## January 31, 2006

### Moerdijk on Orbifolds, II

#### Posted by Urs Schreiber

[Update 1. Feb 2006: I have corrected a couple of typos, improved the discussion of Morita equivalence and added a crucial reference to a a review paper. ]

Here is a transcript of the first of the talks mentioned in the previous entry.

The talk pretty closely followed the review paper

I. Moerdijk
Orbifolds as Groupoids: an Introduction
math.DG/0203100 .

Suggested background reading is the book

I. Moerdijk & J. Mrčun
Introduction to Foliations and Lie Groupoids
Cambridge studies in advanced mathematics 91 (2003).

In that book orbifolds are first described (sections 1 and 2) in terms of local charts, i.e. as spaces locally diffeomorphic to ${ℝ}^{n}/G$ for some $G$ which may change from point to point. In section 5 Lie groupoids are introduced and the relation to orbifolds is briefly discussed in 5.6.

Suggested further reading is the latest paper by Lupercio and Uribe (whose work I mentioned recently)

Ernesto Lupercio, Bernardo Uribe, Miguel A. Xicotencatl
Orbifold String Topology
math.AT/0512658

as well as a an upcoming book

Orbifolds and String Topology
(to appear).

These deal with loop spaces of orbifolds. This will, hopefully, be the topic of tomorrow’s second lecture.

I’ll write down my notes taken in the course of the lecture. Most of this material will probably not be new to experts.

Models for Orbifolds

The original definition of an orbifold goes like this (see this for more technical details)

An orbifold is a space $X$ in which every point has a neighborhood of the form $U/G$, where $U\subset {ℝ}^{n}$ and $G$ is a finite group acting on $U$. Here $n\in ℕ$ is fixed, but $G$ may change from point to point.

Orbifolds in this description were introduced originally by Satake. While this proved to be a very useful concept, it has the following problems:

1) The description in terms of local charts does not provide any global picture.

2) It is not manifest which notion of morphism between such structures would be natural.

Here are some examples of orbifolds:

i) the global orbifold [my terminology]. Let

(1)$X=M/\Gamma$

globally, where $M$ is a manifold and $\Gamma$ a finite group acting by diffeomorphisms on $M$.

In the physics literature, this special case tends to be regarded as the definition of an orbifold.

ii) a first generalization of this example is obtained by considering the action of a Lie group $\Gamma$ acting properly on some manifold $M$, such that the subgroups

(2)${G}_{x}=\left\{\gamma \in \Gamma \mid \gamma x=x\right\}$

which fix a given point of $M$ (the stabilizer groups) are finite. In this case, $X=M/\Gamma$ is an orbifold for which the group $G$ appearing in the above definition in general changes from point to point.

iii) If $\left(M,F\right)$ is a foliated manifold with compact leaves and finite holonomy (see sections 1 and 2 of the above mentioned book for details), then the space of leaves $M/F$ has the structure of an orbifold. This is a consequence of a fact known as Reeb stability, which is explained in section 2.3 of the above mentioned book.

In a foliated space, one naturally has a notion of holonomy or, more generally, of transport as follows.

[ -digression-: Since this example will reappear further below, I’ll sketch some more details.

Given any two points $x$ and $y$ which are sitting on the same leave $L$ of the foliation, choose subspaces ${T}_{x}$ and ${T}_{y}$ which contain $x$ and $y$, respectively, and which are transversal to the leaf $L$. Now choose any path $\gamma$ from $x$ to $y$ which runs entirely inside of $L$. One can show that the homotopy class of such a path defines the germ (in $x$) of a diffeomorphism from ${T}_{x}$ to ${T}_{y}$. If the path is a loop, this defines a homomorphism from the fundamental group of $L$ to the group of germs (in $x$) of diffeomorphisms of ${T}_{x}$. - end of digression - ].

But, in order to study orbifolds as global objects together with maps between them (and possibly with extra stuff and structure, like for instance fiber bundles, over them) the most convenient language is that of Lie groupoids.

A groupoid is a category in which all morphisms are invertible. [As I have said, I am strictly transcribing that lecture… :-)]. A category is [Ok, I’ll make an exception here… ]

A Lie groupoid is a groupoid in the category of smooth spaces, where the space of objects and the space of morphisms are both smooth and source and target maps as well as the composition map are smooth maps.

Example: Consider again a foliated space $\left(M,F\right)$. The holonomy groupoid of a foliation $\left(M,F\right)$ is the groupoid whose space of objects is $M$ and whose morphisms are classes of paths $\gamma$ in leaves $L$ of the foliation, which have the same ‘holonomy’ (really: ‘transport’) in the sense of the above digression. Some work is involved in proving that this groupoid is in fact smooth.

The main message is now that

Orbifolds should be viewed as certain types of Lie groupoids.

Some definitions:

1) Given a Lie groupoid $G$, call $\mid G\mid$ the set of classes of objects of $G$ which are connected by any arrow. $\mid G\mid$ is the set of isomorphism classes of $G$.

2) Given any object $x$ in the groupoid $G$, Call the group ${G}_{x}=\left\{x\stackrel{g}{\to }x\right\}$ the isotropy group of $x$. [Elsewhere this would be called possibly the vertex group at $x$, but ‘isotropy’ is the term that suggests itself once we think of the groupoid as an orbifold.]

3) The groupoid $G$ is called étale if its source and target maps are local diffeomorphisms.

4) The groupoid $G$ is called proper if the map

(3)$\left(s,t\right):\mathrm{Mor}\left(G\right)\to \mathrm{Obj}\left(G\right)×\mathrm{Obj}\left(G\right)$

is a proper map (i.e. closed map with compact fibers).

5) The groupoid $G$ is called a foliation groupoid if all isotropy groups ${G}_{x}$ are discrete.

Now, in terms of Lie groupoids and using the above terminology, one can give the following alternative definition of an orbifold.

Definition. An orbifold structure on a topological space $X$ is given by a proper étale groupoid $G$ together with a homeomorphism

(4)$\mid G\mid \stackrel{\pi }{\to }X\phantom{\rule{thinmathspace}{0ex}}.$

The nice thing is that, once we know what some entity is in terms of abstract nonsense, we immediately know what the right notion of map between two such entites is: a morphism of groupoids is simply a functor. But since we do not just want morphisms between groupoids, but morphisms between smooth groupoids, there is an extra (though familar and well understood) subtlety.

Usually, we would regard a functor ${G}_{1}\to {G}_{2}$ as ‘invertible’ if there is another functor ${G}_{2}\to {G}_{1}$ such that their compositons are naturally isomorphic to the identity. However [if I understood correctly] the problem is that the weak inverse functor, while it may exist at the level of sets, will not in general be smooth. So we use the fact that, at the level of sets, an equivalence is the same as an essentially surjective fully faithful functor and impose a smoothness condition on this:

Definition. A functor between Lie groupoids is called an equivalence if it is a fully faithful and essentially surjective submersion.

Such a functor ${G}_{1}\stackrel{\varphi }{\to }{G}_{2}$ induces a homeomorphism

(5)$\mid {G}_{1}\mid \stackrel{\mid \varphi \mid }{\to }\mid {G}_{2}\mid$

on the spaces of isomorphism classes of these groupoids (which, recall, ‘are’ the orbifolds represented by these groupoids).

We regard

(6)$\pi \circ \mid \varphi \mid :\mid {G}_{1}\mid \stackrel{\mid \varphi \mid }{\to }\mid {G}_{2}\mid \stackrel{\pi }{\to }X$

as defining the same orbifold structure on $X$.

But this notion of equivalence needs to be weakened a little in order to correctly capture the notion of ‘equivalent orbifolds’.

Definition. Two Lie groupoids ${G}_{1}$ and ${G}_{2}$ are Morita equivalent if they are source and target of a span of ordinary equivalences. This means that there is a third Lie groupoid $H$ together with two equivalences from $H$ to ${G}_{1}$ and ${G}_{2}$:

(7)${G}_{1}\stackrel{{\varphi }_{1}}{←}H\stackrel{{\varphi }_{2}}{\to }{G}_{2}\phantom{\rule{thinmathspace}{0ex}}.$

Note: many properties are invariant under Morita equivalence, e.g. that $G$ is proper.

One can establish a dictionary between orbifolds as conceived originally and the corresponding groupoid. For instance

Fact. A groupoid $G$ is a foliation groupoid iff it is Morita equivalent to an étale groupoid.

Many invariants on orbifolds can be studied through propeties of groupoids which are invariant under Morita equivalence.

Example. Question: What would be a covering space of an orbifold?

Answer: A covering space of the orbifold represented by the groupoid $G$ is a covering space $S\to \mathrm{Obj}\left(G\right)$ equipped with a $G$-action $s\to \mathrm{gs}$ covering every morphism $x\stackrel{g}{\to }y$ in $G$.

The category of such covering spaces, $\mathrm{Cov}\left(G\right)$, is equivalent to the category ${\pi }_{1}\left(G,x\right)-\mathrm{Sets}$ for a unique group ${\pi }_{1}\left(G,x\right)$. This group is known as the (Thurston) orbifold fundamental group.

(This fact is supposed to be good news if you already knew what Thurston’s definition was like using non-groupoid language.)

Example. Question: What is a vector bundle over an orbifold? Answer: This is an ordinary vector bundle $E$ over $\mathrm{Obj}\left(G\right)$ together with an isomorphism ${E}_{x}\stackrel{g\left(\cdot \right)}{\to }{E}_{y}$ of fibers covering every morphism $x\stackrel{g}{\to }y$ in $G$.

[So in other words, a vector bundle over an orbifold is a functor from the representing groupoid into vector spaces: $E:G\to \mathrm{Vect}$.]

The category $\mathrm{Vect}\left(G\right)$ of such vector bundles only depends on $G$ up to Morita equivalence. Decategorifying and forming the Grothendieck group, we arrive at $K\left(X\right)$, the orbifold $K$-theory of the orbifold $X$.

Finally, this $K$-theory can be related to the inertia groupoid $\Lambda \left(G\right)$ of $X$.

This groupoid has objects being automorphisms in the groupoid $G$ representing $X$ (i.e. elements of isotropy groups) and morphisms being morphisms in $G$ which relate to such isotropy group elements by conjugation.

One can apparently show that the orbifold K-theory is given by the cohomology of this inertia groupoid:

(8)$K\left(X\right)\simeq {H}^{*}\left(\Lambda \left(G\right),ℂ\right)\phantom{\rule{thinmathspace}{0ex}}.$

[But I realize that I don’t understand this yet. More tomorrow.]

Posted at January 31, 2006 6:50 PM UTC

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### Re: Moerdijk on Orbifolds, II

In non-commutative geometry, you orbifold spaces, sorry algebras, by group actions using crossed products. For those, there is a natural notion of Morita equivalence. The obvious guess is they are related. Can you confirm this? There, you can also show what CFT people have been knowing for long times that you get back to the original space if you orbifold “twice” or at least to a Morita equivalent algebra which represents the same space.

Finally, there is a well studied orbifold which is ${S}^{1}/\tau Z$ for some real number $\tau$. This happens to be the non-commutative torus. Is this an example of what Moerdijk discusses?

Posted by: Robert on January 31, 2006 10:32 PM | Permalink | Reply to this

### Re: Moerdijk on Orbifolds, II

Hi Robert,

I don’t know in detail how crossed products would give rise to groupoids and what that would mean for Morita equivalence. I bet its related. But I’ll try to ask Prof. Moerdijk today.

there is a well studied orbifold which is ${S}^{1}/\tau ℤ$ for some real number $\tau$.

Could you provide further details on that? Is this really supposed to be an ${S}^{1}$ (circle) here?

If you reply before 11:00 today (that’s in half an hour I guess) I’ll have a chance to mention this after the talk.

Posted by: urs on February 1, 2006 9:34 AM | Permalink | Reply to this

### Morita Equivalence and Involution Algebras

[Update: Apparently, something related to the statement below is discussed (proved) in

Crainic & Moerdijk, Foliation groupoids and their cyclic homology, Adv. in Math. 157 (2001), 177-197 . ]

As Robert points out, an alternative way to describe orbifolds is to look at them in terms of their associated ${C}^{*}$-algebras. As there is a well-known notion of Morita equivalence for algebras, one would want to know if that is compatible with the (Morita) equivalence of the groupoids that these algebras come from.

The ${C}^{*}$-algebra associated to any groupoid is essentialy nothing but its category algebra. This is the algebra generated (over some field) by the morphisms in the category, with the product of the generators being their composition if composable and zero otherwise.

There is a different way to formulate this (taking into account the smooth structure on the category that we are dealing with here), which is, if I recall correctly, the way it appears somewhere near the beginning of Connes’ red book on noncommutative geometry.

Given a Lie groupoid $G$, let

(1)${C}_{c}^{\infty }\left(G\right)=\left\{\varphi :\mathrm{Mor}\left(G\right)\to ℂ\right\}$

be the space of smooth functions with compact support on the morphism space of $G$ (with values in the complex numbers, say) and define the product on this space by

(2)$\begin{array}{ccccc}\left(\varphi \cdot \psi \right)& :& \mathrm{Mor}\left(G\right)& \to & ℂ\\ & & g& ↦& \sum _{\left\{k,h\mid k\circ h=g\right\}}\varphi \left(h\right)\overline{\psi }\left(k\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

This is the ${C}^{*}$ algebra encoding the general orbifold associated to the groupoid $G$.

The case which Robert mentioned is the special case of a global orbifold. For a global orbifold $M/\Gamma$, the associated groupoid is just the action groupoid. One can check that in this case the algebra defined above reduces to a crossed product algebra.

Now, the statement is the following:

If

(3)$H\stackrel{\alpha }{\to }G$

is an equivalence of Lie groupoids (an essentially surjective fully faithful submersion) then the corresponding ${C}^{*}$-algebras are Morita equivalent

(4)$...⇒\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{C}_{c}^{\infty }\left(G\right)\stackrel{\mathrm{Morita}}{\simeq }{C}_{c}^{\infty }\left(H\right)\phantom{\rule{thinmathspace}{0ex}}.$

Posted by: urs on February 1, 2006 1:25 PM | Permalink | Reply to this

### Re: Morita Equivalence and Involution Algebras

Ieke Moerdijk and Izak Moerdijk are the same person? I was assuming, from the signature I. Moerdijk, that he was the same mathematician heading two different approaches to foliations: On one side via Sheaf theory (see his book with MacLane) and on other side, later, via the C* algebra of a groupoid. Let me discrepate here with Robert: Connes his book seems to me a adequate place to start learning about action grupoids, most of chapter II is about it, only that it concentrates on getting index theorems (and all that).

Posted by: Alejandro Rivero on October 17, 2006 6:05 PM | Permalink | Reply to this
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