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?