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\begin{document}

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\section*{bisection of a Lie groupoid}

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{bisections_of_lie_groupoids}{}\section*{{Bisections of Lie groupoids}}\label{bisections_of_lie_groupoids}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{abstractly}{Abstractly}\dotfill \pageref*{abstractly} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_lierinehart_algebras}{Relation to Lie-Rinehart algebras}\dotfill \pageref*{relation_to_lierinehart_algebras} \linebreak
\noindent\hyperlink{relation_to_atiyah_groupoids}{Relation to Atiyah groupoids}\dotfill \pageref*{relation_to_atiyah_groupoids} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components}

\begin{defn}
\label{TraditionalDefinition}\hypertarget{TraditionalDefinition}{}
Let $(X_1 \stackrel{(d_0, d_1)}{\to} X_0 \times X_0)$ be a [[Lie groupoid]].

A \textbf{bisection} of is a [[smooth function]] $\sigma : X_0 \to X_1$ such that

\begin{enumerate}%
\item $\sigma$ is a [[section]] of $d_1$;


\item $d_0 \circ \sigma : X_0 \to X_0$ is a [[diffeomorphism]].



\end{enumerate}
Bisections naturally form a [[group]] under pointwise composition in $X$, the \textbf{group of bisections} of the Lie groupoid.

\end{defn}
One can prove that the bisection group is a [[infinite-dimensional Lie group]] in the sense of Milnor (see Neeb's survey) (under some mild assumptions on the underlying Lie groupoid). The infinite-dimensional Lie group of bisections is closely connected to the underlying Lie groupoid (see references below), e.g.

\begin{enumerate}%
\item From the knowledge of the smooth structure of the bisection group and the manifold of units, one can even reconstruct the underlying Lie groupoid (again under some assumptions).


\item The construction is functorial in a suitable sense and extending this one can even relate (smooth) representations of Lie groupoids to smooth representations of its bisection group



\end{enumerate}
\hypertarget{abstractly}{}\subsubsection*{{Abstractly}}\label{abstractly}

Let $\mathbf{H} =$ [[Smooth∞Grpd]]. Let $X \in \mathbf{H}$ be equipped with an \emph{atlas}, hence with an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] $U \to X$ out of a [[0-truncated object]].

We may regard this atlas as an object in the [[slice (∞,1)-topos]] $\mathbf{X} \in \mathbf{H}_{/X}$

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{[[smooth ∞-group]] of bisections} of $\mathbf{X}$ is its [[automorphism ∞-group]]

\begin{displaymath}
\mathbf{BiSect}(X,U) \coloneqq
  \mathbf{Aut}_{/X}(\mathbf{X}, \mathbf{X})
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $X$ a 1-groupoid as above and $U = X_0$, a bisection is precisely a smooth [[natural transformation]] of the form

\begin{displaymath}
\itexarray{
    U &&\stackrel{\simeq}{\to}&& U
    \\
    & \searrow &\swArrow_{\mathrlap{\eta}}& \swarrow
    \\
    && X
  }
  \,.
\end{displaymath}
Here the top morphism is a [[diffeomorphism]] $\phi : X \to X$ and since the diagonal morphisms are identities onto the object manifold the component map of $\eta$ is

\begin{displaymath}
x \mapsto  (x \stackrel{\eta(x)}{\to} \phi(x))  \,.
\end{displaymath}
This is precisely the bisection in the traditional sense of def. \ref{TraditionalDefinition}.

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

\hypertarget{relation_to_lierinehart_algebras}{}\subsubsection*{{Relation to Lie-Rinehart algebras}}\label{relation_to_lierinehart_algebras}

For $U \to X$ a Lie groupoid with atlas as above, write $\mathfrak{g} = Lie(\mathbf{BiSect}(X,U))$ for the [[Lie algebra]] of the group of bisections. Then $(C^\infty(X), \mathfrak{g})$ is the [[Lie-Rinehart algebra]] corresponding to the [[Lie algebroid]] of the Lie groupoid.

\hypertarget{relation_to_atiyah_groupoids}{}\subsubsection*{{Relation to Atiyah groupoids}}\label{relation_to_atiyah_groupoids}

\begin{quote}%
for the moment see at \emph{[[Atiyah groupoid]]} and \emph{[[higher Atiyah groupoid]]}.


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

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Janez Mr?un]], p. 114 of \emph{[[Introduction to foliations and Lie groupoids]]}, Cambridge Studies in Advanced Mathematics \textbf{91}, 2003. x+173 pp. ISBN: 0-521-83197-0


\item [[Alexander Schmeding]], [[Christoph Wockel]], \emph{The Lie group of bisections of a Lie groupoid} (\href{http://arxiv.org/abs/1409.1428}{arXiv:1409.1428})


\item [[Alexander Schmeding]], [[Christoph Wockel]], \emph{(Re)constructing Lie groupoids from their bisections and applications to prequantisation} (\href{https://arxiv.org/pdf/1506.05415.pdf}{arXiv:1506.05415})


\item [[Alexander Schmeding]], [[Christoph Wockel]], \emph{Functorial aspects of the reconstruction of Lie groupoids from their bisections} (\href{https://arxiv.org/abs/1506.05587}{arXiv:1506.05587})


\item Habib Amiri, [[Alexander Schmeding]], \emph{Linking Lie groupoid representations and representations of infinite-dimensional Lie groups} (\href{https://arxiv.org/pdf/1805.03935.pdf}{arXiv:1805.03935})



\end{itemize}
[[!redirects bisections of a Lie groupoid]]

[[!redirects group of bisections]]

[[!redirects bisection]] [[!redirects bisections]]

[[!redirects ∞-bisection]] [[!redirects ∞-bisections]]

[[!redirects ∞-group of bisections]] [[!redirects ∞-groups of bisections]]



\end{document}