Modal Fracture of Higher Groups
David Jaz Myers
TL;DR
The paper develops a cohesive, modal framework for higher groups within homotopy type theory, constructing a modal fracture hexagon that splits a higher group into discrete, infinitesimal, and contractible components via the universal $\infty$-cover and the infinitesimal remainder. It then specializes to ordinary differential cohomology on smooth manifolds, using circle $k$-gerbes with connection and long exact form-classifier sequences to derive a differential cohomology hexagon that mirrors the classical character diagram while highlighting necessary obstructions in general settings. A central theme is the interplay of adjoint modalities $\mathop{\mathrm{\textesh}}$ and $\flat$, enabling a synthetic, internal treatment of differential data through classifying stacks and curvature maps. The work also outlines abstract and combinatorial extensions of differential cohomology, including contractible infinitesimal resolutions and a symmetric simplicial cohesion approach, suggesting broad applicability beyond smooth manifolds. Overall, the approach provides a concise, modular language for higher differential geometry with potential computational and foundational benefits.
Abstract
In this paper, we examine the modal aspects of higher groups in Shulman's Cohesive Homotopy Type Theory. We show that every higher group sits within a modal fracture hexagon which renders it into its discrete, infinitesimal, and contractible components. This gives an unstable and synthetic construction of Schreiber's differential cohomology hexagon. As an example of this modal fracture hexagon, we recover the character diagram characterizing ordinary differential cohomology by its relation to its underlying integral cohomology and differential form data, although there is a subtle obstruction to generalizing the usual hexagon to higher types. Assuming the existence of a long exact sequence of differential form classifiers, we construct the classifiers for circle k-gerbes with connection and describe their modal fracture hexagon.