Meaning
A fiber collects realizations over one specified input. A fibration also supplies coherent ways to move realizations along changes of input.
If π maps each realization to its formal input R, the fiber over R is the inverse image π⁻¹(R): all realizations with that input. That set may be empty. A Grothendieck fibration adds suitable lifts of base arrows and the laws governing their composition. GAT III constructs this stronger structure only on a branch with the required certified continuation data.
Example (illustrative)
Suppose there are realizations over both a coarse input and a fine input. Listing those two collections defines two fibers. To obtain the relevant fibration structure, a fine realization must admit the required certified reindexing over an admitted refinement arrow, with coherent behavior under successive reindexings.
Scope and limits
Nonempty fibers alone do not establish liftability, local triviality, or a fibration. A general correspondence between fibers is also weaker than a chosen reindexing functor.
Source
GAT III — Admissible Yang–Mills Fibrations and Thin Haag–Kastler Reconstruction. GAT 0 inverse-image fiber distinguished from GAT III Grothendieck fibration.