There's a meta-theorem hanging around in the theory of vector bundles that "Any canonical construction in linear algebra gives rise to a geometric version for holomorphic vector bundles." (quote from Huybrecht's Complex Geometry)
In textbooks, this usually done hands on case-by-case. I'm looking for a broader view.
- Which functors $\text{Vect}\to \text{Vect}$ extend to functors in the category of vector bundles over $X$?
Here, an extension of a functor $F:\text{Vect}\to \text{Vect}$ is a functor $\bar F$ between vector bundles such that if $V$ is the typical fiber of $E$ then $F(V)$ is the typical fiber of $F(E)$.
A partial answer: a functor $\text{Vect}\to \text{Vect}$ is smooth if for every $V$ the induced map $F:GL(V)\to GL(FV)$ is smooth. In this case, given cocycles $\psi_{ij}$ defining a bundle, the following cocycles defines an extension of $F$: $$ U_i\cap U_j\xrightarrow{\psi_{ij}} GL(V)\xrightarrow{F} GL(FV) $$
I believe the most common linear algebra constructions on bundles, such as tensors, sums and exterior powers, come from smooth functors.
- Are there non-smooth functors $\text{Vect}\to \text{Vect}$ that extend to vector bundles in the above sense? Can we characterize them?
- Shifting the categorical perspective, does regarding vector bundle as a locally free sheaves brings any insight here? Can these concepts be carried on to algebraic geometry?