The following equivalences are well-established: $$\begin{align} \text{Locally-free sheaves on }X &\leftrightarrow \text{Vector bundles over }X\\ \cap\qquad & \qquad\qquad\textbf{?}\\ \text{Sheaves on }X &\leftrightarrow \text{Étalé spaces over }X\\ \cup\qquad & \qquad\qquad\cup\\ \text{Locally-constant sheaves on }X &\leftrightarrow \text{Covering spaces over }X \end{align}$$
Further, the ways that we construct a sheaf from a vector bundle and from an étalé space are 'identical': we just take the sheaf of sections of the projection map. But it is also known that a non-trivial vector bundle is never étalé. This makes me confused/intrigued about two things:
- Why does the equivalence of categories (sheaves$\leftrightarrow$étalé) not descend to the equivalence (locally-free sheaves$\leftrightarrow$something) in the same way that it does for locally-constant sheaves and covering spaces? (i.e. why are the bottom two rows of this diagram 'nice' while the top two are 'not nice'?)
- Given some vector bundle, we can associate to it (uniquely?) a sheaf, to which we can then associate an étalé space (and vice versa). Does this give us an equivalence of categories (vector bundles$\leftrightarrow$some sort of étalé spaces)?
The top two are different because you're only looking at the module, and forgetting the ring.
The category of sheaves over $X$ doesn't remember how $\mathbb{R}$ looks as a topological space; it only understands the sheaf $C_X$ of continuous real-valued functions, and the ring structure of that sheaf.
The étalé space of $C_X$ doesn't look anything like $\mathbb{R}$.
Under the sheaf-étalé space correspondence, a sheaf of real vector spaces doesn't become a real vector bundle: it becomes a module over the étalé space of $C_X$.