Modal Fracture of Higher Groups

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.

Commuting Cohesions

“…In addition to allowing us to formalize the theorem about Čech nerves of open covers as Theorem 6.1.5, our type theory will be able to handle the equivariant differential cohesion used by Sati and Schreiber in their Proper orbifold cohomology [43], as well as the nested focuses of Schreiber's supergeometric solid cohesion [44]. This extends the work of Cherubini [17] and the first author [34,35,36] of giving synthetic accounts of the constructions of Schreiber [44] and Sati-Schreiber [43].…”