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\begin{document}

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\section*{model structure on dg-coalgebras}

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

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

[[!include model category 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{RelationTodgLieAlgbras}{Relation to dg-Lie algebras}\dotfill \pageref*{RelationTodgLieAlgbras} \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}

A [[model category]] structure on the [[category]] of [[dg-coalgebras]].

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

Let $k$ be a [[field]] of [[characteristic]] 0.

\begin{prop}
\label{TheCETangentAdjunction}\hypertarget{TheCETangentAdjunction}{}
There is a pair of [[adjoint functors]]

\begin{displaymath}
(\mathcal{L} \dashv \mathcal{C})
  \;\colon\;
  dgLieAlg_k
   \underoverset
      {\underset{\mathcal{C}}{\longrightarrow}}
      {\overset{\mathcal{L}}{\longleftarrow}}
      {\bot}
  dgCoCAlg_k
\end{displaymath}
between the category of [[dg-Lie algebras]]and that of dg [[cocommutative coalgebras]], where the [[right adjoint]] sends a dg-Lie algebra $(\mathfrak{g}_\bullet, [-,-])$ to its [[Chevalley-Eilenberg algebras|Chevalley-Eilenberg coalgebra]], whose underlying coalgebra is the [[free graded co-commutative coalgebra]] on $\mathfrak{g}[1]$ and whose differential is given on the tensor product of two generators by the Lie bracket $[-,-]$.

\end{prop}
For (pointers to) the details, see at \emph{\href{model%20structure%20on%20dg-Lie%20algebras#RelationToDgCoalgebras}{model structure on dg-Lie algebras -- Relation to dg-coalgebras}}.

\begin{theorem}
\label{}\hypertarget{}{}
There exists a [[model category]] structure on $dgCoCAlg_k$ for which

\begin{itemize}%
\item the cofibrations are the (degreewise) [[injections]];


\item the weak equivalences are those morphisms that become [[quasi-isomorphisms]] under the functor $\mathcal{L}$ from prop. \ref{TheCETangentAdjunction}.



\end{itemize}
Moreover, this is naturally a [[simplicial model category]] structure.

\end{theorem}
This is (\hyperlink{Hinich98}{Hinich98, theorem, 3.1}). More details on this are in the relevant sections at \emph{[[model structure for L-infinity algebras]]}.

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

\hypertarget{RelationTodgLieAlgbras}{}\subsubsection*{{Relation to dg-Lie algebras}}\label{RelationTodgLieAlgbras}

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

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


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


\item [[model structure on dg-Lie algebras]], [[model structure on simplicial Lie algebras]]


\item [[model structure on coalgebras over a comonad]]


\item [[rational homotopy theory]]



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

In [[characteristic zero]] and in positive degrees the model structure is due 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}
in non-negative degrees in

\begin{itemize}%
\item [[Ezra Getzler]], [[Paul Goerss]], \emph{A model category structure for differential graded coalgebras} (\href{http://www.math.northwestern.edu/~pgoerss/papers/model.ps}{ps}, [[GetzlerGoerss99.pdf:file]])

\end{itemize}
and in unbounded degrees 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})


\item [[Jonathan Pridham]], \emph{Unifying derived deformation theories}, Adv. Math. 224 (2010), no.3, 772-826 (\href{http://arxiv.org/abs/0705.0344}{arXiv:0705.0344})



\end{itemize}
See also

\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 5 of \emph{Rational Homotopy Calculus of Functors} (\href{http://arxiv.org/abs/math/0603336}{arXiv:math/0603336})

\end{itemize}
[[!redirects model stucture on dg-coalgebras]]



\end{document}