lin-0123

9.6 Functorial and descent-compatible repair

Suppose \(\alpha :D\to D'\) is a repair transformation. A tangent-compatible repair has a transported transformation

\[ T\alpha :TD\longrightarrow TD'. \]

If \(D'\) factors through the declared quotient, then a repair advertised as compositionality-preserving must retain that factorization.

For a cover, global and local repairs must also interact with restriction:

\[ \operatorname {res}_i\circ R_G \cong R_i\circ \operatorname {res}_i, \]

with cocycle coherence on overlaps.

Proposition 9.10 Gluing compatible local repairs

Suppose the admissible models form a stack over the chosen cover, the local repairs \(R_iD_i\) agree on overlaps with coherent descent data, and each repaired local model remains admissible. Then the repaired family glues to a global admissible model, unique up to the declared equivalence.

Proof ▼

This is the effectivity condition in the stack semantics applied to the compatible repaired family.

The proposition is deliberately conditional. Without effective descent, local repair compatibility is only a necessary condition for a global repair.