# 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 $\left(f,{f}^{♯}\right):\left(Y,{𝒪}_{Y}\right)\to \left(X,{𝒪}_{X}\right)$ 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 ${𝒪}_{Y}={f}^{♯}\left({𝒪}_{X}\right)/{ℐ}_{f}$. The ring ${f}^{*}\left({𝒪}_{Y}\right)$ 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 $\left(Y,{f}^{♯}\left({𝒪}_{X}\right)/{ℐ}^{n+1}\right)$ is called the $n$-th infinitesimal neighborhood of $Y$ along morphism $f$. Its structure sheaf is called the $n$-th normal invariant of $f$.