Schreiber
∞-Lie algebroid

Idea

The notion of $\mathrm{\infty }$-Lie algebroid is a horizontal and vertical categorification of the notion of Lie algebra.

The Lie algebra $𝔤$ of a Lie group $G$ is the infinitesimal neighbourhood of the neutral element in the Lie group $G$. Equivalently, this is usefully thought of as the collection of morphisms of infinitesimal extension in the delooping Lie groupoid $BG$ of $G$: those infinitesimally close to the identity morphism on the single object.

For a general Lie groupoid $A$, the collection of all morphisms infinitesimally close to any identity morphism forms the Lie algebroid $𝔞$ of $A$.

The notion of $\mathrm{\infty }$-Lie algebroid generalizes this from Lie theory to ∞-Lie theory where Lie groupoids are generalized to ∞-Lie groupoids.

An $\mathrm{\infty }$-Lie algebroid is an infinitesimal ∞-Lie groupoid: an $\mathrm{\infty }$-Lie groupoid all whose k-morphisms have infinitesimal extension.

The archetypical example of an $\mathrm{\infty }$-Lie algebroid is an infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$. For $X$ an ordinary manifold this is known as the tangent Lie algebroid of $X$.

Summary of higher infinitesimal notions

The fundamental definition of infinitesimal objects is at

• infinitesimal object.

Such objects exist in a context called a smooth topos

• smooth topos.

In sufficiently nice smooth toposes every object $X$ comes with its infinitesimal singular simplicial complex of infinitesimal simplices:

• infinitesimal singular simplicial complex.

This simplicial object is usefully thought of as modelling the infinitesimal path $\mathrm{\infty }$-groupoid ${\Pi }^{\inf }(X)$ of $X$:

• infinitesimal path ∞-groupoid.

And moreover, this may be thought of as the archetypical $\mathrm{\infty }$-Lie algebroid (namely the tangent Lie algebroid): an $\mathrm{\infty }$-Lie algebroid is an $\mathrm{\infty }$-Lie groupoid that is built from copies of ${\Pi }^{\inf }(U)$s. See below.

In particular, the Chevalley-Eilenberg algebra of functions on any $\mathrm{\infty }$-Lie groupoid $\mathrm{CE}(A):=N^{•}(C^{\mathrm{\infty }}(A^{•}))$, happens to be garded commutative when $A=𝔞$ is an $\mathrm{\infty }$-Lie algebroid:

• Chevalley-Eilenberg algebra.

Since moreover any graded-commutative dg-algebra is weakly equivalent to a semifree dg-algebra (a Sullivan algebra)

• model structure on dg-algebras

it follows that the Chevalley-Eilenberg algebra of any $\mathrm{\infty }$-Lie algebroid $b𝔞$ having a single 0-cell, is equivalent to (the formal dual of) an $L_{\mathrm{\infty }}$-algebra

• L-∞-algebra.

Definition

Let $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ be a smooth (∞,1)-topos presented by the local model structure on simplicial presheaves $\mathrm{SPSh}(C{)}_{\mathrm{loc}}$ on a site $C$.

Consider the operation

$$\mathrm{Lie}(-):={\Pi }^{\inf }({-}_{{\#}^{\inf }0}):\mathrm{SPSh}(C)\to \mathrm{SPSh}(C)$$Lie(-) := \Pi^{inf}({-}_{#^{inf} 0}) : SPSh(C) \to SPSh(C)

described at ∞-Lie differentiation and integration, that sends an ∞-Lie groupoid $X\in \mathrm{SPSh}(X)$ to its subobject of infinitesimal morphisms.

Definition

($\mathrm{\infty }$-Lie algebroid)

We say that an ∞-Lie groupoid $𝔞\in \mathrm{SPSh}(C)$ is an $\mathrm{\infty }$-Lie algebroid if the canonical morphism

$$\mathrm{Lie}(𝔞)\to 𝔞$$Lie(\mathfrak{a}) \to \mathfrak{a}

is an isomorphism.

Remark

(graded commutativity of $\mathrm{CE}(𝔞)$)

It follows – by the discussion at ∞-Lie differentiation and integration – that the Chevalley-Eilenberg algebra $\mathrm{CE}(𝔞)$ of an $\mathrm{\infty }$-Lie algebroid is graded commutative .

There is an intuitive way to see how the requirement that $\mathrm{CE}(𝔞)$ is (graded) commutative encodes that the cells of $𝔞$ are infinitesimal: by the monoidal Dold-Kan correspondence the product on $N^{•}(C^{\mathrm{\infty }}(𝔞))$ is the cup product on the cosimplicial algebra $C^{\mathrm{\infty }}(𝔞)$. The cup product multiplies two functions on simplices by evaluating them on different faces of a given simplex. If that simplex has a finite extension, then there is no way that the value of this evaluation will, in general, depend on the choice of faces just up to a sign. However, if the faces are infinitesimally small and infinitesimally close then we do know from the example of the infinitesimal path ∞-groupoid – discussed below – that exchanging the order of two faces on which two functions are evaluated will alter the result of taking their product only by a sign. This is essentially the Kock-Lawvere axiom for infinitesimal spaces at work.

Slogan. Graded commutativity of the cup product of functions on simplices encodes the infinitesimal extension of these simplices.

Examples

Propositon

For $X$ an ordinary manifold the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$ is the tangent Lie algebroid $TX$ of $X$.

This means that the normalized Moore cochain complex of functions on ${\Pi }^{\inf }(X)$ is the deRham differential algebra of differential forms ${\Omega }^{•}(X)$.

Proof

The proof is spelled out at ∞-quantity. For more background and literature see differential forms in synthetic differential geometry. It boils down to unwarpping the definition and noticing that it produces item-by-item the synthetic structure that Anders Kock has shown to yield the deRham complex.

Proposition

For $G$ a Lie group, for $BG$ its delooping regarded as an object of $\mathrm{sSh}(C)$, let

$$𝔤:=B{G}_e^{(1)}$$\mathfrak{g} :=\mathbf{B}G_{e}^{(1)}

be the sub-$\mathrm{\infty }$-groupoid which in degree 1 is the first order infinitesimal neighbourhood of the neutral element $e\in G$.

Then the corresponding $\mathrm{\infty }$-quantity of functions $C^{\mathrm{\infty }}(B{G}_e^{(1)})$ is under the monoidal Dold-Kan correspondence the Chevalley-Eilenberg algebra of the Lie algebra $g=\mathrm{Lie}(G)$:

$$N^{•}(C^{\mathrm{\infty }}(B{G}_e^{(1)}))=\mathrm{CE}(g).$$N^\bullet(C^\infty(\mathbf{B}G_e^{(1)})) = CE(g) \,.

Proof

The proof again turns out to involve unwrapping the definition of the left hand side and applying item by item various propositions and theorems by Anders Kock. Details are at Chevalley-Eilenberg algebra in synthetic differential geometry.

Proposition

We have indeed $𝔤=\mathrm{Lie}(BG)$

Proof

This is essentially the characterization of $G$-valued differential forms by Anders Kock.

As an example of this consider additive group $R$. The corresponding $\mathrm{\infty }$-Lie algebroid is the simplicial object

$$𝔲(1)=D(•)=(\cdots D(2)\overrightarrow{\overrightarrow{\to }}D\overrightarrow{\to }*),$$\mathfrak{u}(1) = D(\bullet) = \left( \cdots D(2) \stackrel{\to}{\stackrel{\to}{\to}} D \stackrel{\to}{\to} {*} \right) \,,

where

$$D(n)=\{\stackrel{\rightharpoonup }{x}\in R^n\mid \forall i,j:x_i\cdot x_j=0\}$$D(n) = \{\vec x \in R^n | \forall i,j : x_i \cdot x_j = 0\}

and where face maps are given by forgetting the first and last entry and by adding middle entries, while degeneracy maps are given by inserting 0s.

This is the object that corresponds to the odd line in supergeometry. Indeed, its Chevalley-Eilenberg algebra $\mathrm{CE}(𝔲(1))=ℝ\oplus \theta \cdot ℝ$ is the algebra of dual numbers but with the nilpotent generator $\theta$ in degree 1, not degree 0. This is a $\mathbb{N}$-grading refinement of the ${\mathbb{Z}}_2$-graded object $C^{\mathrm{\infty }}({ℝ}^{0\mid 1})$ in supergeometry.

Discussion

Concerning the terminology at Chevalley-Eilenberg algebra:

Jim Stasheff: This is a very unfortunate abuse of language since the classical Chevalley-Eilenberg algebra is by definition graded commutative. A more illuminating name is called for.

Urs Schreiber: I know what you mean. I was searching for a familiar term that, while necessarily imperfect, is as suggestive as possible. Do you have a suggestion for a better terminology?