Remark. Scalar factorization, localization and gluing [fdip-004Q]
AGENTDRAFTED
The retained-variance example tests scalar
factorization: for specified functions \(c:X\to Y\) and
\(r:X\to \mathbb R\), it asks whether \(r=F\circ c\) for a
function on \(c(X)\). The fiber criterion answers
exactly this question, without supplying continuity or any
categorical structure.
Functorial factorization instead starts with categories and
functors \(U:\mathcal B\to \mathcal D\) and
\(O:\mathcal B\to \mathcal A\), and asks for
\(\overline O:\mathcal D\to \mathcal A\) and a
natural isomorphism
\(O\cong \overline O\circ U\). A localization at a declared
class \(W\) of arrows has an additional universal property: it
inverts those arrows and is universal among functors that do so
[stacks2017stacks, Section 4.27], Tag 04VB.
A generic forgetful functor is not thereby a localization.
Mathlib's CategoryTheory.Localization.lift requires
both a localization functor and a functor that inverts \(W\);
these hypotheses are absent from the scalar example.
Local-to-global descent requires a site with covers,
restriction maps and compatible data on overlaps, together with
an effective gluing statement. For category-valued data, a stack
also requires a fibred category and sheaves of morphisms
[stacks2017stacks, Definition 8.4.1], Tag 0268.
The enriched background groupoid and the scalar
variance calculation supply none of this cover-and-gluing structure.
The groupoid is a category of finite backgrounds, not necessarily a
finite category. No local-to-global descent theorem follows.
