4.7 Weil probes and higher order

Tangent structure admits a useful algebraic classifier. In representable settings, an infinitesimal object supplies the dual-number probe; more generally, Weil algebras index tangent and higher-order prolongations [ Leung , 2017a , MacAdam , 2022 ] .

If \(W\) denotes the first-order dual-number object and

is the classifying functor, then \(T=F(W)\). Iterated tangent structure probes mixed and higher infinitesimal behavior. This allows LINCS to distinguish:

• first-order route sensitivity;

• symmetric acceleration under a declared connection;

• antisymmetric order effects;

• curvature or holonomy; and

• higher jets of a factorization obstruction.

These refinements are optional structure, not part of every first-order INC object.