The n-Category Café

I’m pretty sure that the 2d orbifold we get from a specific wallpaper pattern has a Riemannian metric coming from the usual distance function on the plane.

So they get to be called Euclidean 2-orbifolds.

Hmm, nice looking notes. Looks like you need to put Morse theory into the mix:

We sincerely believe that critical-net and critical-net-on-orbifold crystallographic illustrations will some day be of fundamental importance in crystal chemistry and crystallographic topology. Critical nets are based on the concepts of Morse functions, Morse theory (i.e., critical point analysis), and hyperplane arrangements in topological complexes, which are classic topics in the mathematical topology and global analysis literature. Morse functions on orbifolds constitute a relatively new aspect of equivariant (i.e., group orbit compatible) topology.

I only know two places where the number 17 played an important role in mathematics – the other is much more famous.

The Feller number?

At the orbifold shop, each handle costs 2 collars.

I know geometric topologists talk about collars, but presumably this should be dollars.

Yesterday the Proofs, Programs and Systems group had a little microconference:

Bonjour,

Nous profitons de la visite de John Baez ànotre équipe PPS pour organiser àChevaleret une journée “n-catégories, n-groupo¯des et logique”.

La journée se tiendra jeudi 24 juillet, en salle 6C01.

Le programme de la journée est le suivant:

• 10h – Mark Weber : A tutorial on monads with arities
• 11h – Nicolas Tabareau : Computing free models as Kan extensions
• 14h – John Baez : Groupoidification
• 15h – Dimitri Ara : Weak infinity-categories : Grothendieck versus Batanin
• 16h30 – Samuel Mimram : A tutorial on polygraphs and their applications to semantics
• 17h30 – Jonas Frey : a 2-dimensional adjunction between triposes and toposes

En espérant vous voir nombreux jeudi,

Paul-André Melliès
Mehrnoosh Sadrzadeh

Jamie Vicary came all the way from London to attend, getting up at a frightfully early hour to catch the train to Paris.

It was a lot of fun! Dimitri Ara’s talk was of special historical interest, because his advisor Maltsionitis has pored through Grothendieck’s Pursuing Stacks, extracted a definition of weak $\infty$-groupoids, generalized it to weak $\infty$-categories… and now Ara has compared it with Batanin’s definition, finding it to be equivalent, but phrased in a wholly different language: the language of sketches!

But, I admit that at moments when my attention flagged, I started working out the 2d orbifolds coming from various wallpaper groups. This made me realize how fun the subject actually is… so today I put a bunch more material about this in This Week’s Finds. I’ll post it as a comment below, for those who’ve already read the earlier version.

By the way, my request to compute the symmetry 2-groups of some of the resulting Euclidean 2-orbifolds was mainly directed at David Corfield (since it ties into our Klein 2-Geometry thread) and Eugene Lerman (since it should be easy for him). Maybe some partially worked-out examples will help… though I didn’t do the most interesting examples. In fact, I’m not sure I see any interesting 2-groups! I’m a bit confused, but I only see the 1-groups.

Here’s some stuff I added to week267, which should make 2d orbifolds easier to understand:

Take a wallpaper pattern and count two points as “the same” if they’re related by a symmetry. In other words — in math jargon — take the plane and mod out by the wallpaper group. The result is a 2-dimensional “orbifold”.

In a 2d manifold, every point has a little neighborhood that looks like the plane. In a 2d orbifold, every point has a little neighborhood that looks either like the plane, or the plane mod a finite group of rotations and/or reflections.

Let’s see how this works for a few simple wallpaper groups.

I’ll start with the most boring wallpaper group in the world, p1. If you thought pg was dull, wait until you see p1. It’s the symmetry group of this wallpaper pattern:

   R     R     R     R     R     R    
  RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
   R     R     R     R     R     R    
   R     R     R     R     R     R    
   R     R     R     R     R     R    
  RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
   R     R     R     R     R     R    
   R     R     R     R     R     R    
   R     R     R     R     R     R    
  RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
   R     R     R     R     R     R    

This group doesn’t contain any rotations, reflections or glide reflections - I used the letter R to rule those out. It only contains translations in two directions, the bare minimum allowed by the definition of a wallpaper group.

If we take the plane and mod out by this group, all the points labelled x get counted as “the same”:

   R     R     R     R     R     R    
  RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
   R     R     R     R     R     R    
   R   x R   x R   x R   x R   x R    
   R     R     R     R     R     R    
  RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
   R     R     R     R     R     R    
   R   x R   x R   x R   x R   x R    
   R     R     R     R     R     R    
  RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
   R     R     R     R     R     R    

Similarly, all these points labelled y get counted as “the same”:

   R     R     R     R     R     R    
  RRRRRyRRRRRyRRRRRyRRRRRyRRRRRyRRR
   R     R     R     R     R     R    
   R     R     R     R     R     R    
   R     R     R     R     R     R    
  RRRRRyRRRRRyRRRRRyRRRRRyRRRRRyRRR
   R     R     R     R     R     R    
   R     R     R     R     R     R    
   R     R     R     R     R     R    
  RRRRRyRRRRRyRRRRRyRRRRRyRRRRRyRRR
   R     R     R     R     R     R    

So, when we take the plane and mod out by the group p1, we get a rectangle with its right and left edges glued together, and with its top and bottom edges glued together. This is just a torus. A torus is a 2d manifold, which is a specially dull case of a 2d orbifold.

Now let’s do a slightly more interesting example:

   T     T     T     T     T     T     
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T     T     T     T     T     T    
   T     T     T     T     T     T    
   T     T     T     T     T     T    
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T     T     T     T     T     T    
   T     T     T     T     T     T    
   T     T     T     T     T     T    
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T     T     T     T     T     T    

The letter T is more symmetrical than the letter R: you can reflect it, and it still looks the same. (If you’re viewing this using some font where the letter T doesn’t have this symmetry, switch fonts!) So, the symmetry group of this wallpaper pattern, called pm, is bigger than p1: it also contains reflections and glide reflections along a bunch of parallel lines. So now, all these points labelled x get counted as the same when we mod out:

   T     T     T     T     T     T     
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T     T     T     T     T     T    
   T x x T x x T x x T x x T x x T    
   T     T     T     T     T     T    
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T     T     T     T     T     T    
   T x x T x x T x x T x x T x x T    
   T     T     T     T     T     T    
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T     T     T     T     T     T    

and similarly for all these points labelled y:

   T     T     T     T     T     T     
  TTTyTyTTTyTyTTTyTyTTTyTyTTTyTyTTT
   T     T     T     T     T     T    
   T     T     T     T     T     T    
   T     T     T     T     T     T    
  TTTyTyTTTyTyTTTyTyTTTyTyTTTyTyTTT
   T     T     T     T     T     T    
   T     T     T     T     T     T    
   T     T     T     T     T     T    
  TTTyTyTTTyTyTTTyTyTTTyTyTTTyTyTTT
   T     T     T     T     T     T    

but look how these points labelled z work:

   T     T     T     T     T     T     
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T  z  T  z  T  z  T  z  T  z  T    
   T     T     T     T     T     T    
   T     T     T     T     T     T    
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T  z  T  z  T  z  T  z  T  z  T    
   T     T     T     T     T     T    
   T     T     T     T     T     T    
  TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
   T  z  T  z  T  z  T  z  T  z  T    

There are only half as many z’s per rectangle, since they lie on reflection lines.

Because of this subtlety, this time when we mod out we get an orbifold that’s not a manifold! It’s the torus of the previous example, but now folded in half. We can draw it as half of one of the rectangles above, with the top and bottom glued together, but not the sides:

   TTTT
   T  .
   T  .
   T  .
   TTTT

So, it’s a cylinder… but in a certain technical sense the points at the ends of this cylinder count as “half points”: they lie on reflection lines, so they’ve been “folded in half”.

This particular orbifold looks a lot like a 2d manifold “with boundary”. That’s a generalization of a 2d manifold where some points - the “boundary” points - have a neighborhood that looks like a half-plane. But 2d orbifolds can also have “cone points” and “mirror reflector” points.

What’s a cone point? It’s like the tip of a cone. Take a piece of paper, cut it like a pie into n equal wedges, take one wedge, and glue its edges together! This gives a cone - and the tip of this cone is a “cone point”. We say it has “order n”, because the angle around it is not 2π but 2π/n.

Here’s a more sophisticated way to say the same thing: take a regular n-gon and mod out by its rotational symmetries, which form a group with n elements. When we’re done, the point in the center is a cone point of order n.

We could also mod out by the rotation and reflection symmetries of the n-gon, which form a group with 2n elements. This is harder to visualize, but when we’re done, the point in the center is a “corner reflector of order 2n”.

To see one of these fancier possibilities, let’s look at the orbifold coming from a wallpaper pattern with even more symmetries:

   .     .     .     .     .     .    
  .................................
   .     .     .     .     .     .    
   .     .     .     .     .     .    
   .     .     .     .     .     .    
  .................................
   .     .     .     .     .     .    
   .     .     .     .     .     .    
   .     .     .     .     .     .    
  .................................
   .     .     .     .     .     .    

These are supposed to be rectangles, not squares. So, 90-degree rotations are not symmetries of this pattern. But, in addition to all the symmetries we had last time, now we have reflections about a bunch of horizontal lines. We get a wallpaper group called pmm.

What orbifold do we get now? It’s a torus folded in half twice! That sounds scary, but it’s not really. We can draw it as a quarter of one of the rectangles above:

   ....
   .  .
   ....

Now no points on the edges are glued together. So, it’s just a rectangle. The points on the edges are boundary points, and the corners are corner reflection points of order 4.

In a certain technical sense — soon to be explained — points on the edges of this rectangle count as “half points”, since they lie on a reflection line and have been folded in half. But the corners count as “1/4 points”, since they lie on two reflection lines, so they’ve been folded in half twice!

This is where it gets really cool. There’s a way to define an “Euler characteristic” for orbifolds that generalizes the usual formula for 2d manifolds. And, it can be a fraction!

The usual formula says to chop our 2d manifold into polygons and compute

$$V - E + F$$

That is: the number of vertices, minus the number of edges, plus the number of faces.

In a 2d orbifold, we use the same formula, but with some modifications. First, we require that every cone point or corner reflector be one of our vertices. Then:

• We count edges and vertices on the boundary for 1/2 the usual amount.
• We count a cone point of order n as 1/nth of a point.
• We count a corner reflector of order 2n as 1/(2n)th of a point.

The idea is that these features have been “folded over” by a certain amount, so they count for a fraction of what they otherwise would.

It turns out that if we calculate the Euler characteristic of a 2d orbifold coming from a wallpaper group, we always get zero. And, there are just 17 possibilities!

In fact wallpaper groups are secretly the same as 2d orbifolds with vanishing Euler characteristic! So, they’re not just mathematical curiosities: they’re almost as fundamental as 2d manifolds.

The torus and the cylinder, which we’ve already seen, are two examples. These are well-known to have Euler characteristic zero. Of course, we should be careful: now we’re dealing with the cylinder as an orbifold, so the points on the boundary count as “half points” — but its Euler characteristic still vanishes. A more interesting example is the square we get from the group pmm. Let’s chop it into vertices, edges, and one face, and figure out how much each of these count:

          1/2 
     1/4-------1/4
      |         | 
      |         | 
   1/2|    1    |1/2 
      |         | 
      |         | 
     1/4-------1/4
          1/2

So, the Euler characteristic of this orbifold is

$$(1/4 + 1/4 + 1/4 + 1/4) - (1/2 + 1/2 + 1/2 + 1/2) + 1 = 0$$

This is different than the usual Euler characteristic of a rectangle!

Hi!

I have been praying for years now that John Baez writes a book where all ( or at least some) of this marvelous stuff from his twf columns is tied together.

But I’m afraid this day won’t come.

Hmm, John and David’s discussion has reminded me of a result I once proved:

Suppose we are given a morphism $f: X\to Y$ of orbifolds and a 2-automorphism $\phi: f\Rightarrow f$. Then $\phi$ is trivial if and only if its restriction to any point of $X$ is trivial.

One consequence of this is the following:

Let $X$ be an effective orbifold. Then any self-equivalence of $X$ has no nontrivial automorphisms.

And in particular:

The isometry 2-group of an effective orbifold is equivalent to a group.

What does effective mean? It means that the automorphism group of any point in $X$ acts effectively on its tangent space, and consequently that almost all points of the orbifold have no inertia at all. This includes all of the orbifolds you are discussing. The fact that the isometry 2-groups are all equivalent to groups is apparent in the description I gave earlier: if we have a 2-arrow $g: (f,\phi)\Rightarrow(f,\phi)$ then $g f = f$, so that $g$ is the identity.

Of course, there are many interesting noneffective orbifolds. Some of them are called gerbes. (Gerbes with band a finite group.) Maybe you want to compute their isometry 2-groups? Here’s a fact for you:

The isometry 2-group of the nontrivial $\mathbb{Z}_2$-gerbe over $S^2$ is itself a $\mathbb{Z}_2$-gerbe over $O(3)$, the isometry group of $S^2$. In particular, it is not equivalent to any group.

The gerbe over $O(3)$, however, is trivial. But is there a trivialization that respects the 2-group structure?

In the main twf article, you say

If you let $A$ be all of $X$, you get the fundamental 2-group of $(X,x)$

which unfortunately should read more like “If you let $A = \{x\}$ …”. Compare to the situation of the relative fundamental thingie (=pointed set) $\pi_1(X,A,x)$ Here $x\in A$.

There’s been a lot of discussion of orbifolds here and in another recent post. I thought I’d take this opportunity to enthuse about the current state of the subject, particularly as a response to John’s comment

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.

I think that in fact the `more modern viewpoint’, where people think of orbifolds as groupoids or stacks, is far better served than John’s comment would have us believe.

To see the state of the art you could look at the programme of the Workshop on Topological and Differentiable Stacks that took place at the CRM in Barcelona earlier this year. But right now I really just want to list a few topics to whet your appetite.

1) String topology has been a very popular topic in topology in recent years; it is an example of an HCFT and also arises in various breeds of Floer theory. It has been extended to stacks by Behrend et al. in their paper String topology for stacks. Without the stacky viewpoint it would have been very difficult to describe the necessary loopspaces to develop the theory.

2) One fantastic example of an orbifold is the the moduli stack $\overline{M}_{g,n}$ of stable nodal curves of genus $g$ and with $n$ marked points. Ebert and Giansiracusa’s paper Pontrjagin-Thom maps and the homology of the moduli stack of stable curves develops Pontrjagin-Thom maps for stacks uses this to concoct large families of torsion classes in the homology of $\overline{M}_{g,n}$.

3) For me, the most interesting thing about orbifolds at the moment is the work on the crepant resolution conjecture. A crepant resolution $Y$ of an (algebraic) orbifold $X$ is a `smoothing’ of the orbifold that is somehow `minimal’. The crepant resolution conjecture says that we should be able to predict the topology of $Y$ from an appropriate notion of the `stacky topology’ of $X$. See Coates-Ruan’s Quantum Cohomology and Crepant Resolutions: A Conjecture. This is strongly connected to the McKay correspondence.

In: The discrete groups of Monstrous Moonshine,
Conway, McKay, and Sebbar:Proc. Amer. Math. Soc. 132, 2233–2240, (2004).

describe the 171 discrete groups fixing the MM functions.
It would be very useful to have the orbifold notation for
these groups.