  We explore several notions of $k$-form at a point in a diffeological space,
construct bundles of such $k$-forms, and compare sections of these bundles to
differential forms. As they are defined locally, our $k$-forms can contain more
information than the values of differential forms contain, and we illustrate
this with many examples. To organize our work, we develop the basic theory of
diffeological vector pseudo-bundles, including a detailed understanding of
their limits and colimits, as well as a variety of fibrewise operations such as
products, direct sums, tensor products, exterior powers and dual bundles.
