### Zhu on Lie’s Second Theorem for Lie Groupoids

#### Posted by Urs Schreiber

The last couple of days Chenchang Zhu had been visiting Hamburg. Yesterday she gave a nice colloquium talk on her work:

Chenchang Zhu
*Lie II theorem*

(pdf slides, 57 slides with overlay)

Chenchang’s work revolves around the “oidification” or horizontal categorification of ordinary Lie theory: namely the generalization of the relation between Lie groups and Lie algebras to that between Lie groupoids and Lie algebroids.

The main technical issue here is that the general procedure of Lie integration naturally produces from any Lie $n$-algebroid $g$ for finite $n$ always an $\infty$-Lie groupoid, which I’ll denote by $\Pi_\infty(g)$ for certain reasons. This encodes in degrees greater than $n$ higher homotopy groups which are not supposed to be present in the “true” Lie $n$-groupoid $\Pi_n(g)$ which should integrate $g$.

So $\Pi_n(g)$ is, as the notation indicates, supposed to be obtained from $\Pi_\infty(g)$ by

- *truncating* all nontrivial $(k \gt n)$-morphisms away;

- *quotienting* the existing $n$-morphisms by $(n+1)$-morphisms.

Due to the dichotomy between nice objects and nice categories taking this very quotient tends to carry one out of any category of nice objects that one may want to work internal to.

Chenchang’s work is based on the idea of taking this quotient not internal to Manifolds, but internal to generalized smooth spaces known, equivalently, as

- Lie groupoids up to Morita equivalence;

- differentiable stacks (i.e. stacks (on the site of manifolds) of principal $G$-bundles for $G$ a Lie groupoid).

Accordingly, she shows that every Lie algebroid may be integrated to a stacky Lie groupoid, which is defined to be a Lie groupoid internal to Lie groupoid/differentiable stacks, but with the condition that the Lie groupoid/stack of objects is just an ordinary manifold.

Saying this using the words “Lie groupoid” instead of the pretty much equivalent “differentiable stack” makes already quite plausible one of Chenchang’s theorems, namely that these stacky Lie groupoids are equivalenty certain Lie 2-groupoids.

This means that the remarkable aspect of Lie theory for Lie algebroids is that for $g$ any Lie algebroid one can always safely truncate $\Pi_\infty(g)$ in degree 2 – instead of in degree 1 – and obtain $\Pi_2(g)$ internal to a category of nice space (Banach manifolds), while the truncation down to $\Pi_1(g)$ in general throws one out of the category of nice spaces. Moreover, in the case that $\Pi_1(g)$ exists, $\Pi_2(g)$ is Morita equivalent to $\Pi_1(g)$: the reason is (a generalization to Lie groupoids of) the familiar fact that for every Lie group $G$ the second homotopy group vanishes, $\pi_2(G) = 0$. This means that $\Pi_2(g)$ still does not pick up any of the spurious homotopies which are present in $\Pi_\infty(g)$.

As Pavol Ševera, who originally envisioned much of this, put it in his email greetings to me:

Beware of large homotopies!

In any case, using some of these gymnastics, one obtains the analog of Lie’s third theorem for Lie groups and Lie algebras, generalized to Lie groupoids and Lie algebroids.

Given that, one can continue to develop the theory. Chenchang discusses Lie’s second theorem for Lie groupoids in

Chenchang Zhu,
*Lie II theorem for Lie algebroids via stacky Lie groupoids*

(arXiv)

## Re: Zhu on Lie’s Second Theorem for Lie Groupoids

WHOA!

the familiar fact that for every Lie group G the second homotopy group vanishes, π 2(G)=0.

compact? fin dim?