Schreiber
Chevalley-Eilenberg algebra

Contents

Idea

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

If A=𝔤A = \mathfrak{g} is the infinitesimal object given by the Lie algebra of an ordinary Lie group, then CE(A)=CE(𝔤)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 𝔞\mathfrak{a} an ∞-Lie algebroid, CE(𝔞)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 thC = 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() opCEsPSh(C) proj loc cAlg(\mathbb{R})^{op} \stackrel{\overset{CE}{\leftarrow}}{\underset{}{\to}} sPSh(C)_{proj}^{loc}

where cAlg() opcAlg(\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 CECE is the functor

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

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

Proof

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

That CECE has a right adjoint

A Hom Alg(A ,PSh C(,R)):cAlg() opsPSh(C) 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

Hom sPSh C(X ,Hom Alg(A ,PSh C(,R)) UHom SSet(X (U),Hom Alg(A ,PSh C(U,R))) U,nHom Set(X n(U),Hom Alg(A n,PSh C(U,R))) U,nHom Alg(A n,Hom Set(X n(U),PSh C(U,R))) U,nHom Alg(A n,Hom Set(X n(U),PSh C(U,R))) nHom Alg(A n, UHom Set(X n(U),PSh C(U,R))) nHom Alg(A n,Hom PSh C(X n,R)) Hom Alg Δ(A ,PSh C(X n,R)). \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 sSetsSet-enrichment of the right adjoint is then fixed by adjunction.

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

Now observe that CECE send Cech nerve projections C({U i})UC(\{U_i\}) \to U for covering families {U iU}\{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 CECE sends fibrant objects to locally fibrant objects.

Since cAlg() opcAlg(\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 CECE is a left Quillen functor.

Examples

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

    𝔤=((G×G) (1)G (1)*), \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(𝔤)CE(\mathfrak{g}) is (under passage to normalized cochains) the ordinary Chevalley-Eilenberg algebra of the Lie algebra of GG.

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

  • Let XX 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 Δ inf )=(Ω (X),d dR). CE(X^{\Delta^\bullet_{inf}}) = (\Omega^\bullet(X), d_{dR}) \,.

    Then we have a zig-zag of quasi-isomorphism

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

    in cAlg() opcAlg(\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 )X = (X_\bullet) be a simplicial manifold. Then

    CE(X (Δ inf ))=(Ω (X ),d dR) 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.

Revised on May 20, 2010 06:34:57 by Urs Schreiber (87.212.203.135)