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I am moving the following old query box exchange from orbifold to here.
old query box discussion:
I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely does not view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.
Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?
Urs Schreiber: please, go ahead. It would be appreciated.
end of old query box discussion
I have touched the entry orbifold once more, polishing a bit:
moved some statements from Idea to Properties
reorganized and expanded Idea a bit to read more smoothly;
added references and pointers to references.
(I am still far from happy with the state of this entry. But if I continue to make a tiny edit every six months of so, then maybe in a few years it will be good.)
(I am still far from happy with the state of this entry. But if I continue to make a tiny edit every six months of so, then maybe in a few years it will be good.)
Maybe we can list what is unsatisfactory in this entry.
In the idea-section there is a statement (”There is also a notion of finite stabilizers in algebraic geometry. A singular variety is called an (algebraic) orbifold if it has only so-called orbifold singularities. ”) without reference.
Maybe one could emhasize that to give some reasonable 2-catgory (Lerman mentions that in no case there is a reasonable 1-category of them) of orbifolds requires to take the spans called morita morphisms as morphisms; the same is true for Lie groupoids in general (Hilsum-Skandalis). This somehow seems to be a genuine nlab topic!
Moreover I dont’t know what makes orbifolds categorially interesting since the presentation of orbifolds by proper étale smooth groupoids divides the focus into two different subjects of study and I didn’t yet notice what precisely distinguishes stacks presented by this combination of properties from others ”out there”.
Hi Stephan,
right I should have been more explicit. Here some things that deserve to be improved:
As you notice, the discussion of the relation to algebraic and other kinds of “geometric stacks” needs a section all of its own with a systematic discussion.
The issue of what kind of category or 2-category orbifolds form and the emphasis on Morita morphisms is maybe of historical interest, but should not dominate the conceptual discussion. It is clear that the natural thing to consider is the full sub-(2,1)-category on objects equivalent to orbifolds in the (2,1)-topos $Sh_2(SmthMfd)$ or equivalently in $Sh_\infty(SmthMfd)$ Period. Everything else is a corollary of this. The fact that morphisms in $Sh_2(SmthMfd)$ can be presented by “Morita morphisms” of Lie gorupoids should be (and partly is, I think) dealt with in the entry Lie groupoid and need not be re-iterated here. It only makes a simple concept sound involved and awkward.
The paragraph on orbifold cohomology has no links to anything and needs more context. This should eventually be systematically embedded into the general discussion of cohomology, homotopy, etc. of smooth infinity-groupoids, as discussed there. Somehow the entry defeats its own assertions if it starts out saying “an orbifold really is just” only to then behave as if one needs lots of special notions for orbifolds. No, the entry should explain how all these special notions considered in the literature are but special case of general simple concepts for smooth higher groupoids.
3: Stephan, I do not know the reference, but it is often heard and used. Why would be quoting infinity stacks without a reference be more unsatisfactory than to quote a striking geometric fact that knowing the singularities determines the orbifold structure in algebraic setup. Of course here only the effective orbifolds are meant, what used to be the classical generality; it is definitely not true for noneffective. I am not competent to defend the precise statement, and know it as a folklore.
I do not understand the unanonymous complaints listed in 1. Yes, one should look at 2-category of stacks. While the precise conditions in the case of differentiable version are clarified by Moerdijk, in algebraic situation the stack version is of prevalent usage and “everybody knows” there that geometrically stacks are usually presented by groupoids; but stack is more fundamental point of view than groupoid presentation (e.g. stack does not need further equivalence relation).
I afree with Urs 4.
I do not know the reference, but it is often heard and used. Why would be quoting infinity stacks without a reference be more unsatisfactory than to quote a striking geometric fact that knowing the singularities determines the orbifold structure in algebraic setup.
I would tend to require references in particular for statements or terminology which is not in the primary scope of the nlab. It is quite easy to find out within the nlab what an infinity-stack is but the word ”orbifold singularity” is not explained here.
Of course here only the effective orbifolds are meant, what used to be the classical generality; it is definitely not true for noneffective.
I guess this is no longer the standard. For example Eugene Lerman in ”orbifolds as stacks?” requires just étale- and properness. He remarks to Satake’s definition(s):
”The group actions were required to be effective (and there was a spurious condition on the codimen- sion of the set of singular points). The requirement of effectiveness created a host of problems: there were problems in the definition of suborbifolds and of vector (orbi-)bundles over the orbifolds. A quotient of a manifold by a proper locally free action of a Lie group was not necessarily an orbifold by this definition.”
I added a definition section at orbifold saying just that it is a stack presented by an orbifold groupoid such that the reader can customize the definition to his individual preferences.
Note that people don’t only study smooth orbifolds, but algebraic and topological. It is the properness and etale-ness of the groupoids that is important, as these have (sometimes exact) analogues in categories other than $Manifolds$.
(Also we need to distinguish between effective and non-effective orbifolds. I’m going to bed now else I’d do it (-: )
Note that people don’t only study smooth orbifolds, but algebraic and topological. It is the properness and etale-ness of the groupoids that is important, as these have (sometimes exact) analogues in categories other than $Manifolds$.
I gave pointers in this direction in the idea-section of orbifold groupoid.
I would tend to require references in particular for statements or terminology which is not in the primary scope of the nlab.
Orbifold is a notion quite close to the center of the scope of nLab. It is among the MAIN examples of stacks, it is used widely in mathematical physics and geometry, both of which drive many of us here, especially e.g. Urs, John, David Roberts and me. Finally, if some thing is NEW in the nLab, it is more logical to expect that it starts with lower criteria, and that the higher criteria of polishness be achieved at a LATER stage. Also if on some entry, notion or theorem MANY people work it will be easier to them to make it satisfactory than for a single person, so it is again against common sense to require more from more isolated efforts. Finally, your level of liking of some math or theorem should not discriminate on other’s contributor’s feeling on how to contribute.
Finally I agree that infinity stacks are easier to find, but it is harder to use and understand for most of nLab users, including some of the regular contributors. It is a language which is theoretically both powerful and demanding. There are hundreds of mathematicians who silently use nLab, and most of them do not speak infinity stacks, while speak more classical notions.
David: there are also noncommutative orbifolds, especially in mathematical physics literature.
I added a reference to Adem-Leida-Ruan 2007 to orbifold for the statement that every orbifold is the quotient of an effective almost free action of a compact Lie group on a manifold.
am adding references:
Survey of basic orbifold theory:
Adam Kaye, Two-Dimensional Orbifolds, 2007 (pdf)
Joan Porti, An introduction to orbifolds, 2009 (pdf)
Daryl Cooper, Craig Hodgson, Steve Kerckhoff, Three-dimensional Orbifolds and Cone-Manifolds (pdf)
On orbifolds with Riemannian metric and as singular limits of Riemannian manifolds (such as metric cones):
Christian Lange, Orbifolds from a metric viewpoint (arXiv:1801.03472)
Renato G. Bettiol, Andrzej Derdzinski, Paolo Piccione, Teichmüller theory and collapse of flat manifolds, Annali di Matematica (2018) 197: 1247 (arXiv:1705.08431, doi:10.1007/s10231-017-0723-7)
added pointer to
Has anyone see an account that would discuss curved Riemannian manifolds as limits of flat orbifolds in a limit where the “density of orbifold singularities” grows large?
So there are lots of accounts that consider singular limits of Riemannian manifolds as orbifolds, with one or a handful singularities where the manifolds degenerate. But here I a wondering about a different kind of limit:
Something that looks like a curved Riemannian manifold on large scales, but as one zooms in one sees that the curvature is not evenly spread out, but carried by a “gas of curvature singularities”. If you see what I mean.
Has anyone looked into this?
How do you mean? Take a 2-sphere. Would a case be ever finer triangulations? Vertices there would provide the overall curvature.
Yes, that’s exactly the kind of example that I have in mind: Like triangulations, but such that the result is a flat orbifold, with each vertex an orbifold singularity.
It seems non-trivial to me to see that any given triangulation admits the structure of a flat orbifold, but maybe I am missing something.
In computer graphics they talk about “Orbifold Tutte embeddings” (Aigerman-Lipman 15 web) and the graphics they show as well as their use of the word “orbifold” suggests that they do realize fairly arbitrary triangulations of surfaces as (flat?) orbifolds. But I am not sure if they really mean orbifolds in the standard mathematical sense. Even if, they don’t seem to be after compact flat orbifolds, but after orbifold structures on bounded domains in the plane.
But their pictures at least express the idea that I am asking for: Can we speak of a version of Riemannian geometry which looks like ordinary smooth Riemannian geometry to arbitrary accuracy on large enough scales, but which on small scales is really given by flat orbifolds.
Have to be doing something else, but when Milnor speaks of classifying “all possible flat orbifold structures on the Riemann sphere” on p. 11 here, concluding that there are four types, doesn’t the mean the answer to your question is ’No’?
I guess here you are thinking of the classification of compact flat 2-dimensional orbifolds here. Right, there is only few of them (four) and they have only a handful of singularities. So in 2d and for compact orbifolds there is not much going on. But relaxing these conditions, there should be more. I’d be interested in dimension d=10 and not necessarily a compactness condition.
In any case, I suppose my question is now sufficiently clear. If anyone sees any literature vaguely in this direction, please drop me a note.
I am still trying to figure out the extent to which the following statement is true, and the extent to which this is known in the literature:
Statement: For $n \in \mathbb{N}$ there exists an $n$-dimensional flat orbifold whose underlying topological space is the $n$-sphere.
(For $n=2$ this is classical: There is the pillowcase and three other orbifolds of this kind (here)).
One way to show existence should be to exhibit the $n$-torus as a branched cover over the $n$-sphere, such that the action of the group of deck transformations is smooth. In this case the global homotopy quotient of the $n$-torus by this group should be the desired flat spherical orbifold.
Moreover, for the action of the group of deck transformations of the branched cover to be smooth, it should be sufficient that the singular locus of the branched cover is itself smooth. (Right?)
I was hoping to extract just this statement, for $n = 4$, from
but I am still unsure about the intended precise definitions of some of the technical assumptions there. However, p. 30 of
asserts exactly that this is what Iori-Piergallini prove. So it looks like it should be right.
[edit:
ah, footnote 5 in arXiv:1411.0977 has more on the issue of parsing the result of Iori-Piergallini:
This theorem is stated in the PL category but, as confirmed to us by R. Piergallini, it holds in the smooth category due to general results PL=Smooth in 4-dimensions.
]
made this an MO question, here
Just trivia:
I am looking for more nice graphics illustrating orbifolds, preferably Euclidean orbifolds.
Besides the hand-drawn graphics currently in the page, most of what I keep seeing people have is either simplistic examples of cones (deficit angles) or some variant of a computer rendering of the Kummer orbifold.
Is there more/better graphics available? If anyone further pointers, please drop me a note.
Something from Thurston or his school? I haven’t looked, but I know he was into visualisation in a big way.
@Urs you might have better luck in algebraic geometry literature. Especially things on toric varieties. I remember seeing lots of graphics like that.
I have added graphics of the Kummer orbifold from Snowden11 here.
But to be frank, I am a little unsure how to read the graphics. What’s those boundary-like-looking circles? Is a point on the circle a stand-in for an $S^2$?
(Maybe I should in fact remove the graphics until that’s clarified.)
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