Is there a good notion of morphism between orbifolds?

Question

Following Thurston, an orbifold is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable manifolds, the finite groups being part of the structure.

A fundamental example is the quotient of a manifold by a group acting properly discontinuously - but not necessarily freely. The notion of orbifold seems well-established among mathematicians, and I am wondering if there is also a well-established notion of morphism between these objects.

Is there a good notion of morphism between orbifolds?

A good notion should reasonably include the following maps:

• Coverings between orbifolds (in particular, from manifold to orbifolds),
• Fiberings between orbifolds of different dimension, such as a Seifert 3-manifold fibering over its base 2-orbifold,
• The (at least smooth) maps $[0,1] \to O$ that are used to define the orbifold fundamental group of $O$

Answer 1

I believe this is worked out very nicely in "Geometrization of Three-Dimensional Orbifolds via Ricci Flow" by Bruce Kleiner, John Lott (http://arxiv.org/abs/1101.3733).

An atlas for an $n$-orbifold $\mathcal O$ consists of a Hausdorff paracompact topological space $|\mathcal O|$ together with an open covering $\{U_\alpha\}$, local models $\{(\hat U_\alpha,G_\alpha)\}$ ($U_\alpha$ connected open subset of $\mathbb R^n$) and homeomorphisms $\varphi_\alpha:U_\alpha\to \hat U_\alpha/G_\alpha$ satisfying a compatibility condition. An orbifold is then defined by an equivalence class of such atlas. (See page 6 of Kleiner-Lott.)

A smooth map $f:\mathcal O_1\to\mathcal O_2$ between orbifolds is given by a continuous map $|f|:|\mathcal O_1|\to |\mathcal O_2|$ with the property that for each $p\in |\mathcal O_1|$, there are local models $(\hat U_i,G_i)$ ($i=1$, $2$) and a smooth map $\hat f:\hat U_1\to \hat U_2$ equivariant with respect to a homomorphism $\rho:G_1\to G_2$ such that $\pi_2\circ \hat f = |f|\circ \pi_1$ where $\pi_i:\hat U_i \to U_i$ is the projection ($\rho$ is not required to be injective or surjective). (See page 7 of Kleiner-Lott.)

I think this satisfies your requirements and is in the spirit of Thurston.

Edit: Perhaps I should mention Remark 2.8 in the Kleiner-Lott paper (also in regard to other answers to this post), which recalls that an orbifold can also be seen as a smooth proper étale grupoid (and Morita-equivalent grupoids correspond to equivalent orbifolds). A grupoid morphism gives rise to an orbifold map, but these correspond to a stricter class of maps called good maps. The advantage of these maps is that one can pull back orbi-vector bundles.

Answer 2

Let me make a few remarks.

First of all, with all my respect for Bill Thurston "an orbifold is a topological space which looks locally like a finite quotient of Rn by a finite group of O(n)" is a rather poor definition. It fails to address the (important!) class of non-effective orbifolds. For example, the moduli space of elliptic curves is a non-effective orbifold (because every elliptic curve has a $-1$ automorphism). To adress user76758's comment, I'll note that this is indeed a definition. In that approach, one takes an orbifold to be a topological space $X$ equipped with a maximal atlas $\{\psi_i\}$, where each $\psi_i$ is a homeomorphism between open subsets of $X$ and of $\mathbb R^n/G_i$ for some finite group $G_i$ that depends on $i$. The transition functions should be smooth, which means that they should lift (locally!) to smooth maps from (open subsets) of $\mathbb R^n$ to $\mathbb R^n$. This approach has been worked out in the paper Orbifolds as diffeologies. In that approach, orbifolds form a category, as explained in that paper.

One way to see that the above approach is poorly behaved is that your don't get the correct orbifold fundamental group if you follow your nose and write down the obvious definition using paths and homotopies. The orbifold $S^1/\mathbb Z_2$ (action given by $(x,y)\mapsto (x,-y)$) should have $D_\infty$ as its orbifold fundamental group, but as a diffeological space, its fundamental group is trivial (exercise!).

In a correct definition of orbifolds (namely one that is equivalent to smooth stacks -- see e.g. Topological and Smooth Stacks) it is important to realize that orbifolds form a 2-category, not a category! If you get something that looks like it's a category, you're doing something wrong.

Instead of trying to argue that orbifolds form a 2-category, I'll give an exercise. The goal of this exercise is to classify purely ineffective orbifolds whose coarse moduli space$^\dagger$ is $S^1$. Here, "purely ineffective" means that the isotropy group is everywhere the same (and non-trivial).

Exercise:
1) Make a guess about what the classification of purely ineffective orbifolds with coarse moduli space $S^1$ might looks like. Use the fact that these orbifolds are the mapping cylinders of morphisms from $[pt/\!\!/G]$ to $[pt/\!\!/G]$.

2) Consider the following orbifold with coarse moduli space $S^1$. It is given by $[S^1/\!\!/S_n]$ where the action of a permutation $\sigma\in S_n$ is by $(-1)^\sigma$. The isotropy is everywhere $A_n$, and so one gets a purely ineffective orbifold with coarse moduli space $S^1$ and isotropy group $A_n$.

3) Identify where that orbifold sits in the conjectural classification from part 1).

I used an example closely related to the above exercise in this previous post about orbifolds.

I'll finish by advertizing for my early paper on orbifolds. In it, I gave a definition of orbifolds that is equivalent to the one of smooth stacks but that is not too technical.

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$\dagger$ Other people would call that the "underlying space" of the orbifold, but that terminology is somewhat misleading because it makes you think that an orbifold consists of a space with extra structure, whereas most good definitions of orbifolds do not mention that space at all.

Answer 3

You might look at http://arxiv.org/pdf/math/0203100v1.pdf. Moerdijk defines morphisms of orbifolds on page 8. The key is to link the map on the underlying spaces, with a morphism of the corresponding orfold groupoids. (i leave you to check a few pages earlier for the definition of them.)

Notice that the orbifolds are also stacks, and there is an extensive literature on them! :-)

Answer 4

Concerning orbifolds there are a lot of misunderstandings. The original definition is due to Ishiro Satake in two papers:

[Sat1] On a generalization of the notion of manifold, Proceedings of the National Academy of Sciences 42 (1956), 359–363.

[Sat2] The Gauss-Bonnet theorem for V-manifolds, Journal of the Mathematical Society of Japan, Vol. 9., No. 4 (1957), 464–492.

After this papers Thurston decided to change the (not very sexy) name of V-Manifold for (the more sexy) Orbifolds, and he got some success. Fact is that the notion of orbifold is still, with Satake and Thurston, a space equipped with a smooth structure: Thurston didn't change original Satake definition, he just changed the name.

As spaces with a smooth structure, these orbifolds [are naturally integrated in the category of diffeological spaces][IKZ] and inherit that way all the differential environment.

Later, the concept of orbifold has changed, and has been associated with a groupoid defining in some sense the underlying orbifold structure. [It is like if you wanted to remind the structure of $S^3$ in the smooth structure of $S^2$ because of the Hopf fibration.] That is the direction taken by Haefliger, Moerdijk and his school. Of course then, the various notions of diffeomorphism, homotopy etc. diverge. The notion of orbifold changed (or refine) again then with the apparition of stacks (but here I'm not familiar enough to have an opinion).

But: diffeologically speaking, if you want to isolate the internal structure of the diffeological orbifold you can consider its structural groupoid (the germs of the automorphisms of an admissible generating family). Therefore, you can recover what people consider to be the homotopy of the orbifold as the isotropy groups of this structural groupoid. For example, the cone orbifold ${\cal Q}_m = {\bf C}/({\bf Z}/m{\bf Z})$ is clearly contractible, since the retraction $(t,z) \mapsto tz$ is ${\rm SO}(2)$-equivariant, therefore its homotopy is trivial but the structural groupoid has ${\bf Z}/m{\bf Z}$ as isotropy group at the origin and $\{{\bf Id}\}$ elsewhere, that is the information you were looking for. It is not contained in the homotopy group but in the structural groupoid.

Now, it's up to you to choose which direction fits more your needs.

------ Edit March 2017

Coming back to the heart of the question, the legitimacy of this question on morphisms between orbifolds comes from Satake's construction in [Sat2, p.469], where he said, and I cite:

The notion of $C^\infty$-map thus defined is inconvenient in the point that a composite of two $C^\infty$-maps defined in a different choice of defining families is not always a $C^\infty$ map.

That's because of the required property of equivariance of the local lifting of maps between orbifolds. This problem disapears in Diffeology. Indeed, there exist smooth maps between orbifolds (as diffeologies) that have no local equivariant liftings at all. This is shown in the example 25 of our paper on orbifolds [IKZ]. The function $f \colon \mathbf{C} \to \mathbf{C}$ defines by projectioon a well-defined smooth map between $\mathcal{Q} =\mathbf{C}/\mathbf{Z}_m$ to $\mathcal{Q}_n$ that has no equivarian local liftings on the neighborhood of $0$, any small you take the neighborhood:

$$ f(x,y) = \begin{cases} 0 & \text{ if } r > 1 \text{ or } r = 0 \\ e^{-1/r} \rho_n(r) (r,0) & \text{ if } \frac{1}{n+1} < r \leq \frac{1}{n} \text{ and $n$ is even } \\ e^{-1/r} \rho_n(r) (x,y) & \text{ if } \frac{1}{n+1} < r \leq \frac{1}{n} \text{ and $n$ is odd}, \end{cases} $$

Here $r = \sqrt{x^2+y^2}$, $z=x+iy \in \mathbf{C}$, and $\rho_n$ is a function vanishing flatly outside the interval $]1/(n+1),1/n[$ and not inside.

What is interesting is that, a contrario, a local diffeomorphism between orbifold has always a local equivariant lifting, in any local representation. This is the Lemma 21 of [YKZ].

So, maybe now, I made the point clearer, at least considering the diffeology point of view on orbifolds.

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[IKZ] Yael Karshon, Patrick Iglesias(-Zemmour), Moshe Zadka. Orbifolds as Diffeologies. Trans. Amer. Math. Soc. 362 (2010), 2811-2831 http://math.huji.ac.il/~piz/documents/OAD.pdf

[Sat1] Ishiro Satake. On a generalization of the notion of manifold, Proceedings of the National Academy of Sciences 42 (1956), 359–363.

[Sat2] Ishiro Satake. The Gauss-Bonnet theorem for V-manifolds, Journal of the Mathematical Society of Japan, Vol. 9., No. 4 (1957), 464–492.

Answer 5

Not sure if it's essentially* the same as the already given answer(s), but a useful definition of orbifolds and morphisms thereof (that works also in other categories, e.g. in the algebro-geometric one) is via stacks.

I think in this framework a (topological) orbifold would be a stack on the category of (not necessarily compact) topological manifolds that is locally isomorphic to a quotient stack of the form $[\mathbb{R}^n/G]$ with $G$ a finite group (perhaps embeddable in $O(n)$ for the same $n$, if you want). Note that the notion of sheaf and the analogous notion of "vector bundle" is very naturally available in this framework. There is also a theory of differentiable stacks and in particular differentiable orbifolds, where you can talk about tangent "bundles" and metrics.

${}^*$ Because a groupoid internal to a category of "spaces" is some kind of "presentation" for a stack on the same category, and Morita equivalence corresponds to the right notion of "isomorphism" (equivalence) of stacks.

Answer 6

In case you want to work with morphisms of orbifolds without ever mentioning 2-categories, you need to carefully avoid all morphisms between orbifolds that admit 2-automorphisms (in a 2-category, a 2-automorphism of a morphism is a 2-morphism from the morphism to itself).

First of all, non-effective oribifolds need to avoided, since maps into non-effective oribifolds typically have plenty of 2-automorphisms.

A convenient condition that ensures that a morphism has no 2-automorphisms is that it's $C^1$ and transverse to the singular locus of the target orbifold.
Warning: those morphisms do not form a category: their are not closed under composition.

Such morphisms turn out to be sufficient to define the fundamental group of an effective orbifold.

Here is how you can do it:
Pick a non-singular base point along with a tangent vector at that point and consider the set of paths that are smooth, transverse to the singular locus, and whose intial and final velocities are given by the given tangent vector. Homotopies are also required to be smooth and transverse to the singular locus. This yields the (correct) orbifold fundamental group.

That definition is somewhat inconvenient, for the following reasons:
- It is not visibly functorial: it is not clear that a map of orbifolds induces a map of fundamental groups.
- It relies in an essential way on differentiability, whereas the fundamental group should really be a topological notion.