Can morphisms of vector bundles be expressed in terms of classifying spaces?

Question

A vector bundle on a space $X$ can be encoded as a map $X \to BU(n)$. Does a similar thing occur for morphisms? One very optimistic interpretation would be as follows. If I have two vector bundles on $X$, corresponding to maps $X \to BU(n)$ and $X \to BU(m)$, then the morphisms between the two vector bundles are in one-to-one correspondence with maps $BU(n) \to BU(m)$ making everything commute. It sounds too good to be true, but my example skills are too weak to disprove it.

Answer 1

This isn't true. Note that the space of morphisms between two vector bundles is a vector space, and so is contractible, whereas a homotopy-theoretic construction involving classifying spaces won't be.

You can think of $BU(n)$ as the classifying space of the topological category of $n$-dimensional complex vector spaces. Maps $BU(n) \to BU(m)$ therefore correspond to (a homotopy-ish version of) functors between such categories, e.g. tensor, symmetric, or exterior powers; that isn't what you want.

Homotopy theory can at best talk about isomorphisms, as follows: you can identify isomorphisms between two vector bundles in terms of homotopies between two maps $X \to BU(n)$. Somewhat more explicitly, there is a space of such isomorphisms, and that space has the homotopy type of the loop space of the mapping space $[X, BU(n)]$, based at some vector bundle.

---

To use a category-theoretic analogy that may make things clearer, let's think about a representation of a group $G$ as a functor $BG \to \text{Vect}$ (here $BG$ is analogous to $X$ and $\text{Vect}$ is analogous to $BU(n)$, slightly confusing but hopefully that's clear). Then morphisms between representations are natural transformations of such functors, or equivalently morphisms in the functor category $[BG, \text{Vect}]$; in particular we don't talk at all about functors $\text{Vect} \to \text{Vect}$ in discussing them.