The n-Category Café

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.

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.

But now for an admission:

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’.

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