Meaning
The liftable locus of a base arrow consists of source realizations that have at least one certified physical passage over that arrow.
For f:ξ→ζ, the definition is Lift♯(f)={X: there exists Y with T_f(X,Y) nonempty}. Thus liftability is tested for an actual source realization and a target realization, using the certified comparison collection.
Example (illustrative)
A permitted refinement f may admit a comparison for X₁ but none for X₂ because X₂ lacks the required compatible target data. Then X₁ lies in the liftable locus while X₂ does not.
Scope and limits
Existence of f in the base does not make every source liftable. Existence of some destination does not make it unique or canonical.
Source
GAT VIII — Indexed Admissible Distributors and Selected Cofinal Čech Transport in Semisimple Yang–Mills Theory. GAT VIII, Equation 3.5.