Meaning
A banded 2-transport certificate makes coefficient-line transport coherent along physical passages and their comparison surfaces.
It assigns a coefficient line to each certified passage, compositors to pairs, the prescribed associator to triples, and unitary line maps to comparison surfaces. It also checks the higher relations, changes of band chart, and typed restrictions. The physical passage itself need not be invertible.
Example (illustrative)
Two parallel comparison surfaces Σ₀ and Σ₁ have line maps with matched endpoints. Their ratio tΣ₁⁻¹tΣ₀ is an automorphism of one line and hence a phase. A single open surface does not generally have such a scalar value.
Scope and limits
Invertibility of coefficient-line maps does not invert the underlying physical correspondence or remove a nontrivial associator class.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 20.1.