Meaning
A finite gauge-presentation cover organizes local descriptions of the same proposed gauge object.
It is a finite family of presentations with a compatibility rule: a tuple is admitted when its detector markings, coefficient lines, lift and filling data, Gauss data, and validity conditions have a common allowed domain. Compatible tuples form the gauge Čech nerve, which records pairwise and higher overlaps.
Example (illustrative)
Two gauge presentations can be compared where all their required data share a valid domain. A third presentation belongs to a triple overlap only when the full triple is compatible; pairwise compatibility alone is not a substitute.
Scope and limits
This is the gauge-presentation direction. It is distinct from the spacetime cover, and its index tuples do not themselves supply physical comparison maps.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 5.1.