3.7 Presentation, transport, and sketchability

A sketch is itself a presentation. Two presentations can describe equivalent learning behavior while exposing different paths, auxiliaries, or parameter symmetries. Later chapters therefore quotient presentation-null variation before assigning repair effort.

Maps between learning graphs also require care. For unconstrained categorical schemas, functorial data migration is automatic. Once designated cones and cocones are added, a graph map need not preserve them [ Spivak , 2014 ] . A change of learning sketch must state which equations, universal properties, admissible models, and repair spaces it transports.

Ordinary sketches have a further expressivity boundary: not every natural category of structured objects and structure-preserving morphisms is the category of set-valued models of a sketch [ Barr and Wells , 1992 ] . Higher, enriched, fibrational, or homotopical presentations may be needed when admissibility itself quantifies over structure outside ordinary sketch logic. Section 0.2 fixes the book’s convention: the ordinary sketch records the shared compositional skeleton, while any hom-object enrichment or other semantic structure required by an application must be declared separately.