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\begin{document}

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\section*{higher Klein geometry}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{higher_super_poincar_klein_geometry}{Higher super Poincar\'e{} Klein geometry}\dotfill \pageref*{higher_super_poincar_klein_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\emph{Higher Klein geometry} is the generalization of [[Klein geometry]] from traditional ([[differential geometry|differential]]) [[geometry]] to [[higher geometry]]:

where Klein geometry is about ([[Lie group|Lie]]) [[groups]] and their [[quotient]]s, higher Klein geometry is about ([[smooth infinity-group|smooth]]) [[∞-groups]] and their [[groupoid object in an (∞,1)-category|∞-quotients]].

The way that the generalization proceeds is clear after the following observation.

\begin{prop}
\label{}\hypertarget{}{}
Let $G$ be a [[discrete group]] and $H \hookrightarrow G$ a subgroup. Write $\mathbf{B}G$ and $\mathbf{B}H$ for the corresponding [[delooping]] [[groupoid]]s with a single object. Then the [[action groupoid]] $G//H$ is the [[homotopy fiber]] of the inclusion [[functor]]

\begin{displaymath}
\mathbf{B}H \to \mathbf{B}G
\end{displaymath}
in the [[(2,1)-category]] [[Grpd]]: we have a [[fiber sequence]]

\begin{displaymath}
G//H \to \mathbf{B}H \to \mathbf{B}G
\end{displaymath}
that exhibits $G//H$ as the $G$-[[principal bundle]] over $\mathbf{B}H$ which is classified by the [[cocycle]] $\mathbf{B}H \to \mathbf{B}G$.

Moreover, the [[decategorification]] of the [[action groupoid]] (its [[0-truncated|0-truncation]]) is the ordinary [[quotient]]

\begin{displaymath}
\tau_0 (G//H) = G/H
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This should all be explained in detail at [[action groupoid]].

The fact that a [[quotient]] is given by a [[homotopy fiber]] is a special case of the general theorem discussed at \href{http://ncatlab.org/nlab/show/limit+in+a+quasi-category#WithValInooGrpd}{∞-colimits with values in ∞Grpd}

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
That fiber sequence continues to the left as

\begin{displaymath}
H \to G \to G//H \to \mathbf{B}H \to \mathbf{B}G
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The above statement remains true \emph{verbatim} if [[discrete group]]s are generalized to [[Lie group]]s -- or other \href{http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#InfinGroups}{cohesive groups} -- if only we pass from the [[(2,1)-topos]] [[Grpd]] of [[discrete groupoid]]s to the [[(2,1)-topos]] [[Smooth∞Grpd|SmoothGrpd]] of \emph{smooth groupoids} .

\end{prop}
This follows with the discussion at [[smooth ∞-groupoid -- structures]].

Since the quotient $G/H$ \emph{is} what is called a \emph{[[Klein geometry]]} and since by the above observations we have analogs of these quotients for \href{http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#InfinGroups}{higher cohesive groups}, there is then an evident definition of \emph{higher Klein geometry} :

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

Let $\mathbf{H}$ be a choice of [[cohesive (∞,1)-topos|cohesive structure]]. For instance choose

\begin{itemize}%
\item $\mathbf{H} =$ [[Disc∞Grpd]] for discrete higher Klein geometry (no actual geometric structure);


\item $\mathbf{H} =$ [[ETop∞Grpd]] for continuous higher Klein geometry (with [[topological structure]]);


\item $\mathbf{H} =$ [[Smooth∞Grpd]] for higher Klein geometry based on [[differential geometry]];


\item $\mathbf{H} =$ [[SuperSmooth∞Grpd]] for the [[supergeometry]] version of higher Klein geometry


\item and so on.



\end{itemize}
\begin{defn}
\label{}\hypertarget{}{}
An \textbf{$\infty$-Klein geometry} in $\mathbf{H}$ is a [[fiber sequence]] in $\mathbf{H}$

\begin{displaymath}
G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G
\end{displaymath}
for $i$ any morphism between two [[connected]] objects, as indicated, hence $\Omega i : H \to G$ any morphism of [[∞-group]] objects.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $X$ an object equipped with a $G$-[[action]] and $f : Y \to X$ any morphism, the higher Klein geometry induced by ``the shape $Y$ in $X$'' is given by taking $i : H \to G$ be the [[stabilizer ∞-group]] $Stab(f) \to G$ of $f$ in $X$. See there at \emph{\href{stabilizer+group#KleinGeometry}{Examples -- Stabilizers of shapes / Klein geometry}}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
\begin{itemize}%
\item By the discussion at [[looping and delooping]], and using that a [[cohesive (∞,1)-topos]] has [[homotopy dimension]] 0 it follows that every connected object indeed is the delooping of an [[∞-group]] object.


\item The above says that $G//H$ is the [[principal ∞-bundle]] over $\mathbf{B}H$ that is classified by the [[cocycle]] $i$.

Continuing this [[fiber sequence]] further to the left yields the long fiber sequence

\begin{displaymath}
H \to G \to G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G
\end{displaymath}
This exhibits $G$ indeed as the [[fiber]] of $G//H \to \mathbf{B}H$.



\end{itemize}
\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{higher_super_poincar_klein_geometry}{}\subsubsection*{{Higher super Poincar\'e{} Klein geometry}}\label{higher_super_poincar_klein_geometry}

Let $\mathbf{H} =$ [[SuperSmooth∞Grpd]] be the context for [[SynthDiff∞Grpd|synthetic]] [[higher geometry|higher]] [[supergeometry]].

Write $\mathfrak{sugra}_11$ for the [[super L-∞ algebra]] called the [[supergravity Lie 6-algebra]]. This has a sub-super $L_\infty$-algebra of the form

\begin{displaymath}
\mathbf{B}(\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R})
  \hookrightarrow
  \mathbf{B}\mathfrak{sugra}_11
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\mathfrak{so}(d,1)$ is the [[special orthogonal Lie algebra]];


\item $b^{n-1} \mathbb{R}$ is the [[line Lie n-algebra]].



\end{itemize}
The quotient

\begin{displaymath}
\mathfrak{sugra}_11 / 
  ((\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}))
\end{displaymath}
is the [[super translation Lie algebra]] in 11-dimensions.

This higher Klein geometry is the local model for the [[higher Cartan geometry]] that describes [[11-dimensional supergravity]]. See [[D'Auria-Fre formulation of supergravity]] for more on this.

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

[[!include local and global geometry - table]]

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

That there ought to be a systematic study of higher Klein geometry and [[higher Cartan geometry]] has been amplified by [[David Corfield]] since 2006.

Such a formalization is offered in

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

\end{itemize}
For more on this see at \emph{[[higher Cartan geometry]]} and \emph{[[schreiber:Higher Cartan Geometry]]}.

[[!redirects higher Klein geometries]]



\end{document}