What makes differential forms special

Solution 1:

One important thing to notice is that the assignment $M \mapsto \Gamma(M, TM)$ does not define a functor on the category of smooth manifolds and smooth maps, because tangent vector fields do not behave well under pushforwards or pullbacks by smooth maps. On the other hand, the assignment $M \mapsto \Gamma(M, \bigwedge^k T^*M)$ defines a contravariant functor on the category of smooth manifolds, since forms can be pulled back by smooth maps.

Also, notice that the functor $C^\infty(-)$ which assigns to each smooth manifold its algebra of smooth functions is the contravariant representable functor $\text{Hom}(-, \mathbb{R})$. So if we want a functorial construction assigning to each smooth manifold some algebra that generalizes the smooth function algebra, it should probably be a contravariant construction.

This fits into the more general duality between space and quantity. The heuristic is that a spacial category can often be thought of as the opposite of an algebraic category, so functors that assign some "algebra of quantities" to a space should generally be contravariant.