22.8 Topology of repair spaces

A scalar repair cost orders candidates but hides how qualitatively different repairs are connected. Let \(\mathcal R_{\operatorname {INC}}(D)\) be a category whose objects are admissible base and tangent repairs and whose morphisms are structure-preserving transformations among them. Its nerve

is a simplicial set. Vertices are repairs, edges are transformations, and higher simplices encode coherent chains of transformations.

The geometric realization

turns the repair category into a classifying space. Connected components can distinguish inequivalent repair regimes; loops can record nontrivial cycles; and higher homotopy or homology can detect coherent obstructions invisible to a scalar ranking. This parallels classifying-space and higher algebraic \(K\)-theoretic treatments of causal equivalence [ Mahadevan , 2023 , 2025b ] .

The ordinary nerve forgets infinitesimal organization. The Weil nerve suggests retaining a Weil-indexed family of repair categories [ MacAdam , 2022 ] :

The monoidal unit records base repair topology, the dual-number generator records first-order repair directions, and Weil composites organize higher and mixed prolongations.

A complementary integration problem asks whether an infinitesimal repair algebroid assembles into a groupoid of finite repairs. Failure of the infinitesimal-to-finite comparison would be a new obstruction: a locally coherent repair calculus that cannot be globally executed.