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\begin{document}

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\section*{Haefliger groupoid}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Variants}{Variants and Generalizations}\dotfill \pageref*{Variants} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classification_of_foliations}{Classification of foliations}\dotfill \pageref*{classification_of_foliations} \linebreak
\noindent\hyperlink{universal_characterization}{Universal characterization}\dotfill \pageref*{universal_characterization} \linebreak
\noindent\hyperlink{sheaves_and_stacks_on_the_haefliger_groupoid}{Sheaves and stacks on the Haefliger groupoid.}\dotfill \pageref*{sheaves_and_stacks_on_the_haefliger_groupoid} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$, the \textbf{Haefliger groupoid} $\Gamma^n$ is the [[groupoid]] whose set of [[objects]] is the [[Cartesian space]] $\mathbb{R}^n$ and for which a [[morphism]] $x \to y$ is a [[germ]] of a [[diffeomorphism]] $(\mathbb{R}^n ,x) \to (\mathbb{R}^n ,y)$.

This is regarded as a [[topological groupoid|topological]] or [[Lie groupoid|Lie]] [[étale groupoid]] via the canonical [[topological space|topology]]/[[smooth structure]] on $(\Gamma^n)_0 = \mathbb{R}^n$ and taking $s \colon (\Gamma^n)_1 \to (\Gamma^n)_0$ to be the [[étale space]] \href{etale+space#RelationToSheaves}{associated} to the [[sheaf]] on $\mathbb{R}^n$ (with its canonical [[open cover]] [[Grothendieck topology]]) which is the [[sheafification]] of the presheaf that sends $U \subset \mathbb{R}^n$ to the set of all open embeddings of $U$ into $\mathbb{R}^n$.

The [[smooth stack]] represented by the smooth Haefliger groupoid is also called the \emph{Haefliger stack}.

\end{defn}
\hypertarget{Variants}{}\subsection*{{Variants and Generalizations}}\label{Variants}

\begin{remark}
\label{}\hypertarget{}{}
There is also the full smooth structure on the space of germs of diffeomorphisms. This gives a Lie groupoid whose underlying bare groupoid is the same as that of the Haefliger groupoid, but whose smooth structure is different.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
More generally for a given [[integrable G-stucture]] there is a corresponding Haefliger groupoid, for instance for [[symplectic structures]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Instead of considering [[germs]] of local diffeomorphisms one may consider (just) order-$k$ [[jets]] of these. The resulting Lie groupoids are known as [[jet groupoids]] (see \hyperlink{Lorenz09}{Lorenz 09})

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

\hypertarget{classification_of_foliations}{}\subsubsection*{{Classification of foliations}}\label{classification_of_foliations}

The Haefliger groupoid classifies [[foliations]]. See at \emph{[[Haefliger theorem]]}.

\hypertarget{universal_characterization}{}\subsubsection*{{Universal characterization}}\label{universal_characterization}

Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.

\begin{prop}
\label{HIsTerminalInEtaleEtale}\hypertarget{HIsTerminalInEtaleEtale}{}
The Haefliger stack is a [[terminal object]] in the [[2-category]] of [[étale stacks]] on the site of [[smooth manifolds]] with [[étale morphisms]] between them.

\end{prop}
(\hyperlink{Carchedi12}{Carchedi 12, theorem 3.3.})

This implies (\hyperlink{Carchedi12}{Carchedi 12, 3,2})

\begin{theorem}
\label{StacksOnEtaleSiteAndEtaleStacks}\hypertarget{StacksOnEtaleSiteAndEtaleStacks}{}
There is an equivalence

\begin{displaymath}
\Theta \colon St(SmthMfd^{et}) \simeq EtSt(SmthMfd)^{et}
\end{displaymath}
between [[stacks]] on the [[site]] of smooth manifolds with [[local diffeomorphisms]] between them and [[étale stacks]] with [[étale morphisms]] between them inside all [[smooth stacks]].

\end{theorem}
(\hyperlink{Carchedi12}{Carchedi 12, theorem 1.3})

\begin{remark}
\label{}\hypertarget{}{}
This in turn implies for instance that the Haefliger groupoid for [[complex structures]] (\hyperlink{Carchedi12}{Carchedi 12, p. 38}) is simply the image under the equivalence $\Theta$ in theorem \ref{StacksOnEtaleSiteAndEtaleStacks} of the sheaf on $SmthMfd^{et}$ which sends each smooth manifold to its set of [[complex structures]]. (\ldots{})

\end{remark}
\hypertarget{sheaves_and_stacks_on_the_haefliger_groupoid}{}\subsubsection*{{Sheaves and stacks on the Haefliger groupoid.}}\label{sheaves_and_stacks_on_the_haefliger_groupoid}

Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.

\begin{prop}
\label{HIsTerminalInEtaleEtale}\hypertarget{HIsTerminalInEtaleEtale}{}
The [[category of sheaves]] over $\mathcal{H}$ is equivalently the category of sheaves on the [[site]] of [[smooth manifolds]] with [[local diffeomorphism]] between them.

\end{prop}
(\hyperlink{Carchedi12}{Carchedi 12, theorem 3.1}).

\begin{prop}
\label{}\hypertarget{}{}
The [[2-topos]] over the Haefliger stack is equivalent to the 2-topos over the site $SmthMfd^{et}$ of smooth manifolds with local diffeomorphisms between them:

\begin{displaymath}
St(\mathcal{H}) \simeq St(SmthMfd^{et})
\end{displaymath}
\end{prop}
(\hyperlink{Carchedi12}{Carchedi 12, 3.2}).

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

Original articles include

\begin{itemize}%
\item [[André Haefliger]], \emph{Groupo\"i{}des d'holonomie et espaces classiants} , Ast\'e{}risque 116 (1984), 70-97


\item [[Raoul Bott]], \emph{Lectures on characteristic classes and foliations} , Springer LNM 279, 1-94



\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Janez Mrčun]], \emph{[[Introduction to foliations and Lie groupoids]]}, Cambridge Studies in Advanced Mathematics \textbf{91}, 2003. x+173 pp. ISBN: 0-521-83197-0

\end{itemize}
Discussion in a broader context of [[étale stacks]] is in

\begin{itemize}%
\item [[David Carchedi]], section 2.2, section 3 of \emph{\'E{}tale Stacks as Prolongations} (\href{http://arxiv.org/abs/1212.2282}{arXiv:1212.2282})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Haefliger_structure}{Haefliger structure}}

\end{itemize}
Discussion of [[jet groupoids]] includes

\begin{itemize}%
\item Arne Lorenz, \emph{Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method}, Thesis (\href{http://wwwb.math.rwth-aachen.de/~arne/Dissertation_Lorenz_Arne.pdf}{pdf}) 2009

\end{itemize}
The [[geometric realization]]/[[shape modality]] for Haefliger-type groupoids is discussed in

\begin{itemize}%
\item [[David Carchedi]], \emph{On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces} (\href{http://arxiv.org/abs/1504.02394}{arXiv:1504.02394})

\end{itemize}
[[!redirects Haefliger groupoids]]

[[!redirects Haefliger stack]] [[!redirects Haefliger stacks]]



\end{document}