nLab
Lie integration

Idea

Given an $L_{\mathrm{\infty }}$-algebroid (for instance a Lie algebra, or a Lie algebroid or an $L_{\mathrm{\infty }}$-algebra) $𝔤$ and given a $d$-dimensional manifold $P^d$ diffeomorphic to the $d$-dimensional disk $D^d$, let a $d$-path in $𝔤$ be a morhism of $L_{\mathrm{\infty }}$-algebroids

$$\Sigma :T{P}^d\to 𝔤,$$\Sigma : T P^d \to \mathfrak{g} \,,

where $T{P}^d$ denotes the tangent Lie algebroid of $P^d$ (see the list of examples here).

By restricting such morphisms to parts of the boundary of $P^d$, the collections of all such $L_{\mathrm{\infty }}$-algebroid $d$-paths naturally arrange themselves in diagrams of the form

$$S^{•}(𝔤):=(\cdots \to {\mathrm{Hom}}_{L_{\mathrm{\infty }}\mathrm{Alg}}(T{P}^2,𝔤)\to {\mathrm{Hom}}_{L_{\mathrm{\infty }}\mathrm{Alg}}(T{P}^1,𝔤)\to {\mathrm{Hom}}_{L_{\mathrm{\infty }}\mathrm{Alg}}(T{P}^0,𝔤)).$$S^\bullet(\mathfrak{g}) := ( \cdots \to Hom_{L_\infty Alg}(T P^2, \mathfrak{g}) \to Hom_{L_\infty Alg}(T P^1, \mathfrak{g}) \to Hom_{L_\infty Alg}(T P^0, \mathfrak{g}) ) \,.

In particular

• if $P^d={\Delta }^d$ is the standard $d$-simplex, $d$-paths in $𝔤$ naturally form a simplicial set;

• if $P^d=D^d$ is the standard $d$-disk, $d$-paths in $𝔤$ naturally form a globular set.

Notice that morphisms of $L_{\mathrm{\infty }}$-algebroids are, essentially by definition, dual to morphisms of qDGCAs, so that the above can be equivalently rewritten as

$$S^{•}(mathrakg):=(\cdots \to {\mathrm{Hom}}_{\mathrm{qDGCAs}}(\mathrm{CE}(𝔤),{\Omega }^{•}(P^2))\to {\mathrm{Hom}}_{\mathrm{qDGCAs}}(\mathrm{CE}(𝔤),{\Omega }^{•}(P^1))\to {\mathrm{Hom}}_{\mathrm{qDGCAs}}(\mathrm{CE}(𝔤),{\Omega }^{•}(P^0))).$$S^\bullet(\mathrak{g}) := ( \cdots \to Hom_{qDGCAs}(CE(\mathfrak{g}), \Omega^\bullet(P^2)) \to Hom_{qDGCAs}(CE(\mathfrak{g}), \Omega^\bullet(P^1)) \to Hom_{qDGCAs}(CE(\mathfrak{g}), \Omega^\bullet(P^0)) ) \,.

Here we used that the Chevalley–Eilenberg algebra of the tangent Lie algebroid is the deRham complex of differential forms, see the list of examples here.

In this form these graded (simplicial/globular) sets associated to a qDGCA have been familiar as the Sullivan construction? in rational homotopy theory since the late 1960s. In this context the interpretation of qDGCAs as Chevalley–Eilenberg algebras of $L_{\mathrm{\infty }}$-algebroids is not usually mentioned, however, instead these differential algebras are thought of, equivalently, as models for deRham algebras of certain spaces.

But with the interpretation of qDGCAs as $L_{\mathrm{\infty }}$-algebroids in hand, it is natural to ask for extra properties and structure on $S^{•}(𝔤)$ that would allow to interpret $S^{\mathrm{\infty }}(𝔤)$ as the graded set underlying an infinity-groupoid.

This step, obvious as it may be with hindsight, was only made a few years ago. The general idea has been described first in the article

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

Of course one wants to regard in this context $S^{•}(𝔤)$ not just as a graded set but as a graded space. In

• Ezra Getzler, Lie theory for nilpotent L-infinity algebras, (arXiv)

this $S^{•}(𝔤)$ was realized

• as internal to Diff for the case that $𝔤$ is a nilpotent $L_{\mathrm{\infty }}$-algebra;

• as having the property of being a Kan complex.

This then makes $S^{•}(𝔤)$ a Lie $\mathrm{\infty }$-groupoid, which is regarded as the structure which Lie-integrates $𝔤$ in the context of Lie theory.

Various variations of this theme are possible: in

• André Henriques, Integrating $L_{\mathrm{\infty }}$ algebras,(arXiv)

(whose origin possibly preceeds that of the previous article) $S^{•}(𝔤)$ is realized internal to Banach manifolds, for $𝔤$ an $L_{\mathrm{\infty }}$-algebra.

The main technical issue: truncation and quotienting

The main technical point in these constructions is to ensure that the general construction can be realized internal to a category of well-behaved spaces. In particular, usually one wants to integrate a Lie $n$-algebroid to a Lie $n$-groupoid, which means that $S^{•}(𝔤)$ has to be truncated after degree $n$ and quotienting out $(n+1)-\mathrm{cells}$ by replacing the space of $n$-paths $S^n(𝔤)$ in the $L_{\mathrm{\infty }}$-algebroid with the quotient space $S^n(𝔤)/S^{n+1}(𝔤)$ of $n$-paths modulo $(n+1)$-dimensional homotopy.

By the old dichotomy between nice objects and nice categories, the more well-behaved a space is the less likely is this quotient still to be well behaved.

One can

• either carefully try to cut down $S^{•}(𝔤)$ to something small and well behaved (this is the strategy in Getzler’s work #);

• or one realizes these quotients as suitable generalized spaces (this is the strategy in Zhu’s work #).

Lie integration in low categorical dimension

Unsurprisingly, the properties of this general construction are best studied and understood for $𝔤$ a Lie $n$-algebroid for low $n$. For $n=1$ it reproduces ordinary Lie theory and in particular Lie's three theorems (see there for details), at least if one passes to generalized smooth spaces.

Remark

There is a way to understand Lie integration as being about forming fundamental $\mathrm{\infty }$-groupoids of certain generalized smooth spaces. This is described at Lie theory in the private $n$Lab area.