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Meaning
Admissible transport first meant scale-comparison transport encoded by a 2-lax natural transformation; later GAT formulations organize more general certified physical comparisons.
Earlier meaning: scale-comparison transport
The earlier question was how local realizations at different scales compare under refinement. The construction supplies comparison morphisms between those realizations and higher comparison cells relating a two-step refinement to the direct comparison. Those cells obey compatibility over quadruple refinements. This is the 2-lax natural-transformation formulation: strict agreement is not assumed, but the departure from strictness is itself structured. Nontrivial higher comparison data leave room for vacuum twisting; they do not by themselves prove that a twisting class is nonzero.
Later GAT formulation
In the later GAT VIII formulation, the contexts explicitly retain gauge presentation, spacetime region, scale, and markings. The primitive datum is the comparison groupoid T_f(X,Y) of certified physical passages from X to Y over an admitted base arrow f. It can retain several destinations or no passage at all. Representability is the additional condition that recovers an ordinary transported object. This is the later formulation of the archive term; the earlier 2-lax scale-comparison formulation remains part of its mathematical genealogy. The present entry does not assert an equivalence between the two formal frameworks.
Example (illustrative)
Consider scales i, j, and k. One route compares the realization at i with that at j and then with that at k; another uses the direct i-to-k comparison. In the earlier formulation, a specified higher cell relates those routes after the required restriction of their inputs. Adding a fourth scale requires those cells to fit together coherently. Simply declaring the two routes equal would erase the very comparison information that the 2-lax formulation retains.
Scope and limits
Here lax does not mean an uncontrolled approximation, and non-strict composition is not automatically a failure of coherence. The comparison cells have their own laws. The historical 2-lax datum, the later sector-transport pseudofunctor, and GAT VIII’s distributor-valued comparisons are distinct structures whose relationships must be stated at their proper level. A base arrow alone guarantees neither a physical lift nor an inverse, unitary comparison, or canonical destination.
Source
Historical formulation: Vacuum Characterization and Structural Existence of Physical Yang–Mills in 3 + 1 Dimensions within the Admissible Haag–Kastler Realization Class, §3.2, Definition 3.2 and Remarks 3.3–3.4, printed pp. 10–11; source file Vacuum_Characterization__Universality_of_Admissibility.pdf. Later formulation: GAT VIII, §§3–5 and §16. The earlier source is cited here for the definition’s genealogy, not to promote its other historical theorem claims or release its PDF.