2.2 Prediction before decision

Let \(i:\mathcal A\to \mathcal C_\star \) be a category of admissible probes. The interpretation \(M_\star \) distinguishes a core family among them and types the answers as physical, numerical, spatial, shape, agentive, or social evidence. The response of an object \(x\) to those probes is the restricted nerve

Two objects are observationally equivalent when these presheaves are naturally isomorphic. For dense \(i\), the restricted nerve is fully faithful. Otherwise it deliberately identifies distinctions unavailable to the learner.

The same idea operates one level higher. The Yoneda embedding of the ambient 2-category of structured hypotheses sends

Thus structured category-shaped experiments probe a world through its core-preserving functors and transformations. Forgetting the core structure recovers ordinary category-shaped experiments \(\operatorname {Fun}(\mathcal B,\mathcal C)\). ORACLE learns a structured category through its responses to such probes, just as ordinary Yoneda learns an object through all arrows into it.