I've been reading Gravitation by Misner, Thorne & Wheeler. I've come across the concept of tensor-valued forms recently and am confused. I've had a read through this question: Clarification of vector valued forms (sections and tensor products)
This was nearly sufficient. However, when E becomes a tensor (should it formally be called a tensor algebra?), I get weary. Shouldn't tensors on a manifold depend on the point of interest, and be constructed from the respective tangent and cotangent spaces at that point? I was thinking that, in order to make sense of a tensor-valued (rather than E-valued) differential k-form, E might need to be replaced by a tensor algebra $T(V)$, having the tangent space at the point p as V, i.e. $V=T_p M$. So that $E=\sqcup_{p\in M}{T(T_pM)}$, or would E be $E=T(TM)$. It may be something completely different. So any help is appreciated