nLab
infinitesimal neighborhood

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

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 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.

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

In nonstandard analysis

Revised on May 13, 2015 12:21:53 by Urs Schreiber (195.113.30.252)