Background

Let $\mathfrak{A}$, $\mathfrak{B}$, and $\mathfrak{C}$ be subcategories of the category of Banach spaces (over $\mathbb{R}$). Suppose we have a functor $\lambda:\mathfrak{A}^{op}\times\mathfrak{B}\to \mathfrak{C}$.

Let $f:E'\to E$ be a morphism belonging to $\mathfrak{A}$, and let $g:F\to F'$ be a morphism belonging to $\mathfrak{B}$. (Note: These are morphisms of topological vector spaces).

Then we have a map $$\matrix{Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))\\ (f,g)\mapsto\lambda(f,g)}$$

We say $\lambda$ is of class $C^p$ if for all manifolds $U$, and any two $C^p$ morphisms $U\to Hom(E',E)$ and $U\to Hom(F,F')$, the composition $$U\to Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))$$ is also of class $C^p$. (Note: We can replace $\mathfrak{A}$ and $\mathfrak{B}$ with categories of tuples to generalize this to several variables. In fact, this is what we do below.)

It is not hard to show that this induces a unique functor $$\lambda_X:VB(X, \mathfrak{A})^{op}\times VB(X,\mathfrak{B})\to VB(X,\mathfrak{C}).$$ on vector bundles taking values in the appropriate vector bundle categories over $X$.

We define a tensor bundle of type $\mathbf{\lambda}$ on $X$ to be $\lambda_X(TX)=\lambda_X((TX,\dots,TX),(TX,\dots,TX))$, where $TX$ is the tangent bundle.

However, this doesn't agree with the definition given on Wikipedia or anywhere else I've looked.

Questions

• Is this terminology nonstandard?

• Is the notion itself nonstandard?

• If the terminology is nonstandard, but the notion is standard, does it have a different name?

• Is this definition useful?

• Does this include more vector bundles as tensor bundles than the standard definition?