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Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R. Such algebras form the basis of the Weil approach to differential geometry, pioneered by André Weil in 1953 (see his paper Théorie des points proches sur les variétés différentiables), which essentially amounts to adjoining formal duals of Weil algebras to the category of smooth manifolds, and then developing a form of infinitesimal calculus, similar to what Grothendieck did later for schemes.

The opposite category of Weil algebras can be thought of as a category whose objects are points with infinitesimal fuzz around them (henceforth called Weil points), similar to superpoints in supergeometry. An important distinction between Weil points and superpoints is that the algebra of functions on a Weil point is an ungraded commutative algebra, whereas the algebra of functions on a superpoint is a commutative super algebra.

A canonical example of a Weil algebra is the jet algebra of order k of a finite-dimensional real vector space V, which is defined as the truncated polynomial algebra Sym<kV. In fact, one can prove that the class of Weil algebras coincides with the class of factoralgebras of jet algebras.

One obvious question that arises from the above fact is whether the study of Weil algebras can somehow be reduced to the study of jet algebras. For example, the opposite category of jet algebras can be trivially turned into a site (any nonempty family of morphisms is a covering family), and then one has a canonical embedding of the opposite category of Weil algebras (i.e., the category of Weil points) into the category of sheaves of sets on this site. This embedding is faithful if and only if nonzero elements in Weil algebras can be detected by maps to jet algebras. More precisely, one can ask the following question:

Given an element a≠0 in a Weil algebra A, can we always find a homomorphism f of the form A→Sym<kV (for some k and V) such that f(a)≠0?

If the answer to the above question is negative, can we say anything interesting about Weil algebras that do satisfy the above property?

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Weil algebras (even if you admit more than one idempotent) correspond to product preserving functors from the category of smooth manifolds and smooth mappings into itself. See [Gerd Kainz, Peter W. Michor: Natural transformations in differential geometry. Czechoslovak Math. J. 37 (1987), 584-607, (pdf)], or chapter VIII in here. Product preserving functors on the category of infinite dimensional manifolds correspond to $C^\infty$ algebras (see here) which also play a fundamental role in synthetic differential geometry.

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