A witness system is complete for the theory when
For maps of spaces, this is equivalent to requiring every reconstruction fiber to be contractible. Completeness for one selected datum is weaker than completeness for the whole theory.
Example. One partial field may possess a unique completion even though another field in the same witness reconstruction object has no completion.
Source
Admissible Witness Descent for Higher Gauge Fields, public website edition v1.0. Not peer reviewed.