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22.5 Homotopical repair

Strict factorization can be too rigid for representations, policies, world models, or reasoning graphs whose semantics are invariant under controlled deformation. Strict repair asks for

\[ D=\overline Dq, \]

whereas homotopical repair asks for a coherent deformation

\[ D\simeq \overline Dq. \]

Definition 22.3 Homotopy INC

In a homotopical, model-categorical, or \(\infty \)-categorical enrichment of \(\mathcal C\), define

\[ \operatorname {INC}_{\mathrm h}(D) = \operatorname {Obs}\! \left( \operatorname {Fact}_{\mathbb S}(TD)\text{ up to coherent homotopy} \right). \]

It vanishes when the tangent factorization problem is inhabited up to the declared notion of coherent deformation.

This definition does not turn every approximate equality into a homotopy. The deformation paths, higher coherence cells, and allowed endpoints belong to the declaration. Otherwise “equivalent up to deformation” becomes an unfalsifiable escape from a failed diagram.

Example 22.4 Argument repair up to presentation

Two Toulmin realizations may differ in sentence order, lexical choice, or the number of intermediate propositions while preserving claim, warrant, source, qualifier, and rebuttal structure. A homotopical argument sketch could treat these presentation changes as coherent paths. A deformation that silently weakens a qualifier or disconnects a source would leave the admitted component and remain obstructed.

Admission contract

A homotopical repair must exhibit the deformation and verify its coherence, endpoint semantics, and behavior under restriction. A low distance between representations is not a substitute for this certificate.