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\begin{document}

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\section*{lined topos}

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

\hypertarget{synthetic_differential_geometry}{}\paragraph*{{Synthetic differential geometry}}\label{synthetic_differential_geometry}

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \linebreak
\noindent\hyperlink{constructions_in_lined_toposes}{Constructions in lined toposes}\dotfill \pageref*{constructions_in_lined_toposes} \linebreak
\noindent\hyperlink{path_objects}{Path objects}\dotfill \pageref*{path_objects} \linebreak
\noindent\hyperlink{ContractibleObjects}{Contractible objects}\dotfill \pageref*{ContractibleObjects} \linebreak


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

\begin{udefn}
A \textbf{lined topos} $(\mathcal{T}, R)$ is

\begin{itemize}%
\item a [[ringed topos]] $(\mathcal{T}, k)$

(usually with the [[internalization|internal]] [[ring]] object $(k,+,\cdot)$ assumed to be commutative)


\item and equipped with a choice $(R,+,\cdot)$ of [[internalization|internal]] commutative [[algebra]] object $(R,+,\cdot)$ over $k$ -- the \textbf{line object}.



\end{itemize}
\end{udefn}
\hypertarget{variations}{}\subsection*{{Variations}}\label{variations}

\begin{itemize}%
\item A [[smooth topos]] is a lined topos where the line is required to be a smooth differentiable space with [[infinitesimal object|infinitesimal]] subspaces in a certain way. This is the basic type of object studied in [[synthetic differential geometry]].


\item A [[super smooth topos]] is a lined topos that is a [[smooth topos]] and in which the $k$-algebra structure on $R$ is refined to that of a $k$-[[superalgebra]].



\end{itemize}
\hypertarget{constructions_in_lined_toposes}{}\subsection*{{Constructions in lined toposes}}\label{constructions_in_lined_toposes}

\hypertarget{path_objects}{}\subsubsection*{{Path objects}}\label{path_objects}

The line object $R$ in a lined topos $\mathcal{T}$ canonically has the structure of a [[interval object|cartesian interval object]].

As described there, this canonically induces

\begin{itemize}%
\item a [[cosimplicial object]] $\Delta_R : \Delta \to \mathcal{T}$


\item a functor $\Pi : \mathcal{T} \to S \mathcal{T}$

that sends each object in the topos to a [[simplicial object]]

\begin{displaymath}
X \mapsto X^{\Delta_R^\bullet}
\end{displaymath}
( which may be interpreted as presenting the [[schreiber:path ∞-groupoid|path ∞-groupoid]] of $X$).



\end{itemize}
\hypertarget{ContractibleObjects}{}\subsubsection*{{Contractible objects}}\label{ContractibleObjects}

The following terminology is sometimes useful.

\begin{udefn}
\textbf{(contractible object)}

Let $(\mathcal{T} = Sh(C), R)$ be a lined [[Grothendieck topos]] with respect to a [[site]] $C$.

Call an object $X \in \mathcal{T}$ \textbf{contractible} with respect to the [[interval object]] $R$, if the [[simplicial presheaf|simplicial sheaf]] $\Pi(X) = X^{\Delta_R^\bullet} : C^{op} \to$ [[SSet]] sends each object to a [[contractible space|contractible]] [[simplicial set]].

\end{udefn}
\textbf{Examples}

\begin{itemize}%
\item \textbf{sheaves on topological spaces} Let $Top'$ be a [[small category|small]] version of the category of sufficiently nice [[topological space]]s, for instance connected [[CW complex]]es. The canonical line object in $Sh(Top)$ is ${*} \stackrel{0}{\to} [0,1] \stackrel{1}{\leftarrow} {*}$ the standard topological interval. For $X \in Top$, $\Pi(X) = X^{\Delta_R^\bullet}$ is the [[singular simplicial complex]] of $X$. This is contractible in the above sense precisely if $X$ is a [[contractible space]] in the standard sense.


\item \textbf{sheaves on cartesian spaces} Let [[CartSp]] be the full subcategory of [[Diff]] on [[smooth manifold]]s of the form $\mathbb{R}^n$, for $n \in \mathbb{N}$. The canonical line object in $\mathcal{T} = Sh(CartSp)$ is the real line regarded as an [[interval object]]

\begin{displaymath}
R = ({*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*})
  \,.
\end{displaymath}
\begin{lemma}
\label{}\hypertarget{}{}
In the lined topos $(\mathcal{T} = Sh(CartSp), R = \mathbb{R})$ the [[representable functor|representable]] objects $\mathbb{R}^n$ are contractible with respect to $R$.

\end{lemma}
\begin{proof}
This is not quite as entirely trivial as it may seem on first sight, but follows directly from the [[Tietze extension theorem]] for smooth manifolds:

we check that for all $V \in$ [[CartSp]] every [[boundary of a simplex]] $\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V)$ extends through $\partial \Delta[k] \hookrightarrow \Delta[k]$:

by the \href{http://ncatlab.org/nlab/show/interval+object#FundGeomInftCat}{construction of the cosimplicial object} $\Delta_R : \Delta \to Sh(CartSp)$ we have that morphisms $\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V)$ correspond to smooth maps from the boundary of a $V$-cylinder over the standard $k$-simplex in $\mathbb{R}^k \times V 
\to \mathbb{R}^n$. Since this is a closed subset of $\mathbb{R}^k \times V$, by the [[Tietze extension theorem]] these maps extend (apply the theorem to each of their components) to all of $\mathbb{R}^k \times V$, hence in particular to the standard $k$-simplex inside $\mathbb{R}^k$ defined by the interval object. This constitutes the required extension to a $V$-family of $k$-simplices in $\mathbb{R}^n$

\begin{displaymath}
\itexarray{
    \partial \Delta[n] &\to& (\mathbb{R}^n)^{\Delta_R^\bullet}(V)
    \\
    \downarrow & \nearrow
    \\
    \Delta[n]
  }
  \,.
\end{displaymath}
\end{proof}

\item \textbf{sheaves on cartesian smooth loci} A small variation of the above example leads to [[smooth topos]]es with contractible representables:

let $CartSp_{synth} \subset \mathbb{L}$ be the full [[subcategory]] of [[smooth loci]] on those smooth loci of the form $\mathbb{R}^n \times D_k(r)$, where $D_k(r)$ is the [[infinitesimal space]] of $k$th order infinitesimal neighbours of the origin in $\mathbb{R}^r$.

The line object is again ${*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}$ as in the above example. Crucially, the [[infinitesimal space]]s $D_k(r)$ all have a unique point ${*} \to D_k(r)$. Accordingly, there is also a unique morphism $R^n \to D_k(r)$ for all $n$. It follows that simplices in $R^n \times D_k(r)$ are simplices in $R^n$ as above, and trivial as maps to the $D_k(r)$-factor. Hence the above argument carries over to this case and shows that all the $\mathbb{R}^n \times D_k(r)$ are contractible.



\end{itemize}


\end{document}