ora-0109
8.6 Three strengths of categorical approximation
Let \(\widehat{\mathcal C}\) be a learned world.
It is behaviorally PACC when the guarantee concerns only the answers returned by a declared external query doctrine.
It is structurally PACC when a learned comparison functor, span, or profunctor to the target is also tested for properties such as faithfulness, fullness, essential coverage, and preservation of declared universal constructions.
It is coherently PACC when the probes range over simplices, horns, filler spaces, and higher comparison witnesses, with discrepancy invariant under the declared homotopy equivalences.
Behavioral PACC can be enough for a fixed predictive interface, just as a PSR need not recover a privileged latent state. Structural PACC supports reuse of compositions in new contexts. Coherent PACC is needed when the space of repairs, rather than the mere existence of one answer, matters. None implies causal correctness unless the probe doctrine contains interventions and the usual identification assumptions justify their interpretation.
For universal properties, a zero–one discrepancy may ask whether a comparison of cone categories is an equivalence. A graded discrepancy may measure which probes fail to admit a mediator or how the homotopy type of a filler space differs. This is categorically meaningful only after the answer objects and weak equivalences have been declared.