The String Coffee Table

We started by answering a question by Robert Helling which I forwarded to I. Moerdijk. I have reported on that in the comment section here.

Next came a discussion of some open questions in the theory of groupoid description of orbifolds. Before entering that, however, I will reproduce the discussion of another exmaple for an orbifold.

Example. There is a popular example for a non-local orbifold known as the tear drop or droplet orbifold.

Topolocically the drop orbifold is the 2-dimensional sphere, but with one singular point. In the language of local patches this is obtained as follows.

Pick any symmetric group $S_n$ and let this act on the unit disk $D$ by permutation of the rays originating at the center of the disk. Hence $D/S_n$ looks like a cone. Glue this along its boundary to another disk $D$. This is the droplet.

Interestingly, the same orbifold can be expressed in terms of a global quotient by a Lie group. (Recall that a global orbifold is one of the form $M/\Gamma$ where $\Gamma$ is finite. However, for $G$ Lie the space $M/G$ is an orbifold that locally looks like $U/\Gamma_i$, where $\Gamma_i$ are the subgroups of $G$ that fix a given point in $M$.)

Namely, the tear drop is the global quotient $S^3/S^1$ of $S^3$ by $S^1$ with the following action of $S^1$ on $S^3$.

Identify

and

and define the action of $S^1$ on $S^3$ by

where the $n$ appearing here is the index of the symmetric group $S_n$ menioned above.

This example is important, because it illustrates what I. Moerdijk said is a major open problem in the theory of orbifolds in terms of groupoids:

Problem: “Is every proper étale groupoid $G$ (Morita) equivalent to one coming from a global quotient $M/G$, where $G$ is a compact Lie group acting smoothly and ‘almost freely’ (meaning that its stabilizer groups are all finite)?”

Conjecturing that the answer to this question is ‘yes’ is known as the Global Quotient Conjecture.

(Another open question which was briefly mentioned is if there is a model structure in which Morita morphisms are strictly invertible. That’s as in derived categories, but I will not try to give any more details on this issue.)

As for groups, one is interested in the classifying space $B G$ of any groupoid $G$.

Given any category $C$, we can consider the simplicial set whose

- 0-simplices are objects of $C$

- 1-simplices are morphisms of $C$

- 2 simplices are pairs of composable morphisms in $C$

- and so on.

This simplicial set is called the nerve $\mathrm{N}(C)$ of $C$. (This played a major role in previous entries, for instance here or here.)

Identifying each $p$-simplex in the nerve of $C$ with the standard $p$-simplex in $\mathbb{R}^n$ yields a topological space known as the geometric realization $|\mathrm{N}(C)|$ of the nerve of $C$.

We may regard any group as a groupoid with a single object. The familiar classifying spaces $B G$ for $G$ a group are nothing but the geometric realizations of the nerves of these groups.

The same formula holds, by definition, also for groupoids. So the classifying space of a groupoid is the geometric realization of its nerve.

Fine. Now we have the following

Theorem: If $H \overset{\phi}{\to} G$ is an equivalence of Lie groupoids, then the map between classifying spaces which it induces

is a weak homotopy equivalence.

For some purposes, one might want not to deal with $B G$ itself, but with something closely related, namely with the classifying space of the Čech-groupoid associated to the groupoid $G$.

I had mentioned this beast before in a somewhat naïive way that did not care about smoothness. The more elaborate defintion of a groupoid’s Čech groupoid works as follows.

Given any groupoid $G$, pick a good covering $\mathcal{U}$ of $\mathrm{Obj}(G)$ by open contractible subsets. Then define a new groupoid, called $\mathrm{Emb}_\mathcal{U}(G)$ as follows.

- The objects of $\mathrm{Emb}(G)$ are the open sets $U_i \in \mathcal{U}$.

- The morphisms of $\mathrm{Emb}(G)$ from $U_i$ to $U_j$ are smooth functions

from $U$ to morphisms in $G$ such that these morphisms go from objects in $U$ to objects in $V$. More precisely, we want

Using this definition, one can give another characterization of the isotropy groups of a groupoid.

Let $\mathcal{B}$ be a good covering of the orbifold $\mathrm{Obj}(G) \overset{\pi}{\to}|\mathrm{Obj}(G)| \simeq X$, such that every open set $B_i$ of the orbifold is the projection $B_i = \pi({U_i})$ of some open set on the object space of the representing groupoid. The $B_i$ are the objects of a category of open sets, with morphsims being inclusions $B_i \subset B_j$.

Now, a way to address the isotropy groups of $G$ is to look at the group of automorphisms $\mathrm{Aut}_{\mathrm{Emb}_\mathcal{U}(G)}(U,U)$ of open sets in the Čech groupoid corresponding to $\mathcal{U}$. More precisely, we have the following

Theorem. Let $B_i$ such that $B_i = \pi(U_i)$. Then the map

gives a pseudofunctor from the category of open sets in $\mathcal{B}$ to groups. (The 2-morphisms are given by inner automorphisms. See section 3.8 of the above mentioned paper for details.)

The last part of the talk was about some basic ideas concerning loop spaces of orbifolds. I think I’ll talk about that in a seperate entry.