Definition. Weil algebra [math-T4B2]

A Weil algebra $W$ is an algebra over the rational numbers $\mathbb{Q}$, equipped with a morphism

such that $W$ is a local ring with maximal ideal:

with $I$ a nilpotent ideal, and such that $W$ is a finite dimensional $\mathbb{Q}$-vector space.

The notion of Weil algebra makes sense in any topos $\mathcal{E}$ with a natural numbers object $N$.