ifc-0044

3.2 Weil algebras as probe shapes

The basic first-order infinitesimal algebra is

\[ W=\mathbb N[\varepsilon ]/(\varepsilon ^2). \]

The coefficient rig \(\mathbb N\) belongs to Leung’s universal algebraic presentation. It is not a claim that the semantic tangent fibers are natural-number spaces. A realization functor may send this probe shape into a smooth, synthetic, or other tangent setting; familiar real dual numbers and the Kock–Lawvere object arise only after such semantic structure has been chosen. Its nilpotent generator records a first-order displacement while suppressing higher powers. More elaborate Weil algebras describe several infinitesimal directions and relations among them. Write \(\mathsf{Weil}_1^{\mathrm{tan}}\) for Leung’s category \(\mathbb N\text{-}\mathsf{Weil}_1\): it is generated from the basic algebra \(W\) using the monoidal coproduct and the designated products and foundational pullbacks. Its product powers include algebras of the form

\[ W_n= \mathbb N[\varepsilon _1,\ldots ,\varepsilon _n] /(\varepsilon _i\varepsilon _j\mid 1\leq i\leq j\leq n), \]

together with their structure-preserving maps.

The crucial point is that a Weil algebra is not a model of the whole conceptual space. It is a probe shape. A monoidal functor realizes that shape as an operation on every object of the ambient category.

Theorem 3.2 Leung’s classification, cited

Giving a tangent structure on a category \(\mathcal M\) is equivalent, up to isomorphism, to giving a strong monoidal functor

\[ F:\mathsf{Weil}_1^{\mathrm{tan}}\longrightarrow \operatorname {End}(\mathcal M) \]

that preserves Leung’s designated foundational pullbacks and the equalizer encoding the universality axiom for the vertical lift. The generating algebra \(W\) is sent to the tangent functor \(T\), while maps among Weil algebras induce the structural natural transformations of the tangent category [ Leung , 2017 , Theorem 14.1 ] .

Thus \(F(A)X\) is the realization at \(X\) of the infinitesimal experiment encoded by \(A\). The monoidal product organizes composition of probe shapes: up to the coherence supplied by the strong monoidal structure,

\[ F(A\otimes B)\cong F(A)\circ F(B). \]

Projection, zero, addition, lift, and canonical flip arise from distinguished maps between the relevant Weil algebras. What appears axiomatically as a list of tangent-category operations is thereby organized by a monoidal algebraic theory.