Characterize Representable Admissible Distributors

Existing page: Characterize Representable Admissible Distributors
Status: Open research problem.
Suggested display title: When Does Admissible Comparison Determine a Universal Transported Object?

Catalog summary

A certified continuation need not determine a universal transported object. Find verifiable physical and categorical conditions under which the full comparison distributor is representable, and those representations fit coherently across changes of description.

What the problem concerns

A comparison distributor records the admissible ways a physical realization can continue under a change of description. It may record several continuations, their relations, or no continuation at all. Representability asks whether all those comparisons can be described through one universal target object.

Let f take an index ξ to an index ζ, let X be a physical realization over ξ, and write T_f(X,Y) for the groupoid of certified comparisons from X to a target Y over ζ. A representative is an object over ζ with a natural equivalence

The Hom expression includes the relevant comparison morphisms and their modifications. The equivalence must respect changes of Y; an isolated correspondence of objects is insufficient.

In words, every admissible continuation from X factors through a universal comparison from X to f_!X, essentially uniquely in the groupoid-valued sense. “Essentially uniquely” retains the specified equivalences and their compatibility, rather than requiring literal equality of presentations.

Representability and profunctors are standard categorical notions. The research question concerns their realization for the full physical comparison system constructed in GAT VIII.

What is already established

Section 16 states the universal property and derives ordinary transport when coherent representatives are supplied. Proposition 16.1 shows how the distributor’s composition law induces a comparison between transport along a composite arrow and successive transport along its factors.

The result is oplax transport: the two routes are connected by specified comparison maps. The stronger conclusion that these composition comparisons are equivalences requires equivalence of the distributor compositors. Representability alone does not remove the coefficient gerbe.

What remains open

Find intrinsic, checkable conditions on the source-certified physical comparison system that characterize when universal representatives exist and can be assembled coherently.

There are two distinct questions:

1. Existence of a representative. For a given f and X, does T_f(X,−) have the stated universal property? Which features of the physical comparisons ensure it, and which obstruct it?
2. Coherence of representatives. When representatives exist across a specified part of the base, do their choices respect changes of source realization, units, composition, and the declared comparison cells?

Any criterion must apply to the full physical package. A simpler observable description may admit a representative after forgetting distinctions that remain present in the physical comparison system.

Why continuation alone does not settle it

Finding a target Y and one certified comparison proves that continuation is possible. Choosing continuations consistently along a branch proves more. Neither establishes that every other continuation factors through the chosen target with the required universal property.

The open task is therefore to supply physical and categorical criteria for that property, rather than restate representability as an assumption.

Example

Consider a deliberately simple categorical model. The target category contains exactly two objects, A and B, with only identity arrows. For a fixed source X, suppose the comparison distributor assigns one comparison to A and one to B.

Continuation exists. Yet A cannot represent the distributor: there is no arrow from A to B to account for the comparison to B. B cannot represent it for the same reason. Choosing either continuation does not make it universal.

This is an elementary illustration of nonempty but nonrepresentable comparison data, not a claimed Yang–Mills example. A physical criterion must decide the analogous question using the actual certified comparisons and retained physical structure.

What would count as a resolution

A characterization would give necessary and sufficient conditions, or an equivalent verifiable criterion, for coherent representability on a precisely stated class of physical comparison systems. A sufficient criterion verified on a nontrivial physical class would be a meaningful partial result, but would not alone characterize every admissible distributor.

An obstruction theorem could identify physical comparison data that prevent a universal representative or prevent coherent assembly of locally available representatives. Section 16 also records a possible intermediate situation: representatives available locally and coherently equivalent on overlaps. It does not assert that every nonrepresentable distributor has that local form.

Even a successful representability result would leave arbitrary-cover extension, stronger composition equivalences, and analytic limiting questions to their own hypotheses.

Source

GAT VIII — Indexed Admissible Distributors and Selected Cofinal Čech Transport in Semisimple Yang–Mills Theory, public version v0.4-webpub1: §6; §16, equation (16.1), Proposition 16.1, and Remark 16.2; §17; Appendix A.5. The universal property and conditional recovery result are established source content. The search for intrinsic physical criteria is the open problem recorded by the archive.