infinitesimal neighborhood

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, ….

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, the **monad** or **halo** 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 $p$.

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).

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**

geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|

$\leftarrow$ differentiation | integration $\to$ | |||||||

smooth functions | derivative | Taylor series | germ | smooth function | ||||

curve (path) | tangent vector | jet | germ of curve | curve | ||||

smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||

function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||

arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||

Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||

symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |

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

- wikipedia Monad (non-standard analysis)
- S. S. Kutateladze,
*Leibnizian, Robinsonian, and Boolean valued monads*arxiv/1106.2755 - Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn, Nonstandard methods in stochastic analysis and mathematical physics, Academic Press 1986

Revised on December 18, 2016 03:36:54
by Toby Bartels
(108.167.41.14)