More generally, the analog of differential forms in secondary calculus are the elements of the first term of the so-called C-spectral sequence, and so on.
On the contrary, the most natural language in which to study these objects is differential calculus over commutative algebras.
For instance, it was understood that observables in field theory are classes in horizontal de Rham cohomology which are invariant under the corresponding gauge group and so on.
It is relevant that secondary calculus and cohomological physics, which developed for twenty years independently from each other, arrived at the same results.
Their confluence took place at the international conference Secondary Calculus and Cohomological Physics (Moscow, August 24–30, 1997).