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\begin{document}

%-------------------------------------------------------------------


\section*{pro-manifold}

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

\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{procartesian_spaces}{Pro-Cartesian spaces}\dotfill \pageref*{procartesian_spaces} \linebreak
\noindent\hyperlink{embedding_into_smooth_loci}{Embedding into smooth loci}\dotfill \pageref*{embedding_into_smooth_loci} \linebreak
\noindent\hyperlink{SiteOfTowersOfCartesianSpaces}{The site of towers of Cartesian spaces and pro-morphisms}\dotfill \pageref*{SiteOfTowersOfCartesianSpaces} \linebreak


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

A \emph{pro-manifold} is a [[pro-object]] in a [[category]] of [[manifolds]], i.e. a formal [[projective limit]] of [[manifolds]].

Details depend on what exactly is understood by ``[[manifold]]'', i.e. whether [[topological manifolds]] or [[smooth manifold]], etc.

Typically one wants to mean [[pro-objects]] in manifolds of [[finite number|finite]] [[dimensions]], the point being then that a pro-manifold is like an [[infinite-dimensional manifold]] but with ``mild'' infinite dimensionality, expressed by the very fact that it may be presented as a formal projective limit of finite dimensional manifolds.

To amplify this specification, one should properly speak of ``pro-(finite dimensional smooth manifolds)'', but beware that people often abbreviate to ``pro-manifold'' regardless. Also ``pro-finite manifold'' is in use, which however, strictly speaking, is a misnomer since a ``finite manifold'' is one with a [[finite number]] of points.

An important example of pro-objects in finite-dimensional smooth manifolds are infinite [[jet bundles]]. These are the formal projective limits of the underlying finite-order jet bundles.

\hypertarget{procartesian_spaces}{}\subsection*{{Pro-Cartesian spaces}}\label{procartesian_spaces}

\hypertarget{embedding_into_smooth_loci}{}\subsubsection*{{Embedding into smooth loci}}\label{embedding_into_smooth_loci}

\begin{defn}
\label{proCartSp}\hypertarget{proCartSp}{}
Write [[CartSp]] for the [[full subcategory]] of that of [[smooth manifolds]] on the [[Cartesian spaces]], i.e. on those of the form $\mathbb{R}^n$, for $n \in \mathbb{N}$. Write

\begin{displaymath}
ProCartSp \coloneqq Pro(CartSp)
\end{displaymath}
for its category of [[pro-objects]], the \textbf{pro-Cartesian spaces}.

\end{defn}
\begin{prop}
\label{ProCartSpFullyFaithfulInSmoothLoc}\hypertarget{ProCartSpFullyFaithfulInSmoothLoc}{}
The functor which sends a formal cofiltered limit of Cartesian spaces to its actual [[cofiltered limit]] of [[smooth loci]] is a [[fully faithful functor]], hence constitutes a [[full subcategory]] inclusion of [[pro-Cartesian spaces]] (def. \ref{proCartSp}) into [[smooth loci]]:

\begin{displaymath}
Pro(CartSp) \hookrightarrow SmthLoc
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Since $Pro(\mathcal{C}) \simeq (Ind(\mathcal{C}^{op}))^{op}$ (\href{pro-object#ProObjectAsFormalDualOfIndObject}{remark}) it is sufficient to show that the functor in question is on opposite categories a fully faithful functor of the form

\begin{displaymath}
Ind(CartSp^{op})
    \hookrightarrow
  SmthLoc^{op} = SmthAlg_{\mathbb{R}}
  \,,
\end{displaymath}
where $SmothAlg_{\mathbb{R}}$ is the category of [[smooth algebras]].

Now, there is the [[fully faithful functor]]

\begin{displaymath}
i \;\colon\; CartSp \hookrightarrow SmthLoc
\end{displaymath}
(\href{smooth+locus#SmoothManifoldsFullSubcategoryOfSmoothLoci}{prop.}) hence a fully faithful functor

\begin{displaymath}
i^{op} \colon CartSp^{op} \hookrightarrow SmthAlg_{\mathbb{R}}
  \,.
\end{displaymath}
Moreover, the image of the latter is in [[compact objects]] $i^{op} \colon CartSp^{op} \hookrightarrow (SmthAlg_{\mathbb{R}})_{cpt} \hookrightarrow SmthAlg$, because

\begin{displaymath}
C^\infty(\mathbb{R}^n) \simeq y(\mathbb{R}^n)
  \in 
  SmthAlg_{\mathbb{R}} \simeq Func_\times(CartSp,Set)
\end{displaymath}
is [[representable functor|co-representable]], hence [[compact object|compact]] (by the [[Yoneda lemma]] and since colimits are computed objectwise \href{limits+and+colimits+by+example#LimitsOfPresheaves}{prop.}).

This implies that the composite

\begin{displaymath}
Ind(CartSp^{op})
   \overset{Ind(i^{op})}{\hookrightarrow}
  Ind(SmthAlg_{\mathbb{R}})
    \overset{L}{\longrightarrow}
  SmthAlg_{\mathbb{R}}
\end{displaymath}
is also fully faithful (\href{ind-object#JFIsFullyFaithful}{prop.}).

Here $Ind(i^{op})$ takes formal filtered colimits in $CartSp^{op}$ to the corresponding formal colimits in $SmthAlg_{\mathbb{R}}$ (\href{ind-object#FunctorialityOfInd}{prop.}), while $L$ takes formal filtered colimits to actual [[filtered colimits]] (\href{ind-object#ReflectionToYonedaEmbedding}{prop.}). Hence this is indeed the functor in question.

\end{proof}
\hypertarget{SiteOfTowersOfCartesianSpaces}{}\subsubsection*{{The site of towers of Cartesian spaces and pro-morphisms}}\label{SiteOfTowersOfCartesianSpaces}

\begin{quote}%
under construction


\end{quote}
\begin{defn}
\label{TowersOfCartesianSpaces}\hypertarget{TowersOfCartesianSpaces}{}
Write

\begin{displaymath}
TowCartSp \hookrightarrow ProCartSp
\end{displaymath}
for the [[full subcategory]] of the category of [[pro-Cartesian spaces]] (def. \ref{proCartSp}) on those [[pro-objects]] in [[CartSp]] which are presented as formal [[sequential limits]] of [[tower]] diagrams, i.e. where the indexing category $\mathcal{K} = \mathbb{N}_{\geq}$.

\end{defn}
\begin{defn}
\label{ProGoodOpenCoverOnACartesianSpace}\hypertarget{ProGoodOpenCoverOnACartesianSpace}{}
For $U \in TowCartSp$ a [[tower of Cartesian spaces]] (def. \ref{TowersOfCartesianSpaces}), say that a \textbf{tower of [[good open covers]]} of $U$ is a sequence of morphisms $\{U_i \overset{\phi_i}{\to} U\}$ in $TowCartSp$ such that these are the formal [[sequential limit]] of a cofiltered diagram of [[good open covers]] $\{U_i^k \overset{\phi_i^k}{\to} U^k\}$.

\begin{displaymath}
\itexarray{
    U_i^{k} &\overset{\underset{\longleftarrow}{\lim}^f}{\mapsto}& U_i
    \\
    {}^{\mathllap{\phi_i^k}}\downarrow && \downarrow^{\mathrlap{\phi_i}}
    \\
    U^k &\overset{\underset{\longleftarrow}{\lim}^f}{\mapsto}& U
  }
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The collection of towers of [[good open covers]] on $TowCartSp$, according to def. \ref{ProGoodOpenCoverOnACartesianSpace}, constitutes a [[coverage]].

\end{defn}
\begin{proof}
By the definition of [[coverage]] (\href{coverage#ConditionsOnACoverage}{def.}) we need to check that for every tower of [[good open covers]] $\{U_i \overset{\phi_i}{\to} U\}$ and for every morphism $V \overset{g}{\longrightarrow} U$ in $TowCartSp$, there exists a tower of [[good open covers]] $\{V_j \overset{\psi_j}{\longrightarrow} V\}$ of $V$ such that for each index $j$ we may find an index $i$ and a morphism $V_j \overset{}{\to} U_i$ such as to make a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    V_j &\overset{}{\longrightarrow}& U_i
    \\
    \downarrow 
      &&
    \downarrow^{\mathrlap{\phi_i}}
    \\
    V &\underset{g}{\longrightarrow}& U
  }
  \,.
\end{displaymath}
Now by \href{tower#ProMorphismsBetweenTowerDiagrams}{this prop.} the bottom morphism is represented by a sequence of component morphisms

\begin{displaymath}
V^{n(k)} \overset{}{\longrightarrow} U^k
  \,.
\end{displaymath}
Since ordinary [[good open covers]] do form a [[coverage]] on [[CartSp]] (\href{good+open+cover#GoodOpenCoversFormACoverageOnParacompactSmooothManifolds}{prop.}) each of these component diagrams may be completed

\begin{displaymath}
\itexarray{
    \tilde V^{n(k)}_j &\overset{}{\longrightarrow}& U^k_i
    \\
    \downarrow 
      &&
    \downarrow^{\mathrlap{\phi^k_i}}
    \\
    V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k
  }
\end{displaymath}
by first forming the [[pullback]] [[open cover]] $(g^k)^\ast U^k_i \to V^{n(k)}$ and then refining this to a [[good open cover]] $\tilde V^{n(k)}_j \to V^{n(k)}$. By the [[universal property]] of the [[pullback]], there are morphisms

\begin{displaymath}
\tilde V^{n(k+1)} \longrightarrow (g^k)^\ast U^k_i
\end{displaymath}
that make the evident cube commute

\begin{displaymath}
\itexarray{
    \tilde V^{n(k+1)}_j &\overset{}{\longrightarrow}& U^{k+1}_i
    \\
    \downarrow 
      &&
    \downarrow^{\mathrlap{\phi^{k+1}_i}}
    \\
    V^{n(k+1)} &\underset{g^{k+1}}{\longrightarrow}& U^{k+1}
  }  
  \;\;\;\;\;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  \itexarray{
    (g^{k})^\ast U_i^k &\overset{}{\longrightarrow}& U^k_i
    \\
    \downarrow 
      &&
    \downarrow^{\mathrlap{\phi^k_i}}
    \\
    V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k
  }
\end{displaymath}
Take

\begin{displaymath}
V^{n(0)}_j \coloneqq \tilde V^{n(0)}_j
\end{displaymath}
and then inductively define

\begin{displaymath}
V^{n(k+1)}_j
\end{displaymath}
to be a refinement by a good open cover of the joint refinement of $\{\tilde V^{n(k+1)}_j\}$ with the pullback of $\{V^{n(k)}_j\}$ to $V^{n(k+1)}$.

This refines the above commuting cubes to

\begin{displaymath}
\itexarray{
    V_j^{n(k+1)} &\overset{}{\longrightarrow}& U^{k+1}_i
    \\
    \downarrow 
      &&
    \downarrow^{\mathrlap{\phi^{k+1}_i}}
    \\
    V^{n(k+1)} &\underset{g^{k+1}}{\longrightarrow}& U^{k+1}
  }  
  \;\;\;\;\;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  \itexarray{
    V_j^{n(k)} &\overset{}{\longrightarrow}& U^k_i
    \\
    \downarrow 
      &&
    \downarrow^{\mathrlap{\phi^k_i}}
    \\
    V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k
  }
\end{displaymath}
and hence provides components for the required diagram in $TowCartSp$.

\end{proof}
[[!redirects pro-manifolds]]

[[!redirects promanifold]] [[!redirects promanifolds]]

[[!redirects pro-Cartesian space]] [[!redirects pro-Cartesian spaces]]

[[!redirects tower of Cartesian spaces]] [[!redirects towers of Cartesian spaces]] [[!redirects tower of cartesian spaces]] [[!redirects towers of cartesian spaces]]



\end{document}