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\begin{document}

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\section*{infinity-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{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil 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{ExplicitDefinition}{Explicit definition}\dotfill \pageref*{ExplicitDefinition} \linebreak
\noindent\hyperlink{Topos}{$(\infty,1)$-topos theoretic interpretation}\dotfill \pageref*{Topos} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{transgression_between_invariant_polynomials_and_cocycles_via_chernsimons_elements}{Transgression between invariant polynomials and cocycles via Chern-Simons elements}\dotfill \pageref*{transgression_between_invariant_polynomials_and_cocycles_via_chernsimons_elements} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{Extensions}{Extensions}\dotfill \pageref*{Extensions} \linebreak
\noindent\hyperlink{examples_3}{Examples}\dotfill \pageref*{examples_3} \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}

The notion of \emph{$\infty$-Lie algebra cohomology} generalizes the notion of [[Lie algebra cohomology]] from [[Lie algebra]]s to [[∞-Lie algebra]]s.

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

For $\mathbf{H}$ an [[(∞,1)-topos]] over duals of algebras over an abelian [[Lawvere theory]] $T$, we have by the theory of [[function algebras on ∞-stacks]] a [[reflective (∞,1)-subcategory]]

\begin{displaymath}
\mathbf{L} \stackrel{\leftarrow}{\hookrightarrow}
  \mathbf{H}
\end{displaymath}
obtained as the [[localization of an (∞,1)-category|localization]] of $\mathbf{H}$ at morphisms that induces [[isomorphism]]s in [[cohomology]] with coefficients in the canonical [[line object]] $\mathbb{A}$, where the small objects in $\mathbf{L}$ are modeled by duals of [[cosimplicial algebra]]s.

We may think of $\mathbf{L}$ as the [[(∞,1)-category]] of all [[∞-Lie algebroid]]s inside the [[∞-Lie groupoid]]s which are the objects of $\mathbf{H}$. For instance for $T$ the theory of commutative [[associative algebra]]s over a field, the [[monoidal Dold-Kan correspondence]] identified cosimplicial algebras with [[dg-algebra]]s, which we may think of as the [[Chevalley-Eilenberg algebra]]s of the [[∞-Lie algebroid]]s.

An [[∞-Lie algebra]] $\mathfrak{g}$ is a [[connected]] object in $\mathbf{L}$ and \textbf{$\infty$-Lie algebra cohomology} is the [[cohomology|intrinsic cohomology]] of $\mathbf{H}$ restricted to $\mathbf{L}$.

Typically $\mathbf{L}$ is [[presentable (∞,1)-category|presented]] by the [[opposite category|opposite]] of a [[function algebras on ∞-stacks|model structure on cosimplicial/cochain algebras]]: the [[Chevalley-Eilenberg algebra]]s of the given [[∞-Lie algebroid]]s. In terms of that model cocycle in $\infty$-Lie algebra cohomology have explicit and familiar algebraic expressions. These we discuss in

\begin{itemize}%
\item \hyperlink{ExplicitDefinition}{Explicit definition} .

\end{itemize}
A discussion of details of how exactly this models the general abstract definition is in

\begin{itemize}%
\item \hyperlink{Topos}{(∞,1)-Topos theoretic interpretation}.

\end{itemize}
\hypertarget{ExplicitDefinition}{}\subsubsection*{{Explicit definition}}\label{ExplicitDefinition}

For $\mathfrak{g}$ an [[∞-Lie algebra]] and $n \in \mathbb{N}$, a [[cocycle]] on $\mathfrak{g}$ in degree $n$ with coefficients in the trivial module is a morphism

\begin{displaymath}
\mu : \mathfrak{g} \to b^{n-1}\mathbb{R}
\end{displaymath}
to the [[line Lie n-algebra]].

Dually this is a [[dg-algebra]] morphism

\begin{displaymath}
CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R}) : \mu
\end{displaymath}
of [[Chevalley-Eilenberg algebra]]s. here $CE(b^{n-1} \mathbb{R})$ is the [[semifree dga]] on a single generator in degree $n$ with vanishing differential. So this is equivalently an element

\begin{displaymath}
\mu \in \wedge^n \mathfrak{g}^*
\end{displaymath}
which is closed in $CE(\mathfrak{g})$. For $\mathfrak{g}$ an ordinary Lie algebra, this latter description reproduces the traditional definition of cocycles in [[Lie algebra cohomology]].

For the moment see

\begin{itemize}%
\item [[Chevalley-Eilenberg algebra]]


\item [[Weil algebra]]


\item [[invariant polynomial]]



\end{itemize}
for more.

\hypertarget{Topos}{}\subsubsection*{{$(\infty,1)$-topos theoretic interpretation}}\label{Topos}

We may understand the above definitions of $\infty$-Lie algebra cocycles as a special case of the general notion of the [[cohomology|intrinsic cohomology]] of an [[(∞,1)-topos]] by embedding $\infty$-Lie algebras as [[infinitesimal space|infinitesimal]] [[∞-Lie group]]s into the [[(∞,1)-topos]] $\mathbf{H} =$ [[?LieGrpd]] of [[∞-Lie groupoids]].

For a general recognition principle of [[homotopy fibers]] in the [[model structure for L-infinity algebras]] see also (\hyperlink{FiorenzaRogersSchreiber13}{Fiorenza-Rogers-Schreiber 13, theorem 3.1.13}).

Recall from [[function algebras on ∞-stacks]] that [[∞-Lie algebroids]] form the [[reflective sub-(∞,1)-category]]

\begin{displaymath}
\mathbf{L}
  \stackrel{\leftarrow}{\hookrightarrow}
  \mathbf{H}
\end{displaymath}
of a corresponding [[(∞,1)-topos]] $\mathbf{H}$ of structure $\infty$-groupoids.

As described at [[?LieGrpd]], one realization of this general situation for genuine $\infty$-Lie groupoids is as follows:

Let [[ThCartSp]] be the [[site]] of [[infinitesimal object|infinitesimally]] thickened [[Cartesian space]]s. This is the site for the [[Cahiers topos]]. Then the [[(∞,1)-category of (∞,1)-sheaves]] $\mathbf{H} = Sh(ThCartSp)$ we may take to be the $(\infty,1)$-topos of [[synthetic differential ∞-groupoid]]s. We have then a [[simplicial Quillen adjunction]]

\begin{displaymath}
(C^\infty Alg^{\Delta})^{op} \stackrel{\leftarrow}{\hookrightarrow}
  [ThCartSp^{op}, sSet]_{proj,loc}
\end{displaymath}
between the [[opposite category|opposite]] of the [[function algebras on ∞-stacks|model structure on cosimplicial smooth algebras]]. This models the reflective inclusion of [[∞-Lie algebroid]]s into all synthetic differential $\infty$-groupoids

\begin{displaymath}
\infty LieAlg stackrel{\leftarrow}{\hookrightarrow}
  \infty SDGrpd
  \,.
\end{displaymath}
Details on this are at [[function algebras on ∞-stacks]]. But the [[function algebras on ∞-stacks|model structure on cosimplicial smooth algebras]]s is the [[transferred model structure]] of the [[model structure on cosimplicial rings]], and for the following discussion we can essentially just as well use the analogous Quillen adjunction without the smooth structure originally considered by [[Bertrand Toen]]

\begin{displaymath}
(CAlg^\Delta)^{op} \stackrel{\leftarrow}{\hookrightarrow}
  [CAlg,sSet]_{proj,cov}
\end{displaymath}
that is referenced and reviewed in some detail at [[rational homotopy theory in an (∞,1)-topos]].

Notice that the embedding map is just degreewise the [[Yoneda embedding]].

Notice moreover that by the [[monoidal Dold-Kan correspondence]] (see there for details) we have that the dual Dold-Kan functor $\Xi : Ch^\bullet_+ \to Ab^\Delta$ extends to the right adjoint part in a [[Quillen equivalence]] between the opposite of the [[model structure on dg-algebras]] and the opposite [[model structure on cosimplicial rings|model structure on cosimplicial algebras]]

\begin{displaymath}
\Xi^{op} : dgAlg^{op} \stackrel{\simeq_{Quillen}}{\to}
   (CAlg^\Delta)^{op}
  \,.
\end{displaymath}
In total this gives a right Quillen functor

\begin{displaymath}
R : 
  dgAlg^{op} \stackrel{\Xi^{op}}{\to} (CAlg^\Delta)^{op}
  \to
  [CAlg, sSet]_{proj,cov}
\end{displaymath}
that models the embedding of $\infty$-Lie algebroids into a [[(∞,1)-topos]] of $\infty$-Lie groupoids. When restricted to [[∞-Lie algebra]]s ($\infty$-Lie algebroids over the point) the difference between the sites $CAlg^{op}$ and [[ThCartSp]] plays no role. In fact for that case we could just as well restrict to a site of only [[infinitesimal space]]s, because all homs from a finite non-thickened space into an infinitesimal space are trivial anyway.

Therefor for $\mathfrak{g}$ and $\mathfrak{h}$ $\infty$-Lie algebras, a [[cocycle]] on $\mathfrak{g}$ with values in $\mathfrak{h}$ is just a morphism

\begin{displaymath}
(c : \mathfrak{g} \to \mathfrak{h}) \in \infty LieAlg \subset \infty LieGrpd
\end{displaymath}
and the [[∞-groupoid]] of cocycles is

\begin{displaymath}
\infty LieGrpd(\mathfrak{g}, \mathfrak{h})
  \simeq
  \infty LieAlg(\mathfrak{g}, \mathfrak{h})
  \,.
\end{displaymath}
Such cocycles are modeled by morphisms in $dgAlg^{op}$ from a cofibrant representative of $\mathfrak{g}$ to a fibrant representative of $\mathfrak{h}$. Since in $dgAlg$ all objects are fibrant, in $dgAlg^{op}$ all objects are cofibrant. The cofibrant objects in the [[model structure on dg-algebras]] are the [[Sullivan algebra]]s $CE(\mathfrak{h})$. In particular for $\mathfrak{h} = b^{n-1}\mathbb{R}$ we have that $CE(b^{n-1}\mathbb{R})$ is a Sullivan algebra, so $b^{n-1} \mathbb{R}$ is fibrant in $dgAlg^{op}$.

In summary, this says that morphisms

\begin{displaymath}
CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R})
\end{displaymath}
indeed model the abstract intrinsic $(\infty,1)$-topos theoretic notion of cocycles in $\infty Lie Algd \subset \infty Lie Grpd$.

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

Special cases of $\infty$-Lie algebra cohomology are of course

\begin{itemize}%
\item [[Lie algebra cohomology]]


\item [[nonabelian Lie algebra cohomology]].



\end{itemize}
Specific examples include:

\begin{itemize}%
\item cohomology of [[Poisson Lie algebroids]] is \emph{[[Poisson cohomology]]}.

\end{itemize}
\hypertarget{transgression_between_invariant_polynomials_and_cocycles_via_chernsimons_elements}{}\subsection*{{Transgression between invariant polynomials and cocycles via Chern-Simons elements}}\label{transgression_between_invariant_polynomials_and_cocycles_via_chernsimons_elements}

We recall the procedure by which to an [[∞-Lie algebroid]] [[invariant polynomial]] $\omega$ we associate an [[∞-Lie algebroid]] cocycle $\nu$ that is \emph{in transgression with} $\omega$.

The [[dg-algebra]] of [[invariant polynomial]]s is a sub-dg-alghebra of the [[kernel]] of the canonical morphism $W(\mathfrak{a}) \to CE(\mathfrak{a})$ from the [[Weil algebra]] to the [[Chevalley-Eilenberg algebra]] of $\mathfrak{a}$

\begin{displaymath}
inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) = ker(W(\mathfrak{a}) \to CE(\mathfrak{a}))
  \,.
\end{displaymath}
From the [[short exact sequence]]

\begin{displaymath}
CE(\Sigma \mathfrak{a}) \to W(\mathfrak{a}) \to 
  CE(\mathfrak{a})
\end{displaymath}
we obtain the long exact sequence in [[chain homology and cohomology|cohomology]]

\begin{displaymath}
\cdots 
  \to 
  H^{n+1}(CE(\mathfrak{a}))
  \stackrel{\delta}{\to} H^{n+2}(CE(\Sigma \mathfrak{a}))
  \to \cdots 
  \,.
\end{displaymath}
We say that $\mu \in CE(\mathfrak{a})$ is in transgression with $\omega \in inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a})$ if their classes map to each other under the connecting homomorphism $\delta$:

\begin{displaymath}
\delta : [\mu] \mapsto [\omega]
  \,.
\end{displaymath}
The following spells out in detail how one finds to a given invariant polynomial $\omega$ the cocycle that it is in transgression with.

\begin{enumerate}%
\item We first regard the [[invariant polynomial]] $\omega$ as an element of the [[Weil algebra]] $W(\mathfrak{a})$ under the inclusion $inv(\mathfrak{a}) \hookrightarrow W(\mathfrak{a})$, where, by the very definiton of invariant polynomials, it is closed: $d_{W(\mathfrak{a})} \omega = 0$.


\item then we find an element $cs_\omega \in W(\mathfrak{a})$ with the property that $d_{W(\mathfrak{a})} cs_\omega = \omega$. This is guranteed to exist because $W(\mathfrak{a})$ has trivial cohomology.


\item then we send this element $cs_\omega\in W(\mathfrak{a})$ along the restriction map $W(\mathfrak{a}) \to CS(\mathfrak{a})$ to an elemeent we call $\nu$.



\end{enumerate}
The procedure is illustarted by the following diagram

\begin{displaymath}
\itexarray{
    0 && \omega &\leftarrow & \omega
    \\
    \;\;\uparrow^{\mathrlap{d_{CE(\mathfrak{a})}}}
    &&
    \;\;\uparrow^{\mathrlap{d_{W(\mathfrak{a})}}}
    \\
    \nu &\leftarrow& cs(\omega)
    \\
    \\
    \\
    \\
    CE(\mathfrak{a})
    &\leftarrow&
    W(\mathfrak{a})
    &\leftarrow&
    inv(\mathfrak{a})    
  }
\end{displaymath}
From the fact that all morphisms involved respect the differential and from the fact that the image of $\omega$ in $CE(\mathfrak{a})$ vanishes it follows that

\begin{itemize}%
\item this element $\nu$ satisfies $d_{CE(\mathfrak{a})} \nu = 0$, hence that it is an $\infty$-Lie algebroid cocycle.


\item any two different choices of $cs_\omega$ lead to cocylces $\mu$ that are cohomologous.



\end{itemize}
We say $\nu$ is a cocycle \emph{in transgression with} $\omega$. We may call $cs_{\omega}$ here a \emph{Chern-Simons element} of $\omega$. Because for $A : T X \to \mathfrak{a}$ any collection of [[∞-Lie algebroid valued differential forms]] coming dually from a dg-morphism $\Omega^\bullet(X) \leftarrow W(\mathfrak{a}) : A$ the image $\omega(A)$ of $\omega$ will be a curvature characteristic form and the image $cs_\omega(A)$ its corresponding Chern-Simons form.

In the case where $\mathfrak{g}$ is an ordinary semisimple [[nLab:Lie algebra|Lie algebra]], this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with $\mathfrak{g}$-valued 1-forms. This is described in the section \emph{Semisimple Lie algebras} .

\hypertarget{examples_2}{}\subsubsection*{{Examples}}\label{examples_2}

For $\mathfrak{g}$ an [[semisimple Lie algebra]], the transgression between the [[Killing form]]-[[invariant polynomial]] and the 3-cocycle $\langle -, [-,-] \rangle$ is exhibited by the ``ordinary'' Chern-Simons element, which gives these [[action functional]] of ordinary [[Chern-Simons theory]].

A [[symplectic Lie n-algebroid]] is an [[∞-Lie algebroid]] equipped with a nondegenerate binary invariant polynomial in degree $n+2$.

Examples are

\begin{itemize}%
\item [[symplectic manifold]]s


\item [[Poisson Lie algebroid]]s


\item [[Courant algebroid]]s.



\end{itemize}
The coresponding Chern-Simons elements exhibiting the transgression of these invariant polynomials give [[action functional]]s for generalized [[Chern-Simons theory]] (see the above entries for more details).

\hypertarget{Extensions}{}\subsection*{{Extensions}}\label{Extensions}

In any [[(∞,1)-topos]] with its intrinsic notion of [[cohomology]], a [[cocycle]] $c : X \to \mathbf{B}^{n+1} A$ classifies an \emph{extension} $\mathbf{B}^n A \to \hat X \to X$. This $\hat X$ is nothing but the [[homotopy fiber]] of $c$, or equivalently the $\mathbf{B}^n A$-[[principal ∞-bundle]] classified by $c$.

After embedding [[∞-Lie algebra]]s into the [[(∞,1)-topos]] of [[∞-Lie groupoid]]s as described \hyperlink{Topos}{above}, the same abstract reasoning applies to $\infty$-Lie algebra cocycles and the extensions of $\infty$-Lie algebras that these classify: for $c : \mathfrak{g} \to b^n \mathbb{R}$ a cocycle of $\infty$-Lie algebras, the extension $b^{n-1} \mathbb{R} \to \hat \mathfrak{g} \to \mathfrak{g}$ is the [[homotopy fiber]] of this morphism in [[?LieGrpd]].

\begin{quote}%
a more systematic discussion is now in the section  at [[synthetic differential ∞-groupoid]].


\end{quote}
For $\mathfrak{g}$ an ordinary [[Lie algebra]], this reproduces the ordinary notions of extensions from [[Lie algebra cohomology]] and [[nonabelian Lie algebra cohomology]].

\textbf{Observation}

For $c : \mathfrak{g} \to b^n \mathbb{R}$ an $(n+1)$-cocycle of an $\infty$-Lie algebra $\mathfrak{g}$, the ordinary [[pullback]] in $dgAlg^{op}$

\begin{displaymath}
\itexarray{
    \hat g &\to& inn(b^{n-1}\mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathfrak{g} &\stackrel{c}{\to}& b^n \mathfrak{R}
  }
\end{displaymath}
maps under $R$ to a pullback diagram of simplicial presheaves which exhibits $R(\hat \mathfrak{g})$ as isomorphic to the [[homotopy pullback]] in the [[homotopy category]].

Here the right morphism denotes the dual of the generating cofibration in $dgAlg$, which models the $b^n \mathfrak{R}$-[[universal principal ∞-bundle]].

\textbf{Proof}

Being a right Quillen functor, $R$ preserves fibrations and pullbacks, hence

\begin{displaymath}
\itexarray{
    R \hat g &\to& R inn(b^{n-1}\mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    R \mathfrak{g} &\stackrel{R c}{\to}& R b^n \mathfrak{R}
  }
\end{displaymath}
is a pullback of a fibration. Since $[ThCartSp^{op},sSet]_{proj}$ is a [[proper model category|right proper model category]] this is a [[homotopy pullback]], even if $R \mathfrak{g}$ is possibly not fibrant. (The detailed argument for that is reproduced at [[proper model category]].)

Since [[∞-stackification]] preserves finite [[(∞,1)-limit]]s, this is sufficient to deduce that $R \hat \mathfrak{g}$ represents in the [[homotopy category]] $Ho([ThCartSp, sSet]_{proj,cov})$ the [[homotopy fiber]] of $R c : R \mathfrak{g} \to R b^n \mathbb{R}$.

\hypertarget{examples_3}{}\subsubsection*{{Examples}}\label{examples_3}

\begin{itemize}%
\item For $\mathfrak{g}$ and $\mathfrak{h}$ ordinary [[Lie algebra]]s, and $der(\mathfrak{h})$ the Lie algebra of [[derivation]]s of $\mathfrak{h}$, a morphism $\mathfrak{g}\to der(\mathfrak{h})$ is a cocycle in [[nonabelian Lie algebra cohomology]] and the extension it classifies is an ordinary [[Lie algebra extension]].


\item The [[string Lie 2-algebra]] is the $b \mathbb{R}$-extension of a semisimple [[Lie algebra]] $\mathfrak{g}$ with bilinear [[invariant polynomial]] $\langle -,-\rangle$ corresponding to the 3-cocycle $\langle -,[-,-]\rangle \in CE(\mathfrak{g})$.


\item Similarly for the [[supergravity Lie 3-algebra]] and the [[supergravity Lie 6-algebra]] and all the other extensions in [[schreiber:The brane bouquet]].



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

A comprehensive discusson of an ambient $\infty$-topos in which $\infty$-Lie algebroid cohomology lives is at

\begin{itemize}%
\item [[synthetic differential ∞-groupoid]].

\end{itemize}
Other notions related to $\infty$-Lie algebroid cohomology include

\begin{itemize}%
\item \textbf{$\infty$-Lie algebra cocycle}


\item [[Chern-Simons element]]


\item [[invariant polynomial]]


\item [[signs in supergeometry]]



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

Discussion of cohomology of $L_\infty$-algebras is in

\begin{itemize}%
\item Michael Penkava, \emph{$L_\infty$-algebras and their cohomology} (\href{http://arxiv.org/abs/q-alg/9512014}{arXiv:q-alg/9512014})

\end{itemize}
The relation between $L_\infty$-cohomology and extension of $L_\infty$-algebras is discussed around theorem 3.8 of

\begin{itemize}%
\item [[Andrey Lazarev]], \emph{Models for classifying spaces and derived deformation theory} (\href{http://arxiv.org/abs/1209.3866}{arXiv:1209.3866})

\end{itemize}
The general structure of the threory of $\infty$-Lie algebroid cohomology and [[transgression]] between $\infty$-Lie algebroid [[invariant polynomial]]s and -cocycles via [[Chern-Simons element]] was given in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{$L_\infty$-connections} ()

\end{itemize}
A recognition principle for [[homotopy fibers]] of $L_\infty$-homomorphisms appears as theorem 3.1.13 in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]} (\href{http://arxiv.org/abs/1304.6292}{arXiv:1304.6292})

\end{itemize}
Discussion of extensions of [[super L-∞ algebras]] based on the [[super Poincare Lie algebra]] is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]}

\end{itemize}
The hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor as shown in

\begin{itemize}%
\item [[Ugo Bruzzo]], \emph{Lie algebroid cohomology as a derived functor}, \href{http://arxiv.org/abs/1606.02487}{arxiv/1606.02487}

\end{itemize}
[[!redirects ∞-Lie algebra cohomology]] [[!redirects L-∞-algebra cohomology]] [[!redirects L-∞ algebra cohomology]] [[!redirects Lie ∞-algebra cohomology]]

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[[!redirects L-∞ cocycle]] [[!redirects L-∞ cocycles]] [[!redirects L-infinity cocycle]] [[!redirects L-infinity cocycles]]

[[!redirects L-∞ extension]] [[!redirects L-∞ extensions]] [[!redirects L-infinity extension]] [[!redirects L-infinity extensions]]

[[!redirects Lie algebroid cohomology]] [[!redirects Lie algebroid cocycle]] [[!redirects Lie algebroid cocycles]]

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[[!redirects ∞-Lie algebroid cohomology]] [[!redirects infinity-Lie algebroid cohomology]]

[[!redirects L-∞ algebra extension]] [[!redirects L-∞ algebra extensions]]

[[!redirects L-infinity algebra extension]] [[!redirects L-infinity algebra extensions]]



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