  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.
