Schreiber
Chevalley-Eilenberg algebra

Idea

In a context of ∞-Lie theory the Chevalley-Eilenberg algebra of an ∞-Lie groupoid – which may be an ∞-Lie algebroid – is essentially the algebra of functions on its k-morphisms for all $k\in \mathbb{N}$.

This definition generalizes both the ordinary notion of Chevalley-Eilenberg algebra from Lie algebras to ∞-Lie algebroids as well as that of singular cohomology from path ∞-groupoids to arbitrary ∞-Lie groupoids.

The discussion here is to be joined with that at rational homotopy theory in an (∞,1)-topos.

Definition

Fix $(𝒯=\mathrm{Sh}(C),R)$ by a lined topos of sheaves on a site $C$ with subcanonical topology.

Let $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ be the corresponding hypercomplete (∞,1)-topos, i.e. the one presented by the local model structure on simplicial sheaves $\mathrm{SSh}(C)$.

Remark

In applications to ∞-Lie theory we will furthermore demand that $(𝒯,R)$ is a smooth topos and $H$ a smooth (∞,1)-topos, but the following definitions and constructions depend just on the fact that we have a lined topos.

The line object $R\in 𝒯$ is by definition a $k$-algebra object for some commutative ring object $k\in 𝒯$. We write in the following

$$𝕜:=𝒯(*,k)$$\mathbb{k} := \mathcal{T}({*}, k)

for the corresponding external commutative ring (in Set) and

$$ℝ:=𝒯(*,R)$$\mathbb{R} := \mathcal{T}({*}, R)

for the corresponding $𝕜$-algebra object.

The notation is slightly abusive, but in fact for all the well adapted models we care about indeed this object $ℝ$ is the ordinary real line.

Definition

For $A\in \mathrm{SPSh}(C)$ regarded as an object $A_{•}\in [{\Delta }^{\mathrm{op}},\mathrm{PSh}(C)]$ we say that the Chevalley-Eilenberg algebra of $A$ is

$$\mathrm{CE}(A):=C^{•}(𝒯(A_{•},R)).$$CE(A) := C^\bullet(\mathcal{T}(A_\bullet,R)) \,.

Here on the right we have

• the normalized Moore cochain complex of the cosimplicial algebra over $ℝ$ of (external) degreewise functions on $A$

• regarded as a dg-algebra over $ℝ$ using the cup product as described at monoidal Dold-Kan correspondence.

This extends to a contravariant functor

$$\mathrm{CE}(-):\mathrm{SSh}(C)\to \mathrm{dgAlg}(ℝ).$$CE(-) : SSh(C) \to dgAlg({\mathbb{R}}) \,.

Remark (characterization of $\mathrm{\infty }$-Lie algebroids)

An ∞-Lie algebroid is an ∞-Lie groupoid whose Chevalley-Eilenberg algebra is graded commutative. See there for details.

Properties

Proposition

The functor ${\mathrm{PSh}}_C(-,R):{\mathrm{PSh}}_C\to \mathrm{Alg}(ℝ{)}^{\Delta }$ is the left adjoint functor in a Quillen adjunction

$$𝒯(-,R):\mathrm{SSh}(C{)}_{\mathrm{inj}}^{\mathrm{loc}}\overleftarrow{\to }(\mathrm{Alg}(ℝ{)}^{\Delta }{)}^{\mathrm{op}}$$\mathcal{T}(-,R) : SSh(C)_{inj}^{loc} \stackrel{\leftarrow}{\to} (Alg(\mathbb{R})^{\Delta})^{op}

with the injective local model structure on simplicial sheaves on the left and the op-projective model structure on cosimplicial algebras on the right.

Proof

First observe that $𝒯(-,R)$ has a right adjoint

$$\mathrm{Alg}(ℝ{)}^{\Delta }\to {\mathrm{PSh}}_C$$Alg(\mathbb{R})^{\Delta} \to PSh_C

given by $A^{•}\mapsto {\mathrm{Hom}}_{\mathrm{Alg}}(A^{•},{\mathrm{PSh}}_C(-,R))$. This follows the general pattern of dualizing objects:

$$\begin{array}{rl}{\mathrm{Hom}}_{{\mathrm{SPSh}}_C}(X_{•},{\mathrm{Hom}}_{\mathrm{Alg}}(A^{•},{\mathrm{PSh}}_C(-,R))& \simeq {\int }^U{\mathrm{Hom}}_{\mathrm{SSet}}(X_{•}(U),{\mathrm{Hom}}_{\mathrm{Alg}}(A^{•},{\mathrm{PSh}}_C(U,R)))\\ & \simeq {\int }^{U,n}{\mathrm{Hom}}_{\mathrm{Set}}(X_n(U),{\mathrm{Hom}}_{\mathrm{Alg}}(A^n,{\mathrm{PSh}}_C(U,R)))\\ & \simeq {\int }^{U,n}{\mathrm{Hom}}_{\mathrm{Alg}}(A^n,{\mathrm{Hom}}_{\mathrm{Set}}(X_n(U),{\mathrm{PSh}}_C(U,R)))\\ & \simeq {\int }^{U,n}{\mathrm{Hom}}_{\mathrm{Alg}}(A^n,{\mathrm{Hom}}_{\mathrm{Set}}(X_n(U),{\mathrm{PSh}}_C(U,R)))\\ & \simeq {\int }^n{\mathrm{Hom}}_{\mathrm{Alg}}(A^n,{\int }^U{\mathrm{Hom}}_{\mathrm{Set}}(X_n(U),{\mathrm{PSh}}_C(U,R)))\\ & \simeq {\int }^n{\mathrm{Hom}}_{\mathrm{Alg}}(A^n,{\mathrm{Hom}}_{{\mathrm{PSh}}_C}(X_n,R))\\ & \simeq {\mathrm{Hom}}_{{\mathrm{Alg}}^{\Delta }}(A^{•},{\mathrm{PSh}}_C(X_n,R))\end{array}.$$\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} \,.

By the fact that $({\mathrm{Alg}}_k^{\Delta })$ is a simplicially enriched model category one finds that this is right Quillen for the global model structure on $\mathrm{sPSh}(C)$. For the Quillen adjunction on the local structure it now suffices to observes that the $R$-cohomology of covers $Y\to U$ is an isomorphism, hence that $\mathrm{CE}$ sends covers to quasi-isomorphisms.

(…details…)

Corollary

The functor $\mathrm{CE}(-)$ respects weak equivalences, hence on objects it coincides with its left derived functor

$$𝕃\mathrm{CE}(-):\mathrm{Ho}({\mathrm{SPSh}}_C^{\mathrm{loc}})\to \mathrm{Ho}(\mathrm{Alg}(ℝ{)}^{\Delta }).$$\mathbb{L}CE(-) : Ho(SPSh_C^{loc}) \to Ho(Alg(\mathbb{R})^\Delta) \,.

Examples

For the examples of interest we want to assume now explicitly that $(𝒯,R)$ is not just any lined topos, but a smooth topos, so that we have a sensible notion of infinitesimal objects in $𝒯$.

• Let $G$ be a Lie grouop and

  $$BG=(\cdots G\times G\overrightarrow{\overrightarrow{\to }}G\overrightarrow{\to }*)$$\mathbf{B}G = \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} {*} \right)

  its delooping Lie groupoid. Then $\mathrm{CE}(BG)$ is the complex that computes the Lie group cohomology of $G$.

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

  $$𝔤=(\cdots (G\times G{)}_{(1)}\overrightarrow{\overrightarrow{\to }}{G}_{(1)}\overrightarrow{\to }*),$$\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 $\mathrm{CE}(𝔤)$ is 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 and $\Pi (X)$ its path ∞-groupoid. Then $\mathrm{CE}(\Pi (X))$ is the complex that computes the singular cohomology of $X$.

• Let $TX:={\Pi }^{\inf }(X)=\mathrm{Lie}(\Pi (X))\hookrightarrow \Pi (X)$ be the infinitesimal path ∞-groupoid of the manifold $X$. Then

  $$\mathrm{CE}({\Pi }^{\inf }(X))=({\Omega }^{•}(X),d_{\mathrm{dR}})$$CE(\Pi^{inf}(X)) = (\Omega^\bullet(X), d_{dR})

  is (canonically isomorphic to) the deRham complex of $X$.

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

  $$\mathrm{CE}({\Pi }^{\inf }(X))=({\Omega }^{•}(X_{•}),d_{\mathrm{dR}})$$CE(\Pi^{inf}(X)) = (\Omega^\bullet(X_\bullet), d_{dR})

  is 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.