\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{nonabelian Lie algebra cohomology}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - 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{nonabelian_2cocycles}{Nonabelian 2-cocycles}\dotfill \pageref*{nonabelian_2cocycles} \linebreak
\noindent\hyperlink{schreiers_theory_for_lie_algebras}{Schreier's theory for Lie algebras}\dotfill \pageref*{schreiers_theory_for_lie_algebras} \linebreak
\noindent\hyperlink{AbstractDefinition}{Abstract definition}\dotfill \pageref*{AbstractDefinition} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Abstractly, nonabelian Lie algebra cohomology is the restriction of the general notion of [[∞-Lie algebra cohomology]] to [[cocycle]]s of the form $\mathfrak{g} \to der \mathfrak{h}$, where $\mathfrak{g}$ and $\mathfrak{h}$ are ordinary [[Lie algebra]]s and $der(-)$ denotes the Lie algebra of [[derivation]]s.

Traditionally abelian [[Lie algebra cohomology]] is conceived as the cohomology of the [[Chevalley-Eilenberg complex]] of a Lie algebra and some nonabelian generalizations of this model have been given in the literature. We show below how these definitions are the [[nonabelian cohomology]] special cases of the general abstract definition of [[∞-Lie algebra cohomology]].

The coefficients are not now in a Lie algebra module (which is viewed here as an abelian Lie algebra with action of another Lie algebra), but an arbitrary Lie algebra with something that is action of another Lie algebra up to an inner automorphism.

For example the problem of extensions of Lie algebras by nonabelian Lie algebras leads to 1,2,3 nonabelian cocycles; 2-cocycles are analogues of [[group extension|factor systems]].

Below, in the section \hyperlink{AbstractDefinition}{Abstract definition} we discuss how a nonabelian Lie algebra cocylce is a morphism

\begin{displaymath}
(\psi,\chi) : \mathfrak{g} \to Der(\mathfrak{k})
\end{displaymath}
of [[L-∞-algebra]]s to the [[strict Lie 2-algebra]] of derivations of $\mathfrak{k}$.

A generalization (indeed a [[horizontal categorification]]) is [[nonabelian Lie algebroid cohomology]].

\hypertarget{nonabelian_2cocycles}{}\subsection*{{Nonabelian 2-cocycles}}\label{nonabelian_2cocycles}

Let $F$ be a field. \textbf{Lie algebra factor system} (or a \textbf{nonabelian 2-cocycle}) on a $F$-Lie algebra $\mathfrak{b}$ with coefficients in $F$-Lie algebra $\mathfrak{k}$ is a pair $(\chi,\psi)$ where $\chi: \mathfrak{b}\wedge \mathfrak{b}\to\mathfrak{k}$ and $\psi:\mathfrak{b}\to Der(\mathfrak{k})$ are $F$-linear maps satisfying

\begin{displaymath}
\begin{aligned}
& \chi([b_1,b_2]\wedge b_3)-\chi(b_1\wedge [b_2,b_3])+\chi(b_2\wedge[b_1,b_3])
\\& = \psi(b_3)(\chi(b_1\wedge b_2))-\psi(b_1)(\chi(b_2\wedge b_3))+\psi(b_2)(\chi(b_1\wedge b_3))
\end{aligned}
\end{displaymath}
for all $b_1,b_2,b_3\in B$ and

\begin{displaymath}
[\psi(a),\psi(b)]=\psi([a,b])+ad_{\mathfrak{k}}(\chi(a\wedge b))
\end{displaymath}
where $a,b\in B$ and $ad_{\mathfrak{k}}:\mathfrak{k}\to Int(\mathfrak{k})$ is the canonical map into inner automorphisms $k\mapsto [k,]$.

\hypertarget{schreiers_theory_for_lie_algebras}{}\subsection*{{Schreier's theory for Lie algebras}}\label{schreiers_theory_for_lie_algebras}

[[Otto Schreier]] (1926) and Eilenberg-Mac Lane (late 1940-s) developed a theory of nonabelian [[group extension|extensions of abstract groups]] leading to the low dimensional nonabelian group cohomology. For Lie algebras, the theory can be developed in the same manner. One tries to classify [[Lie algebra extension|extensions of Lie algebras]]

\begin{displaymath}
0\to \mathfrak{k}
\overset{i}\to \mathfrak{g}\overset{p}\to\mathfrak{b}\to 0
\end{displaymath}
\textbf{Theorem.} To every Lie algebra extension as above, and a choice of $F$-linear section $\sigma:\mathfrak{b}\to\mathfrak{g}$ of $p$, one can assign a nonabelian 2-cocycle (factor system) on $\mathfrak{b}$ with values in $\mathfrak{k}$ as follows: set

\begin{displaymath}
\chi(b_1\wedge b_2):=-\sigma([b_1,b_2])+[\sigma(b_1),\sigma(b_2)]
\end{displaymath}
and define $\phi:\mathfrak{g}\to Der(\mathfrak{k})$ by $\phi(g)(k):=[g,k]$. Then set $\psi:=\phi\circ\sigma$. Then $(\chi,\psi)$ is a nonabelian 2-cocycle on $\mathfrak{b}$ with values in $\mathfrak{k}$.

\textbf{Theorem. (cocycle crossed product of Lie algebras)} Let $(\chi,\psi)$ be a factor system as above. Then define a $F$-linear bracket on the $F$-vector space $\mathfrak{b}\oplus\mathfrak{k}$ by

\begin{displaymath}
[(b_1,k_1),(b_2,k_2)] = ([b_1,b_2],\chi(b_1\wedge b_2)+\psi(b_1)(k_2)-\psi(b_2)(k_1)+[k_1,k_2])
\end{displaymath}
Then

(i) $[,]$ is a antisymmetric and satisfies the Jacobi identity, i.e. $\mathfrak{g}:=(\mathfrak{b}\oplus\mathfrak{k},[,])$ is an $F$-Lie algebra.

(ii) $k\mapsto (0,k)$ defines an embedding $i:\mathfrak{k}\to\mathfrak{g}$ of Lie algebras and $(b,k)\mapsto b$ is a surjective homomorphism of Lie algebra $p:\mathfrak{g}\to\mathfrak{b}$ whose kernel is the Lie ideal $i(\mathfrak{k})=0\oplus\mathfrak{k}\subset\mathfrak{g}$. This way $0\to\mathfrak{k}\overset{i}\to\mathfrak{g}\overset{p}\to\mathfrak{b}\to 0$ is an extension of the base Lie algebra $\mathfrak{b}$ by the kernel Lie algebra $\mathfrak{k}$.

(iii) If the 2-cocycle is obtained from a Lie algebra extension $0\to \mathfrak{k}\overset{i_0}\to \mathfrak{g}_0\overset{p_0}\to\mathfrak{b}\to 0$ and an arbitrary $F$-linear section $\sigma_0$ of $p_0$, then the map $can_\sigma:\mathfrak{g}_0\to\mathfrak{g}$ given by $g\mapsto (p(g),-\sigma(p(g))+g)$ is well-defined and a Lie algebra isomorphism such that $can_\sigma\circ i_0=i$, $p_0=p\circ can_\sigma$, hence the two extensions are isomorphic.

In addition to the problem of extensions, nonabelian 2-cocycles appear in a more general problem of liftings of Lie algebras.

\hypertarget{AbstractDefinition}{}\subsection*{{Abstract definition}}\label{AbstractDefinition}

We claim that the above definition of nonabelian Lie algebra cocycles may be understood naturally in terms of the general notion of [[cohomology]] and in particular is the image of the story of [[nonabelian group cohomology]] under Lie differentiation:

The following observation is not in the literature.

\begin{uprop}
Let $\infty Lie$ be the [[(∞,1)-category]] of [[L-∞-algebra]]s. Let $\mathfrak{g}, \mathfrak{k}$ be [[Lie algebra]]s. Then the degree 2 nonabelian Lie algebra cohomology of $\mathfrak{g}$ with coefficients in $\mathfrak{k}$ is

\begin{displaymath}
H^2_{nonab}(\mathfrak{g}, \mathfrak{k})
  \simeq
  \pi_0 \infty Lie(\mathfrak{g}, Der(\mathfrak{k}))
  \,,
\end{displaymath}
where $Der(\mathfrak{k})$ is the [[strict Lie 2-algebra]] of derivations on $\mathfrak{k}$.

More in detail:

\begin{itemize}%
\item nonabelian degree 2 Lie algebra cocycles $(\psi,\xi)$ are in natural bijections with morphisms

\begin{displaymath}
\mathfrak{g} \to Der(\mathfrak{k})
\end{displaymath}

\item coboundaries $\eta$ between cocycles $(\psi_1,\xi_1)$ and $(\psi_2,\xi_2)$ correspond to homotopies between these

\begin{displaymath}
\itexarray{
    & \nearrow\searrow^{\mathrlap{(\psi_1,\chi_1)}}
    \\
    \mathfrak{g}
    &\Downarrow^{\eta}&
    Der(\mathfrak{k})
    \\
    & \searrow\nearrow^{\mathrlap{(\psi_2,\chi_2)}}
  }
\end{displaymath}
and this correspondence is precise if we take the homotopy to be induced from the ``standard [[cylinder object]]'', described below.



\end{itemize}
\end{uprop}
\begin{proof}
Checking this is a straightforward matter of unwinding the definitions of morphisms of $L_\infty$-algebras.

Which is what we indicate.

We model $\infty Lie$ as usual a subcategory of [[dg-algebra]]s of [[semifree dga]]s, by representing each $L_\infty$-algebra $\mathfrak{g}$ by its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$.

For the [[Lie algebra]] $\mathfrak{g}$ itself with Lie bracket $[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}$ this is the [[semifree dga]]

\begin{displaymath}
CE(\mathfrak{g}) =
  (\wedge^\bullet \mathfrak{g}^* , \; d = [-,-]^* )
  \,,
\end{displaymath}
where the differential is on generators the dual of the Lie bracket, $[-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*$ extended as a graded derivation to all of $\wedge^\bullet \mathfrak{g}^*$.

For any [[strict Lie 2-algebra]] coming from a [[differential crossed module]] $(\mathfrak{h}_1 \stackrel{\delta}{\to} \mathfrak{h}_1)$ with action $\rho : \mathfrak{h}_1 \to der(\mathfrak{h}_2)$ -- that we think of in the following as equivalently a linear map $\rho : \mathfrak{h}_1 \otimes \mathfrak{h}_2 \to \mathfrak{h}_2$ -- the [[Chevalley-Eilenberg algebra]] is

\begin{displaymath}
CE(\mathfrak{h}_2 \stackrel{\delta}{\to} \mathfrak{h}_1)
  =
  \left(
    \wedge^\bullet ( \mathfrak{h}_1^* \oplus \mathfrak{h}_2^* )
    ,
    \;
    d_{\delta}
  \right)
\end{displaymath}
with $\mathfrak{h}_1^*$ in degree 1 and $\mathfrak{h}_2^*$ in degree 2, and with the differential given on degree 1 generators by

\begin{displaymath}
d_\delta |_{\mathfrak{h}_1^*} = [-,-]_{\mathfrak{h}_1}^* + \delta^*
  : \mathfrak{h}_1^* \to 
    \mathfrak{h}_1^* \wedge \mathfrak{h}_1^*
    \oplus
    \mathfrak{h}_2^*
\end{displaymath}
and on degree 2 generators by

\begin{displaymath}
d_\delta |_{\mathfrak{h}_2^*} = \rho^*
  : 
  \mathfrak{h}_2^* \to \mathfrak{h}_1^* \otimes \mathfrak{h}_2^*
  \,.
\end{displaymath}
The case of the derivation [[strict Lie 2-algebra]] of a Lie algebra $\mathfrak{k}$ is the special case of this for

\begin{displaymath}
Der(\mathfrak{k}) = (\mathfrak{k} \stackrel{ad}{\to} der(\mathfrak{k}))
  \,.
\end{displaymath}
Now a morphism

\begin{displaymath}
(\psi, \chi) : \mathfrak{g} \to Der(\mathfrak{k})
\end{displaymath}
of $\infty$-Lie algebras is given by a morphism

\begin{displaymath}
CE(\mathfrak{g}) \leftarrow CE(Der(\mathfrak{k})) :
  (\psi^*, \chi^*)
\end{displaymath}
of [[dg-algebra]]s.

Morphisms of [[dg-algebra]]s are given by morphisms of the underlying graded algebras, subject to the respect for the differentials. Morphisms of the underlying graded [[Grassmann algebra]]s are given by grading preserving linear maps on the spaces of generators.

So the underlying maps

\begin{displaymath}
\wedge^\bullet \mathfrak{g}^*
   \leftarrow
  \wedge^\bullet (\mathfrak{h}_1^* \oplus \mathfrak{h}_2^*)
  :
  (\psi^* , \chi^*)
\end{displaymath}
come from linear maps

\begin{displaymath}
\mathfrak{g}^* \leftarrow \mathfrak{h}_1^* : \psi^*
\end{displaymath}
and

\begin{displaymath}
\mathfrak{g}^* \wedge \mathfrak{g}^* \leftarrow \mathfrak{h}_2^* : \chi^*
\end{displaymath}
i.e. form linear maps

\begin{displaymath}
\psi : \mathfrak{g} \to der(\mathfrak{k})
\end{displaymath}
and

\begin{displaymath}
\chi : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{k}
  \,.
\end{displaymath}
This is the underlying data of the nonabelian 2-cocycle. Now the respect for the differentials on the Chevalley-Eilenberg algebras will give the cocycle condition:

let $\omega \in \mathfrak{h}_2^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2)$ be any degree 2 element, then respect for the differential implies that

\begin{displaymath}
\itexarray{
    \omega(\chi([-,-],-))
    =
    \omega(\rho(\psi(-)(\chi(-,-))))
    &\stackrel{(\psi^*, \chi^*)}{\leftarrow}& \omega(\rho(-)(-)))
    \\
    \uparrow^{d_\mathfrak{g} = [-,-]^*} && \uparrow^{\mathrlap{d_{\delta}}}
    \\
    \omega(\xi(-,-))
    &\stackrel{(\psi^*, \chi^*)}{\leftarrow}&
    \omega
  }
  \,.
\end{displaymath}
Since this has to hold for all $\omega$, we get the first part of the cocycle condition:

\begin{displaymath}
\chi([-,-],-) = \rho(\psi(-)\chi(-,-))
\end{displaymath}
(both sides here regarded as elements of a graded Grassmann algebra as indicated above, so with all antisymmetrization on the arguments implicit).

Similarly, for $\lambda \in \mathfrak{h}_1^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2)$ be any degree 1 element, then respect for the differential implies that

\begin{displaymath}
\itexarray{
     \lambda(\psi([-,-]))
     =
     \lambda([\psi(-), \psi(-)])
     +
     \lambda(ad(\chi(-,-)))
     &\stackrel{}{\leftarrow}&  \lambda([-,-]_{\mathfrak{h}_1}) + \lambda(ad(-))
     \\
     \uparrow && \uparrow
    \\
    \lambda(\psi(-))
    &\stackrel{(\psi^* , \chi^*)}{\leftarrow}&
    \lambda
  }
  \,.
\end{displaymath}
Again, this has to hold for all $\lambda$, so we have the auxiliary condition on the cocycle

\begin{displaymath}
\psi([-,-])
  =
  [\psi(-),\psi(-)]
  +
  ad(\xi(-,-))
  \,.
\end{displaymath}
This shows that morphisms $\mathfrak{g} \to Der(\mathfrak{k})$ are in bijection to the nonabelian cocycles.

It remains to show that the homotopies map to coboundaries. For that we may take in $\infty Lie$ the standard [[cylinder object]] of some $CE(\mathfrak{g})$ to be

\begin{displaymath}
CE(\mathfrak{g})\otimes CE(\mathfrak{g})
  \leftarrow
  C^\bullet(\Delta^1)\otimes CE(\mathfrak{g})
  \leftarrow
  CE(\mathfrak{g})
  \,,
\end{displaymath}
where $C^\bullet(\Delta^1)$ is the [[semifree dga]] of cochains on the cellular 1-[[simplex]], i.e.

\begin{displaymath}
C^\bullet(\Delta^1) =
  (\wedge^\bullet (\langle a,b \rangle \oplus \langle c \rangle)
  , d a = - d b = d c
  )
  \,,
\end{displaymath}
with $a,b$ generators in degree 0 and $c$ in degree 1. Using this, write out the data implied by a morphism $\eta$ that is a [[left homotopy]]

\begin{displaymath}
\itexarray{
    CE(\mathfrak{g})
    \\
    \downarrow & \nwarrow^{\mathrlap{(\psi_1^*, \chi_1^*)}}
    \\
    C^\bullet(\Delta^1)\otimes CE(\mathfrak{g})
    &\stackrel{\eta}{\leftarrow}&
    CE(Der(\mathfrak{k}))
    \\
    \uparrow & \swarrow_{\mathrlap{(\psi_2^*, \chi_2^*)}}
    \\
    CE(\mathfrak{g})
  }
\end{displaymath}
along the above lines.

Notice that in $dgAlg^{op}$ every object is cofibrant, so that this is indeed a left homotopy. See [[∞-Lie algebra cohomology]] for more on this.

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

On original source is

\begin{itemize}%
\item G. Hochschild, \emph{Lie algebra kernels and cohomology}, Amer. J. Math. \textbf{76}, n.3 (1954) 698--716.

\end{itemize}
The notation above is from personal notes of Z. \v{S}koda (1997). A systematic theory has been many times partly rediscovered from soon after the Eilenberg--Mac Lane work on group extension, among first by Hochschild and then by many others till nowdays. Here is a recent online account emphasising parallels with [[differential geometry]]:

\begin{itemize}%
\item [[Dmitri Alekseevsky]], [[Peter Michor]], Wolfgang Ruppert, Extensions of Lie algebras (\href{http://arxiv.org/abs/math/0005042}{math.DG/0005042})

\end{itemize}
A more conceptual picture is in a work of [[Danny Stevenson]] which extends also to its [[categorification]], extensions of [[Lie 2-algebra]]s. See

\begin{itemize}%
\item [[Danny Stevenson]], Lie 2-algebras and the geometry of gerbes, Unni Namboodiri Lectures 2006 \href{http://math.ucr.edu/home/baez/namboodiri/stevenson_maclane.pdf}{slides}

\end{itemize}
There is also

\begin{itemize}%
\item N. Inassaridze, E. Khmaladze, and M. Ladra, \emph{Non-abelian Cohomology and Extensions of Lie Algebras} Journal of Lie Theory, Volume 18 (2008) 413--432 (\href{http://www.rmi.acnet.ge/~khmal/PAPERS/LieTheory2008.pdf}{pdf})

\end{itemize}


\end{document}