nLab Lie integration

Idea

Lie integration is a process that assigns to a Lie algebra $\mathfrak{g}$ – or more generally to an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by $\mathfrak{g}$. It is essentially the reverse operation to Lie differentiation, except that there are in general several objects Lie integrating a given Lie algebraic datum, due to the fact that the infinitesimal data does not uniquely determine global topological properties.

Classically, Lie integration of Lie algebras is part of Lie's three theorems, which in particular finds an unique (up to isomorphism) simply connected Lie group integrating a given finite-dimensional Lie algebra.

One may observe that the simply connected Lie group integrating a (finite-dimensional) Lie algebra is equivalently realized as the collection of equivalence classes of Lie algebra valued 1-forms on the interval where two such are identified if they are interpolated by a flat Lie-algebra valued 1-form on the disk. (Duistermaat-Kolk 00, section 1.14, see also the example below).

This path method of Lie integration stands out as having natural generalizations to higher Lie theory (Ševera 01).

In its evident generalization from Lie algebra valued differential forms to Lie algebroid valued differential forms this provides a means for Lie integration of Lie algebroids (e.g. Crainic-Fernandes 01).

In another direction, one may observe that L-∞ algebras are formally dually incarnated by their Chevalley-Eilenberg dg-algebras, and that under this identification the evident generalization of the path method to L-∞ algebra valued differential forms is essentially the Sullivan construction, known from rational homotopy theory, applied to these dg-algebras (Hinich 97, Getzler 04). Or rather, the bare such construction gives the geometrically discrete ∞-group underlying what should be the Lie integration to a smooth ∞-group. This is naturally obtained, as in the classical case, by suitably smoothly parameterizing the ∞-Lie algebroid valued differential forms (Henriques 08, Roytenberg 09, FSS 12).

Both these directions may be combined via the evident concept of ∞-Lie algebroid valued differential forms to yield a Lie integration of ∞-Lie algebroids to smooth ∞-groupoids. (Moreover, the same formula directly generalizes from $L_\infty$-algebroids to A-infinity categories to yield the dg-nerve construction.)

While the construction exists and behaves as expected in examples, there is to date no good general theory providing higher analogs of, say, Lie's three theorems. But people are working on it.

Definition

Throughout, let $\mathfrak{a}$ be an ∞-Lie algebroid (for instance a Lie algebra, or a Lie algebroid or an L-∞-algebra). Write

$$CE(\mathfrak{a}) \in dgAlg$$

for its Chevalley-Eilenberg algebra, a dg-algebra. Notice that, by the discussion at L-∞ algebra and at ∞-Lie algebroid, the Chevalley-Eilenberg dg-algebras $CE(\mathfrak{a})$ is the formal dual of $\mathfrak{a}$, in that the functor

$$CE \colon \infty LieAlgd \hookrightarrow dgAlg^{op}$$

is a fully faithful functor. Indeed, the following definition of Lie integration (being just a smooth refinement of the Sullivan construction) makes sense just as well for any dg-algebra, not necessarily in the essential image of this embedding. But only for dg-algebras in the essential image of this embedding do the examples come out as expected for higher Lie theory.

An ∞-Lie algebra is equivalently a pointed ∞-Lie algebroid whose base space is the point. We write $\mathfrak{b}\mathfrak{g} \in \infty Lie Algd$ for objects of this form (“delooping”)

$$\array{ \infty LieAlgd &\stackrel{CE}{\hookrightarrow}& dgAlg^{op} \\ \uparrow^{\mathrlap{b}} & \nearrow_{\mathrlap{CE}} \\ \infty LieAlg \,. }$$

Notice that this induces some degree shifts that may be a little ambiguous in situations like the line Lie n-algebra: as an L-∞ algebra this is $b^{n-1}\mathbb{R}$, the corresponding ∞-Lie algebroid is $b^n \mathbb{R}$.

Higher dimensional paths in an $\infty$-Lie algebroid

Example

For $X$ a smooth manifold (possibly with boundary and with corners) then its tangent Lie algebroid $T X$ is the one whose Chevalley-Eilenberg algebra is the de Rham complex

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

Definition

For $k \in \mathbb{N}$ write $\Delta^k$ for the standard $k$-simplex regarded as a smooth manifold (with boundary and with corners).

A $k$-path in the $\infty$-Lie algebroid $\mathfrak{a}$ is a morphism of $\infty$-Lie algebroids of the form

$$\Sigma \; \coloneqq \; T \Delta^k \longrightarrow \mathfrak{a}$$

from the tangent Lie algebroid $T \Delta^k$ of the standard smooth $k$-simplex to $\mathfrak{a}$. Dually this is equivalently a homomorphism of dg-algebras

$$\Omega^\bullet(\Delta^k) \longleftarrow CE(\mathfrak{a}) \;\colon\; \Sigma^*$$

from the Chevalley-Eilenberg algebra of $\mathfrak{a}$ to the de Rham complex of $\Delta^d$.

See also at differential forms on simplices.

Remark

A $k$-path in $\mathfrak{a}$, def. , is equivalently

1. a flat $\mathfrak{a}$-valued differential form on $\Delta^k$;

2. a Maurer-Cartan element in $\Omega^\bullet(\Delta^k)\otimes \mathfrak{a}$.

The Lie integration of $\mathfrak{a}$ is essentially the simplicial object whose $k$-cells are the $d$-paths in $\mathfrak{a}$. However, in order for this to be well-behaved, it is possible and useful to restrict to $d$-paths that are sufficiently well-behaved towards the boundary of the simplex:

Definition

Regard the smooth simplex $\Delta^k$ as embedded into the Cartesian space $\mathbb{R}^{k+1}$ in the standard way, and equip $\Delta^k$ with the metric space structure induced this way.

A smooth differential form $\omega$ on $\Delta^k$ is said to have sitting instants along the boundary if, for every $(r \lt k)$-face $F$ of $\Delta^k$ there is an open neighbourhood $U_F$ of $F$ in $\Delta^k$ such that $\omega$ restricted to $U$ is constant in the directions perpendicular to the $r$-face on its value restricted to that face.

More generally, for any $U \in$ CartSp a smooth differential form $\omega$ on $U \times\Delta^k$ is said to have sitting instants if there is $0 \lt \epsilon \in \mathbb{R}$ such that for all points $u : * \to U$ the pullback along $(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k$ is a form with sitting instants on $\epsilon$-neighbourhoods of faces.

Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write $\Omega^\bullet_{si}(U \times \Delta^k)$ for this sub-dg-algebra.

We write $\Omega_{si,vert}^\bullet(U \times \Delta^k)$ for the further sub-dg-algebra of vertical differential forms with respect to the projection $p : U \times \Delta^k \to U$, hence the coequalizer

$$\Omega^\bullet(U) \stackrel{\stackrel{p^*}{\longrightarrow}}{\underset{0}{\longrightarrow}} \Omega^\bullet_{si}(U \times \Delta^k) \to \Omega^\bullet_{si, vert}(U \times \Delta^k) \,.$$

Remark

The dimension of the normal direction to a face depends on the dimension of the face: there is one perpendicular direction to a codimension-1 face, and $k$ perpendicular directions to a vertex.

Examples

• A smooth 0-form (a smooth function) has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it is constant.

  A smooth function $f : U \times \Delta^1 \to \mathbb{R}$ is in $\Omega^0_{\mathrm{vert}}(U \times \Delta^1)$ if there is $0 \lt \epsilon \in \mathbb{R}$ such that for each $u \in U$ the function $f(u,-) : \Delta^1 \simeq [0,1] \to \mathbb{R}$ is constant on $[0,\epsilon) \coprod (1-\epsilon,1)$.

• A smooth 1-form has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it vanishes.

• Let $X$ be a smooth manifold, $\omega \in \Omega^\bullet(X)$ be a smooth differential form. Let

  $$\phi \colon \Delta^k \to X$$

  be a smooth function that has sitting instants as a function: towards any $j$-face of $\Delta^k$ it eventually becomes perpendicularly constant.

  Then the pullback form $\phi^* \omega \in \Omega^\bullet(\Delta^k)$ is a form with sitting instants.

Remark

The condition of sitting instants serves to make smooth differential forms not be affected by the boundaries and corners of $\Delta^k$. Notably for $\omega_j \in \Omega^\bullet(\Delta^{k-1})$ a collection of forms with sitting instants on the $(k-1)$-cells of a horn $\Lambda^k_i$ that coincide on adjacent boundaries, and for

$$p \colon \Delta^k \to \Lambda^{k-1}_i$$

a standard piecewise smooth retract, the pullbacks

$$p^* \omega_i$$

glue to a single smooth differential form (with sitting instants) on $\Delta^k$.

Remark

That $\omega \in \Omega^\bullet(\Delta^k)$ having sitting instants does not imply that there is a neighbourhood of the boundary of $\Delta^k$ on which $\omega$ is entirely constant. It is important for the following constructions that in the vicinity of the boundary $\omega$ is allowed to vary parallel to the boundary, just not perpendicular to it.

Integration to a discrete $\infty$-groupoid

Here we discuss the discrete ∞-groupoids underlying the smooth ∞-groupoids to which an ∞-Lie algebroid integrates.

For $\mathfrak{a}$ an $\infty$-Lie algebroid, the $d$-paths in $\mathfrak{a}$ naturally form a simplicial set as $d$ varies:

$$\begin{aligned} \exp(\mathfrak{a})_{bare} & \coloneqq \left( \cdots Hom_{\infty LieAlgd}(T \Delta^2, \mathfrak{a}) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{\infty LieAlgd}(T \Delta^1, \mathfrak{a}) \stackrel{\longrightarrow}{\longrightarrow} Hom_{\infty LieAlgd}(T \Delta^0, \mathfrak{g}) \right) \\ & = ( \cdots Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^2)) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^1)) \stackrel{\longrightarrow}{\longrightarrow} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^0)) ) \end{aligned} \,.$$

(We are indicating only the face maps, not the degeneracy maps, just for notational simplicity).

If here instead of smooth differential forms one uses polynomial differential forms then this is precisely the Sullivan construction of rational homotopy theory applied to $CE(\mathfrak{a})$. We next realize smooth structure on this and hence realize this as an object in higher Lie theory.

Remark

(spurious homotopy groups)

For $\mathfrak{a}$ a Lie n-algebroid (an $n$-truncated $\infty$-Lie algebroid) this construction will not yield in general an $n$-truncated ∞-groupoid $\exp(\mathfrak{a})$.

To see this, consider the example (discussed in detail below) that $\mathfrak{a} = \mathfrak{g}$ is an ordinary Lie algebra. Then $\exp(\mathfrak{g})_n$ is canonically identified with the set of smooth based maps $\Delta^n \to G$ into the simply connected Lie group that integrates $\mathfrak{g}$ in ordinary Lie theory. This means that the simplicial homotopy groups of $\exp(\mathfrak{g})$ are the topological homotopy groups of $G$, which in general (say for $G$ the orthogonal group or unitary group) will be non-trivial in arbitrarily higher degree, even though $\mathfrak{g}$ is just a Lie 1-algebra. This phenomenon is well familiar from rational homotopy theory, where a classical theorem asserts that the rational homotopy groups of $\exp(\mathfrak{g})$ are generated from the generators in a minimal Sullivan model resolution of $\mathfrak{g}$.

For the purposes of higher Lie theory therefore instead one wants to truncate $\exp(\mathfrak{g})$ to its $(n+1)$-coskeleton

$$\mathbf{cosk}_{n+1}\exp(\mathfrak{a})_{bare} \,.$$

This divides out n-morphisms by $(n+1)$-morphisms and forgets all higher higher nontrivial morphisms, hence all higher homotopy groups.

$\,$

Integration to a smooth $\infty$-groupoid

We now discuss Lie integration of $\infty$-Lie algebroids to smooth ∞-groupoids, presented by the model structure on simplicial presheaves $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ over the site CartSp${}_{smooth}$.

For the following definition recall the presentation of smooth ∞-groupoids by the model structure on simplicial presheaves over the site CartSp${}_{smooth}$.

Definition

For $\mathfrak{a}$ an L-∞ algebra of finite type with Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ define the simplicial presheaf

$$\exp(\mathfrak{a}) \;\colon\; CartSp_{smooth}^{op} \to sSet$$

by

$$\exp(\mathfrak{a}) \;\colon\; (U,[k]) \mapsto Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(U \times \Delta^k)_{si,vert}) \,,$$

for all $U \in$ CartSp and $[k] \in \Delta$.

Remark

Compared to the integration to discrete ∞-groupoids above this definition knows about $U$-parametrized smooth families of $k$-paths in $\mathfrak{g}$.

The underlying discrete ∞-groupoid is recovered as that of the $\mathbb{R}^0 = *$-parameterized family:

$$\exp(\mathfrak{a}) \colon \mathbb{R}^0 \mapsto \exp(\mathfrak{a})_{disc} \,.$$

Proposition

The objects $\exp(\mathfrak{g})$ are indeed Kan complexes over each $U \in$ CartSp.

Proof

Observe that the standard continuous horn retracts $f : \Delta^k \to \Lambda^k_i$ are smooth away from the preimages of the $(r \lt k)$-faces of $\Lambda[k]^i$.

For $\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i)$ a differential form with sitting instants on $\epsilon$-neighbourhoods, let therefore $K \subset \partial \Delta^k$ be the set of points of distance $\leq \epsilon$ from any subface. Then we have a smooth function

$$f : \Delta^k \setminus K \to \Lambda^k_i \setminus K \,.$$

The pullback $f^* \omega \in \Omega^\bullet(\Delta^k \setminus K)$ may be extended constantly back to a form with sitting instants on all of $\Delta^k$.

The resulting assignment

$$(CE(\mathfrak{g}) \stackrel{A}{\longrightarrow} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i)) \mapsto (CE(\mathfrak{g}) \stackrel{A}{\to} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i) \stackrel{f^*}{\to} \Omega^\bullet_{si,vert}(U \times \Delta^n))$$

provides fillers for all horns over all $U \in$ CartSp.

Definition

Write $\mathbf{cosk}_{n+1} \exp(a)$ for the simplicial presheaf obtained by postcomposing $\exp(\mathfrak{a}) : CartSp^{op} \to sSet$ with the $(n+1)$-coskeleton functor $\mathbf{cosk}_{n+1} : sSet \stackrel{tr_n}{\longrightarrow} sSet_{\leq n+1} \stackrel{cosk_{n+1}}{\to} sSet$.

Properties

Homotopy groups

…(Henriques, theorem 6.4)… remark …

Quillen adjunction

The above construction of Lie integraton to smooth ∞-groupoids clearly applies to all differential graded-commutative algebras, not necessarily just those which are Chevalley-Eilenberg algebras of L-∞ algebras. (but up to weak equivalence, there is no difference). With this generalization, the higher Lie integration extends to a Quillen adjunction (Prop. below). In order to state this conveniently, we first make more explicit the functor assigning smooth families of smooth differential forms on simplices (Def. below).

$\,$

Definition

(smooth families of smooth differential forms on simplices with sitting instants)

For $k \in \mathbb{N}$, write $\Delta^k_{mfd}$ for the k-simplex canonically regarded as a smooth manifold with boundaries and corners.

For $n \in \mathbb{N}$, regard $\mathbb{R}^n \times \Delta^k_{mfd} \overset{p_1}{to} \mathbb{R}^n$ as the trivial fiber bundle over the Cartesian space $\mathbb{R}^n$ with fiber that smooth k-simplex.

Write

$$\Omega^\bullet_{vert}(\mathbb{R}^n \times \Delta^k_{mfd}) \hookrightarrow \Omega^\bullet(\mathbb{R}^n \times \Delta^k_{mfd})$$

for the sub-dgc-algebra of the de Rham algebra on the vertical differential forms with respect to this bundle structure.

Moreover, write

$$\Omega^\bullet_{vert, si}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right) \hookrightarrow \Omega^\bullet_{vert}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right)$$

for the further sub-dgc-algebra on those vertical differential forms which have sitting instants towards the boundary of the k-simplex.

Via pullback of differential forms this construction provides a functor

$$\Omega^\bullet_{vert,si} \;\colon\; CartSp \times \Delta \longrightarrow dgcAlg_{\mathbb{R}, conn}^{op}$$

from the product category of the category CartSp of Cartesian spaces and smooth functions between them, with the simplex category, to the opposite of the category of connective dgc-algebras over the real numbers.

(FSS 12, Def. 4.2.1, see Braunack-Mayer 18, Def. 3.1.3)

Proposition

(Lie integration is right Quillen functor to smooth ∞-groupoids)

There is a Quillen adjunction

$$dgcAlg^{op}_{\mathbb{R}, \geq 0, proj} \; \underoverset {\underset{ Spec }{\longrightarrow}} {\overset{ \mathcal{O} }{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\bot_{Qu}\phantom{A}} \; [CartSp^{op},sSet_{Qu}]_{proj,loc}$$

between

1. the projective local model structure on simplicial presheaves over CartSp, regarded as a site via the good open cover coverage (i.e. presenting smooth ∞-groupoids);

2. the opposite projective model structure on connective dgc-algebras over the real numbers

given by nerve and realization with respect to the functor of smooth differential forms on simplices $CartSp \times \Delta \overset{\Omega^\bullet_{vert,si}}{\longrightarrow} dgcAlg_{\mathbb{R}, conn}^{op}$ from Def. :

1. the right adjoint $Spec$ sends a dgc-algebra $A \in dgcAlg_{\mathbb{R},\geq 0}$ to the simplicial presheaf which in degree $k$ is the set of dg-algebra-homomorphism form $A$ into the dgc-algebras of smooth differential forms on simplices $\Omega^\bullet_{si,vert}(-)$ (Def. ):

   $$Spec(A) \;\colon\; \mathbb{R}^n \times \Delta[k] \;\mapsto\; Hom_{dgcAlg_{\mathbb{R}}} \left( A , \Omega^\bullet_{si, vert}(\mathbb{R}^n \times \Delta^k_{mfd}) \right)$$

2. the left adjoint $\mathcal{O}$ is the Yoneda extension of the functor $\Omega^\bullet_{vert,si} \;\colon\; CartSp \times \Delta \to dgcAlg_{\mathbb{R},conn}^{op}$ assigning dgc-algebras of smooth differential forms on simplices from Def. ,

   hence which acts on a simplicial presheaf $\mathbf{X} \in [CartSp^{op}, sSet] \simeq [\CartSp^{op} \times \Delta^{op}, Set]$, expanded via the co-Yoneda lemma as a coend of representables, as

   $$\mathcal{O} \;\colon\; \mathbf{X} \simeq \int^{n,k} y(\mathbb{R}^n \times \Delta[k]) \times \mathbf{X}(\mathbb{R}^n)_k \;\mapsto\; \int_{n,k} \underset{\mathbf{X}(\mathbb{R}^n)_k}{\prod} \Omega^\bullet_{si,vert}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right)$$

(Braunack-Mayer 18, theorem 3.1.10)

Examples

See also at smooth ∞-groupoid – structures the section Exponentiated ∞-Lie algebras.

Interating Lie algebras to Lie groups

Let $\mathfrak{g} \in L_\infty$ be an ordinary (finite dimensional) Lie algebra. Standard Lie theory (see Lie's three theorems) provides a simply connected Lie group $G$ integrating $\mathfrak{g}$.

With $G$ regarded as a smooth ∞-group write $\mathbf{B}G \in$ Smooth∞Grpd for its delooping. The standard presentation of this on $[CartSp_{smooth}^{op}, sSet]$ is by the simplicial presheaf

$$\mathbf{B}G_c \colon U \mapsto N(C^\infty(U,G) \stackrel{\longrightarrow}{\longrightarrow} *) \,.$$

See at smooth infinity-groupoid – structures – Lie groups for more details.

Proposition

The operation of parallel transport $P \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G$ yields a weak equivalence (in $[CartSp^{op}, sSet]_{proj}$)

$$P \exp(\int (-) ) \;\colon\; \mathbf{cosk}_3 \exp(\mathfrak{g}) \;\simeq\; \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_c \,.$$

This follows from the Steenrod-Wockel approximation theorem and the following observation.

Lemma

For $X$ a simply connected smooth manifold and $x_0 \in X$ a basepoint, there is a canonical bijection

$$\Omega^1_{flat}(X,\mathfrak{g}) \simeq C^\infty_*(X,G)$$

between the set of Lie-algebra valued 1-forms on $X$ whose curvature 2-form vanishes, and the set of smooth functions $X\to G$ that take $x_0$ to the neutral element $e \in G$.

Proof

The bijection is given as follows. For $A \in \Omega^1_{flat}(X,\mathfrak{g})$ a flat 1-form, the corresponding function $f_A : X \to G$ sends $x \in X$ to the parallel transport along any path $x_0 \to x$ from the base point to $x$

$$f_A : x \mapsto tra_A(x_0 \to x) \,.$$

Because of the assumption that the curvature 2-form of $A$ vanishes and the assumption that $X$ is simply connected, this assignment is independent of the choice of path.

Conversely, for every such function $f : X \to G$ we recover $A$ as the pullback of the Maurer-Cartan form on $G$

$$A = f^* \theta \,.$$

From this we obtain

Proof of prop. .

The $\infty$-groupoid $\mathbf{cosk}_2 \exp(\mathfrak{g})$ is equivalent to the groupoid with a single object (no non-trivial 1-form on the point) whose morphisms are equivalence classes of smooth based paths $\Delta^1 \to G$ (with sitting instants), where two of these are taken to be equivalent if there is a smooth homotopy $D^2 \to G$ (with sitting instant) between them.

Since $G$ is simply connected, these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with $G$.

Remark

We do not need to fall back to classical Lie theory to obtain $G$ in the above argument. A detailed discussion of how to find $G$ with its group structure and smooth structure from $d$-paths in $\mathfrak{g}$ is in (Crainic).

Integrating to line/circle Lie $n$-groups

Definition

For $n \in \mathbb{N}, n \geq 1$ write $b^{n-1} \mathbb{R}$ for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree $n$ and vanishing differential. We may call this the line Lie n-algebra.

Write $\mathbf{B}^{n} \mathbb{R}$ for the smooth line (n+1)-group.

Observation

The discrete ∞-groupoid underlying $\exp(b^{n-1} \mathbb{R})$ is given by the Kan complex that in degree $k$ has the set of closed differential $n$-forms (with sitting instants) on the $k$-simplex

$$\exp(b^{n-1} \mathbb{R})_{disc} : [k] \mapsto \Omega^n_{si,cl}(\Delta^k)$$

Proposition

The $\infty$-Lie integration of $b^{n-1} \mathbb{R}$ is the circle n-group $\mathbf{B}^{n} \mathbb{R}$.

Moreover, with $\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet]$ the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree $n$ on $C^\infty(-, \mathbb{R})$ the equivalence is induced by the fiber integration of differential $n$-forms over the $n$-simplex:

$$\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \stackrel{\simeq}{\longrightarrow} \mathbf{B}^{n} \mathbb{R}_{chn} \,.$$

Proof

First we observe that the map

$$\int_{\Delta^\bullet} : (\omega \in \Omega^n_{si,vert,cl}(U \times \Delta^k)) \mapsto \int_{\Delta^k} \omega \in C^\infty(U, \mathbb{R})$$

is a morphism of simplicial presheaves $\exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ on CartSp${}_{smooth}$. Since it goes between presheaves of abelian simplicial groups by the Dold-Kan correspondence it is sufficient to check that we have a morphism of chain complexes of presheaves on the corresponding normalized chain complexes.

The only nontrivial degree to check is degree $n$. Let $\lambda \in \Omega_{si,vert,cl}^n(\Delta^{n+1})$. The differential of the normalized chains complex sends this to the signed sum of its restrictions to the $n$-faces of the $(n+1)$-simplex. Followed by the integral over $\Delta^n$ this is the piecewise integral of $\lambda$ over the boundary of the $n$-simplex. Since $\lambda$ has sitting instants, there is $0 \lt \epsilon \in \mathbb{R}$ such that there are no contributions to this integral in an $\epsilon$-neighbourhood of the $(n-1)$-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the $(n+1)$-simplex, as indicated in the following diagram

Since $\lambda$ is a closed form on the $n$-simplex, this surface integral vanishes, by the Stokes theorem. Hence $\int_{\Delta^\bullet}$ is indeed a chain map.

It remains to show that $\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ is an isomorphism on all the simplicial homotopys group over each $U \in CartSp$. This amounts to the statement that

1. a smooth family of closed $n$-forms with sitting instants on the boundary of $\Delta^{n+1}$ may be extended to a smooth family of closed forms with sitting instants on $\Delta^{n+1}$ precisely if their smooth family of integrals over the boundary vanishes;

2. Any smooth family of closed $n \lt k$-forms with sitting instants on the boundary of $\Delta^{k+1}$ may be extended to a smooth family of closed $n$-forms with sitting instants on $\Delta^{k+1}$.

To demonstrate this, we want to work with forms on the $(k+1)$-ball instead of the $(k+1)$-simplex. To achieve this, choose again $0 \lt \epsilon \in \mathbb{R}$ and construct the diffeomorphic image of $S^k \times [1,1-\epsilon]$ inside the $(k+1)$-simplex as indicated in the above diagram: outside an $\epsilon$-neighbourhood of the corners the image is a rectangular $\epsilon$-thickening of the faces of the simplex. Inside the $\epsilon$-neighbourhoods of the corners it bends smoothly. By the Steenrod-Wockel approximation theorem the diffeomorphism from this $\epsilon$-thickening of the smoothed boundary of the simplex to $S^k \times [1-\epsilon,1]$ extends to a smooth function from the $(k+1)$-simplex to the $(k+1)$-ball.

By choosing $\epsilon$ smaller than each of the sitting instants of the given $n$-form on $\partial \Delta^{k+1}$, we have that this $n$-form vanishes on the $\epsilon$-neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the $(k+1)$-ball.

It is now sufficient to show: a smooth family of smooth $n$-forms $\omega \in \Omega^n_{vert,cl}(U \times S^k)$ extends to a smooth family of closed $n$-forms $\hat \omega \in \Omega^n_{vert,cl}(U \times B^{k+1})$ that is radially constant in a neighbourhood of the boundary for all $n \lt k$ and for $k = n$ precisely if its smooth family of integrals vanishes, $\int_{S^k} \omega = 0 \in C^\infty(U, \mathbb{R})$.

Notice that over the point this is a direct consequence of the de Rham theorem (kernel of integration is the exact differential forms): an $n$-form $\omega$ on $S^k$ is exact precisely if $n \lt k$ or if $n = k$ and its integral vanishes. In that case there is an $(n-1)$-form $A$ with $\omega = d A$. Choosing any smoothing function $f : [0,1] \to [0,1]$ (smooth, surjective, non-decreasing and constant in a neighbourhood of the boundary) we obtain an $n$-form $f \wedge A$ on $(0,1] \times S^k$, vertically constant in a neighbourhood of the ends of the interval, equal to $A$ at the top and vanishing at the bottom. Pushed forward along the canonical $(0,1] \times S^k \to D^{k+1}$ this defines a form on the $(k+1)$-ball, that we denote by the same symbol $f \wedge A$. Then the form $\hat \omega := d (f \wedge A)$ solves the problem.

To complete the proof we have to show that this simple argument does extend to smooth families of forms, i.e., that we can choose the $(n-1)$-form $A$ in a way depending smoothly on the the $n$-form $\omega$.

One way of achieving this is using Hodge theory. Fix a Riemannian metric on $S^n$, and let $\Delta$ be the corresponding Laplace operator, and $\pi$ the projection on the space of harmonic forms. Then the central result of Hodge theory for compact Riemannian manifolds states that the operator $\pi$, seen as an operator from the de Rham complex to itself, is a cochain map homotopic to the identity, via an explicit homotopy $P := d^* G$ expressed in terms of the adjoint $d^*$ of the de Rham differential and of the Green operator $G$ of $\Delta$. Since the $k$-form $\omega$ is exact its projection on harmonic forms vanishes. Therefore

$$\begin{aligned} \omega & = (Id-\pi)\omega \\ & = d (P\omega)+P (d\omega) \\ & = d (P\omega). \end{aligned}$$

Hence $A := P\omega$ is a solution of the differential equation $d A=\omega$ depending smoothly on $\omega$.

Integrating the string Lie 2-algebra to the string Lie 2-group

Let $\mathfrak{string} = \mathfrak{g}_\mu$ be the string Lie 2-algebra.

Then $\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu)$ is equivalent to the 2-groupoid $\mathbf{B}String$

• with a single object;

• whose morphisms are based paths in $G$;

• whose 2-morphisms are equivalence class of pairs $(\Sigma,c)$, where

  • $\Sigma : D^2_* \to G$ is a smooth based map (where we use a homeomorphism $D^2 \simeq \Delta^2$ which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of $D^2_*$ is the 0-vertex of $\Delta^2$)

  • and $c \in U(1)$, and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball $\phi : D^3 \to G$ filling them the labels $c_1, c_2 \in U(1)$ differ by the integral $\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z}$,,

where $\theta$ is the Maurer-Cartan form, $\mu(\theta) = \langle \theta\wedge [\theta \wedge \theta]\rangle$ the 3-form obtained by plugging it into the cocycle.

This is the string Lie 2-group. It’s construction in terms of integration by paths is due to (Henriques)

Integrating Lie algebroids to (stacky) Lie groupoids

Unlike finite dimensional Lie algebras, not every Lie algebroid may be integrated to a Lie groupoid. There exists topological obstruction coming from $\pi_2(L)$ of the leaf $L\subset M$ of a Lie algebroid $A\to M$. The integrability criteria is completely classified through the behaviour of certain monodromy groups and that is the achievement of Crainic-Fernandes 01. These monodromy groups, providing obstructions of integration, may be seen as the image of a transgression map of the long exact sequence of Lie algebroid homotopy groups? induced by the natural Lie algebroid fibration? $A|_L \to L$. (see Brahic-Zhu 10 ).

Let us now describe the construction of the universal groupoid for a Lie algebroid $A$. This is a major step contained in Crainic-Fernandes 01 and was earlier discovered in the setting of Poisson manifolds as the phase space of Poisson sigma model in Cattaneo-Felder 00.

Given a Lie algebroid $A\to M$, an $A$-path (a Lie algebroid path), is a Lie algebroid morphism from $T\Delta^1 \to A$, that is, it is a path $a$ in $A$ such that $a=\rho(\gamma)$ where $\gamma=\pi(a)$ is the base path of $a$ (see example ).

An (end-fixing) $A$-homotopy (a Lie algebroid homotopy) is a Lie algebroid morphism from $T\square \to A$ satisfying certain boundary conditions. Let us be more precise: a vector bundle morphism $T\square \to A$ can be denoted by $a dt + b ds$ (see example ). Then the boundary condition is that $b(s,0)=0$ and $b(s,1)=0$. The fact that $adt+bds$ is a Lie algebroid morphism is equivalent to the following PDE:

$$\partial_t b- \partial_s a= \nabla_{\rho \alpha} \beta - \nabla_{\rho \beta} \alpha +[\alpha, \beta]$$

where $\nabla$ is a TM connection on $A$ and $\alpha$ and $\beta$ are certain time dependent sections of $A$ extending $a$ and $b$ respectively. Notice that there is also a way writing down the right hand side independent of choice of sections. See Section 1 of Crainic-Fernandes 01.

Then the universal groupoid associated to $A$ is the space of $A$-paths ($P_a A$) dividing by $A$-homotopies. It is naturally a topological groupoid. But only when the obstruction vanishes, it becomes a Lie groupoid and is the source-simply connected Lie groupoid integrating $A$.

Fortunately $A$-homotopies form finite codimensional foliation $F$, even though the quotient might not be always representible. Thus, $Mon_F(P_aA)$ represents a differentiable stack which becomes a stacky Lie groupoid over $M$. It turns out that there is a one-to-one correspondence between 'etale stacky Lie groupoid and Lie algebroid. This correspondence provides a positive answer to Lie's third theorem for Lie algebroids. Tseng-Zhu 04.

Related concepts

• Kan-fibrant simplicial manifold

• parallel transport, higher parallel transport

• holonomy, higher holonomy

  • nonabelian Stokes theorem

• dg-nerve

• function algebras on infinity-stacks

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

The “path method” of integrating Lie algebras to simply connected Lie groups appears in

• Hans Duistermaat, J. A. C. Kolk, section 1.14 of Lie groups, 2000

The idea of identifying the Sullivan construction applied to Chevalley-Eilenberg algebras as Lie integration to discrete ∞-groupoids appears in

• Vladimir Hinich, Descent of Deligne groupoids, Internat. Math. Res. Notices, 5:223–239, 1997 (doi, preprint ddg.pdf)

and for nilpotent L-infinity algebras:

• Ezra Getzler, Lie theory for nilpotent $L_\infty$ algebras, Annals of Mathematics, 170 (2009), 271-301 [math.AT/0404003]

(whose main point is the discussion of a gauge condition applicable for nilpotent $L_\infty$-algebras that cuts down the result of the Sullivan construction to a much smaller but equivalent model) .

This was refined from integration to bare $\infty$-groupoids to an integration to internal ∞-groupoids in Banach manifolds in

• André Henriques, Integrating $L_\infty$ algebras, Compos. Math. 144 (2008), no. 4, 1017–1045 (doi,math.AT/0603563)

(whose origin possibly preceeds that of Getzler’s article).

For general ∞-Lie algebroids the general idea of the integration process by “$d$-paths” had been indicated in

• Pavol Severa, Some title containing the words "homotopy" and "symplectic", e.g. this one, based on a talk at Poisson 2000, CIRM Marseille Luminy, June 2000 (arXiv:0105080)

• Pavol Ševera, Michal Širaň, Integration of differential graded manifolds, arxiv/1506.04898

Cast along the lines of Lie's third theorem:

• Christopher L. Rogers, Jesse Wolfson: Lie’s Third Theorem for Lie $\infty$-Algebras [arXiv:2409.08957]

Lie integration of dg-modules to smooth parameterized spectra (twisted differential cohomology theories);

• Vincent Braunack-Mayer, section 3.1 of Rational parameterized stable homotopy theory, Zurich 2018 (pdf)

Discussion of Lie integration of Lie algebroids by the path method is due to

• Marius Crainic, Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. 157 2 (2003) 575-620 [arXiv:math.DG/0105033, doi:10.4007/annals.2003.157.575]

reviewed in

• Rui Loja Fernandes, Marius Crainic, Lectures on Integrability of Lie Brackets, Geometry & Topology Monographs 17 (2011) 1–107 [arxiv:math.DG/0611259, doi:10.2140/gtm.2011.17.1]

and following Duistermaat-Kolk (2000 section 1.14), as well as the discussion of the special case of Lie integration of Poisson Lie algebroids to symplectic groupoids in

• Alberto Cattaneo, Giovanni Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, (ed. Klaas Landsman, M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198 (Birkhäuser,

  2001), 61–93. (arXiv:math/0003023)

upgraded to the stacky version by

• Tseng Hsiang-Hua, Chenchang Zhu, Integrating Lie algebroids via stacks, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, arXiv:math/0405003.

A general proof that equivalent $L_\infty$-algebras integrate to equivalent Lie $\infty$-groupoids is in

• Christopher Rogers, Chenchang Zhu, On the homotopy theory for Lie ∞-groupoids, with an application to integrating L∞-algebras (arXiv:1609.01394)

Obstruction interpreted as transgression of a Lie algebroid fibration by

• Olivier Brahic, Chenchang Zhu, Lie algebroid fibrations, Adv. Math. 226 (2011), no. 4, 3105–3135, arXiv:1001.4904.

A description of Lie integration with values in smooth ∞-groupoids regarded as simplicial presheaves on CartSp (and further the Lie integration of L-infinity cocycles) is in

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech Cocycles for Differential characteristic Classes, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735)

Essentially the same integration prescription is considered in

• Dmitry Roytenberg, Differential graded manifolds and associated stacks: an overview (pdf)

The Lie integration- of Lie algebroid representations $\mathfrak{a} \to end(V)$ to morphisms of ∞-categories $A \to Ch_\bullet^\circ$ / higher parallel transport is discussed in

• Camilo Arias Abad, Florian Schätz, The $A_\infty$ de Rham theorem and integration of representations up to homotopy (arXiv:1011.4693)

Application to the problem of Lie integrating ordinary but infinite-dimensional Lie algebras is in

• Christoph Wockel, Chenchang Zhu, Integrating central extensions of Lie algebras via Lie 2-groups (arXiv:1204.5583)

A generalization of Lie integration to conjectural Leibniz groups has been conjectured by J-L. Loday. A local version via local Lie racks has been proposed in

• Simon Covez, The local integration of Leibniz algebras, arXiv:1011.4112; On the conjectural cohomology for groups, arXiv:1202.2269; L’intégration locale des algèbres de Leibniz, Thesis (2010), pdf

Integration from Lie algebroids to groupoids is also studied in the dual language and generality of integration of Lie-Reinhart algebras and commutative Hopf algebroids,

• Alessandro Ardizzoni, Laiachi El Kaoutit, Paolo Saracco, Towards differentiation and integration between Hopf algebroids and Lie algebroids, arXiv:1905.10288

category: Lie theory