ora-0022
1.10 Weak equivalence and localization
Strict isomorphism is often too rigid for learned representations. Two models may encode the same observable decisions through different coordinates, factorizations, or resolutions.
A relative category is a pair \((\mathcal D,W)\), where \(\mathcal D\) is a category and \(W\) is a declared class of weak equivalences containing all identities.
Localization formally makes every map in \(W\) invertible:
The localization is best regarded as an \(\infty \)-category when the paths and higher witnesses relating weakly equivalent presentations matter.
Weak equivalence is semantic data. UODL must state which observables a weak equivalence preserves. Choosing a familiar model structure does not by itself justify identifying two decision systems.