nLab
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.


Context

Formal geometry

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Compact objects

          Contents

          Definition

          Abstractly

          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 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.

          Remark

          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.

          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,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. ) 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.

          Proposition

          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 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.

          Applications

          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.

          References

          cohesion

          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{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& &#643; &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

          Last revised on August 20, 2018 at 08:44:25. See the history of this page for a list of all contributions to it.