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\begin{document}

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\section*{model structure on dg-Lie algebras}

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

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{RectificationResolution}{Relation to $L_\infty$-algebras}\dotfill \pageref*{RectificationResolution} \linebreak
\noindent\hyperlink{RelationToDgCoalgebras}{Relation to dg-coalgebras}\dotfill \pageref*{RelationToDgCoalgebras} \linebreak
\noindent\hyperlink{relation_to_simplicial_lie_algebras}{Relation to simplicial Lie algebras}\dotfill \pageref*{relation_to_simplicial_lie_algebras} \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}

[[dg-Lie algebras]] may be thought of as the ``strict'' [[strong homotopy Lie algebras]]. As such they support a [[homotopy theory]]. The \emph{[[model category]]} structure on dg-Lie algebras is one way to present this homotopy theory. This is used for instance in [[deformation theory]], see at \emph{[[formal moduli problems]]}.

For dg-Lie algebras in positive degree and over the rational numbers this model structure, due to (\hyperlink{Quillen69}{Quillen 69, theorem II}) is one of the algebraic models for presenting [[rational homotopy theory]] (see there) of [[simply connected topological spaces]].

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

\begin{prop}
\label{ProjectiveModelStructureOndgLieAlgebras}\hypertarget{ProjectiveModelStructureOndgLieAlgebras}{}
There exists a [[model category]] structure $(dgLie_k)_{proj}$ on the category $dgLie_k$ of [[dg-Lie algebras]] over a commutative [[ring]] $k \supset \mathbb{Q}$ such that

\begin{itemize}%
\item [[fibrations]] the [[surjective maps]]


\item [[weak equivalences]] the [[quasi-isomorphisms]] on the underlying [[chain complexes]].



\end{itemize}
\end{prop}
For dg-Lie algebras in degrees $\geq n \geq 1$, this is due to \hyperlink{Quillen69}{Quillen 69}. For unbounded dg-Lie algebras this is due to (\hyperlink{Hinich97}{Hinich 97}).

This becomes a [[simplicial category]] with simplicial [[mapping spaces]] given by

\begin{displaymath}
dgLie(\mathfrak{g}, \mathfrak{h})
   \coloneqq
  ([k] \mapsto Hom_{dgLie}(\mathfrak{g} , \Omega^\bullet(\Delta^k) \otimes\mathfrak{h}))
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\Omega^\bullet(\Delta^k)$ is the [[dg-algebra]] of [[polynomial differential forms]] on the $k$-[[simplex]];


\item $\Omega^\bullet(\Delta^k)\otimes \mathfrak{h}$ is the canonical dg-Lie algebra structure on the [[tensor product]].



\end{itemize}
(\hyperlink{Hinich97}{Hinich 97, 4.8.2}, following Bousfield-Gugenheim 76tructure on dg-algebras\#HomComplexes))

This enrichment satisfies together with the model structure some of the properties of a [[simplicial model category]] (\hyperlink{Hinich97}{Hinich 97, 4.8.3, 4.8.4}), but not all of them.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{RectificationResolution}{}\subsubsection*{{Relation to $L_\infty$-algebras}}\label{RectificationResolution}

dg-Lie algebras with this model structure are a \emph{rectification} of [[L-∞ algebras]]: for $Lie$ the [[Lie operad]] and $\widehat Lie$ its standard [[cofibrant resolution]], [[algebras over an operad]] over $Lie$ in chain complexes are [[dg-Lie algebras]] and algebras over $\widehat Lie$ are [[L-∞ algebras]] and by the rectification result discussed at \emph{[[model structure on dg-algebras over an operad]]} there is an induced [[Quillen equivalence]]

\begin{displaymath}
Alg(\widehat Lie) \stackrel{\simeq}{\to} Alg(Lie)
\end{displaymath}
between the [[model structure for L-∞ algebras]] which is [[transferred model structure|transferred]] from the [[model structure on chain complexes]] (unbounded propjective) to the above model structure on chain complexes.

There is also a [[Quillen equivalence]] from the model structure on dg-Lie algebras to the [[model structure on dg-coalgebras]]. This is part of a web of Quillen equivalences that identifies dg-Lie algebra/$L_\infty$-algebras with [[infinitesimal object|infinitesimal]] [[derived stack|derived]] [[∞-stacks]] (``[[formal moduli problems]]''). More on this is at \emph{[[model structure for L-∞ algebras]]}.

Specifically, there is (\hyperlink{Quillen}{Quillen 69}) an [[adjunction]]

\begin{displaymath}
(\mathcal{R} \dashv i) \;\colon\;
  dgLie \stackrel{\overset{\mathcal{R}}{\leftarrow}}{\underset{i}{\to}} dgCoCAlg
\end{displaymath}
between [[dg-coalgebras]] and dg-Lie algebras, where the [[right adjoint]] is the (non-full) inclusion that regards a dg-Lie algebra as a [[differential graded coalgebra]] with co-binary differential, and where the [[left adjoint]] $\mathcal{R}$ (``rectification'') sends a dg-coalgebra to a dg-Lie algebra whose underlying graded [[Lie algebra]] is the [[free Lie algebra]] on the underlying [[chain complex]]. Over a [[field]] of [[characteristic]] 0, this adjunction is a [[Quillen equivalence]] between the [[model structure for L-∞ algebras]] on $dgCoCAlg$ and the model structure on $dgLie$ (\hyperlink{Hinich98}{Hinich 98, theorem 3.2}).

In particular, therefore the composite $i \circ \mathcal{R}$ is a [[resolution]] functor for $L_\infty$-algebras.

\hypertarget{RelationToDgCoalgebras}{}\subsubsection*{{Relation to dg-coalgebras}}\label{RelationToDgCoalgebras}

Via the above relation to $L_\infty$-algebras, dg-Lie algebras are also connected by a composite adjunction to [[dg-coalgebras]]. We dicuss the direct adjunction.

Throughout, let $k$ be of [[characteristic zero]].

\begin{defn}
\label{CEfunctor}\hypertarget{CEfunctor}{}
\textbf{(Chevalley-Eilenberg dg-coalgebra)}

Write

\begin{displaymath}
CE 
    \;\colon\;
  dgLieAlg_{k}
    \longrightarrow
  dgCocAlg_k
\end{displaymath}
for the [[Chevalley-Eilenberg algebra]] functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to the [[dg-coalgebra]]

\begin{displaymath}
CE(\mathfrak{g},\partial,[-,-])
   \;\coloneqq\;
  \left(
    \vee^\bullet \mathfrak{g}[1]  
    ,\;
    D = \partial + [-,-]
  \right)
  \,,
\end{displaymath}
where on the right the extension of $\partial$ and $[-,-]$ to graded [[derivations]] is understood.

\end{defn}
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (\hyperlink{Quillen}{Quillen 69, appendix B, prop 6.2}). For unbounded dg-algebras, this is due to (\hyperlink{Hinich98}{Hinich 98, 2.2.2}).

\begin{defn}
\label{LeftAdjointToCEfunctor}\hypertarget{LeftAdjointToCEfunctor}{}
For $(X,D) \in dgCocalg_k$ write

\begin{displaymath}
\mathcal{L}(X,D)
     \coloneqq
    \left(
       F(\overline{X}[-1]),\; 
       \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1)
    \right)
    \;\in dgLieAlg_k\;
\end{displaymath}
where

\begin{enumerate}%
\item $\overline{X} \coloneqq ker(\epsilon)$ is the [[kernel]] of the [[counit]], regarded as a [[chain complex]];


\item $F$ is the [[free Lie algebra]] functor (as graded Lie algebras);


\item on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra [[derivation]]



\end{enumerate}
\end{defn}
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (\hyperlink{Quillen}{Quillen 69, appendix B, prop 6.1}). For unbounded dg-algebras, this is due to (\hyperlink{Hinich98}{Hinich 98, 2.2.1}).

\begin{prop}
\label{CEAdjunction}\hypertarget{CEAdjunction}{}
The [[functors]] from def. \ref{CEfunctor} and def. \ref{LeftAdjointToCEfunctor} are [[adjoint functor|adjoint]] to each other:

\begin{displaymath}
dgLieAlg_k
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {\bot}
  dgCocAlg_k
  \,.
\end{displaymath}
Moreover, for $X \in dgCocAlg_k$ and $\mathfrak{g} \in dgLieAlg_k$ then the adjoint [[hom sets]] are [[natural isomorphism|naturally isomorphic]]

\begin{displaymath}
Hom(\mathcal{L}(X), \mathfrak{g})
   \simeq
  Hom(X, CE(\mathfrak{g}))
    \simeq
  MC(Hom(\overline{X},\mathfrak{g}))
\end{displaymath}
to the [[Maurer-Cartan elements]] in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$.

\end{prop}
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (\hyperlink{Quillen}{Quillen 69, appendix B, somewhere}). For unbounded dg-algebras, this is due to (\hyperlink{Hinich98}{Hinich 98, 2.2.5}).

\begin{prop}
\label{QuillenAdjunctionCE}\hypertarget{QuillenAdjunctionCE}{}
The adjunction $(\mathcal{L} \dashv CE)$ from prop. \ref{CEAdjunction} is a [[Quillen adjunction]] between then projective [[model structure on dg-Lie algebras]] as the [[model structure on dg-coalgebras]]

\begin{displaymath}
(dgLieAlg_k)_{proj}
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {\bot}
  (dgCocAlg_k)_{Quillen}
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Hinich98}{Hinich 98, lemma 5.2.2, lemma 5.2.3})

Moreover:

\begin{prop}
\label{}\hypertarget{}{}
In non-negatively graded dg-coalgebras, both [[Quillen functors]] $(\mathcal{L} \dashv CE)$ from prop. \ref{QuillenAdjunctionCE} preserve all [[quasi-isomorphisms]], and both the [[adjunction unit]] and the [[adjunction counit]] are quasi-isomorphisms.

\end{prop}
For dg-algebras in degrees $\geq n \geq 1$ this is (\hyperlink{Quillen76}{Quillen 76, theorem 7.5}). In unbounded degrees this is (\hyperlink{Hinich98}{Hinich 98, prop. 3.3.2})

\begin{theorem}
\label{}\hypertarget{}{}
The Quillen adjunctin from prop. \ref{QuillenAdjunctionCE} is a [[Quillen equivalence]]:

\begin{displaymath}
(dgLieAlg_k)_{proj}
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {{}_{\phantom{qu}}\simeq_{Qu}}
  (dgCocAlg_k)_{Quillen}
  \,.
\end{displaymath}
\end{theorem}
(\hyperlink{Hinich98}{Hinich 98, theorem 3.2}) using (\hyperlink{Quillen76}{Quillen 76 II 1.4})

\hypertarget{relation_to_simplicial_lie_algebras}{}\subsubsection*{{Relation to simplicial Lie algebras}}\label{relation_to_simplicial_lie_algebras}

The [[normalized chains complex]] functor from [[simplicial Lie algebras]] constitutes a [[Quillen adjunction]] from the projective [[model structure on simplicial Lie algebras]], see \href{model+structure+on+simplicial+Lie+algebras#QuillenAdjunctionTodgLieAlgebras}{there}.

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

\begin{itemize}%
\item [[model structure on dg-coalgebras]]


\item [[model structure on simplicial Lie algebras]]


\item [[model structure for L-∞ algebras]]



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

The model structure on dg-Lie algebras in [[characteristic zero]] and in degrees $\geq n \geq 1$ goes back to

\begin{itemize}%
\item [[Dan Quillen]], section II.5 and appendix B of \emph{Rational homotopy theory}, Annals of Math., 90(1969), 205--295 (\href{http://www.jstor.org/stable/1970725}{JSTOR}, \href{http://www.math.northwestern.edu/~konter/gtrs/rational.pdf}{pdf})

\end{itemize}
This is extended to a model structure on dg-Lie algebras in unbounded degrees in

\begin{itemize}%
\item [[Vladimir Hinich]], \emph{Homological algebra of homotopy algebras}, Comm. in algebra, 25(10)(1997), 3291--3323 (\href{http://arxiv.org/abs/q-alg/9702015}{arXiv:q-alg/9702015}, \emph{Erratum} (\href{http://arxiv.org/abs/math/0309453}{arXiv:math/0309453}))

\end{itemize}
and the corresponding [[Quillen adjunction]] to the [[model structure on dg-coalgebras]] in unbounded degrees is discussed in

\begin{itemize}%
\item [[Vladimir Hinich]], \emph{DG coalgebras as formal stacks}, Journal of Pure and Applied Algebra Volume 162, Issues 2--3, 24 August 2001, Pages 209--250 (\href{http://arxiv.org/abs/math/9812034}{arXiv:math/9812034})

\end{itemize}
See also section 2.1 of

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

\end{itemize}
Review with discussion of [[homotopy limits]] and [[homotopy colimits]] is in

\begin{itemize}%
\item [[Ben Walter]], section 4 of \emph{Rational Homotopy Calculus of Functors} (\href{http://arxiv.org/abs/math/0603336}{arXiv:math/0603336})

\end{itemize}
[[!redirects model structures on dg-Lie algebras]]



\end{document}