Cocycles in Local Higher Category Theory

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 $H^{1}$.