  Recently, a set of tools has been developed with the purpose of the study of
Quantum Gravity. Until now, there have been very few attempts to put these
tools into a rigorous mathematical framework. This is the case, for example, of
the so called path bundle of a manifold. It is well known that this topological
principal bundle plays the role of a universal bundle for the reconstruction of
principal bundles and their connections. The path bundle is canonically endowed
with a parallel transport and associated with it important types of derivatives
have been considered by several authors: the Mandelstam derivative, the
connection derivative and the Loop derivative. In the present article we shall
give a unified viewpoint for all these derivatives by developing a
differentiable calculus on the path bundle. In particular we shall show that
the loop derivative is the curvature of a canonically defined one form that we
shall called the universal connection one form.

Differential Geometry