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\begin{document}

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\section*{smooth structure on a topos}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[topos]] can be viewed as a generalization of a [[topological space]]. A \emph{smooth structure on a topos} is a corresponding generalization of the notion of [[smooth manifold]], in that to put a smooth structure on $Sh(X)$, for a topological space $X$, is (at least closely related to) putting the structure of a smooth manifold on $X$. However, since it is phrased internally with reference to the (Cauchy and Dedekind) [[real numbers objects]], it is applicable to any topos.

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

Given a [[topos]] $\mathbf{H}$, write

\begin{itemize}%
\item $\mathbb{N} \in \mathbf{H}$ for its [[natural numbers object]];


\item $\mathbb{R}_{C} \in \mathbf{H}$ for its [[Cauchy real numbers object]];


\item $\mathbb{R}_{D} \in \mathbf{H}$ for its [[Dedekind real numbers object]];


\item $\mathbb{R} \in$ [[Set]] for the external [[real numbers]].



\end{itemize}
There are canononical [[subobject]] inclusions

\begin{displaymath}
\mathbb{N} \hookrightarrow \mathbb{R}_C \hookrightarrow \mathbb{R}_D
  \,.
\end{displaymath}
\begin{defn}
\label{StandardFunction}\hypertarget{StandardFunction}{}
(\hyperlink{Fourman75}{Fourman 75, def. 3.6}) A morphism

\begin{displaymath}
g \colon \mathbb{R}_D^{n_1} \longrightarrow \mathbb{R}_D^{n_2}
\end{displaymath}
in the topos $\mathbf{H}$ is called a \textbf{standard function}, precisely if

\begin{enumerate}%
\item it is a [[continuous function]] in the internal sense that

\begin{displaymath}
\underset{\epsilon \gt 0}{\forall} 
  \underset{\delta \gt 0}{\exists}
  \underset{\vec x, \vec y \in \mathbb{R}_D^{n_1}}{\forall}
  \left(   
    \left(
      \underset{i}{max} {\vert x_i - y_i \vert}
      \lt \delta
    \right)
    \Rightarrow
    \left(
      \underset{i}{max} {\vert g(\vec x)_i - g(\vec y)_i \vert}
      \lt \delta
    \right)
  \right)
\end{displaymath}

\item it respects the sub-object of Cauchy reals, in that

\begin{displaymath}
\underset{\vec x \in \mathbb{R}_C^{n_1}}{\forall} 
  \left(
     g(\vec x) \in \mathbb{R}_{C}^{n_2}
  \right)
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
We will furthermore consider \textbf{smooth standard functions}, meaning standard functions that satisfy the internalized ordinary definition of [[smooth function]] (i.e. $C^\infty$).

We define an [[equivalence relation]] on $\mathbb{R}_D^n$ by taking two elements to be equivalent if there are smooth standard functions taking them into each other:

\begin{displaymath}
\vdash 
  (r \simeq s)
  \coloneqq
  \underset{{f,g } \atop {smooth \; standard\;functions}}{\exists}
  \left( 
     \left(
       f(r) = s
     \right)
        \wedge
     \left(
       g(s) = r
     \right)
  \right)
  \,.
\end{displaymath}
\begin{defn}
\label{SmoothStructureDimN}\hypertarget{SmoothStructureDimN}{}
(\hyperlink{Fourman75}{Fourman 75, def. 4.1}) A \textbf{smooth structure of dimension $n$} on a topos $\mathbf{H}$ is an equivalence class for the above equivalence relation on $\mathbb{R}_D^n$. In other words, the object of smooth structures of dimension $n$ is the quotient of $\mathbb{R}_D^n$ by this equivalence relation.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
(\hyperlink{Fourman75}{Fourman 75, def. 4.4}) Given a smooth structure $S$ of dimension $n$ on the topos $\mathbf{H}$ according to def. \ref{SmoothStructureDimN}, then the \textbf{smooth real number object} is the [[subobject]]

\begin{displaymath}
\mathbb{R}_S \hookrightarrow \mathbb{R}_D
\end{displaymath}
defined internally as the set of all images of points in $S\subseteq \mathbb{R}_D^n$ under smooth standard functions $\mathbb{R}_D^n \to \mathbb{R}_D$. In symbols, it is given by

\begin{displaymath}
\mathbb{R}_S
  \coloneqq
  \left\{
    x \in \mathbb{R}_D
    \;|\;
    \underset{{smooth \; standard\;funct.}\atop{\mathbb{R}_D^n \stackrel{f}{\to} \mathbb{R}_D}}{\exists}
    \underset{s \in S}{\exists}
    (x = f(s))  
  \right\}
  \,.
\end{displaymath}
\end{defn}
Note that any function constant at a Cauchy real is standard. Therefore, every Cauchy real is a smooth real.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
(\hyperlink{Fourman75}{Fourman 75, example 4.3 1}) Let $X$ be a [[smooth manifold]] of [[dimension]] $n$, with [[sheaf topos]] $\mathbf{H} \coloneqq Sh(X)$. As shown at [[real numbers object]], $\mathbb{R}_D$ is then the sheaf of \emph{continuous} real-valued functions on $X$.

Let $S\subseteq \mathbb{R}_D^n$ be the sheaf of local [[coordinate systems]], i.e. $S(U)$ is the set of real-valued functions $U\to \mathbb{R}^n$ that are smooth and are locally diffeomorphisms onto their images. Then $S$ is a smooth structure on $Sh(X)$ of dimension $n$, according to def. \ref{SmoothStructureDimN}.

The corresponding object $\mathbb{R}_S$ of smooth reals is the sheaf of \emph{smooth} real-valued functions $X\to \mathbb{R}$. Note that since $X$ is locally connected, the Cauchy real numbers object $\mathbb{R}_C$ in $Sh(X)$ is the sheaf of locally constant real-valued functions, so $\mathbb{R}_S$ sits strictly in between $\mathbb{R}_C$ and $\mathbb{R}_D$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Let $h:1\to \mathbb{R}_D^n$ be any global section of $\mathbb{R}_D^n$. Then the equivalence class of $h$, under the above equivalence relation, is a smooth structure. In particular, if $h$ is a tuple of Cauchy real numbers (such as $0$), then so is every point in $S$, and thus $\mathbb{R}_S = \mathbb{R}_C$. This gives the ``discrete'' smooth structure on $\mathbf{H}$.

For $\mathbf{H}=Sh(X)$, such a global section is a continuous map $h:X\to \mathbb{R}^n$, and this gives the smooth structure ``cogenerated by'' $h$, in that it makes $h$ smooth and only those other functions that must be smooth if $h$ is. The case when $h$ is a Cauchy real corresponds to $h:X\to \mathbb{R}^n$ being locally constant, in which case all smooth functions are locally constant; this is the ``minimal'' smooth structure on a topological space $X$.

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

\begin{itemize}%
\item [[Michael Fourman]], \emph{Comparaison des R\'e{}els d'un Topos - Structures Lisses sur un Topos El\'e{}mentaire} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{16} (1975) pp.233-239. ( \emph{Colloque Amiens 1975 proceedings} ) (p. 18-24 in \href{http://www.numdam.org/item?id=CTGDC_1975__16_3_217_0}{NUMDAM}))

\end{itemize}


\end{document}