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December 3, 2008

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 nn-algebroid gg for finite nn always an \infty-Lie groupoid, which I’ll denote by Π (g)\Pi_\infty(g) for certain reasons. This encodes in degrees greater than nn higher homotopy groups which are not supposed to be present in the “true” Lie nn-groupoid Π n(g)\Pi_n(g) which should integrate gg.

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

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

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

Posted at December 3, 2008 4:48 PM UTC

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11 Comments & 0 Trackbacks

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?

Posted by: jim stasheff on December 5, 2008 6:36 PM | Permalink | Reply to this

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

Yes, sorry.

Posted by: Urs Schreiber on December 5, 2008 6:55 PM | Permalink | Reply to this

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

As Jim surely knows, π 2\pi_2 vanishes for every Lie group, compact or not — where ‘Lie group’ by default means ‘finite-dimensional Lie group’ except in phrases like ‘infinite-dimensional Lie group’, ‘Banach Lie group’, ‘Fréchet Lie group’ and so on.

These phrases are like ‘nonassociative division algebra’ or ‘mock turtle soup’ — the first adjective serves not to further delimit the concept, but quite the opposite.

From Alice in Wonderland:

Then the Queen left off, quite out of breath, and said to Alice, “Have you seen the Mock Turtle yet?”

“No,” said Alice. “I don’t even know what a Mock Turtle is.”

“It’s the thing Mock Turtle Soup is made from,” said the Queen.

Posted by: John Baez on December 7, 2008 1:24 AM | Permalink | Reply to this

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

As you note, a stacky Lie groupoid is morally a Lie 2-groupoid. If — much to our surprise — integrating a Lie algebroid gives a Lie 2-groupoid, what should we get when we integrate a Lie nn-algebroid?

Posted by: John Baez on December 7, 2008 1:28 AM | Permalink | Reply to this

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

a stacky Lie groupoid is morally a Lie 2-groupoid

I’d go as far as saying that it is not only “morally” so. Given that the “differentiable stacks” that we are talking about really are Lie groupoids (the only difference being the precise way in which we say what “equivalent” means) a stacky Lie groupoid is a groupoid internal to Lie groupoid. We shouldn’t feel shy about calling that a Lie 2-groupoid.

If — much to our surprise — integrating a Lie algebroid gives a Lie 2-groupoid, what should we get when we integrate a Lie n-algebroid?

There are several aspects to this question:

one is: what is god-given is to any Lie nn-algebroid a Lie \infty-groupoid integrating it. Nothing else. Man-made is the construction of cutting that down to a Lie nn-groupoid.

As always when mortals enter the business, things begin to depend on their choices.

There are different choices one can male. One choice is:

1) We might want every Lie nn-algebroid to integrate to a Lie nn-groupoid.

That wish can be fulfilled! Work in the category of diffological spaces (aka concrete sheaves). All the required quotients exist and every Lie nn-algebroid thus has naturally an nn-groupoid internal to diffeological spaces associated with it.

But one may have another wish, namely: 2) Retain as much of Lie’s three theorems at degree nn as possible.

As Chenchang rightly emphasizes, taking the quotient at degree nn the level of sheaves instead of at degree nn at the level of stacks (= at degree (n+1)(n+1) at the level of sheaves) may in special cases lead to Lie nn-groupoids whose Lie differential no longer recovers the Lie nn-algebroid one started with:

there may be less differential forms on a quotient space when the quotient is taken as sheaves, as compared to when the quotient is taken as stacks.

On the other hand, this stil begs the question: as taking the quotient at the level of (pre)stacks is really the same as taking the quotient at higher degree at the level of (pre)sheaves, the real question is:

is there some kk such that for every nn the Lie theory of Lie nn-algebroids involves precisely (n+k)(n+k)-groupoids internal to sheaves?

That gives rise to this question: what is the generalization to higher nn of the fact that π 2\pi_2 vanishes for every fin-dim Lie group?

It’s a nice question about fundamental \infty-groupoids of certain generalized smooth spaces…

Posted by: Urs Schreiber on December 7, 2008 8:45 PM | Permalink | Reply to this

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

John wrote:

If — much to our surprise — integrating a Lie algebroid gives a Lie 2-groupoid […]

It may be noteworthy in this context that

a) not every ordinary Lie algebra integrates to a Lie group – many infinite dimensional ones do not: they intergate locally but globally fail to satisfy associativity(!)

b) that recently Christoph Wockel pointed out that these Lie 1-algebras do integrate to Lie 2-groups!

So its not just the passage from Lie algebras to Lie algebroids which introduces what looks like a mismatch between categorical dimension of infinitesimal annd global Lie structures. This mismatch is already there for ordinary Lie algebras.

Posted by: Urs Schreiber on January 28, 2009 10:41 AM | Permalink | Reply to this

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

Milnor wrote a very nice but ?unpublished? paper on inf dim Lie algebras not necessarily having a group

does anyone have a copy?

Posted by: jim stasheff on January 28, 2009 1:27 PM | Permalink | Reply to this

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

Jim Stasheff:

Milnor wrote a very nice but ?unpublished? paper on inf dim Lie algebras not necessarily having a group

does anyone have a copy?

It is published (and very nice):

MR0830252 (87g:22024) Milnor, J.(1-IASP) Remarks on infinite-dimensional Lie groups. Relativity, groups and topology, II (Les Houches, 1983), 1007–1057, North-Holland, Amsterdam, 1984. 22E65 (58B25)

Posted by: Maarten Bergvelt on February 24, 2010 4:16 PM | Permalink | Reply to this

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

Milnor’ works are coming out in a series of volumes. Several unpublished papers are included. E.g. an earlier version of Milnor-Moore with more Milnor/less Moore.

Posted by: jim stasheff on February 25, 2010 2:53 PM | Permalink | Reply to this

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

Concerning my comment above on Wockel’s result I should add maybe that a comment analogous to Chenchang Zhu’s construction applies:

in the context of diffeological spaces one can of course form the quotient of the resulting 2-group(oid) on isomorphism classes and remain with a diffeological group(oid), if desired. But that in general may lose some information.

Posted by: Urs Schreiber on January 28, 2009 2:57 PM | Permalink | Reply to this

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

…that recently Christoph Wockel pointed out that these Lie 1-algebras do integrate to Lie 2-groups!

That’s my math surprise for the week (month?). I didn’t see that one coming.

Posted by: Bruce Bartlett on January 28, 2009 5:31 PM | Permalink | Reply to this

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