A generalization of the exterior product of differential forms combining Hom-valued forms

summary:This article deals with vector valued differential forms on $C^\infty$-manifolds. As a generalization of the exterior product, we introduce an operator that combines $\operatorname{Hom}(\bigotimes^s(W),Z)$-valued forms with $\operatorname{Hom}(\bigotimes^s(V),W)$-valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles