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\section*{Lie 2-group}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

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

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

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{constructions_and_applications}{Constructions and Applications}\dotfill \pageref*{constructions_and_applications} \linebreak
\noindent\hyperlink{DeloopingLie2Groupoid}{Delooping 2-groupoids}\dotfill \pageref*{DeloopingLie2Groupoid} \linebreak
\noindent\hyperlink{principal_2bundles}{Principal 2-bundles}\dotfill \pageref*{principal_2bundles} \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}

The notion of \emph{Lie 2-group} is the generalization of the notion of [[Lie group]] as [[group]]s are generalized to [[2-groups]]:

it is a [[smooth 2-group]] that happens to have a model given by a [[Lie groupoid]] equipped with the structure of a [[group object]] (in general only up to [[homotopy]]).

One general way to make the notion precise is as a special case of an [[smooth ∞-groupoid]], namely a [[1-truncated]] [[∞-group]] object in [[∞-stack]]s over the [[site]] [[CartSp]]/[[SmthMfd]], possibly with some representability condition:

these are [[stack]]s on the [[site]] of [[smooth manifold]]s (representable by [[Lie groupoid]]s and) equipped with group structure: ``group stacks'' or ``gr-stacks''.

Special cases of this have simpler definitions. For instance a [[crossed module]] [[internal category|internal]] to [[Diff]] is a model for a strict and comparatively tame Lie 2-group.

Analogous to how the [[infinitesimal object|infinitesimal]] version of a [[Lie group]] is a [[Lie algebra]], the infinitesimal version of a Lie 2-group is a [[Lie 2-algebra]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Every ordinary [[Lie group]] $G$ is in particular a Lie 2-group with $G$ as its space of [[object]]s and only [[identity morphism]]s.


\item Every [[discrete infinity-groupoid|discrete]] [[2-group]] is a Lie 2-group equipped with [[discrete space|discrete smooth structure]];


\item For $A$ an [[abelian group|abelian]] [[Lie group]] there is a Lie 2-group denoted $\mathbf{B}A$ or $(A \to 1)$, which has a single [[object]] and $A$ as its space of [[morphism]].

For $A = U(1)$ the [[circle group]], the Lie 2-group $\mathbf{B}U(1)$ is the \emph{[[circle n-group|circle 2-group]]} .


\item Every [[crossed module]] of Lie groups $(H \to G )$, $G \to Aut(H)$ gives an example of a [[strict 2-group|strict]] Lie 2-group, using the general relation between crossed modules and [[strict 2-group]]s.

For instance the crossed module $Spin(n) \to O(n)$. Or $(U(1) \to 1)$. Etc.


\item The [[string 2-group]] $String(G)$ has various different but equivalent incarnations as a Lie 2-group. One is given by the [[crossed module]] $\hat \Omega_* G \to P_* G$ internal to [[Frechet manifold|Frechet]] Lie groups.


\item For $X$ a [[Lie groupoid]], its [[automorphism infinity-group]] is a smooth 2-group.



\end{itemize}
\hypertarget{constructions_and_applications}{}\subsection*{{Constructions and Applications}}\label{constructions_and_applications}

\hypertarget{DeloopingLie2Groupoid}{}\subsubsection*{{Delooping 2-groupoids}}\label{DeloopingLie2Groupoid}

By the discussion at [[looping and delooping]], every Lie 2-group $G$ induces a [[delooping]] [[Lie 2-groupoid]] $\mathbf{B}G$: this has a single [[object]], the space of [[morphism]]s is $G_0$, the space of [[2-morphism]]s is $G_1$ and the [[horizontal composition]] is given by the group product.

\hypertarget{principal_2bundles}{}\subsubsection*{{Principal 2-bundles}}\label{principal_2bundles}

For $X$ a [[smooth manifold]] (or itself a [[Lie groupoid]] such as an [[orbifold]], or generally any [[smooth ∞-groupoid]]), morphisms

\begin{displaymath}
g : X \to \mathbf{B}G
\end{displaymath}
of [[smooth ∞-groupoid]]s from $X$ to the \hyperlink{DeloopingLie2Groupoid}{delooping Lie 2-groupoid} $\mathbf{B}G$ classify smooth $G$-[[principal 2-bundle]]s over $X$.

If $G = AUT(H)$ is the [[automorphism 2-group]] of a [[Lie group]] $H$ then these are equivalently smooth $H$-[[gerbe]]s over $X$.

Notice that a morphism of smooth $\infty$-groupoids $X \to \mathbf{B}G$ is presented by an [[infinity-anafunctor|2-anafunctor]] of 2-groupoid valued presheaves, given by a [[span]]

\begin{displaymath}
\itexarray{
     C(U_i) &\stackrel{g}{\to}& \mathbf{B}G
     \\
     \downarrow^{\simeq}
     \\
     X
  }
 \,,
\end{displaymath}
where $C(U_i)$ is the [[Cech nerve]] 2-groupoid of some [[covering]]. The top morphism here encodes degree-1 [[nonabelian cohomology|nonabelian]] [[Cech cohomology|Cech]] [[hypercohomology]] with coefficients in $G$.

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

\begin{itemize}%
\item [[group]], [[Lie group]], [[Lie groupoid]]


\item [[2-group]], \textbf{Lie 2-group}, [[Lie 2-groupoid]]

\begin{itemize}%
\item [[Poisson Lie 2-group]]

\end{itemize}

\item [[∞-group]], [[smooth ∞-group]], [[smooth ∞-groupoid]]



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

An first exposition is in the lecture notes

\begin{itemize}%
\item Alissa Crans, \emph{A survey of higher Lie theory} (\href{http://myweb.lmu.edu/acrans/SurveyHigherLieTheory.pdf}{pdf})

\end{itemize}
A general review of Lie 2-groups, as well as a discussion of the example of the [[string 2-group]] is in

\begin{itemize}%
\item [[John Baez]], Alissa Crans, [[Urs Schreiber]], [[Danny Stevenson]], \emph{From Loop Groups to 2-Groups} (\href{http://arxiv.org/abs/math/0504123}{arXiv:math/0504123})

\end{itemize}
Discussion in a more comprehensive context is in

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

\end{itemize}
with an introduction in section 1.3.1 and a general abstract discussion in 3.3.2.

On the [[cohomology]] of Lie 2-groups:

\begin{itemize}%
\item [[Grégory Ginot]], [[Ping Xu]], \emph{Cohomology of Lie 2-groups} (\href{http://people.math.jussieu.fr/~ginot/papers/CohLie2gp.pdf}{pdf})


\item [[Christoph Wockel]], \emph{Categorified central extensions, \'e{}tale Lie 2-groups and Lies Third Theorem for locally exponential Lie algebras} (\href{http://www.sciencedirect.com/science/article/pii/S0001870811002313}{web})


\item [[Chris Schommer-Pries]], \emph{Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group} (\href{http://arxiv.org/abs/0911.2483}{arXiv})



\end{itemize}
[[!redirects Lie 2-groups]]



\end{document}