group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
∞-Lie theory (higher geometry)
The notion of $\infty$-Lie algebra cohomology generalizes the notion of Lie algebra cohomology from Lie algebras to ∞-Lie algebras.
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
obtained as the localization of $\mathbf{H}$ at morphisms that induces isomorphisms in cohomology with coefficients in the canonical line object $\mathbb{A}$, where the small objects in $\mathbf{L}$ are modeled by duals of cosimplicial algebras.
We may think of $\mathbf{L}$ as the (∞,1)-category of all ∞-Lie algebroids inside the ∞-Lie groupoids which are the objects of $\mathbf{H}$. For instance for $T$ the theory of commutative associative algebras over a field, the monoidal Dold-Kan correspondence identified cosimplicial algebras with dg-algebras, which we may think of as the Chevalley-Eilenberg algebras of the ∞-Lie algebroids.
An ∞-Lie algebra $\mathfrak{g}$ is a connected object in $\mathbf{L}$ and $\infty$-Lie algebra cohomology is the intrinsic cohomology of $\mathbf{H}$ restricted to $\mathbf{L}$.
Typically $\mathbf{L}$ is presented by the opposite of a model structure on cosimplicial/cochain algebras: the Chevalley-Eilenberg algebras of the given ∞-Lie algebroids. In terms of that model cocycle in $\infty$-Lie algebra cohomology have explicit and familiar algebraic expressions. These we discuss in
A discussion of details of how exactly this models the general abstract definition is in
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
to the line Lie n-algebra.
Dually this is a dg-algebra morphism
of Chevalley-Eilenberg algebras. 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
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
for more.
We may understand the above definitions of $\infty$-Lie algebra cocycles as a special case of the general notion of the intrinsic cohomology of an (∞,1)-topos by embedding $\infty$-Lie algebras as infinitesimal ∞-Lie groups 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 (Fiorenza-Rogers-Schreiber 13, theorem 3.1.13).
Recall from function algebras on ∞-stacks that ∞-Lie algebroids form the reflective sub-(∞,1)-category
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 infinitesimally thickened Cartesian spaces. 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 ∞-groupoids. We have then a simplicial Quillen adjunction
between the opposite of the model structure on cosimplicial smooth algebras. This models the reflective inclusion of ∞-Lie algebroids into all synthetic differential $\infty$-groupoids
Details on this are at function algebras on ∞-stacks. But the model structure on cosimplicial smooth algebrass 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
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 algebras
In total this gives a right Quillen functor
that models the embedding of $\infty$-Lie algebroids into a (∞,1)-topos of $\infty$-Lie groupoids. When restricted to ∞-Lie algebras ($\infty$-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 spaces, 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
and the ∞-groupoid of cocycles is
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 algebras $CE(\mathfrak{h})$. 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
indeed model the abstract intrinsic $(\infty,1)$-topos theoretic notion of cocycles in $\infty Lie Algd \subset \infty Lie Grpd$.
Special cases of $\infty$-Lie algebra cohomology are of course
Specific examples include:
We recall the procedure by which to an ∞-Lie algebroid invariant polynomial $\omega$ we associate an ∞-Lie algebroid cocycle $\nu$ that is in transgression with $\omega$.
The dg-algebra of invariant polynomials 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 $\mathfrak{a}$
From the short exact sequence
we obtain the long exact sequence in cohomology
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 $\delta$:
The following spells out in detail how one finds to a given invariant polynomial $\omega$ the cocycle that it is in transgression with.
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: $d_{W(\mathfrak{a})} \omega = 0$.
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.
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 $\nu$.
The procedure is illustarted by the following diagram
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
this element $\nu$ satisfies $d_{CE(\mathfrak{a})} \nu = 0$, hence that it is an $\infty$-Lie algebroid cocycle.
any two different choices of $cs_\omega$ lead to cocylces $\mu$ that are cohomologous.
We say $\nu$ is a cocycle in transgression with $\omega$. We may call $cs_{\omega}$ here a 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 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 Semisimple Lie algebras .
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
The coresponding Chern-Simons elements exhibiting the transgression of these invariant polynomials give action functionals for generalized Chern-Simons theory (see the above entries for more details).
In any (∞,1)-topos with its intrinsic notion of cohomology, a cocycle $c : X \to \mathbf{B}^{n+1} A$ classifies an 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 algebras into the (∞,1)-topos of ∞-Lie groupoids as described 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.
a more systematic discussion is now in the section Cohomology of ∞-Lie algebroids at synthetic differential ∞-groupoid.
For $\mathfrak{g}$ an ordinary Lie algebra, this reproduces the ordinary notions of extensions from Lie algebra cohomology and nonabelian Lie algebra cohomology.
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 $dgAlg^{op}$
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.
Proof
Being a right Quillen functor, $R$ preserves fibrations and pullbacks, hence
is a pullback of a fibration. Since $[ThCartSp^{op},sSet]_{proj}$ is a 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)-limits, this is sufficient to deduce that $R \hat \mathfrak{g}$ represents in the homotopy category $Ho([ThCartSp, sSet]_{proj,cov})$ the homotopy fiber of $R c : R \mathfrak{g} \to R b^n \mathbb{R}$.
For $\mathfrak{g}$ and $\mathfrak{h}$ ordinary Lie algebras, and $der(\mathfrak{h})$ the Lie algebra of derivations 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.
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 $\langle -,[-,-]\rangle \in CE(\mathfrak{g})$.
Similarly for the supergravity Lie 3-algebra and the supergravity Lie 6-algebra and all the other extensions in The brane bouquet.
A comprehensive discusson of an ambient $\infty$-topos in which $\infty$-Lie algebroid cohomology lives is at
Other notions related to $\infty$-Lie algebroid cohomology include
$\infty$-Lie algebra cocycle
Discussion of cohomology of $L_\infty$-algebras is in
The relation between $L_\infty$-cohomology and extension of $L_\infty$-algebras is discussed around theorem 3.8 of
The general structure of the threory of $\infty$-Lie algebroid cohomology and transgression between $\infty$-Lie algebroid invariant polynomials and -cocycles via Chern-Simons element was given in
A recognition principle for homotopy fibers of $L_\infty$-homomorphisms appears as theorem 3.1.13 in
Discussion of extensions of super L-∞ algebras based on the super Poincare Lie algebra is in
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