# nLab infinitesimal neighborhood

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

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

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

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

### In nonstandard analysis

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

### For ringed spaces

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

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ 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 geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

## References

• 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)