This entry is about the notion of Weil algebra as the algebra of functions on an infintiesimally thickened point. For the concept of Weil algebra in Lie theory see Weil algebra.
objects such that commutes with certain colimits
Red(D) \simeq * \,.
A = R \oplus W \,,
where is a module of finite rank over and consisting of nilpotent elements in the algebra .
In the literature on synthetic differential geometry an algebra of this form is also called a Weil algebra. Notice that this is unrelated to the notion of Weil algebra in Lie theory. For more on that, see Weil algebra.
Over more general base fields, this is called an Artin algebra.
The smallest nontrivial example is the space dual to the ring of dual numbers. This is the point with “minimal infintiesimal thickening”.
A class of examples are the spaces of -tuples of infinitesimal neighbours of the origin of , that are each also infinitesimal neighbours of each other. Their Weil algebras of functions are a model for the degree -differential forms. Details on this are at spaces of infinitesimal k-simplices.
The site of definition for the Cahiers topos is the category of spaces that are products of an with the dual of a Weil algebra. So these are infinitesmally thickened Cartesian spaces. These are typically sufficient as test spaces for more general spaces.