In this lecture we generalise differential forms by allowing them to take values in a vector space (these are called vector-valued forms), or a vector bundle (these are called bundle-valued forms).

We then focus on the case of vector-valued forms on a principal bundle, and show that vector-valued forms that are both horizontal and equivariant can be identified with bundle-valued forms on the associated vector bundle.

The main application of this result will come next semester (as we will see, the curvature of a connection on a principal bundle satisfies these hypotheses).