Glossary of Notation

The semantic program of identifying an unknown categorical world from an interaction presentation, up to a declared query equivalence.

Universal Online Categorical Learning: an algorithmic realization of ORACLE through typed probe, selection, assimilation, and repair rules.

A learning paradigm with an explicit hypothesis category, presentation protocol, query doctrine, observational quotient, non-anticipatory update, and success criterion.

A category of formal operations and equations; Lawvere theories use finite products, while PROPs use one-generated symmetric monoidal structure.

A compact presentation by generating objects and arrows, commutative diagrams, and designated cones or cocones.

A world category together with a sketch, preservation doctrine, completed theory, and structure-preserving interpretation of that theory in the world.

UOCL that learns generators, relations, theory completion, and functorial model semantics in addition to an encountered world category.

A tuple \((\mathcal H,I,U,R)\) between categorical learning interfaces: an indexed hypothesis category, implementation, typed update, and backward request or repair-obligation map.

A limit, commonly a pullback, of modality-specific UOCL learners over a shared doctrine; stronger than their parallel monoidal product because it enforces cross-modal compatibility.

Tangent UOCL: categorical identification in a hypothesis class of tangent categories, using both base and tangent-sensitive probes.

Probably Approximately Categorically Correct learning: a statistical UOCL guarantee relative to a declared categorical probe doctrine, answer equivalence, discrepancy, distribution, accuracy, and confidence.

Expected equivalence-invariant discrepancy between a target and learned categorical world under probes drawn from \(P\).

A requirement that the probe distribution assign positive mass to every categorical stratum whose correctness is claimed.

Categorical probe dimension: the largest compatible family of binary categorical probes whose coherence-admissible answer patterns are realized by the hypothesis class.

A tangent hypothesis induced by the category of coalgebras of a differential category.

Identification of a system \(c:X\to F X\) from its presentation, generally only up to the behavioral equivalence detected by its map into a final \(F\)-coalgebra.

Predictive state representation: a controlled-process state given by probabilities of future action–observation tests, rather than by a privileged latent ontology.

A presheaf or sheaf of local, context-indexed predictive models with restriction maps, overlap audits, gluing, provenance, and registered obstructions.

Universal Online Decision Learning: the specialization of UOCL to hypotheses carrying information, action, consequence, consistency, and observer structure.

Left and right Kan extension along the displayed functor.

Ordinary UDL semantics \(\operatorname {Ran}_K\operatorname {Lan}_JF\).

Homotopy-coherent or derived UDL semantics.

Time or revelation category.

Ordinary nerve of a category.

Category of simplices of a simplicial set \(X\).

The \(i\)-th horn in the \(n\)-simplex.

The localization \(\mathcal D[W^{-1}]\) at declared semantic weak equivalences.

Tangent functor; \(T^h\) denotes a derived tangent functor when one exists.

A Weil algebra and its nilpotent maximal ideal.

Infinitesimal probe represented by a Weil algebra \(W\).

Linearized information-dependency operator \(D_u\Phi _{\mathcal M}\) of an intrinsic model.

Declared aggregation of local values over decision sites \(A\); ordinary summation is one specialization.

Intrinsic regret observed between learner information \(\mathcal I\) and comparator information \(\mathcal J\).

A space of compatible horn fillers or repairs.

Homotopy-coherent online-to-offline defect.

A partial decorated simplex together with settled and open obligations, an admissible filler space, and a comparison observer.

A recorded obstruction to repair in the old declaration, a revised declaration with a filler, and witnesses identifying the obligations preserved by the change.

A learned diagram, universal construction and witness, repair certificates, and the observational quotient on which they are valid.

The family of future diagrammatic problems that factor through a learned package with admitted witnesses.

The maximal registered substructure on which learned universal structure, observers, repairs, and safety admission transport coherently across a specified environment path.