nLab
Lie infinity-algebroid

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

An \infty-Lie algebroid is a smooth ∞-groupoid (or rather a synthetic-differential ∞-groupoid) all whose k-morphisms for all kk have infinitesimal extension (are infinitesimal neighbours of an identity kk-morphism).

\infty-Lie algebroids are to ∞-Lie groupoids as Lie algebras are to Lie groups:

Definition

We discuss \infty-Lie algebroids in the cohesive (∞,1)-topos H th:=\mathbf{H}_{th} := SynthDiff∞Grpd of synthetic differential ∞-groupoids. This is an infinitesimal cohesive neigbourhood of the cohesive (,1)(\infty,1)-topos H:=\mathbf{H} := Smooth∞Grpd of smooth ∞-groupoids, which is exhibited by the infinitesimal path (∞,1)-geometric morphism

(Π infDisc infΓ inf):SynthDiffGrpdSmoothGrpd. (\Pi_{inf} \dashv Disc_{inf} \dashv \Gamma_{inf}) : SynthDiff\infty Grpd \to Smooth\infty Grpd \,.

General abstract definition

There is a general abstract definition of formal cohesive ∞-groupoids in infinitesimal cohesive neighbourhood contexts such as Π inf:\Pi_{inf} : SynthDiff∞Grpd \to Smooth∞Grpd:

Definition

An object (𝔞Π inf(X))SynthDiffGrpd(\mathfrak{a} \to \mathbf{\Pi}_{inf}(X)) \in SynthDiff\infty Grpd over the infinitesimal fundamental ∞-groupoid is a formal synthetic-differential \infty-groupoid over XX if Π inf𝔞Π infX\mathbf{\Pi}_{inf} \mathfrak{a} \simeq \mathbf{\Pi}_{inf} X.

An ∞-group object 𝔤SynthDiffGrpd\mathfrak{g} \in SynthDiff\infty Grpd such that its delooping B𝔤\mathbf{B} \mathfrak{g} is a formal ∞-groupoid we call a formal ∞-group .

An \infty-Lie algebroid is supposed to be a formal cohesive ∞-groupoid whose infinitesimal extension is of first order. We now give an explicit presentation of such objects and then show that they do satisfy the abstract def. 1.

Presentation by dg-algebras and simplicial presheaves

We consider presentations of the general abstract definition 1 of \infty-Lie algebroids by constructing in the standard model structure-presentation of SynthDiffGrpdSynthDiff\infty Grpd by simplicial presheaves on CartSp synthdiff{}_{synthdiff} certain classes of simplicial presheaves in the image of semi-free differential graded algebras under the monoidal Dold-Kan correspondence. This amounts to identifying the traditional description of of Lie algebras, Lie algebroids and L-∞ algebras by their Chevalley-Eilenberg algebras as a convenient characterization of the corresponding cosimplicial algebras whose formal dual simplicial presheaves are manifest presentations of infinitesimal smooth ∞-groupoids.

Definition

Let

L AlgddgCAlg op L_\infty Algd \hookrightarrow dgCAlg_{\mathbb{R}}^{op}

be the full subcategory on the opposite category of cochain dg-algebras over \mathbb{R} on those dg-algebras that are

We call this the category of L L_\infty-algebroids.

=–

More in detail, an object 𝔞L Algd\mathfrak{a} \in L_\infty Algd may be identified (non-canonically) with a pair (CE(𝔞),X)(CE(\mathfrak{a}), X), where

  • XSmoothMfdX \in SmoothMfd is a smooth manifold – called the base space of the L L_\infty-algebroid ;

  • 𝔞\mathfrak{a} is the module of smooth sections of an \mathbb{N}-graded vector bundle of degreewise finite rank;

  • CE(𝔞)=( C (X) 𝔞 *,d 𝔞)CE(\mathfrak{a}) = (\wedge^\bullet_{C^\infty(X)} \mathfrak{a}^*, d_{\mathfrak{a}}) is a semifree dga on 𝔞 *\mathfrak{a}^* – a Chevalley-Eilenberg algebra – where

    C (X) 𝔞 *=C (X)𝔞 0 *(𝔞 0 * C (X)𝔞 0 *𝔞 1 *) \wedge^\bullet_{C^\infty(X)}\mathfrak{a}^* = C^\infty(X) \; \oplus \; \mathfrak{a}^*_0 \; \oplus \; ( \mathfrak{a}^*_0 \wedge_{C^\infty(X)} \mathfrak{a}^*_0 \oplus \mathfrak{a}^*_1 ) \; \oplus \; \cdots

    with the kkth summand on the right being in degree kk.

Definition

An L L_\infty-algebroid with base space X=*X = * the point is an L-∞ algebra 𝔤\mathfrak{g}, or rather is the delooping of an L L_\infty-algebra. We write b𝔤b \mathfrak{g} for L L_\infty-algebroids over the point. They form the full subcategory

L AlgL Algd. L_\infty Alg \hookrightarrow L_\infty Algd \,.

We now construct an embedding of L AlgsL_\infty Algs into SynthDiffGrpdSynthDiff\infty Grpd.

The functor

Ξ:Ch + ()Vect Δ \Xi : Ch^\bullet_+(\mathbb{R}) \to Vect_{\mathbb{R}}^{\Delta}

of the Dold-Kan correspondence from non-negatively graded cochain complexes of vector spaces to cosimplicial vector spaces is a lax monoidal functor and hence induces (see monoidal Dold-Kan correspondence) a functor (which we shall denote by the same symbol)

Ξ:dgAlg +Alg Δ \Xi : dgAlg_{\mathbb{R}}^+ \to Alg_{\mathbb{R}}^{\Delta}

from non-negatively graded cochain dg-algebras to cosimplicial algebras (over mathbR\mathb{R}).

Definition

Write

Ξ:L Algd(CAlg Δ) op \Xi : L_\infty Algd \to (CAlg_{\mathbb{R}}^\Delta)^{op}

for the restriction of the above Ξ\Xi along the inclusion L AlgddgAlg opL_\infty Algd \hookrightarrow dgAlg^{op}_{\mathbb{R}}:

for 𝔞L Algd\mathfrak{a} \in L_\infty Algd the underlying cosimplicial vector space of Ξ𝔞\Xi \mathfrak{a} is given by

Ξ𝔞:[n] i=0 nCE(𝔞) i i n \Xi \mathfrak{a} : [n] \mapsto \bigoplus_{i = 0}^{n} CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n

and the product of the \mathbb{R}-algebra structure on the right is given on homogeneous elements (ω,x),(λ,y)CE(𝔞) i i n(\omega,x), (\lambda,y) \in CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n in the tensor product by

(ω,x)(λ,y)=(ωλ,xy). (\omega , x)\cdot (\lambda ,y) = (\omega \wedge \lambda , x \wedge y) \,.

(Notice that Ξ𝔞\Xi \mathfrak{a} is indeed a commutative cosimplicial algebra, since ω\omega and xx in (ω,x)(\omega,x) are by definition in the same degree.)

To define the cosimplicial structure, let {e j} j=0 n\{e_j\}_{j = 0}^n be the canonical basis for n\mathbb{R}^n and consider also the basis {v j} j=0 n\{v_j\}_{j = 0}^n given by

v j:=e je 0. v_{j} := e_j - e_{0} \,.

Then for α:[k][l]\alpha : [k] \to [l] a morphism in the simplex category, set

αv j:=v α(j)v α(0) \alpha v_j := v_{\alpha(j)} - v_{\alpha(0)}

and extend this skew-multilinearly to a map α: k l\alpha : \wedge^\bullet \mathbb{R}^k \to \wedge^\bullet \mathbb{R}^l. In terms of all this the action of α\alpha on homogeneous elements (ω,x)(\omega,x) in the cosimplicial algebra is defined by

α:(ω,x)(ω,αx)+(d 𝔞ω,v α(0)α(x)) \alpha : (\omega, x) \mapsto (\omega, \alpha x) + (d_\mathfrak{a} \omega , v_{\alpha(0)}\wedge \alpha(x))

This is due to (CastiglioniCortinas, (1), (2), (20), (22)).

We shall refine the image of Ξ\Xi to cosimplicial smooth algebras. Let T:=T := CartSp smooth{}_{smooth} be the category of Cartesian spaces and smooth functions between them, regarded as a Lawvere theory. Write

SmoothAlg:=TAlg SmoothAlg := T Alg

for its category of algebras: these are the smooth algebras.

Notice that there is a canonical forgetful functor

U:SmoothAlgCAlg U : SmoothAlg \to CAlg_{\mathbb{R}}

to the category of comutative associative algebras over the real numbers.

Proposition

There is a unique factorization of the functor Ξ:L Algd(CAlg Δ) op\Xi : L_\infty Algd \to (CAlg_{\mathbb{R}}^\Delta)^{op} from def. 4 through the forgetful functor (SmoothAlg Δ) op(CAlg Δ) op(SmoothAlg_{\mathbb{R}}^\Delta)^{op} \to (CAlg_{\mathbb{R}}^\Delta)^{op} such that for any 𝔞\mathfrak{a} over base space XX the degree-0 algebra of smooth functions C (X)C^\infty(X) lifts to its canonical structure as a smooth algebra

(SmoothAlg Δ) op U L Algd (CAlg Δ) op. \array{ && (SmoothAlg^{\Delta})^{op} \\ & \nearrow & \downarrow^{\mathrlap{U}} \\ L_\infty Algd &\to& (CAlg_{\mathbb{R}}^\Delta)^{op} } \,.
Proof

Observe that for each nn the algebra (Ξ𝔞) n(\Xi \mathfrak{a})_n is a finite nilpotent extension of C (X)C^\infty(X). The claim then follows with using Hadamard's lemma to write every smooth function of sums as a finite Taylor expansion with a smooth rest term. See the examples at smooth algebra for more details on this kind of argument.

Definition

Write i:L AlgdSynthDiffGrpdi : L_\infty Algd \to SynthDiff\infty Grpd for the composite (∞,1)-functor

L AlgdΞ(SmoothAlg Δ) opj[CartSp synthdiff op,sSet]PQ([CartSp synthdiff op,sSet] loc) SynthDiffGrpd, L_\infty Algd \stackrel{\Xi}{\to} (SmoothAlg^{\Delta})^{op} \stackrel{j}{\to} [CartSp_{synthdiff}^{op}, sSet] \stackrel{P Q}{\to} ([CartSp_{synthdiff}^{op}, sSet]_{loc})^\circ \simeq SynthDiff\infty Grpd \,,

where the first morphism is the monoidal Dold-Kan correspondence as in prop. 1, the second is the external degreewise Yoneda embedding and PQP Q is any fibrant-cofibrant resolution functor in the local model structure on simplicial presheaves. The last equivalence holds as discussed there and at models for ∞-stack (∞,1)-toposes.

Remark

We do not consider the standard model structure on dg-algebras and do not consider L AlgdL_\infty Algd itself as a model category and do not consider an (∞,1)-category spanned by it. Instead, the functor i:L AlgdSynthDiffGrpdi : L_\infty Algd \to SynthDiff\infty Grpd only serves to exhibit a class of objects in SynthDiffGrpdSynthDiff\infty Grpd, which below in the section Models for the abstract axioms we show are indeed \infty-Lie algebroids by the general abstract definition, 1. All the homotopy theory of objects in L AlgdL_\infty Algd is that of SynthDiffGrpdSynthDiff\infty Grpd after this embedding.

Properties

General

Proposition

The full subcategory category L AlgL AlgdL_\infty Alg \hookrightarrow L_\infty Algd from def. 3 is equivalent to the traditional definition of the category of L-∞ algebras and “weak morphisms” / “sh-maps” between them.

The full subcategory LieAlgdL AlgdLieAlgd \hookrightarrow L_\infty Algd on the 1-truncated objects is equivalent to the traditional category of Lie algebroids (over smooth manifolds).

In particular the joint intersection LieAlgL AlgLie Alg \hookrightarrow L_\infty Alg on the 1-truncated L L_\infty-algebras is equivalent to the category of ordinary Lie algebras.

This is discussed in detail at L-∞ algebra and Lie algebroid.

Models for the abstract axioms

Above we have given a general abstract definition, def. 1, of \infty-Lie algebroids, and then a concrete construction in terms of dg-algebras, def. 4. Here we discuss that this concrete construction is indeed a presentation for objects satisfying the abstract axioms.

As in the discussion at SynthDiff∞Grpd we now present this cohesive (∞,1)-topos by the hypercompletion of the model structure on simplicial presheaves [FSmoothDiff op,sSet] proj,loc[FSmoothDiff^{op}, sSet]_{proj,loc} of formal smooth manifolds.

Lemma

For 𝔞L Algd\mathfrak{a} \in L_\infty Algd and i(𝔞)[FSmoothMfd op,sSet] proj,loci(\mathfrak{a}) \in [FSmoothMfd^{op}, sSet]_{proj,loc} its image in the standard presentation for SynthDiff∞Grpd, we have that

( [k]ΔΔ[k]i(𝔞) k)i(𝔞) \left( \int^{[k]\in \Delta} \mathbf{\Delta}[k] \cdot i(\mathfrak{a})_k \right) \stackrel{\simeq}{\to} i(\mathfrak{a})

is a cofibrant resolution, where Δ:ΔsSet\mathbf{\Delta} : \Delta \to sSet is the fat simplex.

Proof

We have

  1. The fat simplex is cofibrant in [Δ,sSet Quillen] proj[\Delta, sSet_{Quillen}]_{proj}.

  2. The canonical morphism ΔDelta\mathbf{\Delta} \to Delta is a weak equivalence between cofibrant objects in the Reedy model structure [Δ,sSet Quillen] Reedy[\Delta, sSet_{Quillen}]_{Reedy}.

  3. Because every representable FSmoothMfd[FSmoothMfd op,sSet] proj,locFSmoothMfd \hookrightarrow [FSmoothMfd^{op}, sSet]_{proj,loc} is cofibrant, the object i(𝔞) [Δ op,[FSmoothMfd op,sSet] proj,loc] inji(\mathfrak{a})_\bullet \in [\Delta^{op}, [FSmoothMfd^{op}, sSet]_{proj,loc} ]_{inj} is cofibrant.

  4. Every simplicial presheaf is cofibrant regarded as an object Reedy model structure [Δ op,[FSmoothMfd op,sSet] inj] Reedy[\Delta^{op}, [FSmoothMfd^{op}, sSet]_{inj}]_{Reedy}.

Now the coend over the tensoring

[k]Δ()():[Δ,sSet Quillen] proj×[Δ op,[FSmoothMfd op,sSet] proj,loc] inj[FSmoothMfd op,sSet] proj,loc \int^{[k] \in \Delta} (-)\cdot (-) : [\Delta, sSet_{Quillen}]_{proj} \times [\Delta^{op}, [FSmoothMfd^{op}, sSet]_{proj,loc} ]_{inj} \to [FSmoothMfd^{op}, sSet]_{proj,loc}

is a Quillen bifunctor (as discussed there) for the projective and injective global model structure on functors on the simplex category and its opposite as indicated. This implies the cofibrancy.

It is also a Quillen bifunctor (as discussed there) for the Reedy model structures

[k]Δ()():[Δ,sSet Quillen] Reedy×[Δ op,[FSmoothMfd op,sSet] inj] Reedy[FSmoothMfd op,sSet] inj. \int^{[k] \in \Delta} (-)\cdot (-) : [\Delta, sSet_{Quillen}]_{Reedy} \times [\Delta^{op}, [FSmoothMfd^{op}, sSet]_{inj} ]_{Reedy} \to [FSmoothMfd^{op}, sSet]_{inj} \,.

Using the factorization lemma this implies the weak equivalence (this is the argument of the Bousfield-Kan map).

Proposition

Let 𝔤\mathfrak{g} be an L-∞ algebra, regarded as an L L_\infty-algebroid b𝔤L Algdb \mathfrak{g} \in L_\infty Algd over the point by the embedding of def. 3.

Then i(b𝔤)i(b \mathfrak{g}) \in SynthDiff∞Grpd is an infinitesimal cohesive object, in that it is geometrically contractible

Πb𝔤* \Pi b \mathfrak{g} \simeq *

and has as underlying discrete ∞-groupoid the point

Γb𝔤*. \Gamma b \mathfrak{g} \simeq * \,.
Proof

We present now SynthDiff∞Grpd by the model structure on simplicial presheaves [CartSp synthdiff op,sSet] proj,loc[CartSp_{synthdiff}^{op}, sSet]_{proj,loc}. Since CartSp synthdiff{}_{synthdiff} is an ∞-cohesive site we have by the discussion there that Π\Pi is presented by the left derived functor 𝕃lim\mathbb{L} \lim\to of the degreewise colimit and Γ\Gamma is presented by the left derived functor of evaluation on the point.

With lemma 1 we can evaluate

(𝕃lim )i(b𝔤) lim [k]ΔΔ[k](b𝔤) k [k]ΔΔ[k]lim (b𝔤) k = [k]ΔΔ[k]*, \begin{aligned} (\mathbb{L} \lim_\to) i(b\mathfrak{g}) & \simeq \lim_\to \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot (b \mathfrak{g})_{k} \\ & \simeq \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \lim_\to (b \mathfrak{g})_{k} \\ & = \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot * \end{aligned} \,,

because each (b𝔤) nInfPointCartSp smooth(b \mathfrak{g})_n \in InfPoint \hookrightarrow CartSp_{smooth} is an infinitesimally thickened point, hence representable and hence sent to the point by the colimit functor.

That this is equivalent to the point follows from the fact that Δ\emptyset \to \mathbf{\Delta} is an acylic cofibration in [Δ,sSet Quillen] proj[\Delta, sSet_{Quillen}]_{proj}, and that

[k]Δ()×():[Δ,sSet Quillen] proj×[Δ op,sSet Qillen] injsSet Quillen \int^{[k] \in \Delta} (-)\times (-) : [\Delta, sSet_{Quillen}]_{proj} \times [\Delta^{op}, sSet_{Qillen}]_{inj} \to sSet_{Quillen}

is a Quillen bifunctor, using that *[Δ op,sSet Quillen] inj* \in [\Delta^{op}, sSet_{Quillen}]_{inj} is cofibrant.

Similarily, we have degreewise that

Hom(*,(b𝔤) n)=* Hom(*, (b \mathfrak{g})_n) = *

by the fact that an infinitesimally thickened point has a single global point. Therefore the claim for Γ\Gamma follows analogously.

Proposition

Let (𝔞TX)L Algd[CartSp synthdiff,sSet](\mathfrak{a} \to T X) \in L_\infty Algd \hookrightarrow [CartSp_{synthdiff}, sSet] be an L L_\infty-algebroid, def. 2, over a smooth manifold XX, regarded as a simplicial presheaf and hence as a presentation for an object in SynthDiffGrpdSynthDiff \infty Grpd according to def. 5.

We have an equivalence

Π inf(𝔞)Π inf(X). \mathbf{\Pi}_{inf}(\mathfrak{a}) \simeq \mathbf{\Pi}_{inf}(X) \,.
Proof

Let first X=UCartSp synthdiffX = U \in CartSp_{synthdiff} be a representable. Then according to prop. 1 we have that

𝔞^:=( kΔΔ[k]𝔞 k)𝔞 \hat \mathfrak{a} := \left( \int^{k \in \Delta} \mathbf{\Delta}[k] \cdot \mathfrak{a}_k \right) \simeq \mathfrak{a}

is cofibrant in [CartSp synthdiff op,sSet] proj[CartSp_{synthdiff}^{op}, sSet]_{proj} . Therefore by this proposition on the presentation of infinitesimal neighbourhoods by simplicial presheaves over infinitesimal neighbourhood sites we compute the derived functor

Π inf(𝔞) i *i *𝔞 𝕃(()p)𝕃(()i)𝔞 (()ip)𝔞^ \begin{aligned} \mathbf{\Pi}_{inf}(\mathfrak{a}) & \simeq i_* i^* \mathfrak{a} \\ & \simeq \mathbb{L} ((-) \circ p) \mathbb{L} ((-) \circ i) \mathfrak{a} \\ & \simeq ((-) \circ i p ) \hat \mathfrak{a} \end{aligned}

with the notation as used there.

In view of def. 4 we have for all kk \in \mathbb{N} that 𝔞 k=X×D\mathfrak{a}_k = X \times D where DD is an infinitesimally thickened point. Therefore (()ip)𝔞 k=(()ip)X((-) \circ i p ) \mathfrak{a}_k = ((-) \circ i p ) X for all kk and hence (()ip)𝔞^Π inf(X)((-) \circ i p ) \hat \mathfrak{a} \simeq \mathbf{\Pi}_{inf}(X).

For general XX choose first a cofibrant resolution by a split hypercover that is degreewise a coproduct of representables (which always exists, by the discussion at model structure on simplicial presheaves), then pull back the above discussion to these covers.

Corollary

Every L L_\infty-algebroid in the sense of def. 2 under the embedding of def. 5 is indeed a formal cohesive ∞-groupoid in the sense of def. 1.

Cohomology of \infty-Lie algebroids

We discuss the relation between the intrinsic cohomology of L L_\infty-algebroids when regarded as objects of SynthDiffGrpdSynthDiff\infty Grpd, and the ordinary cohomology of their Chevalley-Eilenberg algebras. For more on this see ∞-Lie algebroid cohomology.

Proposition

Let 𝔞L Algd\mathfrak{a} \in L_\infty Algd be an L L_\infty-algebroid. Then its intrinsic real cohomoloogy in SynthDiff∞Grpd

H n(𝔞,):=π 0SynthDiffGrpd(𝔞,B n) H^n(\mathfrak{a}, \mathbb{R}) := \pi_0 SynthDiff\infty Grpd(\mathfrak{a}, \mathbf{B}^n \mathbb{R})

coincides with its ordinary L-∞ algebra cohomology: the cochain cohomology of its Chevalley-Eilenberg algebra

H n(𝔞,)H n(CE(𝔞)). H^n(\mathfrak{a}, \mathbb{R}) \simeq H^n(CE(\mathfrak{a})) \,.
Proof

By this discussion at SynthDiff∞Grpd we have that

H n(𝔞,)H nN (𝕃𝒪)(i(𝔞)). H^n(\mathfrak{a}, \mathbb{R}) \simeq H^n N^\bullet(\mathbb{L}\mathcal{O})(i(\mathfrak{a})) \,.

By lemma 1 this is

H nN ( [k]ΔΔ[k]𝒪(i(𝔞) k)). \cdots \simeq H^n N^\bullet \left( \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \mathcal{O}(i(\mathfrak{a})_k) \right) \,.

Observe that 𝒪(𝔞) \mathcal{O}(\mathfrak{a})_\bullet is cofibrant in the Reedy model structure [\Delta^{op}, (SmoothAlg^\Delta_{proj}})^{op}]_{Reedy} relative to the opposite of the projective model structure on cosimplicial algebras: the map from the latching object in degree n[\Delta^{op}, (SmoothAlg^\Delta_{proj}}n in SmoothAlg Δ) opSmoothAlg^\Delta)^{op} is dually in SmoothAlgSmoothAlg ΔSmoothAlg \hookrightarrow SmoothAlg^\Delta the projection

i=0 nCE(𝔞) i i n i=0 n1CE(𝔞) i i n \oplus_{i = 0}^n CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n \to \oplus_{i = 0}^{n-1} CE(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n

hence is a surjection, hence a fibration in SmoothAlg proj ΔSmoothAlg^\Delta_{proj} and therefore indeed a cofibration in (SmoothAlg proj Δ) op(SmoothAlg^\Delta_{proj})^{op}.

Therefore using the Quillen bifunctor property of the coend over the tensoring in reverse to lemma 1 the above is equivalent to

H nN ( [k]ΔΔ[k]𝒪(i(𝔞) k)) \cdots \simeq H^n N^\bullet \left( \int^{[k] \in \Delta} \Delta[k] \cdot \mathcal{O}(i(\mathfrak{a})_k) \right)

with the fat simplex replaced again by the ordinary simplex. But in brackets this is now by definition the image under the monoidal Dold-Kan correspondence of the Chevalley-Eilenberg algebra

H n(N ΞCE(𝔞)). \cdots \simeq H^n( N^\bullet \Xi CE(\mathfrak{a}) ) \,.

By the Dold-Kan correspondence we have hence

H n(CE(𝔞)). \cdots \simeq H^n(CE(\mathfrak{a})) \,.

Examples

Classes of examples

  • a \infty-Lie algebroid over the point, 𝔞=*\mathfrak{a} = * is an L-∞-algebra;

  • an nn-truncated \infty-Lie algebroid is a Lie nn-algebroid;

  • an \infty-Lie algebroid the differential of whose Chevalley-Eilenberg algebra is “co-binary”, i.e. d:𝔞 *a *a *g *d : \mathfrak{a}^* \to a^* \oplus a^* \wedge g^*, is strict.

So in particular

Lie algebroids regarded as \infty-Lie algebroids

We discuss the traditional notion of Lie algebroids in view of their role as presentations for infinitesimal synthetic differential 1-groupoids.

Smooth loci of infinitesimal simplices

In this section we characterize ordinary Lie algebroids ETXE \to T X as precisely those synthetic differential \infty-groupoids that under the above presentation are locally on any chart UXU \to X of their base space given by simplicial smooth loci of the form

U×D˜(rankE,2)U×D˜(rankE,1)U, \cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} U \times \tilde D(rank E,2)\stackrel{\to}{\stackrel{\to}{\to}} U \times \tilde D(rank E,1) \stackrel{\to}{\to} U \,,

where D˜(k,n)\tilde D(k,n) is the smooth locus of infinitesimal k-simplices based at the origin in n\mathbb{R}^n. (These smooth loci have been considered in (Kock, section 1.2)).

The following definition may be either taken as an informal but instructive definition – in which case the next definition is to be taken as the precise one – or in fact it may be already itself be taken as the fully formal and precise definition if one reads it in the internal logic of any smooth topos with line object RR – which for the present purpose is the Cahiers topos with line object \mathbb{R}. (For an exposition of the latter perspective see (Kock)).

Definition

For k,nk,n \in \mathbb{N}, an infinitesimal kk-simplex in R nR^n based at the origin is a collection (ϵ aR n) a=1 k(\vec \epsilon_a \in R^n)_{a = 1}^k of points in R nR^n, such that each is an infinitesimal neighbour of the origin

a:ϵ a0 \forall a : \;\; \vec \epsilon_a \sim 0

and such that all are infinitesimal neighbours of each other

a,a:(ϵ aϵ a)0. \forall a,a': \;\; (\vec \epsilon_a - \vec \epsilon_{a'}) \sim 0 \,.

Write D˜(k,n)R kn\tilde D(k,n) \subset R^{k \cdot n} for the space of all such infinitesimal kk-simplices in R nR^n.

Equivalently:

Definition

For k,nk,n \in \mathbb{N}, the smooth algebra

C (D˜(k,n))SmoothAlg C^\infty(\tilde D(k,n)) \in SmoothAlg

is the unique lift through the forgetful functor U:SmoothAlgCAlg U : SmoothAlg \to CAlg_{\mathbb{R}} of the commutative \mathbb{R}-algebra generated from k×nk \times n many generators

(ϵ a j) 1jn,1ak (\epsilon_a^j)_{1 \leq j \leq n, 1 \leq a \leq k}

subject to the relations

a,j,j:ϵ a jϵ a j=0 \forall a, j,j' : \;\; \epsilon_a^{j} \epsilon_a^{j'} = 0

and

a,a,j,j:(ϵ a jϵ a j)(ϵ a jϵ a j)=0. \forall a,a',j,j' : \;\;\; (\epsilon_a^j - \epsilon_{a'}^j) (\epsilon_a^{j'} - \epsilon_{a'}^{j'}) = 0 \,.
Remark

In the above form these relations are the manifest analogs of the conditions ϵ a0\vec \epsilon_a \sim 0 and (ϵ aϵ a)0(\vec \epsilon_a - \vec \epsilon_{a'}) \sim 0. But by multiplying out the latter set of relations and using the former, we find that jointly they are equivalent to the single set of relations

(1)a,a,j,j:ϵ a jϵ a j+ϵ a jϵ a j=0. \forall a,a',j,j' : \;\;\; \epsilon_a^j \epsilon_{a'}^{j'} + \epsilon_{a'}^j \epsilon_{a}^{j'} = 0 \,.

In this expression the roles of the two sets of indices is manifestly symmetric. Hence another equivalent way to state the relations is to say

a,a,j:ϵ a jϵ a j=0 \forall a,a', j: \;\;\; \epsilon_a^{j} \epsilon_{a'}^j = 0

and

a,a,j,j:(ϵ a jϵ a j)(ϵ a jϵ a j)=0 \forall a,a',j,j : \;\;\; (\epsilon_a^j - \epsilon_a^{j'})(\epsilon_{a'}^j - \epsilon_{a'}^{j'}) = 0

This appears as (Kock, (1.2.1)).

Since C (D˜(k,n))C^\infty(\tilde D(k,n)) is a Weil algebra in the sense of synthetic differential geometry, its structure as an \mathbb{R}-algebra extends uniquely to the structure of a smooth algebra (as discussed there) and we may think of D˜(k,n)\tilde D(k,n) as an infinitesimal smooth locus.

Example

For n=2n = 2 and k=2k = 2 we have that C (D˜(2,2))C^\infty(\tilde D(2,2)) consists of elements of the form

f+aϵ 1+bϵ 2+(ωϵ 1)(λϵ 2) =f+a 1ϵ 1 1+a 2ϵ 1 2+b 1ϵ 2 1+b 2ϵ 1 2 +(ω 1λ 2ω 2λ 1)12(ϵ 1 1ϵ 2 2ϵ 1 2ϵ 2 1) \begin{aligned} f + a \cdot \epsilon_1 + b \cdot \epsilon _2 + (\omega \cdot \epsilon_1) (\lambda \cdot \epsilon_2) &= f + a_1 \epsilon_1^1 + a_2 \epsilon_1^2 + b_1 \epsilon_2^1 + b_2 \epsilon_1^2 \\ & + (\omega_1 \lambda_2 - \omega_2 \lambda_1) \frac{1}{2}(\epsilon_1^1 \epsilon_2^2 - \epsilon_1^2 \epsilon_2^1) \end{aligned}

for ff \in \mathbb{R} and (a,b,ω,λ( n) *)(a, b, \omega, \lambda \in (\mathbb{R}^n)^*) a collection of ordinary covectors and with “\cdot” denoting the evident contraction, and where in the last step we used the above relations.

It is noteworthy here that the coefficient of the term which is multilinear in each of the ϵ i\epsilon_i is the wedge product of two covectors ω\omega and λ\lambda: we may naturally identify the subspace of C (D˜(2,2))C^\infty(\tilde D(2,2)) on those elements that vanish if either ϵ 1\epsilon_1 or ϵ 2\epsilon_2 are set to 0 as the space 2T 0 * 2\wedge^2 T_0^* \mathbb{R}^2 of 2-forms at the origin of 2\mathbb{R}^2.

Of course for this identification to be more than a coincidence we need that this is the beginning of a pattern that holds more generally. But this is indeed the case.

Let EE be the set of square submatrices of the k×nk \times n-matrix (ϵ i j)(\epsilon_i^j). As a set this is isomorphic to the set of pairs of subsets of the same size of {1,,k}\{1, \cdots, k\} and {1,,n}\{1, \cdots , n\}, respectively. For instance the square submatrix labeled by {2,3,4}\{2,3,4\} and {1,4,5}\{1,4,5\} is

e=(ϵ 1 2 ϵ 4 2 ϵ 5 2 ϵ 1 3 ϵ 4 3 ϵ 5 3 ϵ 1 4 ϵ 4 4 ϵ 5 4). e = \left( \array{ \epsilon_1^2 & \epsilon_4^2 & \epsilon_5^2 \\ \epsilon_1^3 & \epsilon_4^3 & \epsilon_5^3 \\ \epsilon_1^4 & \epsilon_4^4 & \epsilon_5^4 } \right) \,.

For eEe \in E an r×rr\times r submatrix, we write

det(e)= σsgn(σ)ϵ 1 σ(1)ϵ 2 σ(2)ϵ r σ(r)C (D˜(k,n)). det(e) = \sum_{\sigma} sgn(\sigma) \epsilon_{1}^{\sigma(1)} \epsilon_2^{\sigma(2)} \cdots \epsilon_r^{\sigma(r)} \in C^\infty(\tilde D(k,n)) \,.

for the corresponding determinant, given as a product of generators in C (D˜(k,n))C^\infty(\tilde D(k,n)). Here the sum runs over all permutations σ\sigma of {1,,r}\{1, \cdots, r\} and sgn(σ){+1,1}sgn(\sigma) \in \{+1, -1\} \subset \mathbb{R} is the signature of the permutation σ\sigma.

Proposition

The elements fC (D˜(k,n))f \in C^\infty(\tilde D(k,n)) are precisely of the form

f= eEf edet(e) f = \sum_{e \in E} f_e \; det(e)

for unique {f e|eE}\{f_e \in \mathbb{R} | e \in E\}. In other words, the map of vector spaces

|E|C (D˜(k,n)) \mathbb{R}^{|E|} \to C^\infty(\tilde D(k,n))

given by

(f e) eE eEf edet(e) (f_e)_{e \in E} \mapsto \sum_{e \in E} f_e det(e)

is an isomorphism.

Proof

This is a direct extension of the argument in the above example: a general product of rr generators in C (D˜(k,n))C^\infty(\tilde D(k,n)) is

ϵ i 1 j 1ϵ i 2 j 2ϵ i r j r. \epsilon_{i_1}^{j_1} \epsilon_{i_2}^{j_2} \cdots \epsilon_{i_r}^{j_r} \,.

By the relations in C (D˜(k,n))C^\infty(\tilde D(k,n)), this is non-vanishing precisely if none of the ii-indices repeats and none of the jj-indices repeats. Furthermore by the relations, for any permutation σ\sigma of rr elements, this is equal to

=sgn(σ)ϵ i 1 j σ(1)ϵ i 1 j σ(2)ϵ i 1 j σ(r). \cdots = sgn(\sigma) \epsilon_{i_1}^{j_{\sigma(1)}} \epsilon_{i_1}^{j_{\sigma(2)}} \cdots \epsilon_{i_1}^{j_{\sigma(r)}} \,.

It follows that each such element may be written as

=1r!det(e), \cdots = \frac{1}{r!} det(e) \,,

where ee is the r×rr \times r sub-determinant given by the subset {i 1,,i r}\{i_1, \cdots, i_r\} and ({j 1,,j r})(\{j_1, \cdots, j_r\}) as discussed above.

In (Kock, section 1.3) effectively this proposition appears as the “Kock-Lawvere axiom scheme for D˜(k,n)\tilde D(k,n)” when D˜(k,n)\tilde D(k,n) is regarded as an object of a suitable smooth topos.

Proposition

For any k,nk,n \in \mathbb{N} we have a natural isomorphism of real commutative and hence of smooth algebras

ϕ:C (D˜(k,n)) i=0 n( i k)( i n), \phi : C^\infty(\tilde D(k,n)) \stackrel{\simeq}{\to} \oplus_{i = 0}^n (\wedge^i \mathbb{R}^k) \otimes (\wedge^i \mathbb{R}^n) \,,

where on the right we have the algebras that appear degreewise in def. 4, where the product is given on homogeneous elements by

(ω,x)(λ,y)=(ωλ,xy). (\omega, x) \cdot (\lambda, y) = (\omega \wedge \lambda , x \wedge y) \,.
Proof

Let {t a}\{t_a\} be the canonical basis for k\mathbb{R}^k and {e i}\{e^i\} the canonical basis for n\mathbb{R}^n. We claim that an isomorphism is given by the assignment

ϕ:ϵ a i(t a,e i). \phi : \epsilon^i_a \mapsto (t_a , e^i) \,.

To see that this defines indeed an algebra homomorphism we need to check that it respects the relations on the generators. For this compute:

ϕ(ϵ a iϵ a i) =(t at a,e ie i) =(t at a,e ie i) =ϕ(ϵ a iϵ a i). \begin{aligned} \phi(\epsilon_a^i \epsilon_{a'}^{i'}) & = (t_a \wedge t_{a'}, e^i \wedge e^{i'}) \\ & = -(t_{a'} \wedge t_{a}, e^i \wedge e^{i'}) \\ & = -\phi(\epsilon_{a'}^i \epsilon_{a}^{i'}) \end{aligned} \,.

The inverse clearly exists, given on generators by

ϕ 1:(t a,e i)ϵ a i. \phi^{-1} : (t_a, e^i) \mapsto \epsilon_a^i \,.
Corollary

For 𝔞L Alg\mathfrak{a} \in L_\infty Alg a 1-truncated object, hence an ordinary Lie algebroid of rank kk over a base manifold XX, its image under the map i:L Alg(SmoothAlg Δ) opi : L_\infty Alg \to (SmoothAlg^\Delta)^{op}, def. 5, is such that its restriction to any chart UXU \to X is, up to isomorphism, of the form

i(𝔞)| U:[n]U×D˜(k,n). i(\mathfrak{a})|_U : [n] \mapsto U \times \tilde D(k,n) \,.
Proof

Apply prop. 7 in def. 4, using that by definition CE(𝔞)CE(\mathfrak{a}) is given by the exterior algebra on locally free C (X)C^\infty(X) modules, so that

CE(𝔞| U) ( C (U) Γ(U× k) *,d 𝔞| U) (C (U) k,d 𝔞| U). \begin{aligned} CE(\mathfrak{a}|_U) & \simeq (\wedge^\bullet_{C^\infty(U)} \Gamma(U\times \mathbb{R}^k)^*, d_{\mathfrak{a}|_U}) \\ & \simeq (C^\infty(U) \otimes \wedge^\bullet \mathbb{R}^k, d_{\mathfrak{a}|_U}) \end{aligned} \,.

Tangent Lie algebroid

For XX any smooth manifold, there is a standard notion of the Lie algebroid which is the tangent Lie algebroid

𝔞=TX \mathfrak{a} = T X

of XX. We discuss this from the perspective of infinitesimal groupoids.

Definition

For UCartSp synthdiffU \in CartSp_{synthdiff}, the infinitesimal singular simplicial complex X (Δ inf )X^{(\Delta^\bullet_{inf})} is the simplicial smooth locus which in terms in degree nn is the space of (k+1)(k+1)-tuples of pairwise infinitesimal neighbour points in UU

U (Δ inf n)={x i 0,x i nU|r,s:x i rx i s} U^{(\Delta^n_{inf})} = \left\{ x_{i_0}, \cdots x_{i_n} \in U | \forall r,s : x_{i_r} \sim x_{i_s} \right\}

and whose face and degeneracy maps are as for the finite singular simplicial complex.

More explicitly, in terms of the spaces from def. 6 we may identify

U (Δ inf )=(U×D˜(dimU,2)UD˜˜(dimU,1)U), U^{(\Delta^\bullet_{inf})} = \left( \cdots U \times \tilde D(dim U, 2) \stackrel{\to}{\stackrel{\to}{\to}}U \tilde \tilde D(dim U, 1)\stackrel{\to}{\to} U \right) \,,

where in degree nn a generalized element (x,(ϵ a) a=1,,dimU)(x, (\vec \epsilon_a)_{a = 1, \cdots, dim U}) of U×D˜(dimU,n)U \times \tilde D(dim U, n) is thought of as a base point xx and dimUdim U infinitesimal paths starting at that basepoint

(x 0,,x n)=(x,x+ϵ 1,,x+ϵ dimU). (x_0, \cdots, x_n) = ( x, x + \epsilon_1, \cdots, x + \epsilon_{dim U} ) \,.

The dual cosimplicial algebra is read off from this,

C (U (Δ inf ))=(C (U×D˜(dimU,1))d 1 *d 0 *C (U)). C^\infty(U^{(\Delta^\bullet_{inf})}) = \left( \cdots C^\infty(U \times \tilde D(dim U ,1)) \stackrel{\overset{d_0^*}{\leftarrow}}{\underset{d_1^*}{\leftarrow}} C^\infty(U) \right) \,.

For instance for fC (U)f \in C^\infty(U) we have d 1 *f=fd_1^* f = f and (d 2 *f)(x,ϵ 1)=f(x+ϵ 1)=f(x)+fx iϵ 1 i(d_2^* f)(x,\epsilon_1) = f(x + \epsilon_1) = f(x) + \frac{\partial f}{\partial x^i} \epsilon^i_1.

Note

The object X (Δ inf )X^{(\Delta^\bullet_{inf})} is not objectwise a Kan complex: in general the composite of two first order neighbours produces a second order infinitesimal neighbour. Its Kan fibrant replacement may be thought of as the infinitesikmal \infty-groupoid whose morphisms are paths of a finite number of first order infinitesimal steps.

Proposition

The image of TXT X under the embedding ii from def. 5 is the simplicial smooth locus given by the infinitesimal singular simplicial complex

TX=(X (Δ inf 2)X (Δ inf 1)X) T X = \left( \cdots X^{(\Delta^2_{inf})} \stackrel{\to}{\stackrel{\to}{\to}} X^{(\Delta^1_{inf})} \stackrel{\to}{\to} X \right)

of XX.

Moreover, the intrinsic real cohomology of i(TX)i(T X) \in SynthDiff∞Grpd is the de Rham cohomology of XX

H SynthDiff n(i(TX),)H dR n(X) H^n_{SynthDiff}(i (T X), \mathbb{R}) \simeq H^n_{dR}(X)
Proof

The first statement may be checked locally on any chart UXU \to X where it follows from prop. 2. Since the Chevalley-Eilenberg algebra of the tangent Lie algebroid is the de Rham complex

CE(TX)=(Ω (X),d dR) CE(T X) = (\Omega^\bullet(X), d_{dR})

the second statement follows with prop. 5.

Lie algebra

Let GG be a Lie group with Lie algebra 𝔤\mathfrak{g}. We describe how 𝔤\mathfrak{g} looks when regarded as a special case of an \infty-Lie algebroid.

Write

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

for the delooping groupoid of GG, regarded as an an ∞-Lie groupoid modeled by a simplicial smooth space.

We claim that a morphism

ω:TUBG \omega : T U \to \mathbf{B}G

from the tangent Lie algebroid of some UU \in CartSp is flat Lie-algebra valued form and how that can be used to find the Lie algebra 𝔤\mathfrak{g} as the infinitesimal sub-\infty-groupoid

b𝔤BG \mathbf{b}\mathfrak{g} \hookrightarrow \mathbf{B}G

inside BG\mathbf{B}G.

Since BG\mathbf{B}G is 2-coskeletal (being the nerve of a groupoid) a morphism TUBGT U \to \mathbf{B}G is fixed already under its 2-truncation

U×D˜(2,n) ω 2 G×G p 2 p 1 U×D˜(1,n) ω 1 G U *. \array{ U \times \tilde D(2,n) &\stackrel{\omega_2}{\to}& G \times G \\ \downarrow \downarrow \downarrow && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} \\ U \times \tilde D(1,n) &\stackrel{\omega_1}{\to}& G \\ \downarrow \downarrow && \downarrow \downarrow \\ U &\to& * } \,.

It is clear that ω 1\omega_1 factors through the inclusion D˜(1,dim(G))G\tilde D(1,dim(G)) \hookrightarrow G that sends the unique point of D˜(1,dim(G))\tilde D(1, dim(G)) to the neutral element (by respect for the degeneracy maps). Then from that one finds that ω 2\omega_2 factors through the inclusion D˜(2,dim(G))G×G\tilde D(2, dim(G)) \hookrightarrow G \times G that sends the unique point of D˜(2,dim(G))\tilde D(2,dim(G)) to (e,e)G×G(e,e) \in G \times G. And evidently these two factorizations are universal, in that every other factorization will uniquely factor through these

U×D˜(2,n) ω 2 D˜(2,dim(G)) G×G p 2 p 1 p 2 p 1 U×D˜(1,n) ω 1 D˜(1,dim(G)) G U * *. \array{ U \times \tilde D(2,n) &\stackrel{\omega_2}{\to}& \tilde D(2,dim(G)) &\hookrightarrow& G \times G \\ \downarrow \downarrow \downarrow && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} \\ U \times \tilde D(1,n) &\stackrel{\omega_1}{\to}& \tilde D(1, dim(G)) &\hookrightarrow& G \\ \downarrow \downarrow && \downarrow \downarrow && \downarrow \downarrow \\ U &\to& * &\to& * } \,.

The universal object found this way we claim is the Lie algebra 𝔤\mathfrak{g} in its incarnation as an infinitesimal \infty-Lie groupoid

b𝔤 :=InitialObject(TU𝕃 Delta opBG) =(D˜(2,dim(G))D˜(1,dim(G))*). \begin{aligned} b \mathfrak{g} &:= InitialObject( T U\downarrow \mathbb{L}^{Delta^{op}}\downarrow \mathbf{B}G) \\ & = \left( \cdots \tilde D(2,dim(G)) \stackrel{\to}{\stackrel{\to}{\to}} \tilde D(1,dim(G))\stackrel{\to}{\to} * \right) \end{aligned} \,.
Proposition

The normalized cochain complex of the cosimplicial alghebra of functions on this b𝔤b \mathfrak{g} is isomorphic to the ordinary Chevalley-Eilenberg algebra ( 𝔤 *,[,] *)(\wedge^\bullet \mathfrak{g}^*, [-,-]^*) of 𝔤\mathfrak{g}.

Proof

By the above discussion we have that for C (D˜(k,dim(G))) topC (D˜(k,n))C^\infty(\tilde D(k,dim(G)))_{top} \subset C^\infty(\tilde D(k,n)) the subspace of those functions that are in the joint kernel of the co-degeneracy maps is naturally isomorpic to k( dim(G)) *\wedge^k (\mathbb{R}^{dim(G)})^*, so that we have a natural isomorphism of vector spaces

NC (b𝔤) k k𝔤 *. N C^\infty(b \mathfrak{g})_k \simeq \wedge^k \mathfrak{g}^* \,.

By the fact that everything is 2-coskeletal it suffices to check that the differential in first degree

NC (D˜(1,dim(G)))p 1 *+p 2 *() *NC (D˜(2,dim(G))) N C^\infty(\tilde D(1,dim(G))) \stackrel{p_1^* + p_2^* - (\cdot)^*}{\to} N C^\infty(\tilde D(2,dim(G)))

is indeed the dual of the Lie bracket. But the product G:G×GG\cdot_G : G \times G \to G restricted along D˜(2,dim(G))G×G\tilde D(2,dim(G)) \hookrightarrow G \times G to the infinitesimal space D˜(2,dim(G))\tilde D(2, dim(G)) linearizes in each of its arguments: for (x,y)D˜(2,dim(G))(\vec x,\vec y) \in \tilde D(2,dim(G)) we have

x Gy=x x G(0,0)+y y G(0,0)+x xy y G(0,0). \vec x \cdot_G \vec y = \vec x \cdot \nabla_x \cdot_G (0,0) + \vec y \cdot \nabla_y \cdot_G (0,0) + \vec x \cdot \nabla_x \vec y \cdot \nabla_y \cdot_G(0,0) \,.

Since the origin here corresponds to the neutral element of GG and since with one of its arguments the neutral element the operaton G\cdot_G is the identity, and since the double derivative produces the Lie bracket (keeping in mind that x iy j+x jy i=0x^i y^j + x^j y^i = 0 in D˜(2,dim(G))\tilde D(2,dim(G))), this is

=x+y+[x,y]. \cdots = \vec x + \vec y + [\vec x, \vec y] \,.

Accordingly the alternating sum of co-face maps is

d =p 1 *+p 2 * G * =p 1 *+p 2 *(p 1 *+p 2 *+[,] *) =[,] * \begin{aligned} d &= p_1^* + p_2^* - \cdot_G^* \\ & = p_1^* + p_2^* - ( p_1^* + p_2^* + [-,-]^*) \\ & = - [-,-]^* \end{aligned}

as it should be for the Chevalley-Eilenberg algebra of a Lie algebra.

The infinitesimal reasoning involved in this proof is discussed in (Kock, section 6.8).

References

The term “Lie \infty-algebroid” or “L L_\infty-algebroid” as such is not as yet established in the literature, as most authors working with these objects think of them entirely in terms of dg-algebras or NQ-supermanifolds and either ignore the relation to Lie theory or take it more or less for granted.

Possibly the first explicit appearance of the idea of \infty-Lie algebroids recognized in their full Lie theoretic meaning is

  • Pavol Ševera, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv)

which uses “NQ-supermanifolds”. Of course, as this article also points out, in hindsight one finds that much of this is already implicit in the much older theory of Sullivan models in rational homotopy theory, which is concerned with modelling topological spaces by dg-algebras. That these spaces can be regarded as ∞-groupoids and as ∞-Lie groupoids in particular is clear in hindsight, but was possibly first explicitly realized in the above reference. See also Lie integration, rational homotopy theory in an (∞,1)-topos and function algebras on ∞-stacks.

The explicit term \infty-Lie algebroid / L L_\infty-algebroid as such appears in

The term also appears in

  • Andrew James Bruce, From L L_{\infty}-algebroids to higher Schouten/Poisson structures (arXiv:1007.1389)

The dual monoidal Dold-Kan correspondence is discussed in

  • J.L. Castiglioni, G. Cortiñas, Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence , J. Pure Appl. Algebra 191 (2004), no. 1-2, 119–142, (arXiv:math.KT/0306289) .

The smooth spaces of infinitesimal simplices D˜(k,n)\tilde D(k,n) are considered in section 1.2 of

Revised on May 23, 2014 01:41:00 by Urs Schreiber (185.37.147.12)