# Infinitesimal neighbourhoods

## 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

Consider a morphism $(f,f^\sharp):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X)$ of ringed spaces for which the corresponding map $\bar{F}^\sharp:f^*\mathcal{O}_X\to\mathcal{O}_Y$ of sheaves on $Y$ is surjective. Let $\mathcal{I} = \mathcal{I}_f = Ker\bar{f}^\sharp$, 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 $f$.