Meaning
The internal coefficient-band class records the obstruction carried by reassociating coefficient-line compositions.
For three composable coefficient lines, the two bracketings are compared by an associator phase Ω. Its pentagon law is the degree-three cocycle condition. The class b_D=[Ω] in H³(BB_D;U(1)) is taken using the permitted band-local changes of representative. Here U(1) is the group of complex phases of modulus one.
Example (illustrative)
The two composites (L_c⊗L_b)⊗L_a and L_c⊗(L_b⊗L_a) can differ by a specified phase. Changing local tensor identifications changes the phase representative by a coboundary; the class records what survives those permitted changes.
Scope and limits
A visibly nontrivial phase representative need not define a nonzero class. This degree-three class is distinct from both the filling obstruction and the relative degree-two class.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 10.1.