The n-Category Café

Yes, sorry.

As Jim surely knows, $\pi_2$ 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.

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 $n$-algebroid a Lie $\infty$-groupoid integrating it. Nothing else. Man-made is the construction of cutting that down to a Lie $n$-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 $n$-algebroid to integrate to a Lie $n$-groupoid.

That wish can be fulfilled! Work in the category of diffological spaces (aka concrete sheaves). All the required quotients exist and every Lie $n$-algebroid thus has naturally an $n$-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 $n$ as possible.

As Chenchang rightly emphasizes, taking the quotient at degree $n$ the level of sheaves instead of at degree $n$ at the level of stacks (= at degree $(n+1)$ at the level of sheaves) may in special cases lead to Lie $n$-groupoids whose Lie differential no longer recovers the Lie $n$-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 $k$ such that for every $n$ the Lie theory of Lie $n$-algebroids involves precisely $(n+k)$-groupoids internal to sheaves?

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

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

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.