nLab
infinitesimal neighborhood

Idea

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

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 $p$.

References

• 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

For ringed spaces

Consider a morphism $(f,f^{♯}):(Y,{𝒪}_Y)\to (X,{𝒪}_X)$ of ringed spaces for which the corresponding map ${\overline{F}}^{♯}:f^{*}{𝒪}_X\to {𝒪}_Y$ of sheaves on $Y$ is surjective. Let $ℐ={ℐ}_f=\mathrm{Ker}{\overline{f}}^{♯}$, then ${𝒪}_Y=f^{♯}({𝒪}_X)/{ℐ}_f$. The ring $f^{*}({𝒪}_Y)$ has the $ℐ$-preadic filtration which has the associated graded ring ${\mathrm{Gr}}_{•}={\oplus }_n{ℐ}_f^n/{ℐ}_f^{n+1}$ which in degree $1$ gives the conormal sheaf ${\mathrm{Gr}}_1={ℐ}_f/{ℐ}_f^2$ of $f$. The ${𝒪}_Y$-augmented ringed space $(Y,f^{♯}({𝒪}_X)/{ℐ}^{n+1})$ is called the $n$-th infinitesimal neighborhood of $Y$ along morphism $f$. Its structure sheaf is called the $n$-th normal invariant of $f$.

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