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
Introductions
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geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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The magic algebraic facts
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differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
In differential cohesion an object/type is an infinitesimally thickened point if its corresponding reduced object is the terminal object,
hence it is an anti-reduced object.
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 .
on terminology
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 a local Artinian 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
where .
The formal dual smooth locus is the “order- infinitesimal -disk”.
For an -dimensional smooth manifold and a bundle, and the order- infinitesimal -disk (def. ) 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 ) is a quotient of a jet algebra (def. ).
(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 infinitesimally thickened Cartesian spaces. These are typically sufficient as test spaces for more general spaces.
Ivan Kolář, Peter Michor, Jan Slovák, Def. 35.2 of: Natural operations in differential geometry, Springer (1993) [book webpage, doi:10.1007/978-3-662-02950-3, pdf]
David Carchedi, Dmitry Roytenberg, On Theories of Superalgebras of Differentiable Functions (arXiv:1211.6134)
Last revised on September 11, 2024 at 17:44:57. See the history of this page for a list of all contributions to it.