8.8 Weil probes beyond first order

In representable tangent categories, Weil algebras classify first and higher infinitesimal probes [ Leung , 2017a , 2017b ] . A first-order dual-number probe yields the ordinary tangent lift. Products and iterates of Weil objects encode mixed directions and higher jets.

For a finite probe graph \(G\), assign a first-order generator to each vertex and a mixed interaction probe to selected edges. The graph then controls which directional interactions are evaluated without constructing the full dense higher derivative. This is valuable when a model exposes many candidate modules but the sketch declares only sparse interactions.