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\begin{document}

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\section*{Cahiers 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{cohesive_toposes}{}\paragraph*{{Cohesive toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{synthetic_differential_geometry_2}{Synthetic differential geometry}\dotfill \pageref*{synthetic_differential_geometry_2} \linebreak
\noindent\hyperlink{connectedness_locality_and_cohesion}{Connectedness, locality and cohesion}\dotfill \pageref*{connectedness_locality_and_cohesion} \linebreak
\noindent\hyperlink{ConvenientVectorSpaces}{Convenient vector spaces}\dotfill \pageref*{ConvenientVectorSpaces} \linebreak
\noindent\hyperlink{synthetic_tangent_bundles_of_smooth_spaces}{Synthetic tangent bundles of smooth spaces}\dotfill \pageref*{synthetic_tangent_bundles_of_smooth_spaces} \linebreak
\noindent\hyperlink{RelationToSyntheticTangentSpaces}{Synthetic tangent spaces}\dotfill \pageref*{RelationToSyntheticTangentSpaces} \linebreak
\noindent\hyperlink{variants_and_generalizations}{Variants and generalizations}\dotfill \pageref*{variants_and_generalizations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{Cahier topos} is a [[cohesive topos]] that constitutes a [[Models for Smooth Infinitesimal Analysis|well-adapted model]] for [[synthetic differential geometry]] (a ``[[smooth topos]]'').

It is the [[sheaf topos]] on the [[site]] [[FormalCartSp]] of [[infinitesimal object|infinitesimally thickened]] [[Cartesian spaces]].

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

\begin{defn}
\label{}\hypertarget{}{}
Let [[FormalCartSp]] be the [[full subcategory]] of the category of [[smooth loci]] on those of the form

\begin{displaymath}
\mathbb{R}^n \times \ell W
  \,,
\end{displaymath}
consisting of a [[product]] of a [[Cartesian space]] with an [[infinitesimally thickened point]], i.e. a formal dual of a \emph{Weil algebra} .

Dually, the [[opposite category]] is the [[full subcategory]] $FormalCartSp^{op} \hookrightarrow SmoothAlg$ of [[smooth algebras]] on those of the form

\begin{displaymath}
C^\infty( \mathbb{R}^k \times \ell W)
  = 
  C^\infty(\mathbb{R}^k) \otimes W
  \,.
\end{displaymath}
\end{defn}
This appears for instance in \hyperlink{KockReyes}{Kock Reyes (1)}.

\begin{defn}
\label{}\hypertarget{}{}
Define a structure of a [[site]] on [[FormalCartSp]] by declaring a [[covering]] family to be a family of the form

\begin{displaymath}
\{
    U_i \times \ell W \stackrel{p_i \times Id}{\to} U \times \ell W
  \}
\end{displaymath}
where $\{U_i \stackrel{p_i}{\to} U\}$ is an [[open cover]] of the [[Cartesian space]] $U$ by Cartesian spaces $U_i$.

\end{defn}
This appears as \hyperlink{Kock}{Kock (5.1)}.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{Cahiers topos} $\mathcal{CT}$ is the [[category of sheaves]] on this site:

\begin{displaymath}
\mathcal{CT} := Sh(FormalCartSp)  
  \,.
\end{displaymath}
\end{defn}
This site of definition appears in \hyperlink{KockReyes}{Kock, Reyes}. The original definition is due to \hyperlink{Dubuc79}{Dubuc 79}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{synthetic_differential_geometry_2}{}\subsubsection*{{Synthetic differential geometry}}\label{synthetic_differential_geometry_2}

\begin{prop}
\label{ModelForSynthetic}\hypertarget{ModelForSynthetic}{}
The Cahiers topos is a well-adapted model for [[synthetic differential geometry]].

\end{prop}
This is due to \hyperlink{Dubuc79}{Dubuc 79}.

\hypertarget{connectedness_locality_and_cohesion}{}\subsubsection*{{Connectedness, locality and cohesion}}\label{connectedness_locality_and_cohesion}

\begin{prop}
\label{}\hypertarget{}{}
The Cahiers topos is a [[cohesive topos]]. See [[synthetic differential infinity-groupoid]] for details.

\end{prop}
\hypertarget{ConvenientVectorSpaces}{}\subsubsection*{{Convenient vector spaces}}\label{ConvenientVectorSpaces}

\begin{prop}
\label{}\hypertarget{}{}
The [[category]] of [[convenient vector spaces]] with [[smooth functions]] between them embeds as a [[full subcategory]] into the Cahiers topos.

The embedding is given by sending a convenient vector space $V$ to the sheaf given by

\begin{displaymath}
V
  : 
  \mathbb{R}^k \times \ell W
  \mapsto 
  C^\infty(\mathbb{R}^k, V) \otimes W
  \,.
\end{displaymath}
\end{prop}
This result was announced in \hyperlink{Kock}{Kock}. See the corrected proof in (\hyperlink{KockReyes}{KockReyes}).

\begin{remark}
\label{}\hypertarget{}{}
Together with prop. \ref{ModelForSynthetic} this means that the [[differential geometry]] on convenient vector spaces may be treated synthetically in the Cahiers topos.

\end{remark}
\hypertarget{synthetic_tangent_bundles_of_smooth_spaces}{}\subsubsection*{{Synthetic tangent bundles of smooth spaces}}\label{synthetic_tangent_bundles_of_smooth_spaces}

\hypertarget{RelationToSyntheticTangentSpaces}{}\subsubsection*{{Synthetic tangent spaces}}\label{RelationToSyntheticTangentSpaces}

We discuss here induced [[synthetic tangent spaces]] of [[smooth spaces]] in the sense of [[diffeological spaces]] and more general [[sheaves]] on the site of [[smooth manifolds]] after their canonical embedding into the Cahiers topos.

\begin{defn}
\label{}\hypertarget{}{}
Write $SmoothLoc$ for the [[category]] of [[smooth loci]]. Write

\begin{displaymath}
CartSp \hookrightarrow SmoothLoc
\end{displaymath}
for the [[full subcategory]] on the [[Cartesian spaces]] $\mathbb{R}^n$ ($n \in \mathbb{N}$). Write

\begin{displaymath}
InfThPoint \hookrightarrow SmoothLoc
\end{displaymath}
for the [[full subcategory]] on the [[infinitesimally thickened points]], and write

\begin{displaymath}
CartSp_{synthdiff} \hookrightarrow SmoothLoc
\end{displaymath}
for the full subcategory on those smooth loci which are the [[cartesian product]] of a [[Cartesian space]] $\mathbb{R}^n$ ($n \in \mathbb{N}$) and an [[infinitesimally thickened point]].

We regard [[CartSp]] as a [[site]] by equipping it with the [[good open cover]] [[coverage]]. We regard $InfThPoint$ as equipped with the trivial coverage and $CartSp_{synthdiff}$ as equipped with the induced product coverage.

The [[sheaf topos]] $Sh(CartSp)$ is that of \emph{[[smooth spaces]]}. The sheaf topos $Sh(CartSp_{synthdiff})$ is the \emph{Cahier topos}.

\end{defn}
\begin{example}
\label{InfinitesimalInterval}\hypertarget{InfinitesimalInterval}{}
We write

\begin{displaymath}
D \coloneqq \ell(\mathbb{R}[\epsilon]/(\epsilon^2))
  \in 
  InfThPt \hookrightarrow CartSp_{synthdiff}
\end{displaymath}
for the [[infinitesimal space|infinitesimal interval]], the [[smooth locus]] dual to the [[smooth algebra]] ``[[ring of dual numbers|of dual numbers]]''.

\end{example}
\begin{defn}
\label{SyntheticTangentBundle}\hypertarget{SyntheticTangentBundle}{}
For $X \in Sh(CartSp_{synthdiff})$ any object in the Cahier topos, its \textbf{[[synthetic tangent bundle]]} in the sense of [[synthetic differential geometry]] is the [[internal hom]] space $X^D$, equipped with the projection map

\begin{displaymath}
X(\ast \to D) \colon X^D \to X
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The canonical inclusion functor $i \colon CartSp \to CartSp_{synthdiff}$ induces an [[adjoint pair]]

\begin{displaymath}
Sh(CartSp)
   \stackrel{\overset{i_!}{\to}}{\underset{i^\ast}{\leftarrow}}
  Sh(CartSp_{synthdiff})
\end{displaymath}
where $i^\ast$ is given by precomposing a [[presheaf]] on $CartSp_{synthdiff}$ with $i$. The [[left adjoint]] $i_!$ has the interpretation of the inclusion of [[smooth spaces]] as [[reduced objects]] in the Cahiers topos.

\end{prop}
This is discussed in more detail at \emph{[[synthetic differential infinity-groupoid]]}.

\begin{prop}
\label{CharacterizationOfImageInCahiersTopos}\hypertarget{CharacterizationOfImageInCahiersTopos}{}
For $X \in Sh(CartSp)$ a [[smooth space]], and for $\ell(W) \in InfThPoint$ an infinitesimally thickened point, the morphisms

\begin{displaymath}
\ell(W) \to i_! X
\end{displaymath}
in $Sh(CartSp_{synthdiff})$ are in natural [[bijection]] to [[equivalence classes]] of pairs of morphisms

\begin{displaymath}
\ell(W) \to \mathbb{R}^n \to X
\end{displaymath}
consisting of a morphism in $CartSp_{synth}$ on the left and a morphism in $Sh(CartSp)$ on the right (which live in different categories and hence are not composable, but usefully written in juxtaposition anyway). The [[equivalence relation]] relates two such pairs if there is a [[smooth function]] $\phi \colon \mathbb{R}^n \to \mathbb{R}^{n'}$ such that in the diagram

\begin{displaymath}
\itexarray{
    && \mathbb{R}^n
    \\
    & \nearrow & & \searrow
    \\
    \ell(W) && \downarrow^{\mathrlap{\phi}} && X
    \\
    & \searrow && \nearrow
    \\
    && \mathbb{R}^{n'}
  }
\end{displaymath}
the left triangle commutes in $CartSp_{synthdiff}$ and the right one in $Sh(CartSp)$.

\end{prop}
\begin{proof}
By general properties of [[left adjoints]] of functors of [[presheaves]], $i_! X$ is the [[left Kan extension]] of the presheaf $X$ along $i$. By the [[Yoneda lemma]] and the [[coend]] formula for these (as discussed there), we have that the set of maps $\ell(W) \to i_! X$ is naturally identified with

\begin{displaymath}
(i_! X)(\ell(W))
  =
  (Lan_i X)(\ell(W))
  =
  \int^{\mathbb{R}^n \in CartSp}
  Hom_{CartSp_{synthdiff}}(\ell(W), \mathbb{R}^n)
  \times 
  X(\mathbb{R}^n)
  \,.
\end{displaymath}
Unwinding the definition of this [[coend]] as a [[coequalizer]] yields the above description of equivalence classes.

\end{proof}
\hypertarget{variants_and_generalizations}{}\subsection*{{Variants and generalizations}}\label{variants_and_generalizations}

\begin{itemize}%
\item When restricting the [[site]] of infinitesimally thickened Cartesian spaces to that of plain [[Cartesian spaces]] one obtaines the topos discussed at \emph{[[smooth space]]}. This is still a [[cohesive topos]], but no longer a model for [[synthetic differential geometry]].


\item The [[(∞,1)-sheaf (∞,1)-topos]] over $CartSp_{th}$ is disucssed at [[synthetic differential ∞-groupoid]]. It contains that Cahiers topos as the sub-[[(n,1)-topos|(1,1)-topos]] of [[0-truncated]] objects.


\item [[Dubuc topos]]



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

The Cahiers topos was introduced in

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{Sur les mod\`e{}les de la g\'e{}om\'e{}trie diff\'e{}rentielle synth\'e{}tique} [[Cahiers]] de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 20 no. 3 (1979), p. 231-279 (\href{http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0}{numdam}).

\end{itemize}
and got its name from this journal publication. The definition appears in theorem 4.10 there, which asserts that it is a well-adapted model for [[synthetic differential geometry]].

A review discussion is in section 5 of

\begin{itemize}%
\item [[Anders Kock]], \emph{Convenient vector spaces embed into the Cahiers topos}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 27 no. 1 (1986), p. 3-17 (\href{http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0}{numdam})

\end{itemize}
and with a corrected definition of the site of definition in

\begin{itemize}%
\item [[Anders Kock]], [[Gonzalo Reyes]], \emph{Corrigendum and addenda to: Convenient vector spaces embed into the Cahiers topos}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 27 no. 1 (1986), p. 3-17 (\href{http://www.numdam.org/item?id=CTGDC_1987__28_2_99_0}{numdam})

\end{itemize}
It appears briefly mentioned in example 2) on p. 191 of the standard textbook

\begin{itemize}%
\item [[Anders Kock]], \emph{Synthetic differential geometry} (\href{http://home.imf.au.dk/kock/sdg99.pdf}{pdf})

\end{itemize}
With an eye towards [[Frölicher space]]s the site is also considered in section 5 of

\begin{itemize}%
\item Hirokazu Nishimura, \emph{Beyond the Regnant Philosophy of Manifolds} (\href{http://arxiv.org/abs/0912.0827}{arXiv:0912.0827})

\end{itemize}
The [[(∞,1)-topos]] analog of the Cahiers topos ([[synthetic differential ∞-groupoids]]) is discussed in section 3.4 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}


\end{document}