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\begin{document}

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\section*{CartSp}

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

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

[[!include synthetic differential geometry - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_a_small_category_of_objects_with_a_basis}{As a small category of objects with a basis}\dotfill \pageref*{as_a_small_category_of_objects_with_a_basis} \linebreak
\noindent\hyperlink{as_a_site}{As a site}\dotfill \pageref*{as_a_site} \linebreak
\noindent\hyperlink{as_a_category_with_open_maps}{As a category with open maps}\dotfill \pageref*{as_a_category_with_open_maps} \linebreak
\noindent\hyperlink{as_an_algebraic_theory}{As an algebraic theory}\dotfill \pageref*{as_an_algebraic_theory} \linebreak
\noindent\hyperlink{as_a_pregeometry}{As a pre-geometry}\dotfill \pageref*{as_a_pregeometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
Write $CartSp$ for the [[category]] whose

\begin{itemize}%
\item [[object]]s are [[Cartesian space]]s $\mathbb{R}^n$ for $n \in \mathbb{N}$;


\item [[morphisms]] are suitable structure-preserving [[function]]s between these spaces.



\end{itemize}
For definiteness we write

$CartSp_{lin}$ for the category whose objects are [[Cartesian space]]s regarded as [[real number|real]] [[vector space]]s and whose morphisms are [[linear function]]s between these;

\begin{itemize}%
\item $CartSp_{top}$ for the category whose objects are Cartesian spaces regarded as [[topological spaces]] equipped with their [[Euclidean topology]] and morphisms are [[continuous maps]] between them.


\item $CartSp_{smooth}$ for the category whose objects are Cartesian spaces regarded as [[smooth manifolds]] with their standard [[smooth structure]] and morphisms are [[smooth function]]s.



\end{itemize}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_a_small_category_of_objects_with_a_basis}{}\subsubsection*{{As a small category of objects with a basis}}\label{as_a_small_category_of_objects_with_a_basis}

A [[Cartesian space]] carries a lot of structure, for instance [[CartSp]] may be naturally regarded as a [[full subcategory]] of the category $C$, for $C$ (any one of) the category of

\begin{itemize}%
\item [[vector spaces]],
\item [[affine spaces]],
\item [[normed vector spaces]],
\item [[inner product spaces]],
\item [[Euclidean spaces]].

\end{itemize}
In all these cases, the inclusion $CartSp \hookrightarrow C$ is an [[equivalence of categories]]: choosing an [[isomorphism]] from any of these objects to a [[Cartesian space]] amounts to choosing a [[basis]] of a [[vector space]], a [[coordinate system]].

\hypertarget{as_a_site}{}\subsubsection*{{As a site}}\label{as_a_site}

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{itemize}%
\item [[CartSp]]${}_{top}$ for the category whose [[object]]s are Cartesian spaces and whose [[morphism]]s are all [[continuous maps]] between these.


\item [[CartSp]]${}_{smooth}$ for the category whose [[object]]s are Cartesian spaces and whose [[morphism]]s are all [[smooth functions]] between these.


\item [[CartSp]]${}_{synthdiff}$ for the [[full subcategory]] of the category of [[smooth loci]] on those of the form $\mathbb{R}^n \times D$ for $D$ an [[infinitesimal space]] (the formal dual of a Weil algebra).



\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
In all three cases there is the [[good open cover]] [[coverage]] that makes [[CartSp]] a [[site]].

\end{prop}
\begin{proof}
For [[CartSp]]${}_{top}$ this is obvious. For [[CartSp]]${}_{smooth}$ this is somewhat more subtle. It is a folk theorem (see the references at [[open ball]]). A detailed proof is at [[good open cover]]. This directly carries over to $CartSp_{synthdiff}$.

\end{proof}
\begin{prop}
\label{AsDenseSubsite}\hypertarget{AsDenseSubsite}{}
\begin{itemize}%
\item The site $CartSp_{top}$ is a [[dense subsite]] of the site of [[paracompact topological space|paracompact]] [[topological manifold]]s with the [[open cover]] [[coverage]].


\item The site $CartSp_{smooth}$ is a [[dense subsite]] of the [[site]] [[Diff]] of [[paracompact topological space|paracompact]] [[smooth manifold]]s equipped with the [[open cover]] [[coverage]].



\end{itemize}
\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
Equipped with this structure of a site, [[CartSp]] is an [[∞-cohesive site]].

\end{prop}
The corresponding [[cohesive topos]] [[sheaf topos|of sheaves]] is

\begin{itemize}%
\item $Sh_{(1,1)}(CartSp_{smooth})$, discussed at [[diffeological space]].


\item $Sh_{(1,1)}(CartSp_{synthdiff})$, discussed at [[Cahiers topos]].



\end{itemize}
The corresponding [[cohesive (∞,1)-topos]] [[(∞,1)-category of (∞,1)-sheaves|of (∞,1)-sheaves]] is

\begin{itemize}%
\item $Sh_{(\infty,1)}(CartSp_{top}) =$ [[ETop∞Grpd]];


\item $Sh_{(\infty,1)}(CartSp_{smooth}) =$ [[Smooth∞Grpd]];


\item $Sh_{(\infty,1)}(CartSp_{synthdiff}) =$ [[SynthDiff∞Grpd]];



\end{itemize}
\begin{cor}
\label{}\hypertarget{}{}
We have [[equivalences of categories]]

\begin{itemize}%
\item $Sh(CartSp_{top}) \simeq Sh(TopMfd)$


\item $Sh(CartSp_{smooth}) \simeq Sh(Diff)$



\end{itemize}
and [[equivalences of (∞,1)-categories]]

\begin{itemize}%
\item $Sh_{(\infty,1)}(CartSp_{top}) \simeq Sh_{(\infty,1)}(TopMfd)$;


\item $Sh_{(\infty,1)}(CartSp_{smooth}) \simeq Sh_{(\infty,1)}(Diff)$.



\end{itemize}
\end{cor}
\begin{proof}
The first two statements follow by the \hyperlink{AsDenseSubsite}{above proposition} with the \emph{comparison lemma} discussed at [[dense sub-site]].

For the second condition notice that since an [[∞-cohesive site]] is in particular an [[∞-local site]] we have that $Sh_{(\infty,1)}(CartSp)$ is a [[local (∞,1)-topos]]. As discussed there, this implies that it is a [[hypercomplete (∞,1)-topos]]. By the discussion at [[model structure on simplicial presheaves]] this means that it is [[presentable (∞,1)-category|presented]] by the Joyal-Jardine-[[model structure on simplicial sheaves]] $Sh(CartSp)^{\Delta^{op}}_{loc}$. The claim then follows with the first two statements.

\end{proof}
\hypertarget{as_a_category_with_open_maps}{}\subsubsection*{{As a category with open maps}}\label{as_a_category_with_open_maps}

There is a canonical structure of a category with [[open map]]s on $CartSp$ (\ldots{})

\hypertarget{as_an_algebraic_theory}{}\subsubsection*{{As an algebraic theory}}\label{as_an_algebraic_theory}

The category $CartSp$ is (the [[syntactic category]] of ) a [[Lawvere theory]]: the theory for [[smooth algebra]]s.

\hypertarget{as_a_pregeometry}{}\subsubsection*{{As a pre-geometry}}\label{as_a_pregeometry}

Equipped with the above [[coverage]]-structure, [[open map]]-structure and [[Lawvere theory]]-property, $CartSp$ is essentially a [[pregeometry (for structured (∞,1)-toposes)]].

(Except that the pullback stability of the open maps holds only in the weaker sense of [[coverage]]s).

(\ldots{})

$\,$

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

[[!include geometries of physics -- table]]

$\,$

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

In secton 2 of

\begin{itemize}%
\item [[R. Blute]], [[J.R.B. Cockett]], [[Robert Seely]], \emph{Cartesian differential categories}, Theory and Applications of Categories, Vol. 22, 2009, No. 23, pp 622-672. (\href{http://www.tac.mta.ca/tac/volumes/22/23/22-23abs.html}{journal}, \href{http://www.tac.mta.ca/tac/volumes/22/23/22-23.pdf}{pdf})

\end{itemize}
$CartSp$ is discussed as an example of a ``cartesian differential category''.

There are various slight variations of the category $CartSp$ (many of them [[equivalence of categories|equivalent]]) that one can consider without changing its basic properties as a category of test spaces for [[generalized smooth spaces]]. A different choice that enjoys some popularity in the literature is the category of open (contractible) subsets of Euclidean spaces. For more references on this see [[diffeological space]].

The [[site]] $CartSp_{synthdiff}$ of [[formal smooth manifold|infinitesimally thickened]] Cartesian spaces is known as the site for the [[Cahiers topos]]. It is considered in detail in section 5 of

\begin{itemize}%
\item [[Anders Kock]], \emph{Convenient vector spaces embed into the Cahiers topos} (\href{http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0}{numdam})

\end{itemize}
and briefly mentioned in example 2) on p. 191 of

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

\end{itemize}
following the original article

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{Sur les modeles de la geometrie differentielle synthetique} (\href{http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0}{numdam}).

\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}
category: category

[[!redirects CartSp]] [[!redirects Cart Sp]] [[!redirects CartesianSpace]] [[!redirects Cartesian Space]] [[!redirects CartesianSpaces]] [[!redirects Cartesian Spaces]]

[[!redirects SmoothCartSp]]

[[!redirects smooth Cartesian space]] [[!redirects smooth Cartesian spaces]]



\end{document}