Selected Raw Comparison Bicategory

Source result: GAT VIII, Proposition 4.1.
Status: Source-bound proposition in the public preprint.

Catalog summary

The selected certified physical realizations, their full-package comparisons, and invertible comparison modifications form a bicategory. Composition retains the prescribed physical reassociators and defect data, and projects coherently to the selected transport base.

What the proposition says

To describe transport, it is not enough to list possible endpoints. One must retain the passages between them and the allowed comparisons between passages. A bicategory organizes these three levels: objects, arrows, and 2-cells, with composition associative up to specified coherent isomorphisms.

“Raw” means that admissibility certificates and auxiliary enhancements are still recorded. Removing those redundancies is a later step with additional hypotheses.

Hypotheses and scope

Use the selected source-free transport system of GAT VIII §§2–4. A raw object consists of substantive physical realization data, a selected certificate, and an auxiliary enhancement. A raw arrow consists of an admitted base arrow and a full-package source-certified physical comparison. Its 2-morphisms are declared base 2-cells together with invertible full-package modifications.

The frozen finite interface supplies like-typed composition, units, physical coefficient associators, and the required coherence relations. The selected system is closed under those admitted operations. No undeclared base cell or physical comparison is added solely to complete a diagram.

Precise conclusion

These data form a bicategory . Composition is

The projection to the selected transport-oriented base is a normal homomorphism of bicategories: it preserves units in the declared normal convention and respects composition with the prescribed coherent comparison structure.

Why it works

Compose each physical coordinate in its declared type and paste the certified validity cells. The analytic comparison residuals follow their composition law; coefficient reassociation is inserted when parentheses change. The source associator and its pentagon ensure that the resulting compositions fit together coherently.

Example

For three admitted passages , the composites and are related by the prescribed associator. A coefficient contribution in that associator remains part of the physical comparison. Recording the bicategory does not set it to the identity or erase it because the endpoints look alike.

Claim limits

This proposition does not remove certificate redundancy, choose a continuation for every object, prove representability, or establish a positive coercive floor. Certificate localization, enhancement reduction, and the indexed distributor theorem each have their own subsequent hypotheses. The result applies to the selected system, not arbitrary covers.

Source

GAT VIII, §§2–4, especially Proposition 4.1; §5 explains the separate intrinsic reduction. Bicategories are standard mathematical structures; the source-certified physical composition data give this one its content.