19.7 Transporting the old artistic theory

The sketch map \(i:S\to S^+\) induces restriction

Assume that \(\mathcal E\) admits the pointwise colimits required for the Kan extension. Given an old model \(m:S\to \mathcal E\), a left Kan extension then provides a canonical candidate for generative transport:

Its universal property explains how existing semantic material propagates through the new generators and relations [ Mac Lane , 1998 , Riehl , 2016 ] . It does not guarantee that the result preserves the products, limits, enrichment, tangent structure, or empirical integrity required by the visual model doctrine. Those are additional proof and admission obligations.

The universal construction carries its unit

Conservative transport requires this unit to be an isomorphism, equivalence, or registered observational equivalence of the strength demanded by the model doctrine. The existence of the Kan extension alone does not supply that certificate. In a successful extension, established color, composition, object, and rendering knowledge need not be relearned from scratch. The new technique becomes teachable because it is presented as a finite addition to a shared theory, not because a private latent vector happens to reproduce one image.