infinitesimally thickened point

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.


Formal geometry

Synthetic differential geometry

differential geometry

synthetic differential geometry






Compact objects




In differential cohesion an object/type DD is an infinitesimally thicked point if its corresponding reduced object is the terminal object,

(D)*, \Re(D) \simeq * \,,

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 RR-algebra of the form

A=RW, A = R \oplus W \,,

where WW is a module of finite rank over RR and consisting of nilpotent elements in the algebra AA.


on terminology

In the literature on synthetic differential geometry an algebra AA 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,kn , k \in \mathbb{N}, the jet algebra C (𝔻 n(k))C^\infty(\mathbb{D}^n(k)) is the quotient

C (𝔻 n(k))=C ( n)/I k+1, C^\infty(\mathbb{D}^n(k)) = C^\infty(\mathbb{R}^n)/ I^{k+1} \,,

where I=(x 1,,x n)I = (x_1, \cdots, x_n).

The formal dual smooth locus 𝔻 n(k)\mathbb{D}^n(k) is the “order-kk infinitesimal nn-disk”.

For XX an nn-dimensional smooth manifold and EpXE \overset{p}{\to} X a bundle, and 𝔻 n(k)X\mathbb{D}^n(k) \hookrightarrow X the order-kk infinitesimal nn-disk (def. 1) in XX around a point xXx \in X, then a lift

𝔻 n(k) E p X \array{ \mathbb{D}^n(k) && \longrightarrow && E \\ & \searrow && \swarrow_{\mathrlap{p}} \\ && X }

is a kk-jet of a section of EE at xx. The collection of all of these constitutes the order-kk jet bundle of EE.


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 D˜(n,r)\tilde D(n,r) of rr-tuples of infinitesimal neighbours of the origin of R nR^n, that are each also infinitesimal neighbours of each other. Their Weil algebras of functions are a model for the degree rr-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 nR^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.



tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Revised on September 26, 2016 03:28:47 by Urs Schreiber (