nLab
infinitesimal neighborhood

Context

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)

          Infinitesimal neighbourhoods

          Idea

          Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct, oder die Grenze zwischen bestimmten Stellen im Raume; (Fichte 1795, Grundriss §4.IV)

          An infinitesimal neighbourhood is a neighbourhood with infinitesimal diameter. These can be defined in several setups: nonstandard analysis, synthetic differential geometry, ringed spaces, ….

          Definition

          In differential cohesion

          For H\mathbf{H} a context of differential cohesion with infinitesimal shape modality \Im, then for x:*Xx\colon \ast \to X a global point in any object XHX \in \mathbf{H} the infinitesimal disk 𝔻 x XX\mathbb{D}^X_x \to X around that point is the (homotopy) pullback of the unit i:X(X)i \colon X \to \Im(X) of the \Im-monad

          𝔻 x X X i * x (X). \array{ \mathbb{D}^X_x &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{i}} \\ \ast &\stackrel{x}{\longrightarrow}& \Im(X) } \,.

          The collection of all infinitesimal disks forms the infinitesimal disk bundle over XX.

          In nonstandard analysis

          In nonstandard analysis, the monad or halo of a standard point pp in a topological space (or even in a Choquet space) is the hyperset of all hyperpoint?s infinitely close to pp. It is the intersection of all of the standard neighbourhoods of pp and is itself a hyper-neighbourhood of pp, the infinitesimal neighbourhood of pp.

          It is best to avoid the term ‘monad’ for this concept on this wiki, since it has more or less nothing to with the categorial monads that are all over the place here (including elsewhere on this very page).

          For ringed spaces

          Consider a morphism (f,f ):(Y,𝒪 Y)(X,𝒪 X)(f,f^\sharp):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X) of ringed spaces for which the corresponding map f :f *𝒪 X𝒪 Yf^\sharp:f^*\mathcal{O}_X\to\mathcal{O}_Y of sheaves on YY is surjective. Let = f=Kerf \mathcal{I} = \mathcal{I}_f = Ker\,f^\sharp, then 𝒪 Y=f (𝒪 X)/ f\mathcal{O}_Y = f^\sharp(\mathcal{O}_X)/\mathcal{I}_f. The ring f *(𝒪 Y)f^*(\mathcal{O}_Y) has the \mathcal{I}-preadic filtration which has the associated graded ring Gr = n f n/ f n+1Gr_\bullet =\oplus_{n} \mathcal{I}_f^n/\mathcal{I}^{n+1}_f which in degree 11 gives the conormal sheaf Gr 1= f/ f 2Gr_1 = \mathcal{I}_f/\mathcal{I}^2_f of ff. The 𝒪 Y\mathcal{O}_Y-augmented ringed space (Y,f (𝒪 X)/ n+1)(Y,f^\sharp(\mathcal{O}_X)/\mathcal{I}^{n+1}) is called the nn-th infinitesimal neighborhood of YY along morphism ff. Its structure sheaf is called the nn-th normal invariant of ff.

          Examples of sequences of local structures

          geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
          \leftarrow differentiationintegration \to
          smooth functionsderivativeTaylor seriesgermsmooth function
          curve (path)tangent vectorjetgerm of curvecurve
          smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
          function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
          arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
          Lie theoryLie algebraformal grouplocal Lie groupLie group
          symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

          References

          In algebraic geometry (via infinitesimal shape modality)

          • A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie, Publications Mathématiques de l’IHÉS 32 (1967), p. 5-361, numdam

          Discussion in nonstandard analysis is in

          Discussion in differential cohesion is in

          Discussion in differentially cohesive homotopy type theory is in

          Last revised on June 29, 2017 at 07:14:23. See the history of this page for a list of all contributions to it.