Schreiber Chevalley-Eilenberg algebra

Idea

In a context of Lie theory the Chevalley-Eilenberg algebra $CE(A)$ of an ∞-Lie groupoid $A$ is the algebra of functions on the $\infty$-Lie groupoid: a cosimplicial algebra which in degree $k$ is the algebra of functions on the k-morphisms of the $\infty$-Lie groupoid.

If $A = \mathfrak{g}$ is the infinitesimal object given by the Lie algebra of an ordinary Lie group, then $CE(A) = CE(\mathfrak{g})$ coinices (under taking its normalized chains) with the ordinary notion of Chevalley-Eilenberg algebra of a Lie algebra (with values). More generally, for $\mathfrak{a}$ an ∞-Lie algebroid, $CE(\mathfrak{a})$ is the corresponding Chevalley-Eilenberg dg-algebra.

The more general context in which the operation of forming Chevalley-Eilenberg algebras is to be understood is discussed at rational homotopy theory in an (∞,1)-topos.

Definition

Let $C = CartSp_{th}$, the category of infinitesimally thickened Cartesian spaces (see path ∞-groupoid for this notation).

Definition/Proposition

There is an sSet-enriched Quillen adjunction

$$cAlg(\mathbb{R})^{op} \stackrel{\overset{CE}{\leftarrow}}{\underset{}{\to}} sPSh(C)_{proj}^{loc}$$

where $cAlg(\mathbb{R})^{op}$ is the opposite of the category of cosimplicial algebras over $\mathbb{R}$ equipped with the standard model structure on cosimplicial algebras.

The left adjoint $CE$ is the functor

$$X \mapsto Hom_{PSh(C)}(X_\bullet,R) \,,$$

where on the right we take degreewise homs out of the simplicial object $X \in sPSh(C) = [\Delta^{op},PSh(C)]$ into the presheaf $R$ represented by $\mathbb{R}$ and regard the result as an algebra by using the algebra structure on $\mathbb{R}$.

Proof

First notice that $CE$ is indeed an sSet-enriched functor: for $X,Y \in sPSh(C)$, an $n$-cell in the hom-complex $sPSh(X,Y)$ is a morphism $X \cdot \Delta[n] \to Y$. Applying $PSh_C(-,R)$ to that yields a morphism of cosimplicial algebras $PSh(Y,R) \to PSh(X,R) \otimes C^\bullet(\Delta[n])$. This is indeed an $n$-cell in $cAlg(PSh(Y,R), PSh(X,R))$ and the construction is evnidently compatible with composition on both sides

That $CE$ has a right adjoint

$$A^\bullet \mapsto Hom_{Alg}(A^\bullet, PSh_C(-,R)) : cAlg(\mathbb{R})^{op} \to sPSh(C)$$

follows from general reasoning. In end/coend calculus we check

$$\begin{aligned} Hom_{sPSh_C}(X_\bullet, Hom_{Alg}(A^\bullet, PSh_C(-,R)) &\simeq \int_U Hom_{SSet}( X_\bullet(U), Hom_{Alg}(A^\bullet, PSh_C(U,R)) ) \\ & \simeq \int_{U,n} Hom_{Set}( X_n(U), Hom_{Alg}(A^n, PSh_C(U,R)) ) \\ & \simeq \int_{U,n} Hom_{Alg}( A^n, Hom_{Set}(X_n(U), PSh_C(U,R)) ) \\ & \simeq \int_{U,n} Hom_{Alg}( A^n, Hom_{Set}(X_n(U), PSh_C(U,R)) ) \\ & \simeq \int_{n} Hom_{Alg}( A^n, \int_U Hom_{Set}(X_n(U), PSh_C(U,R)) ) \\ & \simeq \int_{n} Hom_{Alg}( A^n, Hom_{PSh_C}(X_n, R) ) \\ & \simeq Hom_{Alg^{\Delta}}( A^\bullet, PSh_C(X_n, R) ) \end{aligned} \,.$$

The $sSet$-enrichment of the right adjoint is then fixed by adjunction.

To see that $CE$ is a left Quillen functor, first observe that a cofibration $X \to Y$ in $sPSh(C)_{proj}^{loc}$ is the same as a cofibration in $sPSh(C)_{proj}$ which is in particular a cofibration in $sPSh(C)_{inj}$, which is a degreewise monomorphism. It follows that $Hom_{PSh(C)}(Y_n,R) \to Hom_{PSh(C)}(X_n,R)$ is a surjection for all $n \in \mathbb{N}$. Hence $CE$ preserves cofibrations.

Now observe that $CE$ send Cech nerve projections $C(\{U_i\}) \to U$ for covering families $\{U_i \to U\}$ to weak equivalences (A proof of this is spelled out for instance in section 8 here ). By the nature of Bousfield localization this implies that the right adjoint of $CE$ sends fibrant objects to locally fibrant objects.

Since $cAlg(\mathbb{R})^{op}$ is evidently left proper, since all its objects are cofibrant, this allows to apply HTT, prop. A.3.7.2, to conclude that $CE$ is a left Quillen functor.

Examples

• Let $\mathfrak{g} := Lie(\mathbf{B}G) \hookrightarrow \mathbf{B}G$ be the ∞-Lie algebroid corresponding to $\mathbf{B}G$, which as a simplicial object in $\mathcal{T}$ is

  $$\mathfrak{g} = \left( \cdots (G \times G)_{(1)} \stackrel{\to}{\stackrel{\to}{\to}} G_{(1)} \stackrel{\to}{\to} {*} \right) \,,$$

  where in each term we have the first order infinitesimal neighbourhood of the neutral element. Then $CE(\mathfrak{g})$ is (under passage to normalized cochains) the ordinary Chevalley-Eilenberg algebra of the Lie algebra of $G$.

  This is discussed at Chevalley-Eilenberg algebra in synthetic differential geometry.

• Let $X$ be an ordinary manifold. Using the notation and results from path ∞-groupoid we have that (under passage to normalized cochains) a canonical isomorphism

  $$CE(X^{\Delta^\bullet_{inf}}) = (\Omega^\bullet(X), d_{dR}) \,.$$

  Then we have a zig-zag of quasi-isomorphism

  $$\array{ CE(\mathbf{\Pi}_{D}(X)) &\to& CE(\mathbf{\Pi}_R(X)) \\ \downarrow \\ CE(X^{\Delta^\bullet_{inf}}) }$$

  in $cAlg(\mathbb{R})^{op}$. This is the de Rham theorem in that it exhibits the equivalence between de Rham cohomology and singular cohomology (details at path ∞-groupoid.)

• Let more generally $X = (X_\bullet)$ be a simplicial manifold. Then

  $$CE(X^{(\Delta^\bullet_{inf})}) = (\Omega^\bullet(X_\bullet), d_{dR})$$

  is isomorphic to the simplicial deRham complex on the right. See there for the proof.

  The above proposition says that the simplicial deRham complexes of weakly equivalent simplicial manifolds are quasi-isomorphic.