Given a topological space $X$, there's a fundamental category equivalence between local homeomorphisms to $X$ and sheaves of sets over $X$. One direction takes a local homeomorphism to its sheaf of sections, and the other takes a sheaf, and constructs the projection from its étalé space, comprised of stalks.
The above restricts to an equivalence between covering maps and locally constant sheaves.
Covering maps being fiber bundles with a discrete fiber, is there perhaps an intermediate equivalence describing sheaves of sections of general fiber bundles? At least, what distinguishing properties do sheaves of sections of fiber bundles possess?
Best Answer
Fiber bundles with non-discrete fibers are not local homeomorphisms. Hence at the very least there is no "intermediate" equivalence.
Interestingly, vector bundles are equivalent to modules over the sheaf of continuous real functions (itself the sheaf of sections of the trivial fiber bundle with fiber $\mathbb R$ over $X$) , so structured fiber bundles admit nice descriptions.
Perhaps fiber bundles can somehow be described using some canonical sheaf on the space, but I don't see how.