## March 29, 2007

### This Week’s Finds in Mathematical Physics (Week 248)

#### Posted by John Baez

In week248 of This Week’s Finds, see movies of coronal mass ejections, auroras, and tornados on the Sun!

Then, continue reading the Tale of Groupoidification — in which we see how spans of groupoids arise naturally in geometry.

The picture above is taken by the recently launched satellite Hinode. It shows filaments of plasma moving along the magnetic field lines around a sunspot.

Here’s some more eye candy: pictures of coronal loops taken by TRACE – the Transition Region and Coronal Explorer:

Posted at March 29, 2007 2:44 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1219

### Re: This Week’s Finds in Mathematical Physics (Week 248)

I’m really liking this Tale you’re spinning. Looking forward to more.

Posted by: John Armstrong on March 29, 2007 4:05 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

I’m glad to see incidence geometry as spans has gotten off the ground. ;)

Posted by: David Roberts on March 29, 2007 6:32 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Very interesting. I’m looking forward to seeing where this is going.

Hope you don’t mind my pointing out a couple of typos: in the definition of “span of groupoids”, you have “it should preserve preserves all the structure…”

Then a few lines later in the example you have “… we can rotate the whole picture and get a new point and a new line containing a new plane”. Not sure if this one is a typo, or just my misunderstanding: shouldn’t it be “contained in a new plane” (i.e. the points of the line are a subset of the points of the plane, so the line is contained within the plane)? Or am I just misreading this sentence?

Posted by: logopetria on March 29, 2007 8:38 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

All the typos you caught are really typos! I’ll fix them. Thanks.

I actually love it when people point out typos and other mistakes in This Week’s Finds — I like to get these things fixed.

Posted by: John Baez on March 29, 2007 5:50 PM | Permalink | Reply to this

### Magmas?

I am confused about the term groupoid. The name suggests something which is almost a group, and that was apparently the original definition: a group but the binary product is not everywhere defined. Wikipedia mentions both this definition and your categorical one. Are they equivalent? There was some discussion about groupoids versus magmas at WP.

### Re: Magmas?

Another thing that’s not clear: I remember discussion about “groupoids” being the thing to use for studying “semi-symmetries”. IIRC, a semi-symmetry was something like the finite pattern “||||||||” being almost symmetric - in the sense of mapping almost onto itself - under translation by one bar-separation, with the issue being that for the finite pattern the end bar doesn’t have anything to map onto. This was related to the fact that you could have a tranlsation morphism (?) T which was defined for all the bars except the final one, and likewise for the inverse T.

Is this a separate concept that also goes under the name groupoid, a different application of the same mathematical gadget “groupoid” or an aspect to the story about spans and groupoids? I’ve noodled about with the framework in the TWF and as far as I can see your morphisms in the definition of a groupoid could fit with the partial symmetry, but then I can’t see how with your F and G functors you don’t end up with “full” symmetries. (Maybe it’s that in your concrete example of incidence geometry all symmetry-type things are full symmetries?)

Whilst trying to see if I understood the concepts properly, I tried to come up with another concrete groupoidfication example. I think if you work with numbers you can define “not being coprime” as the relationship and your witness being one of the non-unity common factors. Then, say, multiplying everything by a value X again gives you a valid relation, ie, it’s a symmetry. But maybe dividing by a value X is a semi-symmetry: when your witness is divisible by X your statement remains valid, but in some cases you witness and all the other possible witnesses won’t be divisible by X, so it’s not a “full symmetry”. Is this on the right lines?

Posted by: dave tweed on March 29, 2007 10:16 AM | Permalink | Reply to this

### Re: Magmas?

Replying to myself about the obvious flaw: whilst multiplying by a value X keeps all existing relationships valid (ie, 4 and 6 with witness 2 becomes 8 and 12 with witness 4 when multiplied by 2), it also makes everything that wasn’t “not coprime” into “not coprime”, so it can’t be a global symmetry.

Posted by: dave tweed on March 29, 2007 10:24 AM | Permalink | Reply to this

### Re: Magmas?

Alan Weinstein’s Groupoids: Unifying Internal and External Symmetry gives a good little introduction to groupoids. You can see that the two Wikipedia definitions Thomas mentions are equivalent. (The other sense of ‘groupoid’ - merely a set with a binary operation - is clearly different.)

A situation similar to dave tweed’s finite number of bars is also treated. This is a finite patch of tiles, whose symmetries are considered in terms of a transformation groupoid (page 746).

Posted by: David Corfield on March 29, 2007 12:28 PM | Permalink | Reply to this

### Re: Magmas?

I am confused about the term groupoid.

The term was originally used by Oystein Ore back in the 1930s to refer to a set with a binary operation. That usage is still quite common; its MSC classification number is 20N02. What is now called just “groupoid” used to be called a “Brandt groupoid”. You can find the noncategorical definition in, for instance, R.H. Bruck’s classic Survey of Binary Systems.

I’m not sure when exactly it happened, but eventually these just came to be called “groupoids”. That might be due to the influence of Bourbaki. The replacement term “magma” for sets with a binary operation is certainly a Bourbakism (Bourbaki-ism?).

Folks who work a lot in quasigroups, loops, and the like (MSC 20N05) still tend to use “groupoid” in its classical sense. I work in that area, but have coauthored a paper with one of the main proponents (Weinstein) of the modern sense of the term, so I switched to “magma”. I still get annoyed referee reports insisting I switch back to “groupoid”.

Posted by: Michael Kinyon on March 29, 2007 5:57 PM | Permalink | Reply to this

### Re: Magmas?

It sounds like you’re talking about two ways to define a groupoid: as a set with partially defined multiplication and totally defined inverses satisfying some axioms, and as a category where morphisms have inverses. As the Wikipedia explains, these definitions are equivalent if one gets the axioms right in the first approach.

There’s also an unrelated, archaic definition of groupoid: ‘a set equipped with an arbitrary binary operation’. I urge everyone to avoid this confusingly different usage. The preferred term for such an entity is ‘magma’, which nicely captures the feel of this raw, primeval concept.

Posted by: John Baez on March 29, 2007 6:02 PM | Permalink | Reply to this

### Re: Magma…?

John Baez wrote:
The preferred term for such an entity is ‘magma’

Yes, but do you prefer “magmas” or “magmata” for the plural? :-)

Posted by: Michael Kinyon on March 29, 2007 6:35 PM | Permalink | Reply to this

### Re: Magma…?

Oh the jokes to be made…

I’ll just say that since I’m a proponent of “topoi”, I’d have to say “magmata”.

Incidentally, while I like the “primal” explanation, I think the original sense of the term actually captures it better. Magma is what’s left over after a semi-liquid substance (say, treacle) has had the water pressed or evaporated away. A magma is what’s left of a semigroup after we boil off the associativity, like a rack is the “wrack and ruin” after we rip composition from a group to leave only conjugation.

Posted by: John Armstrong on March 29, 2007 7:17 PM | Permalink | Reply to this

### Re: Magma…?

So what is left of a category after we boil off the associativity? A reflexive digraph? Maybe it needs a more “primal” name.

Posted by: Michael Kinyon on March 29, 2007 7:33 PM | Permalink | Reply to this

### Re: Magma…?

Michael wrote:

So what is left of a category after we boil off the associativity?

Then you’ve got the multi-object analogue of a magma. Since the multi-object analogue of a group is a groupoid, and similarly for ringoids and algebroids, what you’ve got is a… magmoid!

All part of the general philosophy of oidization.

Posted by: John Baez on March 29, 2007 7:44 PM | Permalink | Reply to this

### Re: Magma…?

JB wrote:
All part of the general philosophy of oidization.

Alan Weinstein and I used to call it “oidification” in our email exchanges, but I won’t quibble.

Posted by: Michael Kinyon on March 29, 2007 8:03 PM | Permalink | Reply to this

### Re: Magma…?

Hmm.. to keep the chemistry theme maybe we should go back and start talking about groupoxides as examples of oxidation?

Posted by: John Armstrong on March 29, 2007 10:13 PM | Permalink | Reply to this

### Re: Magmas?

The campaign to rename groupoids-in-the-other-sense to magmas really bothers me. Almost no one who studies the other kind of groupoids calls them magmas. Since they named them groupoids first, who are we to tell them they can’t call them that anymore? It’s one of the purest exercises of power politics in mathematics that I’ve seen.

Posted by: Walt on April 5, 2007 5:56 PM | Permalink | Reply to this

### Re: Magmas?

Brandt used the term gruppoid in 1926 in our sense of category with inverses, clearly as a variant on gruppe. I wonder when the first translation as groupoid occurs.

Would you be as bothered if those interested in the other sense of groupoid insisted that German speakers use gruppoid their way?

Posted by: David Corfield on April 5, 2007 8:50 PM | Permalink | Reply to this

### Re: Magmas?

I’m happy with both fields using the term groupoid. I’m happy if the people who work with non-Brandt groupoids spontaneously say “Hey, magma is a great term. Let’s use that.” I’m not happy with practitioners of one field dictating the terminology of the other field.

Posted by: Walt on April 6, 2007 4:19 AM | Permalink | Reply to this

### Re: Magmas?

Walt wrote:

Since they named them groupoids first, who are we to tell them they can’t call them that anymore? It’s one of the purest exercises of power politics in mathematics that I’ve seen.

Ah yes — those big bad category theorists, bossing everyone around! Yet another example of how category theorists run the show in American academia. Not.

Seriously, it’s very common in mathematics for people to urge their favorite terminology on other people. However, as long as this takes the form of public rhetoric, it’s the exact opposite of ‘power politics’, because there’s no ‘power’ backing up the argument except the power of persuasion. Anyone can tell anyone else what to do – but in the end, everyone can do whatever they want! I find nothing wrong with this.

We only have a case of ‘power politics’ if, for example, a referee refuses to accept a paper unless the author changes his terminology. This I would find more obnoxious (though sometimes justifiable, e.g. if somebody calls prime numbers ‘composite’.)

As for my little joke about ‘category theorists running the show in American academia’, it’s worth noting that category theory is so downtrodden that the NSF almost completely refuses to fund work on this subject. Which is why you can count the official ‘category theorists’ in the US on one hand. In Europe the situation is different. So: funding is where power politics really comes into play…

Posted by: John Baez on April 5, 2007 9:06 PM | Permalink | Reply to this

### Re: Magmas?

I don’t know of a single other example of where people outside of a field, people who have no interest in that field, have renamed the main concept of the field. Is there really another example? I find that surprising, but I’m prepared to be corrected.

As unpopular it may be with the NSF, category theory is a much more fashionable branch of mathematics than binary operations. (And in my opinion, rightfully so. Groupoids-as-binary-operations is not a field near and dear to my heart.)

In certain hands, rhetoric is power. Bourbaki is so influential, that people will accept the term magma as the rightful term out of ignorance. Look at the discussion on Wikipedia about whether to call them magmas or groupoids – it mainly consists of people who know nothing about the existing literature on the subject, but who’ve heard that Bourbaki or someone influenced by Bourbaki called them magmas, so we should call them magmas. (Someone actually used the argument that if it was good enough for Serre, then it’s good enough for them. How is that not power?)

In your own comment, you treat the name change as a fait accompli. If you knew nothing about the subject other than your comment, you would come away with the impression that the actual name that everyone uses now is magma, and that calling them a groupoid is “archaic”. You are an influential figure. You don’t think you have the power to shape perception of mathematical reality?

Posted by: Walt on April 6, 2007 4:15 AM | Permalink | Reply to this

### Re: Magmas?

Walt wrote:

In certain hands, rhetoric is power.

[…]

You don’t think you have the power to shape perception of mathematical reality?

I hope I do, and I’m certainly not shy of trying to do it. It isn’t the kind of power that I associate with your phrase ‘pure power politics’. I’m not rejecting papers or withholding funding from people who use the term ‘groupoid’ in some way I don’t like. I’m just advocating my views! But you’re right, I shouldn’t just tell people to use ‘groupoid’ the way I like. I should explain my reasons.

I’ve done it before — but I’ll do it again:

I think the term ‘groupoid’ is a poor descriptor of ‘a set with a binary operation’. A set with a binary operation is so much less than a group, less than a semigroup or even a monoid, less than a loop or even a quasigroup, that calling it a ‘groupoid’ — something like a group — is misleading. A set with a binary operation is really not very much like a group at all! It’s something very basic, though. ‘Magma’ seems as good a term as any.

A category with inverses, on the other hand, really is very much like a group, so the name ‘groupoid’ is very fitting. Many basic results about groups extend nicely to groupoids! Furthermore, the term fits into a nice set of analogies:

group groupoid

ring ringoid

algebra algebroid

which has the potential to become a major pattern in mathematics: replacing structures that are secretly one-object categories by their many-object versions.

Posted by: John Baez on April 6, 2007 6:29 PM | Permalink | Reply to this

### Re: Magmas?

I don’t think there’s anything wrong with calling Brandt groupoids just groupoids, just I don’t have any problem with the fact that ring theorists mean one thing by “algebra”, and universal algebraists mean another. I object to the idea that the terminology of a field can be imposed from the outside. For example, algebraists have normal subgroups while differential geometers have normal bundles. It would bother me if group theorists announced “we’ve renamed normal bundles to volcanic bundles”. It would bother me more the more influential algebraists were relative to differential geometers.

Sets-with-binary-operations is unfashionable a field as I can imagine. I just picture someone who is unfortunate enough to fall in love with such a field, who ends up in a non-tenure track job at Podunk U, and then is told, “We don’t think your field is very interesting. Plus, we changed all of your terminology.”

Posted by: Walt on April 7, 2007 4:33 AM | Permalink | Reply to this

### Re: Magmas?

Walt wrote:

Sets-with-binary-operations is unfashionable a field as I can imagine. I just picture someone who is unfortunate enough to fall in love with such a field, who ends up in a non-tenure track job at Podunk U, and then is told, “We don’t think your field is very interesting. Plus, we changed all of your terminology.”

For what it’s worth, I’m far more interested in sets-with-binary-operations than most mathematicians. So, my desire to call sets-with-binary-operations ‘magmas’ is simply due to a desire for appropriate terminology. And, I don’t think the first name something is given is likely to be the best name it’ll ever get! We need to throw out old names occasionally — if we didn’t, we’d still be using Roman numerals. How do we decide when to do this? The right approach is to let people argue about what’s right and see what happens.

Posted by: John Baez on April 7, 2007 7:38 AM | Permalink | Reply to this

### category theoretic funding

I have to say, judging from Urs’ reports, category theoretic mathematical physics seems to be blooming in Europe. And I remember John Power telling us in Minneapolis that things were better for category theoretic computer science in Europe.

What evidence is there on the pure front? And is there an explanation for this continental difference? The dominance of your pragmatic philosophy?

Posted by: David Corfield on April 6, 2007 11:00 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Presumably at some point one will need to generalise from the two-ness of spans. E.g., if you’re interested in a type of flag composed of figures in several dimensions. Is there a term for, say, a three-legged span? Then would a degroupoidification form a kind of three-dimensional matrix?

This span business doesn’t seem totally unrelated to what statisticians are doing when they study the correlation of two variables, e.g., a sample of people and a pair of maps to weights and heights. Again one can look at higher-order correlations.

Posted by: David Corfield on March 29, 2007 12:49 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Then would a degroupoidification form a kind of three-dimensional matrix?

Perhaps ‘tensor’ would have been better.

Posted by: David Corfield on March 29, 2007 1:12 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

David wrote:

Perhaps ‘tensor’ would have been better.

You got it. Jim, Todd and I use the awkward term ‘multispan’ to mean an object $S$ equipped with $n$ different morphisms

$f: S \to X_i$

By the process of degroupoidification, multispans of groupoids give rank-$n$ tensors. These turn out to be very useful.

The curious thing is that normally, rank-$n$ tensors are covariant in some slots and contravariant in others. When we groupoidify, the distinction between covariant and contravariant disappears.

I’ll let you guess why.

But you’re making me give away some of the fun to come…

Posted by: John Baez on March 29, 2007 6:12 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

This might be more “giving away some of the fun”, but there’s an asymmetry in the definition of groupoidification given so far that I don’t quite understand. Specifically, I don’t see why we’re only replacing relations between X and Y with spans S:X->Y, and not cospans C:X->Y (i.e. in the case of sets, a set C with functions from X and Y).

In one of the examples last week, you said “every Russian has exactly one favorite Frenchman and one favorite Englishwoman” making that Russian a witness to a relation between that Frenchman and that Englishwoman. But likewise, we may suppose that every Frenchman and every Englishwoman has a favourite Russian; if Gérard and Emma are both fans of Dostoevsky, doesn’t he witness a relation between them? [Moreover, if we change “favourite Russian” into “favourite weepy movie” it might end up being a very important relation between them!]

So does everything work out the same if we swap cospans for spans? I strongly suspect it doesn’t, but I haven’t worked out why not yet. If not, and if ‘multispans’ (and presumably ‘multicospans’) are worth studying, can we also get anything interesting out of things that have arrows to some of the sets and from the others?

Of course, if this is all part of the fun that you want to save for later, I’d be content with an infuriatingly tantalising hint instead of more detail!

Posted by: logopetria on March 29, 2007 8:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Well cospans are just spans in the opposite category, so when we decategorify we should get the same set, no? In some cases (knot theory) cospans seem to be more natural, but I don’t think it actually changes much. Of course, I could be wrong…

Posted by: John Armstrong on March 29, 2007 10:11 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Logopetria wrote:

So does everything work out the same if we swap cospans for spans? I strongly suspect it doesn’t, but I haven’t worked out why not yet.

Ah, interesting! Cospans play a conceptually very different role in this theory than spans, but they’re also important. We’ll see them fairly soon, I hope.

Of course, if this is all part of the fun that you want to save for later, I’d be content with an infuriatingly tantalising hint instead of more detail!

Okay, that’s what I’ll do:

We’ll often want to work with groupoids ‘over a base’: that is, groupoids equipped with a morphism to a fixed groupoid $B$. A span in the category of groupoids over a base will involve both a span of groupoids and a cospan of groupoids! Imagine a commutative diamond with downwards-pointing arrows, with $B$ at the base.

Why work with groupoids ‘over a base’? I claim that when we study groupoids on which a fixed group (or groupoid) acts, we’re studying groupoids over a base.

The details of how this works may serve as a fun puzzle, if you’re not already familiar with it.

By the way: one reason I’m dropping so many ‘infuriatingly tantalizing hints’ in response to questions is that if I blow all my expository energy answering questions, I’ll never finish telling the Tale!

Posted by: John Baez on March 31, 2007 3:15 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

David wrote:

Think of all the fun to be had groupoidifying equations such as $A V_k = S_{i j} T_{i j k}$, plugging indices together by equivalence of groupoid, is it?

Yes, it’s lots of fun! Next, imagine that all the coolest sorts of linear algebra — like the theory of simple Lie algebras, quantum groups, and their representations, turns out to groupoidify to give a fascinating new view on incidence geometry! Even more fun!

More technically:

Indices are plugged together by the groupoidified version of the ‘inner product’ — for a quick explanation see page 7 of Jeff Morton’s talk at CT ‘07.

This, by the way is the answer to my puzzle about why we don’t need to worry about the difference between covariant and contravariant tensors: the vector spaces obtained by degroupoidification are all Hilbert spaces, so we can raise and lower indices at will — so it’s no sin to leave them all lowered, as you’ve done.

But we’ll have to be patient.

You can be impatient if you like: knowing someone’s impatient makes it a little easier, psychologically at least, to find more time to write This Week’s Finds.

And perhaps when done, Klein 2-geometry will appear quite straightforward.

The groupoidification project, which was largely pushed through by Jim and Todd, has a curious similarity to our Klein 2-geometry project. You’ll keep seeing this more and more as I go on. But, the groupoidification project has been a lot more sucessful so far in making contact with other branches of math. Maybe that’s because it had a more solid conceptual motivation. In our Klein 2-geometry work, we were just messing around, trying to categorify everything in sight — sort of a scattershot approach.

Posted by: John Baez on April 1, 2007 10:45 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

In our Klein 2-geometry work, we were just messing around, trying to categorify everything in sight — sort of a scattershot approach.

But it was scattershot at a barn door from twenty paces, wasn’t it? Or at leasting finding sand in the Sahara.

Posted by: David Corfield on April 6, 2007 11:08 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

I just caught one of those dizzying glimpses you sometimes offer us. Think of all the fun to be had groupoidifying equations such as $A.V_k = S_{ij}T_{ijk}$, plugging indices together by equivalence of groupoid, is it?

But we’ll have to be patient. And perhaps when done, Klein 2-geometry will appear quite straightforward.

Posted by: David Corfield on March 30, 2007 12:12 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Regarding my last paragraph, what should we make of efforts to introduce probabilistic ideas in topics such as algebraic topology. E.g., Betti numbers of random manifolds looks at the expectations of Betti numbers. It begins

In various fields of applications, such as topological robotics, configuration spaces of mechanical systems depend on a large number of parameters, which typically are only partially known and often can be considered as random variables. Since these parameters determine the topology of the configuration space, the latter can be viewed in such a case as a random topological space or a random manifold. To control such a system one has to understand geometry, topology and control theory of random manifolds.

One of the most natural notion to investigate is the mathematical expectation of the Betti numbers of random manifolds. Clearly, these average Betti numbers encode valuable information for engineering applications; for instance they provide an average lower bound for the number of critical points of a Morse function (i.e. observable) on such manifolds.

If measures can be placed on Grassmanians, as in Klain and Rota’s wonderful lectures, might there be scope in the groupoidified version?

Posted by: David Corfield on April 2, 2007 9:29 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Talking of symmetry, and weather on astronomical bodies, have you seen the giant hexagon on Saturn?

Posted by: Dan Piponi on March 29, 2007 10:16 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

I saw this linked from Bad Astronomy in one of the comments. Am I horribly off base to assume it’s likely from a spherical harmonic with roughly hexagonal symmetry? It really does look a lot like some of their graphs that I’ve seen.

Posted by: John Armstrong on March 29, 2007 10:50 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Rossby waves on Earth are vaguely similar, and apparently have 4-6 ‘lobes’. But it’s hard to imagine something like this maintaining a regular 6 sided figure over 25000km for (at least) tens of years.

Posted by: Dan P on March 30, 2007 6:03 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

I’d always assumed those spectacular pictures of coronal loops as something that required billions of dollars worth of equipment to see, and indeed the pictures you show did. So I was amazed to find that you can buy a hydrogen-alpha telescope for about \$500 and see them for yourself. Obviously it’s not quite the same as a view from a satellite, but through a H-α filter you can clearly see solar prominences and as often as not you can also see that they form faint wispy loops. It’s quite amazing to see these things with your own eyes!

Posted by: Dan Piponi on March 29, 2007 10:31 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

I’ve been on a solar binge since reading Week 248, and Dan P’s telescope comment. If any cafe goers are interested the New York amateur astronomy soc will have some H-alpha telescopes set up in Central Park next Saturday (weather permitting).

The Sun is indeed quite having its day in itself. There was an interesting BBC article today too. (And some nice sound samples to download)

Posted by: Allan E on April 21, 2007 4:47 AM | Permalink | Reply to this

### Wilson lines

I think I got the groupoid idea. The set of open Wilson lines is a groupoid, because two Wilson lines can only be glued if they have a common endpoint. In contrast, the set of gauge transformations is a group. So one can have a group acting on a groupoid.

### Re: This Week’s Finds in Mathematical Physics (Week 248)

It would have been nice if material on E8 had been covered in week 248 rather than week 247: it would have been a rather spooky coincidence, but as it was it wasn’t to be.

Posted by: Ian Stopher on March 30, 2007 10:25 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Maybe you didn’t read all the way to the end of week248?

Posted by: John Baez on March 31, 2007 3:28 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Groupoids, functors and natural transformations form a 2-category. Presumably spans of groupoids form some sort of a weak 2-category?

If the groupoids are topological groupoids or Lie groupoids, then choosing the arrows to be functors is much too restrictive. The problem is that fully faithful and essentially surjective functors may have no continous inverses. So one has to localize at such functors (variously called essential equivalences or weak equivalences). The most common solutions seems to be: replace functors by bibundles. The price is that one loses associativity of 1-arrows. That is, groupoids, bibundles and equivariant maps are a weak 2-category… I presume this is coming up in the new installments?

Posted by: Eugene Lerman on March 31, 2007 2:54 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

If we are going to demand composition of arrows, can anyone explain this puzzle in Sloane’s? It is asbout semigroups, not groupoids, so there is more structure, not less, but still puzzling.

A071536 Extensions to a semigroup of the (categorical) composition of arrows in the complete directed graph on n labeled nodes.

1, 1, 3, 3, 147

OFFSET

0,3

COMMENT

Terms obtained, especially the 147, lend some support to the peculiar conjecture that the composition in any category can be extended to a total, associative operation, that is to a semigroup (not, of course, a monoid).

EXAMPLE

For n=2, arrows 0:0->0, 1:1->1, f:0->1, g:1->0, the self-dual solution (commutes with reversing arrows) has multiplication table (rows and columns indexed by 0, 1, f, g in order) with rows: 0 f f 0; g 1 1 g; 0 f f 0; g 1 1 g.

KEYWORD

hard,nonn,nice

AUTHOR

F. Lockwood Morris (lockwood(AT)ecs.syr.edu), May 29 2002

Posted by: Jonathan Vos Post on March 31, 2007 5:31 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

Incidence is very interesting indeed. I am looking forward to the relation between incidence and Hecke operators/algebra. Especially since I heard that these operators are closely related to more common objects like combinatorial laplacians (discrete laplacian operators on a graph, depending on the indidence matrix!). So while laplacians (on discrete or continuus domains) are very relevant (and understood) for graphs, physics and spectral geometry (in noncommutative geometry length is even defined using the spectrum of the Dirac operator), Hecke operators seem (to me…) to be analogous to this in arithemetic and algebraic geometry?? At least, this seems to be suggested in James Arthur: The Principle of Functoriality”, Bulletin of the AMS v.40 no. 1 October, 2002. So, if this is true I hope we might get some basic intuitive understanding of arithmetic/algebraic theories (like Langlands program, Mordell-Faltings etc.) using concepts better known to ordinary applied mathematicians (graphs, laplacians and geometry). Just posing a very naive question: in geometry/graph theory the idea of hyperbolicity (genus of the surface>1 ) is very important and it is related to the (spectrum of the) laplacian, I believe that in arithmetic/algebraic geometry (e.g. Mordell-Faltings theorem) the same phenomenon pops up? Genus>1 means a significant change in the behavior (numbers of solutions) of the algebraic solutions. Is this also related to the incidence structure and a relevant Hecke algebra? Just curious….

Posted by: lauret on March 31, 2007 4:25 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 248)

This relation to Hecke operators should be pretty interesting. I once grew fond of the fact that lots of “duality” transformations in the literature are all examples of 2-linear maps i.e. (morphisms of 2-vector space) obtained by having a vector bundle on a span (also, now, known as a bi-brane) and performing pull-push through this correspondence: I once listed some examples in Fourier-Mukai, T-Duality and other linear 2-Maps. The Hecke transformation in geometric Langlands is among these, at least in its “classical limit”.

And in fact that makes good sense and fits very nicely into the very big picture, parts of which is the picture drawn by Kapustin/Witten, which becomes, quite clear once you match it to the general theory of 2-vectorial duality in 2-dimensional QFT, as hinted at in Duality and defects in rational conformal field theory.

Posted by: urs on March 31, 2007 10:17 PM | Permalink | Reply to this
Read the post The n-Café Quantum Conjecture
Weblog: The n-Category Café
Excerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
Tracked: June 8, 2007 6:26 PM

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