The dg-algebra$\Omega^\bullet_{vert}(P)$ of vertical differential forms on $P$ is the quotient of the de Rham complex dg-algebra $\Omega^\bullet(P)$ of all forms on $P$, by the dg-ideal of horizontal differential forms, hence 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$d \pi : T P \to T X$.

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