14 Homotopy and Tangent Repair

From obstruction spaces to infinitesimal stability and convergence

Chapter 9 introduced inner horns as unresolved online composites and mapping spaces as repair profiles. Here the aim is a general obstruction theory. An empty filler space records impossibility; a disconnected filler space records qualitatively different repair families; a contractible filler space records coherent uniqueness; and a nontrivial higher homotopy type records residual ambiguity among repairs.

The change of viewpoint is decisive. A failed computation normally returns an error flag or a scalar residual. A failed universal construction returns a shape: the boundary that was supplied, the face that is missing, the space of possible fillers, and the comparisons that prevent a filler from being admitted. Repair is therefore not an untyped perturbation of a model. It is completion of a registered partial diagram subject to explicit preservation obligations.