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\begin{document}

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\section*{Lie&#39;s three theorems}

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

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

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

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

\noindent\hyperlink{lies_three_theorems}{Lie's three theorems}\dotfill \pageref*{lies_three_theorems} \linebreak
\noindent\hyperlink{restriction_to_simply_connected_lie_groups}{Restriction to simply connected Lie groups}\dotfill \pageref*{restriction_to_simply_connected_lie_groups} \linebreak
\noindent\hyperlink{generalization_of_lies_theorems_to_lie_groupoids}{Generalization of Lie's theorems to Lie groupoids}\dotfill \pageref*{generalization_of_lies_theorems_to_lie_groupoids} \linebreak
\noindent\hyperlink{motivation_for_generalized_smooth_groupoids}{Motivation for generalized smooth groupoids}\dotfill \pageref*{motivation_for_generalized_smooth_groupoids} \linebreak
\noindent\hyperlink{generalization_of_lies_theorems_to_stacky_lie_groupoids}{Generalization of Lie's theorems to stacky Lie groupoids}\dotfill \pageref*{generalization_of_lies_theorems_to_stacky_lie_groupoids} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{lies_three_theorems}{}\subsection*{{Lie's three theorems}}\label{lies_three_theorems}

There is an obvious [[functor]]

\begin{displaymath}
Lie : Lie Gp \to Lie Alg
\end{displaymath}
which sends every [[Lie group]] to its [[Lie algebra]] and every homomorphism of Lie groups to the corresponding homomorphism of Lie algebras.

Lie's three theorems can be understood as establishing salient properties of this functor. More exactly, Lie's theorems provide a foundation establishing an [[equivalence]] between [[local Lie groups]] and Lie algebras; subsequent work by [[Elie Cartan]] and others extended the theorems to give information on (global) Lie groups via the functor $Lie$.

\begin{enumerate}%
\item \textbf{Lie's first theorem} is purely local; see the \href{http://www.encyclopediaofmath.org/index.php/Lie_theorem}{Encyclopedia of Math} for a statement. (Here one lacks a good notion of [[differentiable manifold]] for extending this to a global result.)


\item \textbf{Lie II} Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak{g} = Lie(G)$ and $\mathfrak{h} = Lie(H)$, such that $G$ is [[simply connected space|simply connected]]. If $f : \mathfrak{g} \to \mathfrak{h}$ is a morphism of Lie algebras, then there is a unique morphism $F : G \to H$ of Lie groups lifting $f$, i.e. such that $f = Lie(F)$.


\item \textbf{Lie III} (Cartan-Lie theorem) The functor $Lie$ \textbf{is essentially surjective on objects}: for every finite dimensional real Lie algebra $\mathfrak{g}$ there is a real Lie group $G$ such that $\mathfrak{g} \cong Lie(G)$. Moreover, there exists such $G$ which is simply connected.



\end{enumerate}
\begin{uremark}
In his third theorem, Lie proved only the existence of of a [[local Lie group]], but not the global existence (nor simply connected choice) which were established a few decades later by [[Elie Cartan]]. Hence the full theorem is properly called the Cartan-Lie theorem. From an [[nPOV]], the third Lie theorem establishes the essential surjectivity of the functor $Lie$ from the category of \emph{local Lie groups} to the category of finite dimensional real Lie algebras, and similarly the second Lie theorem establishes that this functor is fully faithful (so the two together establish that this functor is an equivalence). The historically incorrect naming of the Cartan-Lie theorem as the ``third Lie theorem'' is largely due to the influence of a book based on lectures of [[Jean-Pierre Serre]] (Lie algebras and Lie groups, W.A. Benjamin, 1965).

\end{uremark}
\hypertarget{restriction_to_simply_connected_lie_groups}{}\subsection*{{Restriction to simply connected Lie groups}}\label{restriction_to_simply_connected_lie_groups}

Let $LieGroups_{simpl}$ be the [[full subcategory]] of $LieGroups$ consisting of simply connected Lie groups. Then the above implies that restricted to $LieGroups_{simpl}$, the functor $Lie$ becomes an [[equivalence of categories]].

\hypertarget{generalization_of_lies_theorems_to_lie_groupoids}{}\subsection*{{Generalization of Lie's theorems to Lie groupoids}}\label{generalization_of_lies_theorems_to_lie_groupoids}

The [[horizontal categorification]] of Lie's theorems for Lie groups leads to analogous statements for [[Lie groupoid|Lie groupoids]]. In other words, there are analogous properties for the differentiation functor

\begin{displaymath}
diff : LieGroupoids \to LieAlgebroids
  \,.
\end{displaymath}
from [[Lie groupoid|Lie groupoids]] to [[Lie algebroid|Lie algebroids]].

In the case of Lie groupoids, the condition of a group being simply connected which plays an important role in the above statements is generalized to the condition that \emph{source fibers} of the Lie groupoid (the preimages $s^{-1}(x)$ of the source map $s : C_1 \to C_0$ at every object $x \in C_0$ of the Lie groupoid $C$) are simply connected. One says

\begin{displaymath}
(C is source-simply connected)
  \Leftrightarrow
  (\forall x \in C_0 : 
    \pi_1(s^{-1}(x)) = 0
  )
  \,.
\end{displaymath}
\textbf{\emph{Lie II} for Lie groupoids} now reads exactly as \emph{Lie II} for Lie groups, with ``simply connected'' replaced by ``source simply connected''.

Lie II for Lie groupoids was proven in

\begin{itemize}%
\item K. C. H. Mackenzie and P. Xu, \emph{Integration of Lie bialgebroids}, Topology, 39(3):445-467

\end{itemize}
and

\begin{itemize}%
\item I. Moerdijk and J Mrun, \emph{On integrability of infinitesimal actions}, Amer. J. Math. 124(3):567-593, 2002

\end{itemize}
\textbf{\emph{Lie III} for Lie groupoids} does \emph{not} hold in direct generalization:

by the general mechanism of [[Lie integration]] the space of morphisms of the source simply-connected \emph{topological} groupoid $G$ integrating a Lie algebroid $\mathfrak{g}$ is a \emph{quotient} space. This quotient may fail to be a \emph{manifold} due to singularities.

The precise conditions under which Lie algebroids do have Lie groupoids integrating them were found in

\begin{itemize}%
\item Crainic and Fernandes, \emph{Integrability of Lie brackets} (\href{http://arxiv.org/abs/math.DG/0105033}{arXiv}).

\end{itemize}
A comprehensive review is in

\begin{itemize}%
\item Rui Loja Fernandes, Marius Crainic, \emph{Lectures on Integrability of Lie Brackets}, (\href{http://aps.arxiv.org/abs/math.DG/0611259}{arxiv})

\end{itemize}
A review of Lie theory of Lie groupoids in on pages 3-5 of

\begin{itemize}%
\item Marius Crainic, Rui Loja Fernandes Lectures on Integrability of Lie Brackets (\href{http://aps.arxiv.org/abs/math.DG/0611259}{arxiv})

\end{itemize}
and in the introduction of

\begin{itemize}%
\item Chenchang Zhu, \emph{Lie II theorem for Lie algebroids via stacky Lie groupoids}, (\href{http://arxiv.org/abs/math/0701024}{arXiv}).

\end{itemize}
\hypertarget{motivation_for_generalized_smooth_groupoids}{}\subsection*{{Motivation for generalized smooth groupoids}}\label{motivation_for_generalized_smooth_groupoids}

This failure of Lie III for [[Lie groupoid|Lie groupoids]], i.e. for [[internal category|internal groupoids]] in [[Diff]] seems to suggest that the category of manifolds is not the natural home for general [[Lie theory]]. More concretely, it seems to suggest that [[Lie theory]] ought to be practiced internal to some category of [[generalized smooth space]]s.

One such choice is given by replacing manifolds by [[differentiable stack]]s.

\hypertarget{generalization_of_lies_theorems_to_stacky_lie_groupoids}{}\subsection*{{Generalization of Lie's theorems to stacky Lie groupoids}}\label{generalization_of_lies_theorems_to_stacky_lie_groupoids}

The generalization of Lie's theorems from Lie groups to to [[Lie theory for stacky Lie groupoids|stacky Lie groupoids]] is discussed in

\begin{itemize}%
\item Chenchang Zhu, \emph{Lie II theorem for Lie algebroids via stacky Lie groupoids}, (\href{http://arxiv.org/abs/math/0701024}{arXiv}).

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

\begin{itemize}%
\item [[Hans Duistermaat]], J. A. C. Kolk, chapter 1 \emph{Lie groups}, 2000

\end{itemize}
[[!redirects Lie's theorem]] [[!redirects Lie's theorems]] [[!redirects Lie's first theorem]] [[!redirects Lie's second theorem]] [[!redirects Lie's third theorem]]



\end{document}