Cocycles in Local Higher Category Theory

TL;DR

This paper develops a new model structure for presheaves of enriched categories, enabling the application of cocycle categories to study homotopy classes and recover classical non-abelian cohomology in a local higher category setting.

Contribution

It introduces a local Bergner model structure on presheaves of enriched categories, connecting it to existing models and applying cocycle theory to analyze homotopy classes.

Findings

Established a right proper model structure on presheaves of enriched categories.

Connected the new model to Joyal and Rezk local models via Quillen equivalences.

Applied cocycle categories to recover classical non-abelian H^1 results.

Abstract

We develop a model structure on presheaves of small simplicially enriched categories on a site $\mathscr{C}$, for which the weak equivalences are 'stalkwise' weak equivalences for the Bergner model structure. This model structure is right proper, and can be connected via a zig-zag of Quillen equivalences to local analogues of the Joyal and Rezk model structures. Because the local Bergner model structure is right proper, we can apply Jardine's cocycle categories to study its homotopy category. As an application of the cocycle theory, we describe the maps $[*, X]$ in the homotopy category of the Jardine model structure as the path components of a category of torsors, where $X$ is a presheaf of Kan complexes. In the case where $X$ is obtained by applying the nerve construction sectionwise to a sheaf of groups, this description recovers classical non-abelian $H^{1}$.