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\begin{document}

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\section*{V-manifold}

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

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

[[!include higher geometry - contents]]

\begin{quote}%
under construction


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{VManifolds}{$V$-Manifolds}\dotfill \pageref*{VManifolds} \linebreak
\noindent\hyperlink{FrameBundles}{Frame bundles}\dotfill \pageref*{FrameBundles} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In the [[axiom|axiomatics]] of [[differential cohesion]] one may [[synthetic differential geometry|synthetically]] formulate a concept of \emph{[[manifolds]]} locally modeled on a [[group object]] $V$. In the [[interpretation]] in an [[differential cohesion|differentially]] [[cohesive (infinity,1)-topos]] these are [[étale infinity-groupoids]].

For exposition see at \emph{[[geometry of physics -- manifolds and orbifolds]]} and \emph{[[geometry of physics -- supergeometry]]}.

\hypertarget{VManifolds}{}\subsection*{{$V$-Manifolds}}\label{VManifolds}

\begin{defn}
\label{LocalDiffeomorphismsInDifferentialCohesion}\hypertarget{LocalDiffeomorphismsInDifferentialCohesion}{}
Given $X,Y\in \mathbf{H}$ then a morphism $f \;\colon\; X\longrightarrow Y$ is a \emph{[[local diffeomorphism]]} if its naturality square of the [[infinitesimal shape modality]]

\begin{displaymath}
\itexarray{
    X &\longrightarrow& \Im X
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\longrightarrow& \Im Y
  }
\end{displaymath}
is a [[homotopy pullback]] square.

\end{defn}
Let now $V \in \mathbf{H}$ be given, equipped with the structure of a [[group]] ([[∞-group]]).

\begin{defn}
\label{VManifold}\hypertarget{VManifold}{}
A \emph{[[V-manifold]]} is an $X \in \mathbf{H}$ such that there exists a \emph{$V$-[[atlas]]}, namely a [[correspondence]] of the form

\begin{displaymath}
\itexarray{
    && U
    \\
    & \swarrow && \searrow
    \\
    V && && X
  }
\end{displaymath}
with both morphisms being [[local diffeomorphisms]], def. \ref{LocalDiffeomorphismsInDifferentialCohesion}, and the right one in addition being an [[epimorphism]], hence an [[atlas]].

\end{defn}
\hypertarget{FrameBundles}{}\subsection*{{Frame bundles}}\label{FrameBundles}

\begin{defn}
\label{InfinitesimalDisk}\hypertarget{InfinitesimalDisk}{}
For $X \in \mathbf{H}$ an object and $x \colon \ast \to X$ a point, then we say that the \emph{[[infinitesimal neighbourhood]] of}, or the \emph{[[infinitesimal disk]] at} $x$ in $X$ is the [[homotopy fiber]] $\mathbb{D}^X_x$ of the [[unit of a monad|unit]] of the [[infinitesimal shape modality]] at $x$:

\begin{displaymath}
\itexarray{
    \mathbb{D}^X_x &\longrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    \ast &\stackrel{x}{\longrightarrow}& \im X
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{FormalDiskBundle}\hypertarget{FormalDiskBundle}{}
For $X$ any object in differential cohesion, its \emph{infinitesimal disk bundle} $T_{inf} X \to X$ is the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    T_{inf} X &\stackrel{ev}{\longrightarrow}& X
    \\
    \downarrow^{\mathrlap{p}} && \downarrow
    \\
    X &\longrightarrow& \Im X
  }
\end{displaymath}
of the [[unit of a monad|unit]] of its [[infinitesimal shape modality]] along itself.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
By the [[pasting law]], the [[homotopy fiber]] of the infinitesimal disk bundle, def. \ref{FormalDiskBundle}, over any point $x \in X$ is the infinitesimal disk $\mathbb{D}^X_x$ in $X$ at that point, def.\ref{InfinitesimalDisk}. Nevertheless, for general $X$ the infinitesimal disk bundle need not be an [[fiber ∞-bundle]] with typical fiber (the infinitesimal disks at different points need not be equivalent, and even if they are, the bundle need not be locally trivial). Below in prop. \ref{FormalDiskBundleOfRegularManifoldsTrivializesOverCover} we see that for $X$ a $V$-manifold modeled on a group object $V$, then its infinitesimal disk bundle is indeed an [[fiber ∞-bundle]], and hence is the [[associated ∞-bundle]] to some [[principal ∞-bundle]]. That principal bundle is the [[frame bundle]] of $X$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The [[Atiyah groupoid]] of $T_{inf} X$ is the [[jet groupoid]] of $X$.

\end{remark}
\begin{lemma}
\label{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle}\hypertarget{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle}{}
If $\iota \colon U \to X$ is a local diffeomorphism, def. \ref{LocalDiffeomorphismsInDifferentialCohesion}, then

\begin{displaymath}
\iota^\ast T_{inf} X \simeq T_{inf}U
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
By the definition of local diffeos and using the [[pasting law]] we have an equivalence of [[pasting diagrams]] of [[homotopy pullbacks]] of the following form:

\begin{displaymath}
\itexarray{
     \iota^\ast T_{inf} X &\longrightarrow& T_{inf} &\longrightarrow& X
     \\
     \downarrow && \downarrow && \downarrow
     \\
     U &\longrightarrow& X &\longrightarrow& \Im X
  }
  \;\;\;\;
  \simeq
  \;\;\;\;
  \itexarray{
      T_{inf} U &\longrightarrow& U &\longrightarrow& X
       \\
       \downarrow && \downarrow && \downarrow
       \\
       U &\longrightarrow& \Im U &\longrightarrow& \Im X
  }
\end{displaymath}
\end{proof}
\begin{defn}
\label{Framing}\hypertarget{Framing}{}
For $V$ an object, a \textbf{[[framing]]} on $V$ is a trivialization of its infinitesimal disk bundle, def. \ref{FormalDiskBundle}, i.e. an object $\mathbb{D}^V$ -- the typical [[infinitesimal disk]] or [[formal disk]], def. \ref{InfinitesimalDisk}, -- and a (chosen) [[equivalence]]

\begin{displaymath}
\itexarray{
    T_{inf} V && \stackrel{\simeq}{\longrightarrow} && V \times \mathbb{D}^n
    \\
    & \searrow && \swarrow_{\mathrlap{p_1}}
    \\
    && V
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{GeneralLinearGroup}\hypertarget{GeneralLinearGroup}{}
For $V$ a framed object, def. \ref{Framing}, we write

\begin{displaymath}
GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V)
\end{displaymath}
for the [[automorphism ∞-group]] of its typical [[infinitesimal disk]]/[[formal disk]].

\end{defn}
\begin{remark}
\label{OrderOfInfinitesimalDisks}\hypertarget{OrderOfInfinitesimalDisks}{}
When the [[infinitesimal shape modality]] exhibits first-order infinitesimals, such that $\mathbb{D}(V)$ is the first order [[infinitesimal neighbourhood]] of a point, then $\mathbf{Aut}(\mathbb{D}(V))$ indeed plays the role of the [[general linear group]]. When $\mathbb{D}^n$ is instead a higher order or even the whole [[formal neighbourhood]], then $GL(n)$ is rather a [[jet group]]. For order $k$-jets this is sometimes written $GL^k(V)$ We nevertheless stick with the notation ``$GL(V)$'' here, consistent with the fact that we have no index on the [[infinitesimal shape modality]]. More generally one may wish to keep track of a whole tower of infinitesimal shape modalities and their induced towers of concepts discussed here.

\end{remark}
This class of examples of framings is important:

\begin{prop}
\label{DifferentialCohesiveInfinityGroupIsCanonicallyFramed}\hypertarget{DifferentialCohesiveInfinityGroupIsCanonicallyFramed}{}
Every differentially cohesive [[∞-group]] $G$ is canonically framed (def. \ref{Framing}) such that the horizontal map in def. \ref{FormalDiskBundle} is given by the left action of $G$ on its [[infinitesimal disk]] at the neutral element:

\begin{displaymath}
ev \;\colon\; T_{inf}G \simeq G \times \mathbb{D}^G_e \stackrel{\cdot}{\longrightarrow} G
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the discussion at \emph{[[Mayer-Vietoris sequence]]} in the section \emph{\href{Mayer-Vietoris%20sequence#OverAGroupObject}{Over an ∞-group}} and using that the [[infinitesimal shape modality]] preserves group structure, the defining [[homotopy pullback]] of $T_{inf} G$, def. \ref{FormalDiskBundle}, is equivalent to the pasting of pullback diagrams

\begin{displaymath}
\itexarray{
    T_{inf} G &\stackrel{}{\longrightarrow}&
    \mathbb{D}^G_e
    &\stackrel{}{\longrightarrow}& \ast
    \\
    \downarrow &&  \downarrow && \downarrow
    \\
    G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G
    &\stackrel{}{\longrightarrow}&
    \Im G
  }
\end{displaymath}
where the right square is the defining pullback for the [[infinitesimal disk]] $\mathbb{D}^G$. Finally for the left square we find by \href{Mayer-Vietoris%20sequence#HTTArgumentForPullback}{this proposition} that $T_{inf} G \simeq G\times \mathbb{D}^G$ and that the top horizontal morphism is as claimed.

\end{proof}
By lemma \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} it follows that:

\begin{prop}
\label{FormalDiskBundleOfRegularManifoldsTrivializesOverCover}\hypertarget{FormalDiskBundleOfRegularManifoldsTrivializesOverCover}{}
For $V$ a framed object, def. \ref{Framing}, let $X$ be a $V$-manifold, def. \ref{VManifold}. Then the infinitesimal disk bundle, def. \ref{FormalDiskBundle}, of $X$ canonically trivializes over any $V$-cover $V \leftarrow U \rightarrow X$ , i.e. there is a [[homotopy pullback]] of the form

\begin{displaymath}
\itexarray{
     U \times \mathbb{D}^V &\longrightarrow& T_{inf} X
     \\
     \downarrow && \downarrow
     \\
     U &\longrightarrow& X
  }
  \,.
\end{displaymath}
This exhibits $T_{inf} X\to X$ as a $\mathbb{D}^V$-[[fiber ∞-bundle]].

\end{prop}
\begin{prop}
\label{ModulatingMapOfFormalDiskBundle}\hypertarget{ModulatingMapOfFormalDiskBundle}{}
By \href{fiber+infinity-bundle#Properties}{this discussion} this fiber [[fiber ∞-bundle]] is the [[associated ∞-bundle]] of an essentially uniquely determined $\mathbf{Aut}(\mathbb{D}^V)$-[[principal ∞-bundle]] $Fr(X)$, i.e. there exists a [[homotopy pullback]] diagram of the form

\begin{displaymath}
\itexarray{
     T_{inf} X \simeq 
     & 
   Fr(X) \underset{\mathbf{Aut}(\mathbb{D}^V)}{\times} \mathbb{D}^V 
     &\longrightarrow&
     V//\mathbf{Aut}(\mathbb{D}^V)
     \\
     & \downarrow && \downarrow
     \\
     & X &\stackrel{}{\longrightarrow}& \mathbf{B}\mathbf{Aut}(\mathbb{D}^V)
  }
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{FrameBundleMap}\hypertarget{FrameBundleMap}{}
Given a $V$-manifold $X$, def. \ref{VManifold}, for framed $V$, def. \ref{Framing}, then its \emph{[[frame bundle]]}

\begin{displaymath}
\itexarray{
     Fr(X)
     \\
     \downarrow
     \\
     X 
  }
\end{displaymath}
is the $GL(V)$-[[principal ∞-bundle]] given by prop. \ref{FormalDiskBundleOfRegularManifoldsTrivializesOverCover} via remark \ref{ModulatingMapOfFormalDiskBundle}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
As in remark \ref{OrderOfInfinitesimalDisks}, this really axiomatizes in general [[higher order frame bundles]] with the order implicit in the nature of the [[infinitesimal shape modality]].

\end{remark}
\begin{remark}
\label{FrameBundlesFunctorial}\hypertarget{FrameBundlesFunctorial}{}
By prop. \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} the construction of frame bundles in def. \ref{FrameBundleMap} is functorial in [[formally étale maps]] between $V$-manifolds.

\end{remark}
This provides all the necessary structure to now set up an axiomatic theory of [[G-structure]] and [[higher Cartan geometry]]. This is discussed further at \emph{[[geometry of physics -- G-structure and Cartan geometry]]}.

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

The concept is due to

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


\item [[Igor Khavkine]], [[Urs Schreiber]], \emph{[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations Part I -- Jets and comonad structure]]} (\href{https://arxiv.org/abs/1701.06238}{arXiv:1701.06238})



\end{itemize}
Formalization in [[modal type theory|modal]] [[homotopy type theory]] is in

\begin{itemize}%
\item [[Felix Wellen]], \emph{[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]}, 2017 (\href{http://www.math.kit.edu/iag3/~wellen/media/diss.pdf}{pdf})


\item [[Felix Wellen]], \emph{Cartan Geometry in Modal Homotopy Type Theory} (\href{https://arxiv.org/abs/1806.05966}{arXiv:1806.05966})



\end{itemize}
[[!redirects V-manifolds]]



\end{document}