Model Categories and the Higher Riemann-Hilbert Correspondence

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.