This entry is about the notion of Weil algebra as the algebra of functions on an infinitesimally thickened point. For the concept of Weil algebra in Lie theory see Weil algebra.
synthetic differential geometry, deformation theory
infinitesimally thickened point
In differential cohesion an object/type $D$ is an infinitesimally thicked point if its corresponding reduced object is the terminal object,
hence it is is an anti-reduced object.
An infinitesimally thickened point is – under Isbell duality – the formal dual of an $R$-algebra of the form
where $W$ is a module of finite rank over $R$ and consisting of nilpotent elements in the algebra $A$.
on terminology
In the literature on synthetic differential geometry an algebra $A$ 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 a local Artin algebra.
The smallest nontrivial example is the space dual to the ring of dual numbers. This is the point with “minimal infinitesimal thickening”.
This is a special case of the following
For $n , k \in \mathbb{N}$, the jet algebra $C^\infty(\mathbb{D}^n(k))$ is the quotient
where $I = (x_1, \cdots, x_n)$.
The formal dual smooth locus $\mathbb{D}^n(k)$ is the “order-$k$ infinitesimal $n$-disk”.
For $X$ an $n$-dimensional smooth manifold and $E \overset{p}{\to} X$ a bundle, and $\mathbb{D}^n(k) \hookrightarrow X$ the order-$k$ infinitesimal $n$-disk (def. 1) in $X$ around a point $x \in X$, then a lift
is a $k$-jet of a section of $E$ at $x$. The collection of all of these constitutes the order-$k$ jet bundle of $E$.
(e.g. Carchedi-Roytenberg 12, prop. 4.43)
A class of examples are the spaces $\tilde D(n,r)$ of $r$-tuples of infinitesimal neighbours of the origin of $R^n$, that are each also infinitesimal neighbours of each other. Their Weil algebras of functions are a model for the degree $r$-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 $R^n$ 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.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$