nLab
vertical differential form

Let $\pi :P\to X$ be a bundle in the category Diff of smooth manifolds.

The dg-algebra ${\Omega }_{\mathrm{vert}}^{•}(P)$ of vertical differential forms on $P$ is the quotient of the de Rham complex dg-algebra ${\Omega }^{•}(P)$ of all forms on $P$, by the dg-ideal of all those forms that vanish when any one vector in their arguments is a vertical vector field in that it is in the kernel of the differential $d\pi :TP\to TX$.

For a trivial bundle $P=X\times F$ the underlying complex of ${\Omega }_{\mathrm{vert}}^{•}(P)$ is ${\wedge }_{C^{\mathrm{\infty }}(X\times F)}^{•}\Gamma (T^{*}F)$.