nLab infinitesimally thickened point

Contents

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.

Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

Contents

Definition

Abstractly

In differential cohesion an object/type $D$ is an infinitesimally thickened point if its corresponding reduced object is the terminal object,

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

hence it is an anti-reduced object.

In the standard type of model

An infinitesimally thickened point is – under Isbell duality – the formal dual of an $R$-algebra of the form

$A = R \oplus W \,,$

where $W$ is a module of finite rank over $R$ and consisting of nilpotent elements in the algebra $A$.

Remark

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.

Examples

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

Definition

For $n , k \in \mathbb{N}$, the jet algebra $C^\infty(\mathbb{D}^n(k))$ is the quotient

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

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. ) in $X$ around a point $x \in X$, then a lift

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

is a $k$-jet of a section of $E$ at $x$. The collection of all of these constitutes the order-$k$ jet bundle of $E$.

Proposition

Every Weil algebra (remark ) is a quotient of a jet algebra (def. ).

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.

Applications

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.

cohesion

tangent cohesion

differential cohesion

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