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\begin{document}

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\section*{Lie differentiation}

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

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

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

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration theory - 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{ForDeformationContexts}{For deformation contexts}\dotfill \pageref*{ForDeformationContexts} \linebreak
\noindent\hyperlink{ForCohesiveContexts}{For cohesive contexts}\dotfill \pageref*{ForCohesiveContexts} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{examples_of_contexts_for_lie_differentiation}{Examples of contexts for Lie differentiation}\dotfill \pageref*{examples_of_contexts_for_lie_differentiation} \linebreak
\noindent\hyperlink{dggeometry}{dg-Geometry}\dotfill \pageref*{dggeometry} \linebreak
\noindent\hyperlink{ExampleContextSyntheticDifferential}{Synthetic-differential $\infty$-groupoids}\dotfill \pageref*{ExampleContextSyntheticDifferential} \linebreak
\noindent\hyperlink{examples_of_lie_differentiation}{Examples of Lie differentiation}\dotfill \pageref*{examples_of_lie_differentiation} \linebreak
\noindent\hyperlink{of_a_lie_group}{Of a Lie group}\dotfill \pageref*{of_a_lie_group} \linebreak
\noindent\hyperlink{of_a_lie_groupoid}{Of a Lie groupoid}\dotfill \pageref*{of_a_lie_groupoid} \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{Lie differentiation} is the process reverse to [[Lie integration]]. It sends a [[Lie group]] to its [[Lie algebra]] and more generally a [[Lie groupoid]] to its [[Lie algebroid]] and a [[smooth ∞-group]] to its [[L-∞ algebra]].

\begin{itemize}%
\item For the moment, for ordinary [[Lie theory]] see at \emph{[[Lie's three theorems]]}.


\item For [[infinity-Lie theory]] see at \emph{\href{synthetic+differential+infinity-groupoid#LieDifferentiation}{synthetic differential infinity-groupoid -- Lie differentiation}} .



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

A formalization of the notion Lie differentiation in [[higher geometry]] has been given in (\hyperlink{Lurie}{Lurie}), inspired by and building on results discussed at \emph{[[model structure for L-∞ algebras]]}. This we discuss in

\begin{itemize}%
\item \hyperlink{ForDeformationContexts}{For deformation contexts}

\end{itemize}
We then specialize this to those \emph{[[deformation contexts]]}, def. \ref{DeformationContext}, that arise in the formalization of [[higher differential geometry]] by [[differential cohesion]]:

\begin{itemize}%
\item \hyperlink{ForCohesiveContexts}{For cohesive contexts}

\end{itemize}
This is the context in which one has a natural formulation of ordinary Lie differentiation of ordinary [[Lie groups]] to [[Lie algebras]] and its generalization to the Lie differentiation of [[smooth ∞-groups]] to [[L-∞ algebras]]. See the discussion of \emph{\hyperlink{Examples}{Examples}} below for more.

\hypertarget{ForDeformationContexts}{}\subsubsection*{{For deformation contexts}}\label{ForDeformationContexts}

\begin{defn}
\label{DeformationContext}\hypertarget{DeformationContext}{}
A \textbf{[[deformation context]]} is an [[(∞,1)-category]] $Sp_*$ such that

\begin{enumerate}%
\item it is a [[presentable (∞,1)-category]];


\item it contains an [[initial object in an (∞,1)-category|initial object]]



\end{enumerate}
and

\begin{itemize}%
\item equipped with a [[set]] of [[objects]]

\begin{displaymath}
\{E_\alpha \in Stab(Sp_*^{op})\}
\end{displaymath}
in the [[stabilization]] of the [[opposite (∞,1)-category]] $Sp_*^{op}$.



\end{itemize}
\end{defn}
This is (\hyperlink{Lurie}{Lurie, def. 1.1.3}) together with the assumption of a terminal object in $Sp_*^{op}$ stated on p.9 (and later implicialy used).

\begin{remark}
\label{}\hypertarget{}{}
Definition \ref{DeformationContext} is meant to be read as follows:

We think of $Sp_*$ as an [[(∞,1)-category]] of [[pointed object|pointed]] [[spaces]] in some [[higher geometry]]. The point is the initial object. We think of the formal duals of the objects $\{E_\alpha\}_\alpha$ as a set of generating [[infinitesimally thickened points]] (points in [[formal geometry]]).

\end{remark}
The following construction generates the ``[[jets]]'' induced by the generating infinitesimally thickened points.

\begin{defn}
\label{SmallObjects}\hypertarget{SmallObjects}{}
Given a [[deformation context]] $(Sp_*, \{E_\alpha\}_\alpha)$, we say

\begin{itemize}%
\item a [[morphism]] in $Sp_*^{op}$ is an \textbf{elementary morphism} if it is the [[homotopy fiber]] to a map into $\Omega^{\infty -n}E_\alpha$ for some $n \in \mathbb{Z}$ and some $\alpha$;


\item a morphism in $Sp_*^{op}$ is a \textbf{small morphism} if it is the composite of [[finite set|finitely]] many elementary morphisms.



\end{itemize}
We write

\begin{displaymath}
Sp_*^{inf} \hookrightarrow Sp_*
\end{displaymath}
for the full [[sub-(∞,1)-category]] on those objects $A$ for which the essentially unique map $A \to *$ is small.

\end{defn}
(\hyperlink{Lurie}{Lurie, def. 1.1.8})

\begin{defn}
\label{FormalModuliProblem}\hypertarget{FormalModuliProblem}{}
Given a [[deformation context]] $(Sp_*, \{E_\alpha\}_\alpha)$, def. \ref{DeformationContext}, the [[(∞,1)-category]] of \textbf{[[formal moduli problems]]} over it is the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-presheaves]] over $Sp_*^{inf}$

\begin{displaymath}
FormalModuli^{Sp_*}
  \hookrightarrow
  PSh_\infty(Sp_*^{inf})
\end{displaymath}
on those [[(∞,1)-functors]] $X \colon (Sp_*^{inf})^{op} \to \infty Grpd$ such that

\begin{enumerate}%
\item over the [[terminal object]] they are [[contractible]]: $X(*) \simeq *$;


\item they sends [[(∞,1)-colimits]] in $Sp_*^{inf}$ to [[(∞,1)-limits]] in [[∞Grpd]].



\end{enumerate}
\end{defn}
(\hyperlink{Lurie}{Lurie, def. 1.1.14})

(\hyperlink{Lurie}{Lurie, remark 1.1.7}).

\begin{remark}
\label{}\hypertarget{}{}
This means that a ``formal deformation problem'' is a [[space]] in [[higher geometry]] whose geometric structure is detected by the ``test spaces'' in $Sp_*^{inf}$ in a way that respects gluing ([[descent]]) in $Sp_*^{inf}$ as given by [[(∞,1)-colimits]] there. The first condition requires that there is an essentially unique such probe by the point, hence that these higher geometric space have essentially a single global point. This is the condition that reflects the infinitesimal nature of the deformation problem.

\end{remark}
We will often just write $Sp_*$ for a deformation context $(Sp_*, \{E_\alpha\}_\alpha)$, when the objects $\{E_\alpha\}$ are understood.

\begin{prop}
\label{FormalModuliIsPresentableCategory}\hypertarget{FormalModuliIsPresentableCategory}{}
The [[(∞,1)-category]] $FormalModuli^{Sp_*}$ of [[formal moduli problems]] is a [[presentable (∞,1)-category]]. Moreover it is a [[reflective sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-presheaves]]

\begin{displaymath}
FormalModuli^{Sp_*}
  \stackrel{\leftarrow}{\hookrightarrow}
  PSh_{\infty}(Sp_*)
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{LieDifferentiationFunctor}\hypertarget{LieDifferentiationFunctor}{}
Given a [[deformation context]] $Sp_*$, the restricted [[(∞,1)-Yoneda embedding]] gives an [[(∞,1)-functor]]

\begin{displaymath}
Lie \;\colon\; Sp_* \to FormalModuli^{Sp_*}
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
For $(X,x) \in Sp_*$, the object $Lie(X,x)$ represents the \emph{[[formal geometry|formal neighbourhood]]} of the basepoint $x$ of $X$ as seen by the infinitesimally thickened points dual to the $\{E_\alpha\}$.

Hence we may call this the operation of \textbf{Lie differentiation} of spaces in $Sp_*$ around their given base point.

In the archetypical implementation of these axiomatics, discuss \hyperlink{Examples}{below}, there is an [[equivalence of (∞,1)-categories]] of [[formal moduli problems]] with [[L-∞ algebras]] and the Lie differentiation of the [[delooping]]/[[moduli ∞-stack]] $\mathbf{B}G$ of a [[smooth ∞-group]] $G$ is its [[L-∞ algebra]] $\mathfrak{g}$: $Lie(\mathbf{B}G) \simeq \mathbf{B}\mathfrak{g}$.

\end{remark}
\begin{prop}
\label{LieDifferentiationPreservesLimits}\hypertarget{LieDifferentiationPreservesLimits}{}
The Lie differentiation functor

\begin{displaymath}
Lie 
  \;
  \colon
  \;
  Sp_* \to FormalModuli^{Sp_*}
\end{displaymath}
of prop. \ref{LieDifferentiationFunctor} preserves [[(∞,1)-limits]].

\end{prop}
\begin{proof}
By prop. \ref{FormalModuliIsPresentableCategory} the [[(∞,1)-limits]] in $FormalModuli^{Sp_*}$ may be computed in $PSh_\infty(Sp_*)$. There the statement is that of the [[(∞,1)-Yoneda embedding]], or rather just the statement that the [[derived hom space|(∞,1)-hom (∞,1)-functor]] $Sp_*(D,-)$ preserves $(\infty,1)$-limits.

\end{proof}
\hypertarget{ForCohesiveContexts}{}\subsubsection*{{For cohesive contexts}}\label{ForCohesiveContexts}

\begin{quote}%
under construction


\end{quote}
Let $\mathbf{H}_{th}$ be a [[cohesive (∞,1)-topos]] $(&#643; \dashv \flat \dashv \sharp)$ equipped with [[differential cohesion]] $(Red \dashv &#643;_{inf} \dashv \flat_{inf})$.

\begin{defn}
\label{ExhibitingInfinitesimalCohesion}\hypertarget{ExhibitingInfinitesimalCohesion}{}
A set of objects $\{D_\alpha \in \mathbf{H}_{th}\}_\alpha$ is said to \textbf{exhibit the differential structure} or \textbf{exhibit the infinitesimal thickening} of $\mathbf{H}_{th}$ if the [[localization of an (∞,1)-category|localization]]

\begin{displaymath}
L_{\{D_\alpha\}} \mathbf{H}_{th}
  \stackrel{\leftarrow}{\hookrightarrow}
  \mathbf{H}_{th}
\end{displaymath}
of $\mathbf{H}_{th}$ at the morphisms of the form $D_\alpha \times X \to X$ is exhibited by the [[infinitesimal shape modality]] $&#643;_{inf} \coloneqq i^* i_*$

\begin{displaymath}
\mathbf{H} 
  \simeq 
  L_{\{D_\alpha\}} \mathbf{H}_{th}
  \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}
  \mathbf{H}_{th}
  \,.
\end{displaymath}
\end{defn}
\begin{Remark}
\label{}\hypertarget{}{}
Def. \ref{ExhibitingInfinitesimalCohesion} expresses the [[infinitesimal cohesion|infinitesimal]] analog of the notion of objects exhibiting [[cohesion]], see at , hence an infinitesimal notion of [[A1-homotopy theory]].

\end{Remark}
\begin{prop}
\label{ObjectsExhibitingDifferentialCohesionAreUniquelyPointed}\hypertarget{ObjectsExhibitingDifferentialCohesionAreUniquelyPointed}{}
If objects $\{D_\alpha \in \mathbf{H}_{th}\}$ exhibit the differential cohesion of $\mathbf{H}_{th}$, then they are essentially uniquely pointed.

\end{prop}
\begin{proof}
The localizing objects are in particular themselves [[local objects]] so that $&#643;_{inf} D_\alpha \simeq *$. By the $(Red \dashv &#643;_{inf})$-[[adjunctions]] this means that

\begin{displaymath}
\begin{aligned}
    \mathbf{H}_{th}(*, D_\alpha)
    & \simeq \mathbf{H}_{th}(Red(*), D_\alpha)
    \\
    & \simeq \mathbf{H}_{th}(*, &#643;_{inf} D_\alpha)
    \\
    & \simeq \mathbf{H}_{th}(*, *)
    \\
    & \simeq *
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
We now consider $(\mathbf{H}_{th}^{\ast/}, \{D_\alpha\})$ as a [[deformation context]], def. \ref{DeformationContext}.

\begin{defn}
\label{LieDifferentiationInCohesion}\hypertarget{LieDifferentiationInCohesion}{}
Write

\begin{displaymath}
Lie 
  \;\colon\;
  \mathbf{H}_{th}^{\ast/}
  \to
  FormalModuli^{\mathbf{H}_{th}^{*/}}
  \hookrightarrow
  PSh_\infty(\mathbf{H}_{th}^{*/})
\end{displaymath}
for the Lie differentiaon [[(∞,1)-functor]], def. \ref{LieDifferentiationFunctor}, which sends $(x \colon * \to X) \in \mathbf{H}_{th}$ to

\begin{displaymath}
Lie(X,x) 
   \;\colon\;
  D
  \mapsto \mathbf{H}^{\ast/}(D,(X,x))
  \,.
\end{displaymath}
\end{defn}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{examples_of_contexts_for_lie_differentiation}{}\subsubsection*{{Examples of contexts for Lie differentiation}}\label{examples_of_contexts_for_lie_differentiation}

\hypertarget{dggeometry}{}\paragraph*{{dg-Geometry}}\label{dggeometry}

[[dg-geometry]] (the running example in (\hyperlink{Lurie}{Lurie})).

\hypertarget{ExampleContextSyntheticDifferential}{}\paragraph*{{Synthetic-differential $\infty$-groupoids}}\label{ExampleContextSyntheticDifferential}

\begin{itemize}%
\item [[SynthDiff∞Grpd]]

\end{itemize}
(\ldots{})

\emph{\href{synthetic+differential+infinity-groupoid#LieDifferentiation}{synthetic differential infinity-groupoid -- Lie differentiation}}

(\ldots{})

\hypertarget{examples_of_lie_differentiation}{}\subsubsection*{{Examples of Lie differentiation}}\label{examples_of_lie_differentiation}

\hypertarget{of_a_lie_group}{}\paragraph*{{Of a Lie group}}\label{of_a_lie_group}

(\ldots{})

\hypertarget{of_a_lie_groupoid}{}\paragraph*{{Of a Lie groupoid}}\label{of_a_lie_groupoid}

Given a Lie groupoid $G_1\Rightarrow G_0$, we take the vector bundle $ker Ts|_{G_0}$ restricted on $G_0$, then we show that there is a [[Lie algebroid]] structure on $A:=ker Ts|_{G_0} \to G_0$. First of all, the anchor map is given by $ker Ts_{G_0} \xrightarrow{Tt} G_0$. Secondly, to define the Lie bracket, one shows that a section $X$ of $A$ may be right translated to a vector field on $G_1$, which is right invariant. Then Jacobi identity implies that right invariant vector fields are closed under Lie bracket. Thus Lie brackets on vector fields on $G_1$ induces a Lie bracket on sections of $A$.

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

[[!include infinitesimal and local - table]]

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

Lie differentiation of [[Lie n-groupoids]] was first considered in generality in

\begin{itemize}%
\item [[Pavol ?evera]], \emph{$L_\infty$ algebras as 1-jets of simplicial manifolds (and a bit beyond)} (\href{http://arxiv.org/abs/math/0612349}{arXiv:0612349})

\end{itemize}
See also theorem 8.28 of

\begin{itemize}%
\item Du Li, \emph{Higher Groupoid Actions, Bibundles, and Differentiation} (\href{http://arxiv.org/abs/1512.04209}{arXiv:1512.04209})

\end{itemize}
Lie differentiation \hyperlink{ForDeformationContexts}{in deformation contexts} is formulated in section 1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Formal moduli problems]]}

\end{itemize}


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