June 30, 2008

Lerman on Orbifolds

Posted by David Corfield

Eugene Lerman gave us his views on orbifolds in a discussion beginning here. Now you can read a whole paper of his on the subject Orbifolds as Stacks?:

Abstract. The goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps need to be thought of as a 2-category. This 2-category may be either taken to be the weak 2-category of groupoids, bibundles and equivariant maps between bibundles or the strict 2-category of geometric stacks represented by proper étale Lie groupoids. While nothing in this paper is strictly speaking new, it is hoped that differential geometers unfamiliar with groupoids, bibundles and stacks would find it a useful introduction to the subject.

Posted at June 30, 2008 10:40 AM UTC

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Re: Lerman on Orbifolds

Hi David,

The link you provided points to this very page instead of the article itself.

Posted by: Daniel de França MTd2 on June 30, 2008 1:10 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

Fixed, thanks.

Posted by: David Corfield on June 30, 2008 1:24 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

I’m not clear what problem this is supposed to solve. Is it that the category of orbifolds not closed under nice operations, or is the author after something else?

Posted by: Walt on July 2, 2008 5:41 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

Before you could say that the category of orbifolds is not closed under nice operations, you have to decide what the category of orbifolds *is*. And one of the answers (the one I find most convincing) is that it’s not a category.

Posted by: Eugene Lerman on July 3, 2008 6:35 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

The point, very naively, is that an orbifold is more like a groupoid than a set: it has ‘built-in symmetries’. Sets form a category, but groupoids form a 2-category… and so do orbifolds! Trying to pretend orbifolds form a mere category does violence to their true nature.

The issue is made subtler by the fact that an orbifold comes along with a topology (and in fact a smooth structure). So, instead of being a mere groupoid, an orbifold is a kind of stack.

This, at least, is my impression before having actually read Eugene’s paper. It’s on my list of papers to read and discuss in This Week’s Finds.

Posted by: John Baez on July 5, 2008 2:14 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

The point, very naively, is that an orbifold is more like a groupoid than a set

This isn’t really that naive. Little harm is done by thinking “orbifold = smooth groupoid”. The only point being that one wants to be precise about what “smooth” means.

But on the other hand, if you have something that you think qualifies as a smooth groupoid which does however not correspond to an orbifold, maybe the fault is with the definition of orbifold!

So, instead of being a mere groupoid, an orbifold is a kind of stack.

This is not a perspective that I like. Stacks may be a way to talk about smooth groupoids just as sheaves are a way to talk about diffeological spaces. But the orbifold is a groupoid much rather than it is a stack. The stack associated with it rather is the collection of probes of the orbifold by manifolds. This is one way to characterize the smoothness condition.

Posted by: Urs Schreiber on July 6, 2008 7:25 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

Hi Urs,

I can’t help commenting here :-)

Stacks may be a way to talk about smooth groupoids just as sheaves are a way to talk about diffeological spaces. But the orbifold is a groupoid much rather than it is a stack. The stack associated with it rather is the collection of probes of the orbifold by manifolds. This is one way to characterize the smoothness condition.

I agree that if one adopts the “sections over the open sets of $X$” way of looking at stacks, then my feelings go toward your approach - that the fundamental geometric object is perhaps more naturally thought of as $X$ rather than its sections.

But I prefer the other way of looking at stacks and sheaves… namely where you “piggyback” onto the already-existing notion of manifold, via the “Grothendieck site” philosophy.

In this viewpoint, a stack is a sheafy thing over the site Diff of smooth manifolds. And the major difference between this and the other viewpoint of stacks, is that it allows one to consider “what the stack assigns to the point” - which we couldn’t even talk about in the open-sets school of thought.

And this allows for a paradigm shift ;-)

Namely, if someone asks me, “What is a stack?”, I will now answer: “A stack is a smooth groupoid”.

Then they ask me, “What do you mean by smooth?”

I would reply, “A groupoid is smooth if you can make sense of smooth families of objects and smooth families of morphisms”.

And if they ask again, “What do you mean by that?”, I would reply, “A groupiod $\mathcal{G}$ is smooth if you can construct a stack $\mathcal{X}$ over Diff such that $\mathcal{X}(pt) = \mathcal{G}$.”

So I guess what I’m saying is that to me, the word “stack” is just a way of making sense of the word “smooth”. I am with you in that I regard the raw groupoid (the thing the stack assigns to the point) as the most fundamental thing. I am just pointing out that saying “G is a stack” doesn’t mean one is moving away from the fundamental geometric object… it’s a matter of language I guess.

It’s like if someone says, “$X$ is a manifold”. The thing which registers in my brain is that “$X$ is firstly and primarily a set, with some extra gismos (the charts) to make sense of smoothness”.

Similarly if someone says, “$X$ is a stack over the site of smooth manifolds”, my brain would interpret this as saying “$X$ is firstly and primarily a groupoid, with some extra gismos (the stack stuff) to make sense of smoothness”.

I hope my understanding of stacks is not flawed in some subtle way.

Posted by: Bruce Bartlett on July 7, 2008 2:28 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

Sorry, typo: in the first paragraph above, where I said “that the fundamental geometric object is perhaps more naturally thought of as $X$ rather than its sections”, one should replace “$X$” by “the bundle-y thing over $X$”.

Posted by: Bruce Bartlett on July 7, 2008 2:33 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

Hi Bruce,

yes, we agree. We know that. I suppose my comment came across a bit more drastic than intended.

I keep having that quibble though, similar to our discussion about gerbes. But it’s not really all that important. I like stacks.

(Well, what I really like currently are rectified stacks. Rectified $\infty$-stacks, even, i.e. $\omega$-category-valued presheaves with descent condition. After a long struggle with finding the right concept, I was so amazed to find how immensely far we can get into the world of $\infty$-stacks with $\omega$-category valued presheaves. Thanks to Ross Street.)

Posted by: Urs Schreiber on July 7, 2008 3:47 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

Oh no! Just when I was excited about groupoids because they provide a nice framework for talking about discrete stuff, I now learn you guys are already trying to smooth them out?! What is the world coming to?! :)

PS: Don’t mind me. I haven’t had my coffee yet…

Posted by: Eric on July 7, 2008 3:55 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

John wrote:

So, instead of being a mere groupoid, an orbifold is a kind of stack.

Urs wrote:

This is not a perspective that I like. Stacks may be a way to talk about smooth groupoids just as sheaves are a way to talk about diffeological spaces. But the orbifold is a groupoid much rather than it is a stack.

Point taken. My sentence was too terse to be very precise. I was trying to add some information to my first sentence — ‘an orbifold is more like a groupoid than a set’ — by pointing out that orbifolds have a smooth structure. But I was also trying to hint (without actually saying it out loud) that this smooth structure may need to be dealt with more subtly than by straightforward internalization, which would suggest using the 2-category of

[smooth groupoids, smooth functors, smooth natural transformations]

This is where the ‘stacky’ perspective starts sneaking in: in his abstract, Eugene Lerman says

… the collection of orbifolds and their maps need[s] to be thought of as a 2-category. This 2-category may be either taken to be the weak 2-category of groupoids, bibundles and equivariant maps between bibundles or the strict 2-category of geometric stacks represented by proper etale Lie groupoids.

So, my sentence was meant to be an abbreviated version of something like this:

So, instead of merely being a groupoid with a discrete space of objects and a discrete space of morphisms, an orbifold is a kind of smooth groupoid — but the correct notion of morphism between these is subtler than you might first think, and here it’s helpful to think of them as stacks.

I figured that if I wrote this, everyone would fall asleep before the period.

I’ve been very reluctant to master the technology of stacks, orbifolds, bibundles, Lie algebroids, Courant algebroids, NQ-manifolds, Hilsum–Skandalis maps, and other crud lying at the interface of differential geometry and higher category theory — in part because so much of this stuff has been set up to live in the category of manifolds, while I wanted to spend my life in a more convenient category of smooth spaces.

But now that I’ve got a more convenient category of smooth spaces, I feel psychologically ready to do something with it…

… and that requires mastering the usual stuff, along with generalizing it. So, I have a lot of catching up to do.

Finishing our paper on higher gauge theory will be one way to do some of that catching up.

But I also have other projects in mind.

I’m afraid some of this may be more like writing a book or wiki, trying to explain what other people have done in a way that I can understand, rather than ‘new research’.

Posted by: John Baez on July 21, 2008 6:22 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

If orbifolds are groupoids, you’d expect their Morse theory to involve ‘Morse functors’. Is that just what is going on in Richard Hepworth’s paper? I.e., is $\mathbb{R}$ being treated as a discrete groupoid?

Could you imagine another groupoid of values?

Google can find nothing for “Morse functor”.

Posted by: David Corfield on July 21, 2008 12:32 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

I’m not surprised nobody talks about ‘Morse functors’, because not many people seem to think of orbifolds as groupoids — not yet, anyway. They seem to think of orbifolds as manifold-like things where some points have neighborhoods that look, not like $\mathbb{R}^n$, but like a quotient of $\mathbb{R}^n$ by a finite group action. A space of orbits, in other words — hence the term ‘orbifold’.

The more modern viewpoint, which Eugene is pushing, amounts roughly to saying: “Don’t just take the quotient of $\mathbb{R}^n$ by a finite group action; take the weak quotient, and get a groupoid!” This viewpoint tends to force itself upon people, even the unwilling, as soon as they try to figure out the right notion of morphism between orbifolds.

(Let category theory stick its foot in the door, and before you know it, you’ll be talking about weak quotients and 2-categories!)

But anyway, David, you’re right: since an ordinary Morse function is a morphism between manifolds, we can expect the things Eugene is pondering to become important as soon as we attempt some sort of Morse theory for orbifolds.

But actually, I think we’ll see similar issues turning up whenever we try to do ‘equivariant Morse theory’ for manifolds equipped with a group action. So maybe you should search under ‘equivariant Morse theory’.

Posted by: John Baez on July 21, 2008 6:53 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

John wrote:

But anyway, David, you’re right: since an ordinary Morse function is a morphism between manifolds, we can expect the things Eugene is pondering to become important as soon as we attempt some sort of Morse theory for orbifolds.

I agree! In my paper on Morse theory for orbifolds I chose to use the framework of stacks in order to work out how to take the gradient flow of a Morse function. The reason for moving away from Lie groupoids is the following. Lie groupoids are to stacks just what atlases are to manifolds. If you tried to take the flow of a vector field on a manifold by working in your favourite atlas, you’d quickly find yourself pouring off the edge of a chart; you need to work in the manifold as a whole. For the same reason, if you want to take the flow of a vector field on a Lie groupoid then it helps to work on the underlying stack.

Of course, if you’d rather work with Lie groupoids then Eugene allows us to translate anything about stacks back into that language.

Posted by: Richard Hepworth on July 22, 2008 2:33 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

If orbifolds are groupoids, you’d expect their Morse theory to involve ‘Morse functors’. Is that just what is going on in Richard Hepworth’s paper? I.e., is ℝ being treated as a discrete groupoid?

Well, I work with stacks, and for me a “Morse function” is just a morphism into ℝ that satisfies an appropriate Morse condition.

Thanks to Eugene, we could translate this into the language of (proper, etale) Lie groupoids. Then a “Morse function” amounts to a Lie groupoid morphism (smooth functor if you like) into ℝ, regarded as a Lie groupoid which only has identity arrows.

The `objects’ part of this Lie groupoid morphism is just a real-valued function on a manifold, and the Morse condition then states that this should be a Morse function in the usual sense.

Regardless of whether you like this definition of “Morse function”, the resulting Morse theory works very well: a generic function is Morse and (once you work out how to take the gradient flow) you can prove the Morse inequalities just as if you were working with a manifold.

Posted by: Richard Hepworth on July 22, 2008 2:00 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

So, reformulating things this way with orbifolds forming a 2-category, is nothing to be gained by considering a category of Morse morphisms from an orbifold into the Lie groupoid $\mathbb{R}$, as the latter has only trivial morphisms?

Posted by: David Corfield on July 22, 2008 4:31 PM | Permalink | Reply to this

Re: Lerman on Orbifolds

That’s correct. Since ℝ has no nontrivial morphisms, the Morse functions form a set rather than a category.

Posted by: Richard Hepworth on July 23, 2008 9:35 AM | Permalink | Reply to this

Re: Lerman on Orbifolds

There is now an entry differentiable stack on the $n$Lab. Somebody should fill in some more references and the like.

Posted by: Urs Schreiber on December 3, 2008 7:48 PM | Permalink | Reply to this
Read the post Smooth Structures in Ottawa II
Weblog: The n-Category Café
Excerpt: A summary of some talks at the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics.
Tracked: May 9, 2009 9:06 PM

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