Modal Fracture of Higher Groups

TL;DR

This paper explores the modal structure of higher groups in Cohesive Homotopy Type Theory, revealing a fracture hexagon that decomposes them into fundamental components, and applies this to differential cohomology and circle k-gerbes.

Contribution

It introduces a modal fracture hexagon for higher groups, providing a synthetic construction of differential cohomology and classifiers for circle k-gerbes with connections.

Findings

Higher groups fit into a modal fracture hexagon decomposing into discrete, infinitesimal, and contractible parts.

Reconstruction of the differential cohomology character diagram from the fracture hexagon.

Construction of classifiers for circle k-gerbes with connection based on the modal fracture framework.

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.