Meaning
A selector-comparison certificate supplies the coherent comparison data needed to turn selected fillings into band loops.
For each selected filling, it provides two invertible band arrows L and R with the same endpoints. It also supplies the two-cells for changes of representative, their composition and reassociation laws, higher modifications, and restriction compatibility. These data make R⁻¹L a well-defined selected comparison loop.
Example (illustrative)
Two permitted constructions of the same boundary comparison may give L and R. To compare another representative of the filling, one needs cells relating both L and R to their new representatives; simply subtracting their names is meaningless.
Scope and limits
This certificate is extra realized comparison data. It does not follow solely from the existence of a cochain map or a filling.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 11.1.