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\begin{document}

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\section*{infinitesimal neighborhood}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{infinitesimal_neighbourhoods}{}\section*{{Infinitesimal neighbourhoods}}\label{infinitesimal_neighbourhoods}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_differential_cohesion}{In differential cohesion}\dotfill \pageref*{in_differential_cohesion} \linebreak
\noindent\hyperlink{in_nonstandard_analysis}{In nonstandard analysis}\dotfill \pageref*{in_nonstandard_analysis} \linebreak
\noindent\hyperlink{for_ringed_spaces}{For ringed spaces}\dotfill \pageref*{for_ringed_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\begin{quote}%
Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuit\"a{}t hat, nicht aber ein blosser Punct, oder die Grenze zwischen bestimmten Stellen im Raume; (\href{Grundriss+des+Eigenth&#252;mlichen+der+Wissenschaftslehre#4IVUnendlichKleinsterTeilDesRaumes}{Fichte 1795, Grundriss \S{}4.IV})


\end{quote}
An infinitesimal neighbourhood is a [[neighbourhood]] with [[infinitesimal]] diameter. These can be defined in several setups: [[nonstandard analysis]], [[synthetic differential geometry]], [[ringed spaces]], \ldots{}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_differential_cohesion}{}\subsubsection*{{In differential cohesion}}\label{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|homotopy]]) [[pullback]] of the [[unit of a monad|unit]] $i \colon X \to \Im(X)$ of the $\Im$-monad

\begin{displaymath}
\itexarray{
    \mathbb{D}^X_x &\longrightarrow& X
    \\
    \downarrow && \downarrow^{\mathrlap{i}}
    \\
    \ast &\stackrel{x}{\longrightarrow}& \Im(X)
  }
  \,.
\end{displaymath}
The collection of all infinitesimal disks forms the [[infinitesimal disk bundle]] over $X$.

\hypertarget{in_nonstandard_analysis}{}\subsubsection*{{In nonstandard analysis}}\label{in_nonstandard_analysis}

In [[nonstandard analysis]], the \textbf{monad} or \textbf{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 \textbf{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).

\hypertarget{for_ringed_spaces}{}\subsubsection*{{For ringed spaces}}\label{for_ringed_spaces}

Consider a morphism $(f,f^\sharp):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X)$ of [[ringed space]]s 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 \textbf{infinitesimal neighborhood} of $Y$ along morphism $f$. Its structure sheaf is called the $n$-th normal invariant of $f$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[jet groupoid]]

\end{itemize}
[[!include infinitesimal and local - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

In [[algebraic geometry]] (via [[infinitesimal shape modality]])

\begin{itemize}%
\item [[A. Grothendieck]], \emph{\'E{}l\'e{}ments de g\'e{}om\'e{}trie alg\'e{}brique (r\'e{}dig\'e{}s avec la collaboration de Jean Dieudonn\'e{}) : IV. \'E{}tude locale des sch\'e{}mas et des morphismes de sch\'e{}mas, Quatri\`e{}me partie}, Publications Math\'e{}matiques de l'IH\'E{}S \textbf{32} (1967), p. 5-361, \href{http://www.numdam.org/item?id=PMIHES_1967__32__5_0}{numdam}

\end{itemize}
Discussion in nonstandard analysis is in

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Monad_%28non-standard_analysis%29}{Monad (non-standard analysis)}
\item S. S. Kutateladze, \emph{Leibnizian, Robinsonian, and Boolean valued monads} \href{http://arxiv.org/abs/1106.2755}{arxiv/1106.2755}
\item [[Sergio Albeverio]], Jens Erik Fenstad, Raphael Hoegh-Krohn, Nonstandard methods in stochastic analysis and mathematical physics, Academic Press 1986

\end{itemize}
Discussion in [[differential cohesion]] is in

\begin{itemize}%
\item [[Igor Khavkine]], [[Urs Schreiber]], \emph{[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]} (\href{https://arxiv.org/abs/1701.06238}{arXiv:1701.06238})

\end{itemize}
Discussion in differentially cohesive [[homotopy type theory]] is in

\begin{itemize}%
\item [[Felix Wellen]], \emph{[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]}, 2017

\end{itemize}
[[!redirects infinitesimal neighborhood]] [[!redirects infinitesimal neighborhoods]] [[!redirects infinitesimal neighbourhood]] [[!redirects infinitesimal neighbourhoods]]

[[!redirects monad in nonstandard analysis]] [[!redirects monads in nonstandard analysis]] [[!redirects monad in non-standard analysis]] [[!redirects monads in non-standard analysis]]

[[!redirects halo]] [[!redirects halos]] [[!redirects haloes]]

[[!redirects infinitesimal disk]] [[!redirects infinitesimal disks]]



\end{document}