  Associated to a differential character is an integral cohomology class,
referred to as the characteristic class, and a closed differential form,
referred to as the curvature. The characteristic class and curvature are equal
in de Rham cohomology, and this is encoded in a commutative square. In the
Hopkins--Singer model, where differential characters are equivalence classes of
differential cocycles, there is a natural notion of trivializing a differential
cocycle. In this paper, we extend the notion of characteristic class,
curvature, and de Rham class to trivializations of differential cocycles. These
structures fit into a commutative square, and this square is a torsor for the
commutative square associated to characters with degree one less. Under the
correspondence between degree 2 differential cocycles and principal circle
bundles with connection, we recover familiar structures associated to global
sections.
