Model Categories and the Higher Riemann-Hilbert Correspondence

TL;DR

This paper develops a new model structure on dg presheaves over a topological space, linking algebraic and geometric categories through Quillen equivalences, and extends these ideas to complex and singular settings.

Contribution

It constructs a novel model structure on dg presheaves and establishes new Quillen equivalences relating dg modules over de Rham and Dolbeault algebras to presheaves, including a singular analogue.

Findings

Established a zig-zag of Quillen equivalences for smooth manifolds.

Extended results to complex manifolds with Dolbeault algebra.

Introduced a singular analogue relating contramodules over cochain algebra to presheaves.

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.