lin-0119
9.2 Localizing factorization failures
Let \(J=\operatorname {Path}(S)\) index a learning sketch and let \(\mathcal U=\{ J_i\hookrightarrow J\} _{i\in I}\) be a cover by subsketches. A global obstruction can be restricted to every \(J_i\), producing local obstruction data.
A localization of \(\mathcal O(D)\) over \(\mathcal U\) consists of local objects
together with restriction maps on overlaps and any coherence needed to compare them.
Localization answers more than “which parameter has a large gradient?” It identifies declared subdiagrams in which a factorization, limit, colimit, or closure obligation fails. A claim of smallest support additionally requires a refinement order and a cover capable of detecting minimal witnesses.
Suppose \(T\) preserves the restrictions and overlap constructions used by \(\mathcal U\), every restricted tangent model is admissible, and the obstruction assignment is natural under these restriction isomorphisms. Then tangent lifting commutes with localization:
By the preservation hypotheses, \((TD)|_{J_i}\cong T(D|_{J_i})\). The two sides therefore present the same restricted tangent factorization problem, and the obstruction assignment transports the isomorphism.
The hypotheses matter. Automatic differentiation of a global program does not by itself show that a designated equalizer, pullback, or descent object is preserved by the chosen tangent realization.