises from the study of the homotopy theory underlying higher Riemann-Hilbert correspondence theorems, as developed by Chuang, Holstein, and Lazarev. Let $X$ be a smooth manifold. We prove the existence of a zig-zag of Quillen equivalences between the category of dg modules over the de Rham algebra and the category of dg presheaves of vector spaces over $X$. In the case where $X$ is a complex manifold, we obtain an analogous result, where the de Rham algebra is replaced by the Dolbeault algebra. In both settings, we equip the categories of modules with model structures of the second kind, whose homotopy categories are, in general, finer invariants than those given by quasi-isomorphisms. Finally, we introduce a singular analogue of this equivalence, stating it as a zig-zag of Quillen equivalences between the category of dg contramodules over the singular cochain algebra $C^{*}(X)$ and dg presheaves. At the level of homotopy categories, this establishes an equivalence between the contraderived category of $C^{*}(X)$-contramodules and the homotopy category of dg presheaves.

Theorems & Definitions (123)

• Theorem
• Theorem
• Theorem
• Definition 2.1.1
• Definition 2.1.2
• Definition 2.1.3
• Remark 2.1.4
• Theorem 2.1.5
• Definition 2.1.6
• Theorem 2.1.7