3.4 Three declarations, three characteristic failures

Path agreement.

For parallel paths \(p,q:x\to y\), the declaration \(p\sim q\) asks whether \(D(p)=D(q)\). Equivariance, Bellman consistency, direct-versus-extended prediction, and argument-role transport all have this form.

Limit realization.

A designated cone can say that a representation is reconstructible from compatible views, that a shared state is an equalizer, or that a database apex is a pullback. The characteristic failure is not merely disagreement of two arrows; it can be failure of existence or uniqueness of the mediating map.

Colimit realization.

A designated cocone can describe empirical aggregation, quotient identification, or gluing of local pieces. Here the failure may be that local objects cannot be assembled into a global realization with the declared universal property.