Differential Cohomology as Diffeological Homotopy Theory

TL;DR

This paper introduces a new approach to differential cohomology using skeletal diffeologies, connecting homotopy theory with differential characters and enabling potential extensions to non-abelian cases.

Contribution

It develops a skeletal diffeological homotopy framework that recovers differential cohomology and relates to existing models, facilitating non-abelian generalizations.

Findings

Reconstruction of differential cohomology via skeletal diffeologies

Establishment of isomorphism with Cheeger-Simons differential characters

Connection to Gajer's geometric loop group model

Abstract

Thin homotopies, introduced by Caetano-Picken, serve to axiomatize the holonomy of connections on principal bundles. This approach has been generalized to higher non-abelian bundles with connection through transport functors and higher holonomies, at least in dimension two and partially in dimension three. In this thesis, we introduce a new variant of thin homotopy based on the definition of skeletal diffeologies introduced recently by Kihara and show that, in the abelian setting, ordinary differential cohomology can be completely recovered in terms of the homotopy theory of skeletal diffeological spaces. Specifically, to any smooth manifold and non-negative integer $k$, we associate a $k$-skeletal simplicial presheaf such that its $k$th cohomology with values in the circle group is isomorphic to the abelian group of Cheeger-Simons differential characters. Further, we relate these higher holonomies to the only other existing model in the literature, developed by Gajer based on the geometric loop group. The main motivation for us is to provide a model where the differential refinement is established by refining the underlying space, rather than the coefficient object. This has the advantage of possible generalizations to non-abelian differential cohomology.