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
objects such that commutes with certain colimits
In differential cohesion an object/type is an infinitesimally thicked point if its corresponding reduced object is the terminal object,
hence it is is an anti-reduced object.
In the standard type of model
An infinitesimally thickened point is – under Isbell duality – the formal dual of an -algebra of the form
where is a module of finite rank over and consisting of nilpotent elements in the 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 , the jet algebra is the quotient
The formal dual smooth locus is the “order- infinitesimal -disk”.
For an -dimensional smooth manifold and a bundle, and the order- infinitesimal -disk (def. 1) in around a point , then a lift
is a -jet of a section of at . The collection of all of these constitutes the order- jet bundle of .
Every Weil algebra (remark 1) is a quotient of a jet algebra (def. 1).
(e.g. Carchedi-Roytenberg 12, prop. 4.43)
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.
graded differential cohesion