0.10 Weil algebras as probe shapes

The elementary first-order infinitesimal algebra is the dual-number algebra

Here \(\mathbb N\) is the initial commutative rig used in Leung’s algebraic classification. This object should not be confused with the familiar smooth dual numbers \(\mathbb R[\varepsilon ]/(\varepsilon ^2)\), or with the Kock–Lawvere infinitesimal object \(D=\{ d\in R\mid d^2=0\} \) internal to a smooth topos. The constructions are related through their first-order role, but they live at different semantic levels: one presents a universal probe shape, while the others realize infinitesimals in a chosen smooth setting [ Cockett and Cruttwell , 2014 ] . The nilpotent generator retains first-order displacement while suppressing higher powers. More elaborate Weil algebras encode several directions and relations among them. In Leung’s formulation, an appropriate monoidal category \(\mathsf{Weil}_1^{\mathrm{tan}}\) acts through a strong monoidal functor

with the generating algebra realized as the tangent functor [ Leung , 2017 , 2018 ] .

For a first reading, one may treat \(\varepsilon \) as a tagged perturbation: evaluate what changes linearly when the tag is present, while deliberately ignoring effects that require two or more copies of the same tag. The categorical construction makes that probing discipline compositional.

The important intuition is experimental. A Weil algebra is not the theory of the domain; it is a probe shape. First-order probes test sensitive directions. Paired probes test interaction. Iterated probes can expose order-effects or curvature-like behavior. A finite implementation selects a probe family rather than observing the entire infinitesimal structure.

Boundary: Local diagnosis and the two Weil constructions. Local diagnosis does not uniquely construct a global theory. A tangent direction may fail to integrate, and many finite theory extensions can share the same local profile. Infinitesimal probes constrain a creative proposal; they do not eliminate the abductive jump.

Two different constructions below carry the name Weil algebra. Leung’s \(\mathsf{Weil}_1^{\mathrm{tan}}\) is a monoidal theory classifying tangent structure. Meinrenken–Pike’s \(\mathcal W_{\mathrm{Lie}}^{\bullet ,\bullet }(\mathbb D)\) is a bigraded cochain algebra attached to a classical double vector bundle. They must not be identified. Involution algebroids provide an established first-order bridge: every ordinary Lie algebroid induces an involution algebroid, while an involution algebroid induces a Lie bracket on sections (and, given a suitable unit object, the Leibniz law) [ Burke and MacAdam , 2019 , Sections 4.2.1–4.2.4 ] ; the corresponding double bridge is a construction proposed in Chapter 3. That chapter formulates the candidate DIAL object as a research target and states the classical realizations and coherence conditions that any completed construction must recover.

The internal involution is more than a change of notation for a bracket. Burke and MacAdam impose a Yang–Baxter-style flip law on the prolongation of an anchored bundle; a Lie bracket satisfying Jacobi is then induced on its sections. This is distinct from Drinfel’d’s classical Yang–Baxter equation for \(r\)-matrices and Poisson–Lie geometry, although both express a three-way coherence of local composition [ Drinfel’d , 1983 , Burke and MacAdam , 2019 ] . In the proposed double theory, each DIAL side must satisfy this internal law before the compatibility between the two sides is tested.