## August 13, 2011

### Geometries: Diffeomorphism Classes vs Quilts

#### Posted by John Baez

What follows is a guest post by Greg Weeks. If your memory extends back before the formation of this blog to the glory days of sci.physics.research, you should remember Greg.

guest post by Greg Weeks

Our topic: general relativity, manifolds vs geometries, and a conceivably new-and-improved definition of geometries

The new definition is so simple and obvious that I suspect that it is flawed or of no practical value. But, so far, I find it conceptually pleasing.

Our starting point is a Riemannian manifold $(M, g)$. (We exclude additional fields for simplicity.) $M$ is a point-set. The atlas making $M$ into a manifold is implicit. And you know what $g$ is.

The established definition of the geometry of $(M, g)$ is the set of all results of applying an active diffeomorphism, i.e.:

$\{ (M, \varphi_*(g)) : \varphi is \; \mathrm{a} \; diffeomorphism \; of M \}$

($\varphi_*$ is the universal “push forward” operator on tensor fields, induced by the diffeomorphism $\varphi$.)

The established definition is algebraic and lacks a point-set. The following definition is analytical and has a derived point-set.

Let $x:U \to \mathbb{R}^n$ be a chart on $(M, g)$. Then the Riemannian manifold

$(x(U), x_* (g|U))$

is isomorphic to $(U, g|U)$.

I call $(x(U), x_*(g|U))$ a “patch”. A patch describes a piece of the geometry of $(M, g)$. To obtain the entire geometry, we simply need to sew overlapping patches together.

That’s all there is to it! The rest is just details.

But before the details, I should stress the following:

Patches (and the sewing described below) live entirely on $\mathbb{R}^n$. The geometry makes no reference to $M$ or $g$. There is no a priori point-set and no diffeomorphism gauge group. On the other hand, there will be a derived point-set.

Now for the details.

First, we derive patches from all of the charts — for the same reason that manifolds are defined to have maximal atlases.

Unfortunately, patches derived from different charts can be identical, so we must tag them to keep them unique. (Yucch!) The tag values are irrelevant, so long as they are different for each chart. If we denote the tag associated with the chart $x$ by $tag(x)$, then the derived patch is the triple $(x(U), x_*(g|U), tag(x))$.

Second, we must sew overlapping patches together. For any pair of charts $x:U \to \mathbb{R}^n$ and $y:V \to \mathbb{R}^n$ and their derived patches

$p_x = (x(U), x_*(g|U), tag(x))$

and

$p_y = (y(V), y_*(g|V), tag(y)),$

we define the (possibly empty) overlap bijection

$B(p_x, p_y): x(U \cap V) ➝ y(U \cap V)$

by $B(p_x, p_y)(c) = y(x^{-1}(c))$.

We’re done. $B$ handles all of our sewing. The geometry is therefore $(P, B)$, where P is the set of all patches.

The geometry of $(M,\varphi_*(g))$ had better be the same as the above! And it is. The patches and overlap bijection derived from $x \circ \varphi^{-1}:φ(U)\to \mathbb{R}^n$ and $y \circ \varphi^{-1}:\varphi(V)\to \mathbb{R}^n$ on $(M, \varphi_*(g))$ are the same as those derived from $x:U\to\mathbb{R}^n$ and $y:V\to \mathbb{R}^n$ on $(M, g)$ — provided that you kindly choose the same tags.

Here’s an attempt to define geometries without any reference to manifolds.

A geometry is a pair $(P,B)$ that satisfies #1-5 below. Please keep in mind that #1-5 are requirements, not implications.

1. P is a set of patches. A patch is a triplet: an open subset of $\mathbb{R}^n$, a (smooth) Riemannian metric on the open set, and a unique tag. Given a patch $p$, we will use the notation $p.S$, $p.g$, and $p.tag$ for the three components.

2. $B$ is a function-valued function on $(P \times P)$. For every pair of patches $p_1$ and $p_2$, $B(p_1, p_2)$ is a (possibly empty) smooth bijection between open subsets of $p_1.S$ and $p_2.S$.

Suppose that $c_1 \in p_1.S$ and $c_2 \in p_2.S$. We refer to the pairs $(p_1, c_1)$ and $(p_2, c_2)$ as patch points. If $B(p_1, p_2)(c_1) = c_2$, then we say that the two patch points are “the same”. (This is a handy abuse of language. “the same” will always appear in quotes.)

Being “the same” is required to be an equivalence relation:

• $B(p, p)$ = the identity map on $p.S$
• $B(p_1, p_2) = B(p_2, p_1)^{-1}$
• If $B(p_1, p_2)(c_1) = c_2$ and $B(p_2, p_3)(c_2) = c_3$, then $B(p_1, p_3)(c_1) = c_3$.

3. Given “the same” patch points $(p_1, c_1)$ and $(p_2, c_2)$, then the components of $p_1.g(c_1)$ and $p_2.g(c_2)$ are related tensorially, using the partial derivatives of $B(p_1, p_2)$ and its inverse.

4. Given $p_1$ and $p_2$, if $p_1.S = p_2.S$ and $B(p_1, p_2)$ is the identity on $p_1.S$, then $p_1 = p_2$.

5. No further patches can be “added” — you know what I mean — whose patch points are all “the same” as pre-existing patch points. (This requirement is intended to achieve maximality.)

While the above definition of geometries does not have an a priori point-set, it does have a derived point-set. The points of the geometry are equivalence classes of patch points that are “the same”.

I suspect that we can create a manifold from a geometry by “pulling up” a topology, atlas, and Riemannian metric from the patches to the derived point-set.

Above, the tensor field $g$ could be replaced with any desired set of tensor fields. General relativity is implemented on a geometry by requiring that $G = \kappa T$ holds for every patch. I would hope that physically equivalent solutions are identical (modulo tags).

I would also hope that there is no Cauchy ambiguity. Take a complete solution and tear off and discard its later half. (Please make that well-defined somehow.) Then the equations of motion should uniquely grow back the discarded half (modulo tags), like a lizard growing back its tail.

So I hope.

Posted at August 13, 2011 8:27 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2424

### Re: Geometries: Diffeomorphism Classes vs Quilts

Above, the tensor field $g$ could be replaced with any desired set of tensor fields.

In particular it could be discarded altogether… in which case the discussion is about the definition of smooth manifolds as such. My understanding is that the proposal here is to observe that instead of saying

“A manifold with an atlas is a set equipped with a topology and a bunch of patches with charts such that…”

we could say

“A manifold with an atlas is a collection of patches equipped with transition functions, such that… “.

without mentioning the set of points. Because, as you observe, that set of points is directly recovered from the second statement as the set of equivalence classes of points in the patches.

I am not sure I see what the big difference between the two formulations is. (But maybe I misunderstand the proposal after all.)

I am also not sure yet that I understand what the proposal has to do with what John said in the introduction:

if we take the space of Riemannian metrics on a fixed manifold M and mod out by the action of Diff(M), we’re left with a rather nasty space. […] In the following post, Greg asks our opinion of a possible alternate approach.

What is the “alternate approach” to talk about diffeomorphism classes of Riemannian manifolds? I don’t see this discussed in the above. Already the equivalence relation between different choices of atlases on the same manifold seems not to be discussed yet!?

Posted by: Urs Schreiber on August 13, 2011 11:42 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I should not have written that introductory passage, since I don’t exactly understand what Greg is trying to do or how he hopes to do it. I’ll delete that stuff and let him explain things in his own words.

Posted by: John Baez on August 13, 2011 3:58 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I’m probably misunderstanding what the overall goal is here. It sounds like you are trying to build manifolds in terms of Cech cocycles’ that take values in local diffeomorphisms of R^n. Since you say this might be “a conceivably new-and-improved definition”, let me just mention that this construction is considered to be well-known. I can give a reference or two if you’d like.

Posted by: Chris on August 13, 2011 5:01 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

It sounds like you are trying to build manifolds in terms of Cech cocycles’ that take values in local diffeomorphisms of $\mathbb{R}^n$. […] I can give a reference or two if you’d like.

Maybe unsurprisingly this is discussed in a bit of detail on the $n$Lab in the entry on Smooth manifolds : after the traditional definition there is a section called general abstract geometric definition which is intended to spell out in detail how manifolds are precisely the “locally representable” objects in the topos over CartSp (precisely the locally representable and 0-truncated smooth ∞-groupoids, if one wishes).

The Čech-nerve description of manifolds is in Smooth manifolds as locally representable objects of Sh(CartSp) .

Posted by: Urs Schreiber on August 13, 2011 6:38 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Thanks for the pointers. Chris, if you have some to add, I’d like to
see them.

BTW, if you read my second comment, you are likely to be certain whether
your pointers are indeed aimed at my topic. If you then need to update

Posted by: Greg Weeks on August 13, 2011 8:25 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

As a warm up for the Cech nerve description Urs is referring to, you could look at section 3.1 of Conlon’s Differentiable Manifolds (2nd ed). What he calls a “structured cocycle” is what I think you want a “quilt” to be (but without the local metric data). However, I’m not qualified to comment on how useful this point of view is for the physics you’re interested in.

Posted by: Chris on August 13, 2011 10:52 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

If it isn’t obvious, I’m not a functorite – to use a term that
differential geometer and non-functorite Michael Spivak used in his epic
differential geometry textbook. Just FYI.

Now, to get my feet wet, I strolled around in

>>>> http://ncatlab.org/nlab/show/smooth%20manifold

which at least includes the main topic covered in show/CartSp.

What I saw appeared to be increasingly deep (as in: over my head)
descriptions of differentiable manifolds, often viewed as special
cases of increasingly general structures.

That’s all well and good, but manifolds are not my goal. And if I had
to guess, I’d guess that none of the generalizations in
show/smooth%20manifold are my goal. In other words, I’m guessing that
Cech whatevers are not what I’m seeking. I can’t say for sure, but
that’s what I’d wager. So, tentatively, onward.

Before I posted, John suggested that I might be talking about
“∞-stacks”. And, indeed, show/space seems to strike nearer to my mark.
The section with the terrifying heading

>>>> “concrete spaces co-probeable by model spaces: structured
>>>> (∞, 1)-toposes”

begins with the on-target paragraph:

>>>> “Spaces probeable by G in the above sense can be very general. They
>>>> need not even have a *concrete underlying space*, even for general
>>>> definitions of what that might mean.”

Good. The elimination of the underlying space *is* what I’m seeking.
It is also gratifying to read that what I’m after requires something so
“very” general that it “even” has no “*underlying space*”.

Nevertheless, it will be much harder for me to grasp “∞-stacks” (to see
if they’re what I want) than for you to grasp “quilts” (and thus provide
a definitive statement of where they fit in existing mathematics).
Unfortunately, “quilts” can grab your attention (and effort) only on
physical, not mathematical grounds. And that hasn’t happened, I
suspect.

So, if you would, (re-)read my second top-level comment, beginning with:

>>>> [Here is a full description of what I’m talking about in the article.

That comment should have been my article, I suspect, with only the
briefest description of “quilts”. Also, please note that the second and
final paragraphs of the comment *frame* the discussion but do not
*belong* to it. Just ignore them for now, please.

Posted by: Greg Weeks on August 14, 2011 1:49 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

[This comment is an experiment. Will the text lines be broken both by
the right-hand text-window edge *as well* as by my line breaks,
resulting in an ugly mess? If so, my apologies, and I’ll avoid single
line breaks in the future.]

If even John wonders what I’m talking about, I’m in trouble.

Okay. I *am* talking about something (I suspect), but it may take a bit
of work to untangle. For starters:

It seems that I have made a mistake “in the small”. I suspected that:

1. Wheeler or DeWitt or ADM or *somebody* had defined 4-geometries as
(active) diffeomorphism equivalence classes of manifolds (with fields).

2. #1 was well-known.

But my googling today doesn’t support either #1 or #2.

So, is either true? (John, I hate to ask you in particular, but it
might be apt.)

My next comment will assume the worst case: that #1 and #2 are false.

Posted by: Greg Weeks on August 13, 2011 6:23 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

It seems that I have made a mistake “in the small”. I suspected that:

1. Wheeler or DeWitt or ADM or somebody had defined 4-geometries as

(active) diffeomorphism equivalence classes of manifolds (with fields).

1. #1 was well-known.

The second point is certainly true. I am not sure who exactly was the first to fully understand the modern precise concept of isomorphism classes of Riemannian manifolds – because that’s what you seem to be talking about. It must have been somewhere around Hilbert, I guess. In a non-precise sense it was apparently clear already to Riemann, of course.

Posted by: Urs Schreiber on August 13, 2011 6:51 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

[Here is a full description of what I’m talking about in the article. I
should have mentioned, BTW, that I’m discussing *classical* field
theory.]

SR has the (active) Poincaré group. GR has the (active) diffeomorphism
group. Offhand, the latter simply generalizes the former. That is what
Raymond Streater asserts, for example. This needs resolution. But
first, let’s focus on GR and adopt the prevailing (and correct) notion
that the (active) diffeomorphism group is effectively a gauge symmetry.

Consequently, a solution to the GR equations is only one of an
infinitude of physically equivalent solutions. In that case, how are we
to view reality?

1. As the set of gauge-invariants?

2. As the diffeomorphism equivalence class of solutions – ie, as a
structure *other than* a manifold that captures *in unique form* the
common *geometry* of the multiple solutions?

3. As a structure other than a manifold *and* other than #2 that
captures in unique form the common geometry of the multiple solutions?

Re #1: Gauge-invariants are hard to come by. (There are no local
gauge-invariants.) So let’s give up on this one.

adopted this view and had called the result “a geometry”. A geometry is
what all of the diffeomorphic manifolds (with fields) have in common.
It is a *unique* object representing a reality in classical GR.

I definitely like the idea of a unique object representing reality. I
hope that you all do as well. I applaud #2. Geometries and manifolds
are different. Reality is a geometry, not a manifold.

Re #3: I’d like to think that space-time does have points, and #2
doesn’t support this. Also, #2 isn’t all *that* unique, since the
abstract point-set of its manifolds could be replaced with some other
abstract point-set. Finally, #2 is an algebraic thing that contains
*more* information than a manifold. I would hope to find an analyst’s
definition of a geometry that contains *less* information than a
manifold. THAT IS WHAT THE ARTICLE IS ABOUT. (Pardon my caps.)

For clarity, I’ll refer to the analytical definition of a geometry as a
quilt – a bunch of overlapping patches sewn together. Here are some
points of interest (to me, anyway).

1. A quilt is not a manifold. It contains less information. In
particular, it does not contain an abstract point-set M.

2. I strongly suspect that a manifold can be *derived* from a quilt.
But please don’t identify the derived manifold with the quilt. For
example, an active diffeomorphism on the derived manifold will destroy
the connection between the structure on the manifold and the structure
on the quilt.

3. A quilt, alas, is not a *unique* representation of a geometry. If
you replace all of the tags in (P, B) with a different set of tags, the
geometry is unchanged. I’d love to avoid this somehow. (I’ve tried.)

4. Quilts, IMO, are kind of neat. For example, I suspect that an
initial-valued strip of quilt – a 4-dimensional Cauchy time-slice –
can be extended uniquely (modulo tags, sigh) to a maximal quilt by the
equations of motion. No Cauchy ambiguity. (A quilt, unlike a manifold,
can “grow” points dynamically as patches are added.)

Putting the tags ambiguity aside, I view quilts as the most satisfying
descriptions of classical space-times. Perhaps that is their only
value. (The structure B is rather unwieldy, after all.)

PS: To wrap up this comment, I should mention that SR space-time is a
unique (modulo tags) quilt as well. The Poincaré group is associated
with *physical* inertial coordinate frame-works – which alas don’t seem
to work in GR. Hence the qualitative difference. IMO.

Posted by: Greg Weeks on August 13, 2011 7:58 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

A quilt is not a manifold.

As far as I can tell from what you write it wants to be the definition of an atlas of a manifold.

Posted by: Urs Schreiber on August 14, 2011 12:45 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Emphatically not. But I clearly need to say more, and I will.

Posted by: Greg Weeks on August 14, 2011 1:24 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I should respond to your comment more specifically. I don’t want the
charts. I want the _images_ (x(U),x*(g|U)) of all the charts. And then
I want to sew them together – all without any reference to M or g.

Posted by: Greg Weeks on August 14, 2011 2:07 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

As others have said, what you have defined/discovered is the Cech groupoid associated to an atlas.

Since Urs is prone to using phrases like ‘Smooth manifolds as locally representable objects of $Sh(CartSp)$’, let me explain it in detail:

* When you say ‘tags’, it is only to ensure that you take the disjoint union, instead of the regular union as subsets of Euclidean space. I shall be clear and talk about disjoint union when I mean it, and I will replace tags by the name of the bijection associated with that chart, e.g. the tag of $x:U \to \mathbb{R}^n$ will be ‘$x$’ However, charts, since they are bijections with open subsets of $\mathbb{R}^n$, I will take then to be open submanifolds of $\mathbb{R}^n$, equipped with a metric. I will still denote by $x,y$ etc. the inclusion $x:U \hookrightarrow \mathbb{R}^n$ etc. But this means we can take the intersection of $U$ and $V$ as submanifolds of Euclidean space.

* Your 1. 2. and 4. define a groupoid internal to the category of manifolds, and this groupoid is such that there is at most one arrow from any object to any other object (let us agree to call such a thing a Cech groupoid - abusing notation somewhat). The ($n$-dimensional) manifold of objects is the disjoint union $\coprod_{P} S$ and the manifold of arrows is $\coprod_{P\times P} S_1 \cap S_2$. The bijection $B$ is absorbed into this definition. “Patch points” are objects of this groupoid. The source of an arrow is given by the map $s$ arising from all the inclusions $S_1 \cap S_2 \hookrightarrow S_1$, and the target of an arrow is given by the map $t$ arising from all the inclusions $S_1 \cap S_2 \hookrightarrow S_2$. Notice that there is a metric on $\coprod_{P} S$ given on each component by the metric on the patch.

* Your definition of “the same” patch points boils down to: “there exists an arrow from the first patch point to the second”. We could consider the quotient of the underlying set of $\coprod_{P} S$ under the equivalence relation “the same”, and this is the underlying set of the manifold. However, let us not do this (I’ll explain why later).

* Your 3. means that there is a metric on $\coprod_{P\times P} S_1 \cap S_2$ compatible with the metric on $\coprod_{P} S$ via $s$ and $t$. Thus the groupoid is a groupoid internal to the category of smooth manifolds equipped with a metric. In fact we can say more: it is a groupoid internal to the category with objects disjoint unions of open submanifolds of $\mathbb{R}^n$ equipped with metrics, and smooth functions for morphisms. (An easier way to say this is that it is a sheaf on the category of metric-equipped open submanifolds of Euclidean space, which is a coproduct of representables - but that is just jargon.)

* Your 5. means that if the quilt arises from a manifold, then the charts are an open cover of the manifold. But this condition is meaningless without some external manifold with which to compare the quilt, because one can always add new patches all of whose patch points are disjoint from all other patch points.

It is not true that a quilt doesn’t have a point set. What you mean is that the point set the quilt has (which is the disjoint union of all the sets underlying all the charts $U$), is not the same as the point set of the underlying manifold, should you define the quilt from the manifold.

As you have noticed, you can recover the manifold from the quilt (modulo questions of Cauchy ambiguity), by equipping the set $\coprod_P S /('the same')$ with the manifold structure arising from $\coprod_P S$. (I’m guessing you can get the metric too, but I haven’t time to check at the moment.)

As regarding uniqueness, you can’t do that (as you’ve seen), as one can always take a Morita/weakly equivalent Cech groupoid, and this presents the same geometry, in that you get a diffeomorphic manifold arising from the construction in the previous paragraph. If the metric descends, then you can be assured that the diffeomorphism is compatible with the induced metrics.

Posted by: David Roberts on August 14, 2011 2:43 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Many thanks. It sounds like my guess was wrong, simply from the
definitiveness of your response. Alas, I’m not going to grasp it
quickly. (Is all mathematics in the language of categories now, or is
it just this group? Back in 1979 …)

(Yes, I know that this is the n-Category Café. But John suggested this
as the place for general math and physics discussions.)

For the moment, one simple comment: #5 is not meaningless in the absence
of a manifold. It doesn’t prohibit the adding of patches. It prohibits
the adding of a patch whose points are all “the same” as pre-existing
patch points. (And you can’t just add an existing patch with a new tag,
because of #4.)

Posted by: Greg Weeks on August 14, 2011 4:14 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Greg wrote, of David’s response:

Alas, I’m not going to grasp it quickly.

There’s no rush: math is eternal. It’s worth deeply pondering David Roberts’ remarks, because they explain how your construction is related to one that’s quite familiar, namely the Cech groupoid.

To help out, I have tried to write shorter, easier explanation. This has Wikipedia links for all the definitions you need to understand. Also, while David focuses on how to get a quilt from a Riemannian manifold together with an atlas, my account focuses on the concept of quilt itself, which seems to be your main concern.

Is all mathematics in the language of categories now, or is it just this group?

If you’re serious about studying symmetry and spacetime, you need category theory. I suggested posting your comment here because I could tell that your quilts are the kind of thing people use category theory to study. Indeed you are already doing category theory; you just don’t know it yet.

Posted by: John Baez on August 14, 2011 6:05 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Ah, demanding no Cauchy ambiguity requires that spacetime is connected, so as you say, one cannot just add disjoint patches.

Also, your maximality condition (let me state it in terms of associating a quilt to a manifold) is rather a minimality condition - sort of. You are asking that we remove charts that are covered by other charts, if we start from an arbitrary atlas.

One thing I do find very interesting is the idea of generating the future of a Cauchy spacelike slice in the context of Lie groupoids as John described them (thanks John - and thanks for the kind words too). A large proportion of my thesis was dedicated to thinking about surfaces in topological groupoids (a generalisation of what you have here), and deformations of said surfaces. Describing these (for me) is clear. What isn’t clear is how to generate the future from Einstein’s equations if we throw it away. Especially if one has a change in topology (such as the collapse of a massive spherical body to a singularity), because then it looks like one might need to ‘make new patches’ on the fly.

Posted by: David Roberts on August 14, 2011 11:26 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

[Even though patch tags currently make me sick, I still feel obliged to
clarify what I’ve written.]

> You are asking that we remove charts that are covered by other charts,
> if we start from an arbitrary atlas.

There will be no charts to remove. If you *do* start with a manifold
(with maximal atlas), there will be a one-to-one correspondence between
charts and patches. The abstract geometry/quilt definition is intended
to maintain that one-to-one correspondence, in this case with the
manifold that I presume can be constructed by pulling the patches
structure up onto the derived point-set.

Anyway, my goal in defining an abstract geometry was a one-to-one
correspondence between charts and patches wherever that comparison makes
sense (assuming a maximal atlas).

Posted by: Greg Weeks on August 15, 2011 6:51 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I (Greg) wrote:

> Many thanks. It sounds like my guess was wrong …

Cancel that comment. If I had read more closely, I’d have seen that the
uniquification of different patches of geometry that happen to look the
same was achieved via *charts*. Quilts, or, rather, the concept that I
was shooting for with quilts, must be defined without reference to a
manifold. “The Cech groupoid associated to an atlas” is not that
concept.

Oh, isn’t there a *canonical* way to uniquify patches of geometry that
just happen to look the same? Must a (potentially) nice idea die from
being pierced by a so fine a point?

Posted by: Greg Weeks on August 15, 2011 5:37 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Urs wrote:

As far as I can tell from what you write it wants to be the definition of an atlas of a manifold.

I don’t think so. For one thing, Greg already knows what an atlas is. For another, a quilt just isn’t the same as an atlas.

Urs also wrote:

The Čech-nerve description of manifolds is in Smooth manifolds as locally representable objects of Sh(CartSp) .

This remark is more helpful. Unfortunately, there’s a 0% chance that Greg will understand it. And if he attempts to read the first sentence in the passage this remark links to, he’ll feel the need to look up the definition of ‘concrete object’, ‘sheaf topos’, ‘diffeological space’, ‘coproduct’ and ‘effective epimorphism’. And if he then clicks on the links to learn what those terms mean, each will ask him to learn 5 more terms, etcetera. The nLab is structured in this sort of way: it’s not easy to learn about anything without learning a bit about everything. It’s a great resource but it’s a bit like a dictionary: if someone doesn’t know English, handing them an English dictionary won’t quickly cure that problem. So, I estimate it would take Greg 5-10 years of hard study on the $n$Lab to comprehend that sentence you just wrote.

David wrote:

As others have said, what you have defined/discovered is the Čech groupoid associated to an atlas.

That’s good - and your detailed comment should be extremely helpful to Greg.

But if I understand it correctly, Greg’s concept of ‘quilt’ (actually he called it ‘geometry’) can be defined this way:

Def. - A quilt is a Lie groupoid $s,t: Q_1 \to Q_0$ which is an equivalence relation, such that:

1. the manifold of objects $Q_0$ is a disjoint union of open subsets of $\mathbb{R}^n$,
2. both the manifold of objects $Q_1$ and the manifold of morphisms $Q_0$ are equipped with Riemannian metrics, and
3. the source and target maps $s,t: Q_1 \to Q_0$ are local diffeomorphisms and also isometries.

One way to built a quilt is to start with a Riemannian manifold together with an atlas. Then its Čech groupoid is a quilt in the above sense.

However, I think Greg is entertaining the idea that we could start with a quilt and then build from it a Riemannian manifold equipped with an atlas…. or perhaps just such a thing up to diffeomorphism. My guess is that he’s seeking a description of a diffeomorphism equivalence class of Riemannian manifolds, not as an equivalence class, but as a ‘thing in its own right’. However, I don’t think quilts accomplish that particular goal.

Posted by: John Baez on August 14, 2011 6:08 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

First off, John, thanks for the hand-holding.

Also, thanks for noting that I preferred “geometry” to “quilt”, although
I’m using the latter term at this point.

And especially, thanks for these words:

> One way to built a quilt is to start with a Riemannian manifold
> together with an atlas. Then its Čech groupoid is a quilt in the
> above sense.
>
> However, I think Greg is entertaining the idea that we could start
> with a quilt … My guess is that he’s seeking a description of a
> diffeomorphism equivalence class of Riemannian manifolds, not as an
> equivalence class, but as a ‘thing in its own right’.

All correct. I *started* the article by extracting a quilt from a
manifold, but that was A) to provide a feel for quilts and B) to note
that different manifolds can have the same quilt.

But quilts, or something like them, need to stand on their own feet,
divorced from manifolds. (Is this possbile with Čech groupoids?) I
seek an anlysts’s view of space-time as an object, and quilts are the
best I’ve been able to find so far. Also, quilt dynamics – which don’t
really exist yet, but I have hopes – ought to avoid the Cauchy
ambiguity.

> However, I don’t think quilts accomplish that particular goal.

Why not? Except for the nonuniqueness due to tags, I haven’t seen any
awful problem.

Posted by: Greg Weeks on August 14, 2011 12:03 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

John wrote:

However, I don’t think quilts accomplish that particular goal.

Greg wrote:

Why not? Except for the nonuniqueness due to tags, I haven’t seen any awful problem.

Right: that’s the awful problem.

Basically, you’re trying to make up:

1) a definition of ‘geometry’

and:

2) a procedure that associates to any Riemannian manifold a geometry of this sort,

and then prove:

3) this procedure maps isomorphic Riemannian manifolds to equal geometries.

You’ve done 1) with the definition of ‘quilt’. You’ve sort of done 2), except that your ‘procedure’ involves an arbitrary choice, namely a choice of ‘tags’. Even if we bite the bullet and accept that, this will prevent you from proving 3). If you don’t believe me, you should try to prove 3).

As I’ve explained here (using too much technical jargon I’m afraid), I believe the best you can do is find a sensible notion of ‘isomorphism’ between quilts and show that isomorphic Riemannian manifolds give ‘canonically isomorphic’ quilts.

But since isomorphic Riemannian manifolds already give canonically isomorphic Riemannian manifolds, you could have accomplished this by defining a geometry to be a Riemannian manifold!

In short, I’m afraid you haven’t cracked the nut you’re trying to crack. But to the extent that you’ve reinvented the concept of the Čech groupoid of an atlas for a manifold, and described what these groupoids are like without reference to manifolds, you certainly deserve a hearty round of applause from the jaded mandarins gathered here tonight.

Posted by: John Baez on August 14, 2011 12:36 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

As I’ve said before, John, you are kind. But you can’t say that I’ve
reinvented “the Čech groupoid of an atlas for a manifold”. That doesn’t
sound pedestrian, and pedestrian is what I’ve been. And I wish to
continue plodding along and sucking brains.

(Besides, I insist that quilts are independent of manifolds.)

Regarding your comment: The tags are awful; I have indeed replaced one
isomorphism class with another; and your #3 is indeed false.

All along, I have vaguely supposed that, for my purposes, “modulo tags”
is qualitatively less offensive than “modulo diffeomorphisms”. But I
don’t know if that can be made meaningful/true.

For now anyway, I accept your comment as it stands.

(My hunch, incidentally, is that the proper notion of isomorphism for
quilts is tag-renaming *and nothing else*. But that’s not important
right now.)

————————————————————————

Issues remain!

1. Is there anyone out there with even a small fire in your belly about
creating a notion of geometry that:

- is common to all manifolds in a diffeomorphism class

- is indeed canonical (as quilts are not)

- has a derived point-set (as a diffeomorphism class does not)

- neatly avoids the Hole_Argument/Cauchy_ambiguity (and in particular
supports the notion of a Cauchy strip of geometry that grows points
dynamically)

- would be of pedagogical value (since even revered physicists disagree
profoundly about basic issues in general relativity!)

2. Being shot down by tags hurts. Is there no elegant way to give
mathematical entities “identity”? (Ironically, in computer science, it
is troublesome to define tuples that *don’t* have identity, since tuples
are most naturally referenced by their address in memory.)

Or can the patch “identity crisis” be solved without tags? For example,
I tried putting the overlap information into the patches themselves –
the sort of thing that a computer scientist might do – and ended up
with self-referential nonsense. But there may still be a way.

3. Is it true that the quilt definition *in the absence of any
reference to manifolds or atlases* is a known mathematical structure?
Eg, “concrete spaces co-probeable by model spaces: structured
(∞,1)-toposes”?

“the Čech groupoid of an atlas for a manifold” does of course reference
manifolds and atlases. *Is this avoidable?* If not, the patches get
their identity from the charts – Right? – which provides us with no
insight into avoiding the patch identity crisis.

4. In short, is there any way to achieve what I’d hoped to achieve with
quilts, either by, uh, patching the idea or with an alternative?

————————————————————————

Finally, John, did I correctly hear you say some months ago that people
*have* tried to achieve what I’m trying to achieve (more or less). I
mean, it isn’t just *me*.

Posted by: Greg Weeks on August 15, 2011 5:48 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I wrote:

As far as I can tell from what you write [quilt] wants to be the definition of an atlas of a manifold.

John says:

I don’t think so. For one thing, Greg already knows what an atlas is. […] A quilt is a Lie groupoid.

That’s not really apparent from what Greg writes. But okay, let’s say a “quilt” is not the atlas $\{U_i \to X\}$ itself, but its Cech groupoid $\left(\coprod_{i j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i \right)$.

The difference between the two is a bit academic, and in fact inessential in a precise way: every effective epimorphism $\coprod_i U_i \to X$ (an atlas) gives rise to its Cech nerve and is recovered from its Cech nerve as its colimiting cocone.

In either case: the notion is evidently equivalent to the notion of manifold! And so it does not affect anything about Cauchy problems in general relativity which definition one uses, nor does it change anything about isomorphism classes of manifolds or their description. Of course atlases and their Čech groupoids are very useful. That’s why people use them all the time.

Posted by: Urs Schreiber on August 14, 2011 12:08 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Urs wrote approximately:

But okay, let’s say a “quilt” is not the atlas itself, but its Cech groupoid […] This notion is evidently equivalent to the notion of manifold!

Right, that’s the problem. I tried to say the same thing in different words in this comment, written while you were writing yours.

For Greg’s sake, I should add that ‘equivalent’ is being used in a technical sense here, which makes precise the intuitive notion, but unlike the intuitive one can be the subject of theorems. Once we know what isomorphic quilts are (and we can already guess), we can settle the question of whether the groupoid of Riemannian manifolds is equivalent to the groupoid of quilts. And if they are, that means quilts are more like Riemannian manifolds than diffeomorphism equivalence classes of Riemannian manifolds.

Posted by: John Baez on August 14, 2011 12:23 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I’m afraid I’ll have to step around your proof and state what seems
clear to me.

> In either case: the notion is evidently equivalent to the notion of
> manifold! And so it does not affect anything about Cauchy problems
> in general relativity which definition one uses,

The notion of “quilt” certainly isn’t *equivalent* to the notion of
“manifold”. The most that you can say is that given a quilt, you can
generate a manifold of a most unusual kind – ie, with derived points
rather than a priori points. Furthermore, if you apply an active
diffeomorphism to this manifold, then you have discarded the quilt,
since the point of, say, maximum curvature of the quilt will not be the
point of maximal curvature of the manifold.

If you *don’t* discard the quilt, then you have an object with no
abstract point-set, no diffeomorphism gauge group, and no Cauchy
ambiguity.

Posted by: Greg Weeks on August 14, 2011 12:38 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

The notion of “quilt” certainly isn’t equivalent to the notion of “manifold”.

The problem I see is that every commenter here who tried to extract a definition from what you write arrives at either the notion of atlas or the notion of Čech nerve of an atlas. If none of these is what you mean, we have two problems: first is: we do not know what you mean. Second is: we do not see which aspect of the standard definition of manifold you feel so unhappy about that you want to change it.

The most that you can say is that given a quilt, you can generate a manifold of a most unusual kind – ie, with derived points rather than a priori points.

This is a weird thing to say. If we start with an atlas and then glue the patches to a manifold, that’s as good a manifold as any other. In fact all manifolds are of this form. This is the very definition of manifold! A point of a manifold does not know whether you regard it as a “derived point” or not, whether you care to make a distinction between an “abstract set of points” and an isomorphic equivalence class of points.

What you say sounds like you are struggling a bit with the abstract language that enters the definition of manifold. Maybe you are not, but that’s what it sounds like. I think the problem that you are trying to go away is not in the theory of manifolds, but in your reading of it.

The only way to find out is to go through this exercise: 1. try to give a precise definition of “quilt”. 2. try to define the notion of equivalence of two quilts.

Then we can see what it is.

Posted by: Urs Schreiber on August 14, 2011 1:00 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

The definition of quilts is in the article – which uses the word
“geometry” rather than “quilt”.

I hereby declare that the only isomorphism is tags replacement.

Responses to the other issues that you raise are scattered about,
unfortunately. But I’ll summarize for you recent events.

- A comment from John pushed my distaste for patch “tags” over the edge.
I haven’t given up on “geometries” – a *concept* that is championed by
both John and me at various points in these comments – but I don’t see
my quilt definition as a satisfactory implementation.

- I asked several questions targeted at a suitable definition of
“geometries”. However:

- John said that a suitable definition is generally thought to be
unattainable.

Posted by: Greg Weeks on August 15, 2011 6:31 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> In either case: the notion is evidently equivalent to the notion of manifold!

If you *don’t* discard the quilt, then you have an object with no abstract point-set…

I think what people are saying is that you go ahead and derive a manifold anyway whose underlying point-set is the set of equivalence classes of patch points, and with an atlas given by quilt data, etc. The manifold with this atlas structure carries essentially the same information as the quilt, and in that sense the notions are equivalent.

John Baez pointed out a problem with trying to read about stuff in the nLab, and I think he’s right, but that’s a topic for another day. But in slightly less high-powered language than that of the article Urs linked to: the derived manifold construction is a certain colimit construction (intuitively, that’s what colimits are: taking disjoint unions of things and then sewing them together in some prescribed fashion).

Posted by: Todd Trimble on August 14, 2011 1:44 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Greg wrote:

The notion of “quilt” certainly isn’t equivalent to the notion of “manifold”.

It depends on what you mean by ‘equivalent’. The modern approach to this question is to formulate a category of quilts, a category of manifolds, and check to see if they’re equivalent as categories.

Indeed, it’s enough for you to tell us your concept of isomorphism for quilts. Once you do that, we get a category with those isomorphisms as morphisms. A category with only isomorphisms as its morphisms is called a groupoid. That’s why I focused on groupoids in my analysis of your idea here.

If the groupoid of quilts is equivalent to the groupoid of Riemannian manifolds, that gives a precise sense in which the notion of quilt is equivalent to the notion of Riemannian manifold. To understand why takes a bit of experience with category theory. But one nice thing is that it’s a yes-or-no question, which one can settle with a theorem.

If instead you want the set of quilts to be isomorphic to the set of isomorphism classes of Riemannian manifolds, you should prove that. But I don’t believe that’s true under any interpretation or modification of your ideas that I can come up with.

Posted by: John Baez on August 15, 2011 1:42 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

There is no way that I can avoid asking this:

You defined a quilt as a “Lie groupoid …”. Is *your* definition
ruined by tags? Or is it non-canonical in some other way? Or is it
that “holy grail” that you mentioned?

Posted by: Greg Weeks on August 15, 2011 7:16 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Defining a quilt as a Lie groupoid (which an entirely equivalent way to talk about them) : it is certainly not canonical. And indeed, the naive (2-)category of Lie groupoids does not have enough morphisms - the first chapter of my thesis was about (a vast generalisation of) this problem!

Posted by: David Roberts on August 16, 2011 12:22 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> As others have said, what you have defined/discovered is the Cech
> groupoid associated to an atlas.
> …
> I will replace tags by the name of the bijection associated with that
> chart, e.g. the tag of x:U→Rⁿ will be ‘x’

Yay! This answers a question that I have strewn about in various places
in this discussion. (And it is answered early on in the thread. The
perils of skimming.)

The “quilts” notion *must* stand independently of an associated
manifold. In particular, the uniquification of patches of different
regions of the geometry that happen to look the same must be achieved by
a means *other* than charts.

Consequently, “the Cech groupoid associated to an atlas” is of *no use*
in achieving what I tried – and, with 95% certainty, failed – to
achieve with “quilts”.

It is a worthy goal, according to John Baez (and me). But, says John,
the general consensus is that it is impossible to achieve.

Posted by: Greg Weeks on August 15, 2011 5:22 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I am bit late to the discussion and haven’t read all of the comments. So apologies if I am making a point already made elsewhere.

Greg Weeks wrote:

The “quilts” notion must stand independently of an associated manifold. In particular, the uniquification of patches of different regions of the geometry that happen to look the same must be achieved by a means other than charts.

Consequently, “the Cech groupoid associated to an atlas” is of no use in achieving what I tried — and, with 95% certainty, failed — to achieve with “quilts”

I think you simply want the notion of a proper etale Lie groupoid with several further conditions:

1. The connected components of its space of objects are open subsets of some fixed $\mathbb{R}^n$ together with Riemannian metrics;

2. The local diffeos defined by the arrows are isometries;

3. The only arrow starting and ending at a given point is the identity arrow (no nontrivial discrete symmetries).

By the way, you do want proper. Otherwise the associated manifold (which is the orbit space/colimit of this groupoid) is not Hausdorff.

I believe that this notion of a Riemannian proper etale Lie groupoid modelled on subsets of $\mathbb{R}^n$ stands independently of any notion of a Riemannian manifold — the space of arrows of such a groupoid is also a disjoint union of subsets of the same $\mathbb{R}^n$. You probably don’t want to put any metric on the space of arrows (here I am thinking out loud).

I think there was a discussion here about a similar concept a few years ago. Metzler in his paper on differentiable stacks talks about geometric stacks over open subsets of $\mathbb{R}^n$’s. These stacks include manifolds.

Another comment: the term “quilts” is taken by Katrin Wehrheim and Chris T. Woodward.
See, for example arXiv:0905.1369.

And one more comment: “geometries” may be confusing for someone who thinks of the Riemannian geometry as just one type of a geometry.

Posted by: Eugene Lerman on August 15, 2011 6:35 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Consequently, a solution to the GR equations is only one of an infinitude of physically equivalent solutions. In that case, how are we to view reality?

I’ve never really understood why people get worked up about this. It seems to me just like mathematicians bothering about whether “the” natural numbers are defined as von Neumann ordinals or whatever else have you. Maybe physicists just haven’t really understood the notion of structural mathematics? Or maybe I just don’t understand what they’re worried about?

We model spacetime by a Lorentzian manifold. An isomorphic Lorenztian manifold could just as well have been used. So what? When we use a particular manifold $M$ to model spacetime, that manifold has points which represent some aspect of spacetime. If we transfer that model along an isomorphism $\phi\colon M\cong N$, then the points of $N$ represent the analogous aspects of spacetime: the point $x\in N$ represents whatever $\phi^{-1}(x)$ represented in the first model. If we instead transfer our original model along a different isomorphism $\psi\colon M\cong N$, then instead $x\in N$ will represent whatever $\psi^{-1}(x)$ represented in the first model. So what?

I definitely like the idea of a unique object representing reality. I hope that you all do as well.

Maybe this is the point: I’m perfectly happy when something is unique up to unique canonical isomorphism. I think to be satisfied with this is one of the important lessons of 20th century mathematics.

Posted by: Mike Shulman on August 14, 2011 5:42 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

[Below, I use φ* for the universal push-forward induced by the
diffeomorphism φ. So does Robert Wald, among others.]

Viewing classical reality as a manifold is not intolerable. Indeed, it
may be ideal. The question is open for exploration, and I am exploring
it. Why? Well…

Define classical electrodynamics to be Maxwell’s equations, the Lorentz
force equation, and some charged fluid, say, φ, for which both electric
currents and a response to the Lorentz force make sense. Would I be
happy saying that a universe consists of *a* pair (A, φ) when I could
say instead that the universe consists of *the* pair (F, φ)? No. When
it is *easy* to eliminate a gauge-symmetry, it’s something to consider.

And when it is easy to eliminate the Hole_Argument/Cauchy_Ambiguity,
that’s something to consider too. It might *conceivably* shift our
perspective in a way that would make the theory easier to quantize.

And quilts *are* easy.

————————————————————————

Finally, consider a favorite among imbedded manifolds, an automobile
fender sitting (physically!!) in R³. If (M, g) is the manifold, and
i:M–>R³ is an isometric imbedding, then the manifold may be represented
(faithfully) as the triple (i(M), i*(g), i⁻¹). i(M) is a fender-shaped
point-set. i*(g) is redundant, being the same as R³’s <,> on their
common domain. And i⁻¹(c) tells you what point p ∈ M is physically
sitting at location c ∈ R³.

Similarly (M, φ*(g)) may be represented as (i(M), i*(g), (φ o i⁻¹)).
Note the triviality of the difference in the representations. You just
move the (point-like) “atoms” of the fender around. What a silly thing
to have to bother about! And what a silly way to represent the geometry
of a fender – to include a bunch of abstract points that don’t mean
anything.

If (i(M), i*(g), i⁻¹) represents the Riemannian manifold (M, g), then
(i(M), i*(g)) represents – what? The answer ought to be “a geometry”.
“Quilts” are my first attempt at defining a geometry.

————————————————————————

And that’s why I’m exploring whether manifolds or geometries better
describe classical reality.

Posted by: Greg Weeks on August 15, 2011 12:18 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Mike wrote:

I’ve never really understood why people get worked up about this. It seems to me just like mathematicians bothering about whether “the” natural numbers are defined as von Neumann ordinals or whatever else have you. Maybe physicists just haven’t really understood the notion of structural mathematics? Or maybe I just don’t understand what they’re worried about?

I think you don’t understand what we’re worried about. Physicists often talk about this issue in the context of quantum gravity, but I think it’s best to start by thinking about in the context of classical gravity — general relativity. The same issues show up, but in a context of a theory that exists, rather than one people are trying to build.

If I give you initial data for the wave equation, there’s a unique solution at future times. If I give you initial data for Einstein’s equation — the basic equation obeyed by the metric on spacetime in general relativity — the solution at future times is unique only up to diffeomorphism. Unfortunately, nobody knows a practical way to name a ‘metric modulo diffeomorphism’ except by giving a specific metric, a representative of its equivalence class. So, physicists who numerically solve Einstein’s equation always ‘gauge-fix’ it, i.e., give a recipe that systematically picks out one of the solutions.

It’s hard to gauge-fix Einstein’s equations without causing problems of one sort or another. Thanks in part to this issue, but also others, the LIGO experiment for detecting gravitational waves was held back for many years by our inability to calculate what gravitational waves to expect when two black holes spiral into each other and collide. For a long time, despite an NSF-funded Binary Black Hole Grand Challenge aimed at this problem, people couldn’t accurately simulate such a system for more than a single orbit. That’s not good enough, since we get roughly one wave of gravitational radiation per orbit!

This passage from Wikipedia may give you more of a sense of how these issues play out:

In the puncture method the solution is factored into an analytical part, which contains the singularity of the black hole, and a numerically constructed part, which is then singularity free. […] Until 2005, all published usage of the puncture method required that the coordinate position of all punctures remain fixed during the course of the simulation. Of course black holes in proximity to each other will tend to move under the force of gravity, so the fact that the coordinate position of the puncture remained fixed meant that the coordinate systems themselves became “stretched” or “twisted,” and this typically lead to numerical instabilities at some stage of the simulation.

In 2005 researchers demonstrated for the first time the ability to allow punctures to move through the coordinate system, thus eliminating some of the earlier problems with the method. This allowed accurate long-term evolutions of black holes. By choosing appropriate coordinate conditions and making crude analytic assumption about the fields near the singularity (since no physical effects can propagate out of the black hole, the crudeness of the approximations does not matter), numerical solutions could be obtained to the problem of two black holes orbiting each other, as well as accurate computation of gravitational radiation (ripples in spacetime) emitted by them.

See? From a lofty conceptual perspective it doesn’t matter which coordinates we use to describe a solution of general relativity: the physics being described is the same. The coordinates of the ‘puncture’ representing a black hole aren’t physically observable, so it seems ridiculous to ask whether the puncture is ‘moving’ or not. But in numerical calculations, we (seemingly) need to choose some coordinate system, and a poorly adapted coordinate system will cause lots of numerical errors.

Here’s a bit of the more recent history, which I haven’t been keeping up with:

The Lazarus project (1998—2005) was developed as a post-Grand Challenge technique to extract astrophysical results from short lived full numerical simulations of binary black holes. It combined approximation techniques before (post-Newtonian trajectories) and after (perturbations of single black holes) with full numerical simulations attempting to solve General Relativity field equations. All previous attempts to numerically integrate in supercomputers the Hilbert-Einstein equations describing the gravitational field around binary black holes led to software failure before a single orbit was completed.

The Lazarus approach, in the meantime, gave the best insight into the binary black hole problem and produced numerous and relatively accurate results, such as the radiated energy and angular momentum emitted in the latest merging state, the linear momentum radiated by unequal mass holes, and the final mass and spin of the remnant black hole. The method also computed detailed gravitational waves emitted by the merger process and predicted that the collision of black holes is the most energetic single event in the Universe, releasing more energy in a fraction of a second in the form of gravitational radiation than an entire galaxy in its lifetime.

Posted by: John Baez on August 15, 2011 2:24 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Thanks for the explanation! But it does sound, from your explanation, as though this is a purely practical problem (even if a very difficult one), not a conceptual one; is that right? I feel like I have frequently heard this problem discussed as if it were conceptual. For instance, Greg’s question “how are we to view reality?” sounds, to me, unequivocally like a conceptual question. Whereas it sounds to me like you’re saying the real question is not “how are we to view reality?” but “how are we to represent our models of reality in a computationally tractable way?”

Posted by: Mike Shulman on August 15, 2011 4:52 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Mike wrote:

Thanks for the explanation! But it does sound, from your explanation, as though this is a purely practical problem (even if a very difficult one), not a conceptual one; is that right?

This problem confused Einstein for about a decade, and led to endless philosophical discussions which continue to this day — see the nice review article on the Stanford Encyclopedia of Philosophy. So, plenty of people including Einstein think or at least thought that there’s a conceptual problem.

But I think that conceptual problem has been solved by now… at least in classical general relativity.

In quantum gravity, however, it’s a different story. First, it’s much harder to know where conceptual problems end and ‘purely practical’ problems start, because we don’t have a working theory yet. Second, in quantum gravity we can contemplate something like a quantum superposition of two solutions; we have to choose some attitude for how the superposition principle interacts with diffeomorphism-invariance. Should our Hilbert space of states come with a representation of some diffeomorphism group? Should we seek a Hilbert space of diffeomorphism-invariant states? Should we entirely abandon the assumption that spacetime is a manifold, except in some sort of classical limit? Etcetera. Each choice brings with it a pile of conceptual and practical problems.

Posted by: John Baez on August 15, 2011 6:52 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Mike is completely right: there is no conceptual problem at all with theories of physics with gauge symmetries, hence those whose configuration spaces are actually groupoids.

There is no conceptual problem with a state of a system being defined up to isomorphism. The “hole argument” that Einstein was still worried about is trivially no problem if you think of the groupoid of spacetimes.

This is routine since the second half of the 20th century. The Lie algebroid of this Lie groupoid is called the BV-BRST complex and has been recognized to be the right way to speak about gauge theories ever since Feynman introduced the “ghost” field in

Quantum theory of gravitation Acta physica polonica (1963)

While I find it a bit worrisome to mention quantum gravity at all in a context of much confusion about basics, there is absolutely no conceptual problem with discussing it in the sense of effective quantum field theory using these methods.

None of the problems that Greg is worried about exist as open problems and the solutions are well known, standard and old. But maybe knowledge about them is not as wide-spread as it could be. The same is true for the definition of manifold and the basics of differential geometry, though.

We should have here a discussion on these points. But I think the “quilts” are standing in the way. We should interrupt the quiltologoy and instead have something like a small course on the conceptual basics of general relativity.

Posted by: Urs Schreiber on August 15, 2011 10:44 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Urs wrote:

The Lie algebroid of this Lie groupoid is called the BV-BRST complex and has been recognized to be the right way to speak about gauge theories ever since Feynman introduced the “ghost” field in Quantum theory of gravitation Acta physica polonica (1963).

which Lie groupoid?

‘ghosts’ in BRST were recognized in terms of the Chevalley-Eilenberg complex much later

and BV came in in the 80s

Posted by: jim stasheff on August 15, 2011 12:46 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

which Lie groupoid?

The one whose objects are configurations of the physical system, and whose morphisms are gauge transformations between these: the configuration groupoid of the physical system.

In the simple case of a gauge group $G$ acting globally on a space $S$ of configurations, this is the action groupoid $S//G$. The Lie algebroid $S//\mathfrak{g}$ of that has as Chevalley-Eilenberg algebra $CE(S//\mathfrak{g})$ the corresponding BRST complex.

If one takes care and replaces “Lie groupoid” by a suitably more general notion of “smooth groupoid” then this applies to physically interesting setups, such as the group $Diff(X)$ of diffeomorphisms on a smooth manifold $X$ acting on the space $Met(X)$ of (pseudo)Riemannian metrics on $X$. The resulting action groupoid $Met(X)//Diff(X)$ can be regarded as the configuration groupoid of gravity on $X$.

‘ghosts’ in BRST were recognized in terms of the Chevalley-Eilenberg complex much later

Sure, Feynman didn’t speak about Lie algebroids of symmetries. Nevertheless, his idea of ghosts introduced the structures that today go by this name, and ever since his work (and that of his colleagues) do physicists know how to deal with physical systems with gauge symmetries. Today we even have a beautiful conceptual picture of the original ideas that makes it all appear utterly non-mysterious and completely transparent.

Posted by: Urs Schreiber on August 15, 2011 2:42 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

what about Fade’ev-Popov ghosts? did they build on Feynman or vice versa or independent?

Posted by: jim stasheff on August 16, 2011 12:51 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Good question. I was told that Fadeev and Popov were unaware of
Feynman’s work. On the other hand, Feynman said that he was aware of
the Fadeev-Popov notion but didn’t consider it worth mentioning.

In that case, we have two different takes on the subject. And
Fadeev-Popov certainly is worth mentioning if you’re trying to
perturbatively evaluate a Euclidean-space functional integral to obtain
Schwinger functions of *gauge-invariants* – which is one way, if not
the way, to address gauge theories.

Posted by: Greg Weeks on August 16, 2011 7:25 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> This is routine since the second half of the 20th century. The Lie
> algebroid of this Lie groupoid is called the BV-BRST complex and has
> been recognized to be the right way to speak about gauge theories ever
> since Feynman introduced the “ghost” field in
>
> Quantum theory of gravitation Acta physica polonica (1963)
> …
> We should have here a discussion on these points. But I think the
> “quilts” are standing in the way. We should interrupt the quiltologoy
> and instead have something like a small course on the conceptual
> basics of general relativity.

Do. But include other gauge theories while you’re at it. I’m afraid
that many standard QFT texts since 1963 have not recognized the right
way to speak about gauge theories.

(That sounded sarcastic, but in fact it merely says what it says.)

MEANWHILE, quiltology is dead to the poster of the article (me) *with
the exception* of a short post beginning with “One last try:”.

Posted by: Greg Weeks on August 15, 2011 8:21 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I wrote:

We should interrupt the quiltologoy and instead have something like a small course on the conceptual basics of general relativity.

Greg replied:

Do.

All right!

But include other gauge theories while you’re at it.

Discuss the configuration space groupoid, its infinitesimal approximation by the BRST complex, the critical locus of the Einstein-Hilbert action functional that carves out the covariant phase space from the configuration space, then in turn its infinitesimal approximation by the BV-BRST complex. All this slowly, in little bite-sized pieces.

In doing so, we can make use of the fact that in three dimensions the gauge theoretic aspects of gravity are particularly well accessible and thereby in the same go have a discussion of all of the above in general 3-d gauge theory, with gravity always being a special case to look at.

I imagine that I would make little bite sized posts once per week along these lines and each time you all jump in at the comment section and throw around questions in an uninhibited manner such as to try to sort things out. Making some notes along the way we could be able to produce a few nice $n$Lab entries this way, even.

I’m afraid that many standard QFT texts since 1963 have not recognized the right way to speak about gauge theories.

Not sure, the better ones certainly do.

There is of course the standard monograph:

Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems , Princeton University Press 1992. xxviii+520 pp.

In Weinberg’s The quantum theory of fields it is in section 15. The ghosts are in 15.6, the “BRST symmetry” in section 15.7.

Or in the lectures collected in Quantum fields and strings it appears here and there. For instance in lecture 3 of Witten’s contribution (p. 1159 of volume 2).

What I think is true is that the nice higher Lie-theoretic meaning of these formalisms which makes it all conceptually become very clear and transparent is not generally made very clear in the available texts. We’ll improve on that.

Posted by: Urs Schreiber on August 16, 2011 1:39 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> Mike is completely right: there is no conceptual problem at all with
> theories of physics with gauge symmetries, hence those whose
> configuration spaces are actually groupoids.

I’m not surprised that there are no conceptual problems at all with
theories of physics with gauge symmetries, although I don’t understand
“resolution”, since there is no problem to resolve.) I personally am
fairly happy conceptually with diffeomorphism equivalence classes
(although I don’t know if alternate point-sets should be allowed).

One of the many things that I hope to do in this lifetime *is* to
understand your take on the matter.

But, as John Baez has pointed out, conceptual okay-ness is not the
issue. He mentioned numerical integration and quantization. I will

I do acknowledge that I would get a warm, fuzzy feeling from the
description of geometric objects that I am seeking. But I don’t *need*
it conceptually. If my earlier comments indicated otherwise, they were
wrong.

Posted by: Greg Weeks on August 16, 2011 8:29 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

But, as John Baez has pointed out, conceptual okay-ness is not the issue. He mentioned numerical integration and quantization.

I am slightly perplexed by this. So you dreamed up your “quilts” with the motivation that it would help you do numerical integration and quantization of gravity?

I will simply add: especially quantization.

I am generally a bit puzzled by this attitude: on the one hand you repeat that the motivation for the discussion is nothing less than quantum gravity. On the other hand John tells me (here) that there is no chance that we can use terms like “coproduct” or “sheaf” in the discussion and the discussion itself revolves around the definition of smooth manifolds.

This does not match! This seems weird to me. And counterproductive. This is as if I said to you:

“Can you help me with this screwdriver here? No, not the power tool, I don’t want to handle that!

What I am going to do with the screwdriver? I want to construct a jumbo jet!

Because I think I know a better way to do it than the guys at Boeing and Airbus do. I really think they use the wrong tools, that’s probably why planes crash every now and then.

So can I have the screwdriver now?”

I’d rather wish we could soberify the discussion here.

Posted by: Urs Schreiber on August 17, 2011 12:24 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> So you dreamed up your “quilts” with the motivation that it would help
> you do … quantization of gravity?

Yes. I was hoping to replace the Cauchy-ambiguous manifold dynamics
with a Cauchy-unambiguous crystal-growth-y dynamics – which might be
easier to quantize.

Posted by: Greg Weeks on August 17, 2011 1:08 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Thanks, John and Urs; I think I understand: now that we understand manifolds and groupoids, the conceptual problem for classical general relativity, and for effective quantum gravity theories, is not a problem, but there are computational problems in both cases, and a conceptual problem remains for non-effective quantum gravity. (What is the antonym of “effective” when applied to a field theory?)

Posted by: Mike Shulman on August 16, 2011 1:13 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

now that we understand manifolds and groupoids, the conceptual problem for classical general relativity, and for effective quantum gravity theories, is not a problem, but there are computational problems in both cases,

Yes!

and a conceptual problem remains for non-effective quantum gravity.

Yes!

(What is the antonym of “effective” when applied to a field theory?)

One says: UV-complete theory . Here “UV” (“ultraviolet”) is the usual poetic term for “at high energies”.

For instance electroweak theory is a (partial) UV-completion of Fermi’s effective theory of weak interactions.

Or the string perturbation series gives a UV-completion of the effective Einstein-Hilbert theory of gravity (plus other fields).

Posted by: Urs Schreiber on August 16, 2011 1:52 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

(What is the antonym of “effective” when applied to a field theory?)

One says: UV-complete theory.

Now there’s a bit of jargon that’s easily comprehensible to an outsider!! (-:

(Thanks.)

Posted by: Mike Shulman on August 17, 2011 4:18 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

In particle physics ‘UV’ and ‘IR’ mean ‘ultraviolet’ and ‘infrared’, which in turn are buzzwords meaning ‘short-distance, high-frequency’ and ‘long-distance, low-frequency’, respectively.

So, an ‘UV-complete theory’ is one whose behavior is completely specified at arbitrary short distances, while an ‘effective theory’ is a theory that makes good approximate sense at sufficiently long distances, but may reveal itself to be a limiting case of some other theory at shorter distances.

Posted by: John Baez on August 17, 2011 5:36 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Actually I don’t think my definition of quilt above catches Greg’s ‘maximality’ condition (condition 5). But apart from that I think it matches his concept.

The problem is that I think Greg wants isomorphic Riemannian manifolds to give the same quilt, but I think they only give isomorphic quilts.

This is related to the problem of ‘tags’: does his procedure give a specific quilt from a Riemannian manifold, or just a quilt up to canonical isomorphism? It seems like the latter, since there’s an arbitrary choice of ‘tags’ in his construction. Given this, there’s no way isomorphic Riemannian manifolds can be guaranteed to give the same quilt. At best, we can hope they’re canonically isomorphic.

But of course, all this can be systematically studied only after choosing a notion of isomorphism for quilts, which Greg has not provided.

I’m afraid if we choose the most natural notion, we’ll find that the groupoid of quilts is equivalent via an anafunctor to the groupoid of Riemannian manifolds. In short, roughly: it’s just a reformulation of the same idea, with isomorphic Riemannian manifolds giving not the same quilt, but just canonically isomorphic ones.

Posted by: John Baez on August 14, 2011 12:15 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Actually I don’t think my definition of quilt above catches Greg’s ‘maximality’ condition (condition 5). But apart from that I think it matches his concept.

Then take the Čech groupoid of a maximal atlas. No problem.

The problem is that I think Greg wants isomorphic Riemannian manifolds to give the same quilt, but I think they only give isomorphic quilts.

The Čech groupoids $C(\{U^1_i\})$ and $C(\{U^1_i\})$ of any two atlases $\{U^1_i \to X\}$ and $\{U^2_i \to X\}$ are in general not isomorphic, but only equivalent as smooth groupoids in the right sense (anafunctors, stack morphisms): let $\{U^3_i \to X\}$ be any common refinement of $\{U^1_i\}$ and $\{U^2_i\}$, then there is a span of weak equivalences

$\array{ C(\{U^1_i\}) &\stackrel{\simeq}{\leftarrow}& C(\{U^3_i\}) &\stackrel{\simeq}{\to}& C(\{U^2_i\}) \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\simeq}} & \swarrow_{\mathrlap{\simeq}} \\ && X } \,.$

And so the equivalence class of $C(\{U_i\})$ in the category of smooth groupoids is precisely the isomorphism class of the manifold $X$ in the category of smooth manifolds: it’s diffeomorphism class.

Same holds true when we equip all of this with metrics or other structure.

(A quick way to see this is to note that equipping everything with Riemannian metrics here is the same as slicing the ambient topos of smouth spaces over the sheaf $Met$ of Riemannian metric. Then a manifold $X$ equipped with a Riemannian metric is a morphism $g : X \to Met$, and a Cech groupoid equipped with a metric is similarly a morphism $C(\{U_i\}) \to Met$. The above equivalences then sit over $Met$, but apart from this everything is as before.)

Posted by: Urs Schreiber on August 14, 2011 12:37 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

John wrote:

The problem is that I think Greg wants isomorphic Riemannian manifolds to give the same quilt, but I think they only give isomorphic quilts.

Urs wrote:

The Čech groupoids $C(\{U^1_i\})$ and $C(\{U^1_i\})$ of any two atlases $\{U^1_i \to X\}$ and $\{U^2_i \to X\}$ are in general not isomorphic, but only equivalent as smooth groupoids in the right sense (anafunctors, stack morphisms)…

That’s true, and it’s important. But, just to redeem my honor, I should say I was assuming Greg was getting a Čech groupoid from a maximal atlas on a manifold. Using this procedure, isomorphic manifolds give isomorphic Lie groupoids.

Posted by: John Baez on August 14, 2011 2:44 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

…assuming Greg was getting a Čech groupoid from a maximal atlas on a manifold. Using this procedure, isomorphic manifolds give isomorphic Lie groupoids

Actually I was confused a little bit on this point also. Actually what Greg called a ‘maximality’ condition is a minimality condition. The condition implies that in getting a quilt from charts on a manifold, if one has a chart that is covered by a bunch of other charts, throw it out. This of course is a massive source of non-canonicity!

Posted by: David Roberts on August 14, 2011 11:00 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Thanks for the clarification, David! I guess insofar as a minimum is a greatest lower bound, it’s also a maximum.

I’m not sure what throwing out ‘unnecessary’ charts achieves. It would be great if it made the set of quilts isomorphic to the set of isomorphism classes of Riemannian manifolds. However, I’m pretty sure it doesn’t achieve that.

As noted here, a nice direct description of the set of isomorphism classes of Riemannian manifolds, or Riemannian metrics on a fixed manifold, is a kind of ‘holy grail’ in general relativity — a holy grail that most people have decided is unattainable.

Here’s one failed but still interesting attempt: given a compact Riemannian manifold, let $S$ be the set of eigenvalues of the Laplacian on that manifold. Isomorphic compact Riemannian manifolds give the same set $S$. Can we recover a compact Riemannian manifold up to isomorphism from its set $S$?

No. For example, these two regions in the plane give the same set $S$:

Click for details.

Of course these aren’t compact manifolds, since they have boundary and even corners. But a long time before the above clever example was found, way back in 1964, Milnor found two different 16-dimensional tori with the same set $S$. Readers familiar with heterotic string theory will instantly and correctly guess that these tori are $\mathbb{R}^{16}$ modulo the two even unimodular lattices in 16 dimensions: $E_8 \times E_8$ and $D_{16}^+$.

Later, in 1980, Vignéras found a counterexample in 2 dimensions: two different Riemann surfaces of constant negative curvature with the same set $S$. His construction uses some fairly serious number theory!

So, we can’t recover a compact Riemannian manifold from the ‘spectrum of its Laplacian’, $S$.

A further attempt along this direction uses the Dirac operator instead of the Laplacian: this idea has been heavily pushed by Connes. I doubt you can completely recover a compact Riemannian spin manifold up to isomorphism from the spectrum of its Dirac operator, but I don’t know for sure… and anyway, people can do some very interesting things with this approach. If we can get ‘all the physics we need’, perhaps that’s good enough.

Posted by: John Baez on August 15, 2011 4:11 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> Finally, John, did I correctly hear you say some months ago that
> people *have* tried to achieve what I’m trying to achieve (more or
> less). I mean, it isn’t just *me*.

before I read this (by John):

> As noted _here_, a nice direct description of the set of isomorphism
> classes of Riemannian manifolds, or Riemannian metrics on a fixed
> manifold, is a kind of ‘holy grail’ in general relativity — a holy
> grail that most people have decided is unattainable.

So, you can ignore my question.

If it really is unattainable – ouch, ouch, ouch – then I agree with
Mike Schulman. Classical reality is a manifold. (Caveat, caveat,
caveat, caveat.) Pretty confusing for novices, though.

Posted by: Greg Weeks on August 15, 2011 6:05 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

A further attempt along this direction uses the Dirac operator instead of the Laplacian: this idea has been heavily pushed by Connes. I doubt you can completely recover a compact Riemannian spin manifold up to isomorphism from the spectrum of its Dirac operator,

But in Connes’ setup you remember not just the Laplace operator or Dirac operator, but also the algebra of functions. Together they form what is called a spectral triple. And that Connes has famously shown allows to reconstruct the Riemannian geometry.

Posted by: Urs Schreiber on August 15, 2011 10:07 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

wrong pronoun for vignéras!

Posted by: anon on August 15, 2011 2:37 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Anon wrote:

wrong pronoun for vignéras!

Whoops!

Posted by: John Baez on August 15, 2011 3:27 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

a nice direct description of the set of isomorphism classes of Riemannian manifolds, or Riemannian metrics on a fixed manifold, is a kind of ‘holy grail’ in general relativity — a holy grail that most people have decided is unattainable.

My impression from this discussion here is that it is the last statement above that has caused some confusion: it seems to have been read as saying that it is somehow conceptually difficult to capture the moduli space of Riemannian metrics and that some new ideas are needed to even grasp the concept.

I have briefly recorded the definition and some references at moduli space of Riemannian metrics.

While conceptually easy, of course these moduli spaces are highly complicated in detail. Quite a bit is known about them for 2-dimensional manifolds. In higher dimensions less is known. There are some results on the homotopy grous of moduli spaces of metrics of positive Ricci curvature (see the references at the above link).

Posted by: Urs Schreiber on August 15, 2011 10:08 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

JB wrote:

If you’re serious about studying symmetry and spacetime, you need category theory. I suggested posting your comment here because I could tell that your quilts are the kind of thing people use category theory to study. Indeed you are already doing category theory; you just don’t know it yet.

I have a knee jerk reaction to ‘need’ and ‘must’. Other than the emphasis on sorting out iso versus equiv versus…, why cat theory? Do you need English to participate in this forum?
M. Bourgeois Gentihomme

I like ‘quilt’ cf. Spencer’s ? description of deformation theory of complex manifolds as resewing grandma’s quilt.

Where in Greg’s definition is there something that sewing the quilt together produces a Hausdorff manifold?

Posted by: jim stasheff on August 14, 2011 1:31 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

John wrote:

If you’re serious about studying symmetry and spacetime, you need category theory.

Jim replied:

Other than the emphasis on sorting out iso versus equiv versus…, why cat theory?

Having spent two decades answering precisely this question in all my writings, including a major fraction of the 300 issues of This Week’s Finds in Mathematical Physics, how could I have anything more to say about this now?

I summarized my thoughts here, and then I quit working on this stuff. Now I work on environmental issues and theoretical biology. That’s where all my zeal has gone.

But anyway, my remark was mainly aimed at Greg Weeks, who was reinventing the theory of Čech groupoids, and could profit from learning what’s known about these.

Posted by: John Baez on August 14, 2011 2:22 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Uh, um,

6. Given patch points pp1 and pp2 that are not “the same”, there exist
patches p1 and p2 such that:

- B(p1, p2)(.) is the empty set

- p1 contains a patch point “the same” as pp1

- p2 contains a patch point “the same” as pp2

I just made that up, of course.

Posted by: Greg Weeks on August 15, 2011 3:59 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Is this 6. meant to be in conjunction to 1.-5.? It seems a bit contradictory to 5. Consider the manifold $S^1$, covered with some charts (or even a generic compact manifold). According to 6., we need a pair of disjoint charts to separate any two given points (so need a infinite number of charts), but by compactness of $S^1$, 5. means we should have at most a finite number of charts, because any open cover (say by charts) has a finite subcover.

Posted by: David Roberts on August 15, 2011 5:15 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> Is this 6. meant to be in conjunction to 1.-5.?

Yes. I *do* want Hausdorff geometries.

> It seems a bit contradictory to 5.

I botched #5 by using the word “can” to mean possiblity, while most
readers have interpreted “can” as permission. This is explained in a
top-level comment with the words “wretched communication problem”.

A should probably mention that, IMO (influenced by John Baez), the
uniquifying tags ruin “quilts”, and I’m looking for a fix or
replacement. (John called my goal a holy grail, and I haven’t given up
on it.)

Posted by: Greg Weeks on August 15, 2011 4:24 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Instead of tacking on more clauses in the definition of ‘quilt’ I’d find it a lot more helpful to know what main theorem is supposed to follow from this definition. That would let me understand the definition and see if it’s ‘working’.

I thought the theorem was supposed to be

‘quilts are in 1-1 correspondence with diffeomorphism equivalence classes of Riemannian manifolds’,

but maybe it’s supposed to be

‘isomorphism classes of quilts are in 1-1 correspondence with diffeomorphism equivalence classes of Riemannian manifolds’

which makes sense given definitions of ‘quilt’ and ‘isomorphism between quilts’.

Until I know the theorem Greg is shooting for, my interest in the subject will rapidly wane, since it’s like archery without a target.

Posted by: John Baez on August 15, 2011 7:48 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

John Baez (“you” henceforward) wrote:

> Until I know the theorem Greg is shooting for, my interest in the
> subject will rapidly wane, since it’s like archery without a target.

I am shooting for this, which you wrote recently:

> As noted _here_, a nice direct description of the set of isomorphism
> classes of Riemannian manifolds, or Riemannian metrics on a fixed
> manifold, is a kind of ‘holy grail’ in general relativity — a holy
> grail that most people have decided is unattainable.

I’m a bit confused by your confusion. You already wrote a rather
definitive comment including “I’m afraid you haven’t cracked the nut
that you’re trying to crack”. You knew what nut I was trying to crack,
and it hasn’t changed. The only difference is that I now seek a fix or
replacement for “quilts” instead of “quilts” themselves.

“quilts” as if I still had hope for them in their original form? I did
that because I feel obliged to clarify any misconceptions that I have
created. And thus I may have created a misconception in you. Irony!

————————————————————————-

Permit me to summarize *everything* that I’ve said.

1. I am seeking the holy grail that you described.

2. I had hoped that “quilts” were it.

3. I felt that the noncanonicalness of tags was relatively harmless,
given that their sole purpose is to uniquify.

4. You’ve 95% convinced me that I was wrong about #3.

5. I am now, with faint hope, seeking a fix or replacement for
“quilts”, along various directions. Can the basic idea be implemented
without tags? If the basic idea matches some pre-existing mathematical
entity, then does that suggest a fix? (This assumes that manifolds are
*not* involved in the pre-existing entity, because, if manifolds *are*
involved, the odds are good that the *charts* provide the identity of
the patches, which doesn’t help at all.) Is there any hope in in the
lovely words “Spaces probeable by G in the above sense can be very
general. They need not even have a concrete underlying space” from
http://ncatlab.org/nlab/show/space? (That was your original hunch some
weeks ago in private correspondence. Could it do the job?)

John, does this clarify what I’ve been up to?

Posted by: Greg Weeks on August 15, 2011 3:51 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

A wretched communication problem:

I wrote the geometry/quilt requirement/axiom #5:

> 5. No further patches can be “added” — you know what I mean — whose
> patch points are all “the same” as pre-existing patch points. (This
> requirement is intended to achieve maximality.)

I just realized that this is ambiguous and has been generally
misinterpreted. Does “can” mean permission or possibility? I quite
regret the confusion that has resulted. Here is #5 again:

————————————————————————
5. It is a requirement on (P, B) that it be *impossible* to add a patch
to (P, B) whose patch points are all the same as pre-existing points.
————————————————————————

The idea is that (P, B) is maximal: Once you’ve got enough patches to
fully specify the desired geometric information – which for simple
algebraic topologies is very few patches indeed – you continue adding
patches, consistent with the existing geometrical specification, until
you can’t add any more. It is quite analogous to maximizing an atlas.

Finally: Adding a patch p is a nontrivial operation. First you include
it in P, creating P_new. Then you augment the domain of B to become
(P_new x P_new), adding overlap bijections consistent with the rest of
the axioms. But hopefully that was obvious!

Again, I quite regret the initial ambiguity.

Posted by: Greg Weeks on August 15, 2011 3:06 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

A summary!

1. “The Cech groupoid associated with an atlas’ is not “quilts”.

2. “Quilts” are not the particular “holy grail” in general relativity that
John Baez mentioned. (You can search for “holy grail”.)

3. In my initial article, I took for granted that every reader would be
aware of this holy grail. (Wrong!!!!)

4. In my initial article, I had hopes that quilts would be this holy
grail. (Wrong!)

There still remains much of interest for me to read in the comments.
However:

>>>> Errors #3 and #4 above have basically doomed the discussion.

To undoom the discussion, you fine contributors would need to first
understand the holy grail that John (and I) have discussed and then
want to find it.

I would like for this to happen :-)

Posted by: Greg Weeks on August 15, 2011 6:16 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

One last try:

I think that we all understand the idea of describing a geometric object
by representing chunks of it with isomorphic chunks of Rⁿ (with fields!)
– and then sewing the overlapping representations together. Several
commenters have said that this notion is merely such-and-such. Good!

(Note, BTW, that the above doesn’t mention a point-set or an atlas.
It’s not a manifold. Done properly, it shouldn’t be “equivalent” to a
manifold. And it is a good thing to have. A holy grail, even.)

The problem is:

>>>> How do we canonically uniquify chunks of Rⁿ (with fields) that are
>>>> equal but represent different parts of the geometric object?

Solve this problem and I believe that you will have done something
wonderful, something that certain people want and have not been unable
to obtain.

Do any of the such-and-suches solve this problem? Any other ideas?

Posted by: Greg Weeks on August 15, 2011 8:00 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> something that certain people … have not been unable to obtain

Aw, crap.

Posted by: Greg Weeks on August 15, 2011 8:04 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Greg Weeks wrote:

The problem is:

How do we canonically uniquify chunks of Rⁿ (with fields) that are equal but represent different parts of the geometric object?

Am I correct to think that you want certain geometric objects to be “equal” and “distinct” at the same time? And if my understanding is correct, would you be willing to interpret “equal” as “isomorphic”? What I am trying to say is that category theory is more thoughtful in handling the issue of identity than “everything is a set with structure” kind mathematics that most people are taught.

Posted by: Eugene Lerman on August 16, 2011 1:31 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

You are on-target when you speak of the “everything is a set with
structure” kind of mathematics that most people are taught. That is my
kind of mathematics, and I am failing to get what I (and many others)
want.

It may be helpful for you know that the comment “One last try:” to which
you responded is self-contained and all-inclusive of one person’s
concerns (mine). The rest of the thread is what *led* me to “One last
try:”. But that’s not important right now.

In particular, the “I think” sentence is precise¹ (except for the notion
of “sewing overlapping representations together”). I envision a set C
of Rⁿ chunks (with fields) and a somewhat messy function-valued function
F that describes how the various Rⁿ chunks “overlap”. (Later, if you
want the specifics of F, see the top-level article, where chunks are
“patches”, the set C is “P”, and the function F is “B”.)

I hope that you fully grasp this idea for the description of geometric
objects – a simple, grubby analyst’s idea. If you don’t, then I fear
that you won’t be able to help.

The problem is that I want to use the “everything is a set with
structure” POV when I compare two descriptions (C1,F1) and (C2,F2). On
the other hand, two Rⁿ chunks that represent different chunks of the
geometric object must be distinct, *even if the two Rⁿ chunks are equal
in the “everything is a set with structure” sense*.

I originally tried to solve the problem by attaching arbitrary unique
tags to the Rⁿ chunks in C. But that renders the description
noncanonical (and ugly). The description of the geometric object must
be canonical.

Is the problem now clear?

Can you fix it? :-)

————————————————————————

The only hint that I’ve seen personally is the paragraph

>>>> Spaces probeable by G in the above sense can be very general. They
>>>> need not even have a *concrete underlying space*, even for general
>>>> definitions of what that might mean.

from “http://ncatlab.org/nlab/show/space”. This is hopeful because the
description of geometric objects above does not have an underlying
space. Indeed, this led John Baez to point me toward the paragraph
above. But I don’t remotely understand the web page.

————————————————————————

Footnote 1: Okay, neither “geometric object” nor “isomorphic” are
precise. But they are as precise as they can be under the
circumstances.

Posted by: Greg Weeks on August 16, 2011 7:10 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Can you fix it? :-)

It depends on what you would accept as a fix.

Let’s forget for the moment about geometric structures/fields since they confuse the issue a bit.

With this proviso (no fields), it seems that you want to redefine a manifold as a bunch of chunks of $\mathbb{R}^n$ sewed together by (partial) diffeomorphisms. For exampe $\mathbb{R}\ni x\mapsto \frac{1}{x} \in \mathbb{R}$ would be your version of $S^1$. In this case you need two copies of $\mathbb{R}$ which are and are not the same.

I originally tried to solve the problem by attaching arbitrary unique tags to the Rⁿ chunks in C. But that renders the description noncanonical (and ugly). The description of the geometric object must be canonical.

One low tech version of a fix is to redefine “chunk of $\mathbb{R}^n$” to mean any set $X$ together with a bijection $f : X \to U$, where $U$ is an open subset of $\mathbb{R}^n$,. Then your collection of “chunks” is a lot bigger, but they are no longer literally open subsets of the standard $\mathbb{R}^n$. Would you accept this? In the example above you could have $\mathbb{R}\times \{0\}$ and $\mathbb{R}\times \{1\}$ as your two chunks, neither of which is literally $\mathbb{R}^1$,together with the obvious maps $f_i: \mathbb{R}\times \{i\}\to \mathbb{R}$, $i=0,1$.

You could then say when two such chunks are isomorphic, i.e., “equal.” Thus I am attaching tags to subsets of $\mathbb{R}^n$: you may think of $f:X\to U$ as $U$ being tagged with the pair $(X, f)$. And $f:X\to U$ equals $g:Y\to U$ if $f\circ g^{-1} :U\to U$ is a diffeomorphism.

A higher tech version of the fix was mentioned previously —a class of proper etale geometric stacks over the category open subsets $\mathbb{R}^n$. It will amount to the same thing once the definition is expanded. It may take a few pages.

You may object that I am going back to your original proposal:

I originally tried to solve the problem by attaching arbitrary unique tags to the $\mathbb{R}^n$ chunks in C. But that renders the description noncanonical (and ugly). The description of the geometric object must be canonical.

In effect I am. But I am not bothered by two things that bothered you: “ugliness” and “noncanonical.”

Let’s leave ugliness aside. “Noncanonical” is a feature, not a bug — any one of your manifolds would have many atlases and all of them have equal rights. So we cannot single out one of the atlases. But the objects (“Cech cover groupoids”) the atlases define should be all isomorphic and they are, once you define isomorphisms correctly.

Posted by: Euene Lerman on August 16, 2011 8:47 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Thanks for taking a shot it it. However:

> “Noncanonical” is a feature, not a bug

Not for numerical integration. See:
http://golem.ph.utexas.edu/category/2011/08/geometries_diffeomorphism_clas.html#c039122

And quite possibly not for quantization.

(By the way, not only do I see “noncanonical” as a bug, but I also fail
to see it as a feature. Take manifolds, for instance. The atlas is
usually required to be maximal. That’s a *good* thing, I would say.
Does anyone disagree?)

Posted by: Greg Weeks on August 17, 2011 12:52 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

(By the way, not only do I see “noncanonical” as a bug, but I also fail to see it as a feature. Take manifolds, for instance. The atlas is usually required to be maximal. That’s a good thing, I would say. Does anyone disagree?)

Once you go from manifolds as “sets with structure” to manifolds as “Cech cover groupoids” it becomes a feature.

Or to make the same point slightly differently — manifolds are special cases of orbifolds. Orbifolds, properly defined as stacks, don’t have maximal atlases because they are not sets with structure. And the Cech groupoid point of view takes manifolds as stacks. Hence my claim that it’s a feature.

As far as this comment goes: it is interesting that

the solution at future times is unique only up to diffeomorphism.

It reminds me of Richard Hepworth’s work on integrating vector fields on geometric stacks: their flowlines are unique only up to an isomorphism.

I don’t know if anyone tried to integrate numerically vector fields on stacks, but it sounds like a toy version of the problem you’re dealing with.

Posted by: Eugene Lerman on August 17, 2011 2:47 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> > Take manifolds, for instance. The atlas is usually required to be
> > maximal. That’s a good thing, I would say. Does anyone disagree?)
>
> Once you go from manifolds as “sets with structure” to manifolds as
> “Cech cover groupoids”[, noncanicalness] becomes a feature.

Interesting. The nPOV can clash *directly* with the analyst’s POV. (And
in this milieu, the latter is simply wrong.) Again, interesting.

Posted by: Greg Weeks on August 17, 2011 4:06 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

The atlas is usually required to be maximal. That’s a good thing, I would say. Does anyone disagree?

Have you ever tried to write down a maximal atlas explicitly? I think you’ll pretty much always end up giving some non-maximal atlas and then saying “and add in all the charts compatible with these”. Given that we only care about manifolds up to diffeomorphism anyway, and we can define smooth maps and diffeomorphisms easily based on just giving an arbitrary (non-maximal) atlas, why bother throwing in all that mess of extra charts?

Posted by: Mike Shulman on August 17, 2011 4:16 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Precisely so that a manifold was not defined as an equivalence class - no?

Posted by: jim stasheff on August 17, 2011 12:38 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

why bother throwing in all that mess of extra charts?

Precisely so that a manifold was not defined as an equivalence class - no?

It is more practical simply to check that the two atlases have a common refinement, and not go directly to the maximal atlas. I for one do not “usually require” the atlas to be maximal. I regard the insistence on this requirement as mostly a matter of personal aesthetics; I don’t see how it’s really important in mathematical practice.

Another way of saying it: two atlas structures on a space $M$ produce the same manifold structure if $id: M_1 \to M_2$ is a diffeomorphism.

Posted by: Todd Trimble on August 17, 2011 1:50 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Or, you could simply not demand that “manifold structures on a given set” form a poset, but be content with a preorder. What’s the difference, really, between two atlases on a set defining “the same manifold structure”, versus the identity map being a diffeomorphism between them?

Posted by: Mike Shulman on August 17, 2011 5:53 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I am interested in this and may like to try to contribute some thoughts, but I wonder if Greg might first consider writing a follow up post summarizing what he’s learned and what he’s currently thinking?

I tend to view discussions like this with a tilt toward abstract finitary geometry. Why begin with a continuum of points? Could you consider quilts in terms of some kind of Hasse diagram somehow? Maybe there is some graph theory lurking in here.

Posted by: Eric on August 17, 2011 3:35 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> I am interested in this and may like to try to contribute some
> thoughts, but I wonder if Greg might first consider writing a follow
> up post summarizing what he’s learned and what he’s currently
> thinking?

I’d love to.

The main problem is explaining my goal (which is the goal of others as
well). (I will try to do better below. Please be patient.) John Baez
tried to help. But, to my knowledge, no other person sees the goal as
desirable. Consequently, the ambient brain-power around here has not
been directed toward the goal.

Quilts were an *attempt* at the goal. I had hopes that they were a
successful attempt, but John Baez convinced me otherwise (95% anyway).

Several people pointed out what quilts actually were, both in the
context of being derived from a manifold and being stand-alone. But
once John “killed” quilts, it didn’t matter.

John wanted me to rigorize the construction of quilts from manifolds and
the construction of manifolds from quilts, and to show that the
construction mapped isomorphism classes to isomorphism classes. But I
didn’t know what he wanted, so even *he* was disappointed. Anyway,
since he “killed” quilts, it no longer matters except as an exercise.

So, finally, the goal: The ultimate goal is quantum gravity. As John
pointed out, quantization is sensitive to how you describe classical
reality. I’ll mention, for example, that you can get different quantum
mechanics from very simple classical systems simply by transforming your
(q,p) variables! Thus we are forced to *choose* how we represent our
classical systems based on vague “niceness” criteria. Don’t laugh.
That’s how it is!

In the context of Einsteinian gravity, I wanted to replace the
Cauchy-ambiguous manifold dynamics with a Cauchy-unambiguous
crystal-growth-y dynamics. (Why? The “niceness” criteria.)

To obtain Cauchy-unambiguity, you must get rid of the a-priori point-set
of manifolds, which are the source of active diffeomorphisms. Quilts
did that. In order to satisfy the “niceness” criteria, you want to have
a concrete analytical object at hand, rather than a huge equivalence
class of some kind. Quilts did that. In order to satisfy *my*
“niceness” criteria, the geometric object should have points, although
derived, not a-priori. This way we can talk about a “Cauchy strip” and
“points being *grown* by the dynamics”. This gets me my
crystal-growth-y dynamics. Quilts did that.

I hope you’re getting the idea. I wanted space-time to be represented
by a canonical geometrical object that could be ripped apart along a
Cauchy “surface” and then grow back the identical geometric object via
the dynamics of Einsteinian gravitation. Then I would consider
quantization.

But quilts failed because of the tags. My geometric objects are not
canonical, because of the tags. And the dynamics are not
Cauchy-unambiguous, because of the tags.

Here, then, is one path toward “my” goal: Implement the idea behind
quilts – ie, describing geometric objects by sewing together chunks of
Rⁿ (with fields) that are isomorphic to chunks of the object – without
the tags. In particular, make the description canonical.

The other path to “my” goal is to replace quilts with something entirely
different that achieves the goal above: gravitational dynamics as
(canonical) “crystal growth” from (canonical) initial Cauchy strips.

I should point out that John has said that most people feel that the
above goal – which is an instance of a somewhat broader goal – is
unachievable. I, though, am driven by the thought that quilts were so
simple and seemed so close to the goal that there must be a way.

Now that the goal is, I hope, reasonably clear, my guess is that it
isn’t quite your cup of tea. But there’s a chance.

Posted by: Greg Weeks on August 17, 2011 5:23 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> Now that the goal is, I hope, reasonably clear, my guess is that it
> isn’t quite your cup of tea.

That was a good guess. Well, I misunderstood a couple of things when
I posted.

First, I interpreted “n-Category Café” as meaning “a group blog on math,
physics, and philosopy” rather than “a group blog on math, physics, and
philosopy from the nPOV”. Yes, the name was a hint. But it wasn’t
explicit enough for me.

My second misunderstanding was that the “tags” in “quilt” “patches” were
tolerable.

So get rid of the tags. Now, pick a patch and start exploring it. At
any point in your exploration, there will be a set of overlapping
patches to which you can switch if you like. So make the switch, and
add the second patch, along with its particular overlap with the first
patch, to your domain of exploration. And continue expanding your
domain of exploration, patch by patch.

So, a “geometry” – the term that I preferred all along, will be a set
of “patches” and a set of “domains of exploration” such that:

1. Every patch is a domain of exploration.

2. For every domain of exploration, there is a well-defined set of
additional overlapping patches that can be explored, any one of which
can be added to obtain an expanded domain of exploration.

Obviously, this *at best* needs a lot of work.

Posted by: Greg Weeks on August 22, 2011 2:02 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

Here are some comments. Most of them start with the word “No”. Imagine me saying it in a polite tone. But it’s still a no.

To obtain Cauchy-unambiguity, you must get rid of the a-priori point-set of manifolds, which are the source of active diffeomorphisms.

No, that makes no sense! Several comments here have tried to explain to you that your worries about “a-priori point sets of manifolds” are based on a misunderstanding of what a manifold is. Or, I think, even what a set with structure is.

Quilts did that.

No, they certainly did not.

In order to satisfy the “niceness” criteria, you want to have a concrete analytical object at hand, rather than a huge equivalence class of some kind. Quilts did that.

No, they did not! At best they made the situation worse, as they introduced lots of extra choices.

In order to satisfy my “niceness” criteria, the geometric object should have points, although derived, not a-priori.

This distinction does not sensibly exist.

This way we can talk about a “Cauchy strip” and “points being grown by the dynamics”. This gets me my crystal-growth-y dynamics. Quilts did that.

This is really a weird statement.

Then I would consider quantization. But quilts failed because of the tags.

No. Quantum gravity is far remote from what you are discussing here. But not just because of “tags”.

Here, then, is one path toward “my” goal: Implement the idea behind quilts – ie, describing geometric objects by sewing together chunks of $\mathbb{R}^n$ (with fields) that are isomorphic to chunks of the object – without the tags.

As was pointed out here by several commenters: such a sewing of manifolds by $\mathbb{R}^n$s is the most standard operation in the world. Several equivalent (and sensible) ways of going about it have been suggested here. And standard textbooks discuss it, of course.

gravitational dynamics as (canonical) “crystal growth” from (canonical) initial Cauchy strips.

Statements like this do sound crack-pottish. You oscillate in this discussion here between being mixed up about the definition of a manifold and indulging in unfounded speculations about quantum gravity. And you ignore explanations posted here.

Where is this conversation headed? I see it going in circles.

Posted by: Urs Schreiber on August 22, 2011 10:12 AM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

I wonder if people realise just how extraordinarily unlikely it is that they will be able to use some ideas they’ve worked up by themselves to make a significant contribution to mathematics or mathematical physics. There are simply too many things that have to go right and so much prior thought to cross check.

In this case, a mastery of existing work on smooth spaces is necessary for starters.

Posted by: David Corfield on August 22, 2011 1:25 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

My feeling about this is that ‘points’ are bad as they are not observable (so you throw out the manifolds), what are you left with …. the observations! Those and the interrelationships between them are the building blocks so surely the idea of maximal atlases is a red herring. Maximal atlases can not be observed. The Cech groupoid type of construction can as that encodes how the observations interact. (Long live finitary geometric models. :-))

Posted by: Tim Porter on August 17, 2011 5:48 PM | Permalink

### Re: Geometries: Diffeomorphism Classes vs Quilts

> observable (so you throw out the manifolds), what are you left with
> …. the observations!

What you say is true of classical mechanics. But quantum mechanics,
alas, is not based on what we, in reality, observe. At least in special
relativtiy, quantum mechanics is based on what we might *conceivably*
observe, quite independent of the actual state of things. In classical
mechanics, I can talk about the curvature at the tip of my nose right
now. In quantum mechanics, that is not an observable. Or, rather, the
formulation of the quantum theory uses only general-purpose
observables, which can be measured in any state.

And in general relativity, those are hard to come by. References to my
nose are not general-purpose.

(BTW, I’m still for points, but many people say what you say about them.)

Posted by: Greg Weeks on August 17, 2011 6:05 PM | Permalink