An admitted witness system is a small indexing category
It specifies exactly which patches, overlaps, higher overlaps, arrows, and coherences are retained as reconstruction evidence.
Example. Keeping every patch and pairwise overlap but omitting one tetrahedral coherence is a witness system. It is not equivalent to retaining the full descent diagram merely because it contains many objects.
Source
Admissible Witness Descent for Higher Gauge Fields, public website edition v1.0. Not peer reviewed.