## March 23, 2007

### This Week’s Finds in Mathematical Physics (Week 247)

#### Posted by John Baez

In week247 of This Week’s Finds, read about symmetry — from the appearance of quasicrystals in medieval Islamic tile patterns:

to Tale of Groupoidification.

The first picture, of the Darb-i Imam shrine in Isfahan, Iran, is from the source material for this paper:

The second picture, of the $\mathrm{E}_8$ root system, was created by John Stembridge.

Posted at March 23, 2007 11:30 PM UTC

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### Re: This Week’s Finds in Mathematical Physics (Week 247)

I am really excited by the “Tale of Groupoidification”. Can’t wait for the next TWF!

Posted by: Bruce on March 24, 2007 12:29 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Let $X$ be the set of people on Earth. Let $T$ be the $X \times X$ matrix corresponding to the relation “is the father of”. Why does the matrix $T^2$ correspond to the relation “is the grandfather of”?

Sorry to be pedantic, but I guess that should be “is the paternal grandfather of”.

Posted by: David Corfield on March 24, 2007 10:55 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Not pedantic — accurate! I’ll fix that.

Posted by: John Baez on March 24, 2007 9:45 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

I’ve read up on E8 and about symmetry in general over the last couple of days. Leaving me with a question I can’t seem to find much information about. Is there a relation between symmetry and duality? If so can we n-ify the concept of duality to n-dimensions and do symmetry groups play a role in this n-ification.

Excuse me if this sounds like a stupid question I am but a self-taught computer programmer, with a like to read stuff that is beyond my comprehension.

Posted by: niels on March 25, 2007 6:48 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

‘Duality’ means about a hundred different things. For example, there’s:

and many more.

So, I’m not sure exactly what you mean when you ask:

Is there a relation between symmetry and duality?

but the answer is surely yes! All these different dualities are related to the group with two elements, usually called $\mathbb{Z}/2$ — ‘the integers mod 2’. That’s because a duality is a kind of operation such that if you do it twice, you get back where you started!

By the way, when I say there are lots of kinds of duality, that doesn’t mean they’re unrelated. There are actually lots of fascinating relations. A project I’d love to do someday (though I’ll probably never have time) is write a paper about all these dualities, and try to unify them into one single sort of duality.

Or, even better: two sorts of duality, which are related by a duality!

Anyway, there is a lot of interesting stuff to say about $n$-categories and the generalizations of the concept of ‘dual vector space’, and we talk about this a lot here. In fact, I need to finish answering a question Bruce Bartlett asked about this subject! I think I’ve made some progress on it…

Posted by: John Baez on March 25, 2007 11:49 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Or, even better: two sorts of duality, which are related by a duality!

You mean, “a braiding”? :D

Posted by: John Armstrong on March 26, 2007 12:07 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

John said “That’s because a duality is a kind of operation such that if you do it twice, you get back where you started!” Just to point out the (probably well-known) fact that some “concrete” dualities (by which I mean precise recipe for going to the dual space) double duality gives you an “object” isomorphic to your original “object”. A well-known signal processing example is the Fourier transform (and its discrete relatives), (eg, this page). One thing I’ve wondered about: is there in general any structure to the sets of objects which can be related by some number of applications of a concrete double-duality? (beyond this double-dual defintional property, of course). Probably not.

Posted by: dave tweed on March 26, 2007 4:56 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Thanks you for your response, John. As I understand you well, there is as of yet no unified notion of duality, despite relations between known dualities. This kind of answers the other question as well, doesn’t it? I was looking for a higher (dimensional?), (order?) notion of duality, leading to triality, quadrality etc. Googling for this only lead me to Spinors and Trialities of your own hand and an article from 1949 with the title The n-ality Theory of Rings. Is there a more or less introductory text on the subject?

Posted by: niels on March 26, 2007 6:00 PM | Permalink | Reply to this

### Duality of dualities

Or, even better: two sorts of duality, which are related by a duality!

An example of this is discussed in the string theory literature:

The “fundamental string” (i.e. that string which gives string theory its name) and the “solitonic string”,also known as the D1-brane (which arises as a localized boundary condition for the fundamental string) can be dual to each other in suitable backgrounds.

If this is the case, one finds that the T-duality of the fundamental string corresponds to the S-duality of the D1-string.

This phenomenon is addressed as a duality of dualities.

See for instance table 1 on p. 5 of

I gather that there is a way to understand this duality of dualities as an instance of what is called U-duality, but I don’t know how this works.

A more mathematical discussion of this phenomenon can be found here (see towards the end of p. 21).

Posted by: urs on March 26, 2007 6:23 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

The dual space of the dual space of a Banach space (bidual or double-dual) may properly contain the original space. Or more correctly (apropos of Dave Tweed’s comment) the bidual contains a canonically isometric copy of the original space, which may or may not be the entire bidual. I wonder whether this sort of phenomenon arises for any of the sorts of duality more likely to come up in this blog.

Posted by: Mark on March 27, 2007 3:49 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

I wonder whether this sort of phenomenon arises for any of the sorts of duality more likely to come up in this blog.

Oh surely. In fact, it’s everywhere you look!

Consider Galois theory: he extension field of elements fixed by the members of the group fixing the elements of a given extension field may be a strictly larger field. Or in intuitionistic logic, not-not-P may not imply P. Of course, You’ve already given the example of a double-dual for infinite-dimensional vector spaces.

In general, this sort of thing happens when you’ve got adjoints floating around, and (IIRC) all three of these are instances of a special kind of adjoint called a “Galois correspondence”.

Posted by: John Armstrong on March 27, 2007 5:59 AM | Permalink | Reply to this

### Duality of dualities

dual space of the dual space of a Banach space (bidual or double-dual)

By the way, I’d say this is, while interesting, not what one would understand under “duality of dualities”.

Rather, if $\array{ & \nearrow\searrow^d \\ A && B }$ is a duality between $A$ and $B$ and if $\array{ A && B \\ & \searrow\nearrow_{d'} }$ is another duality between $A$ and $B$, then a “duality of these dualities” would be something relating $d$ and $d'$, i.e. a 2-morphism $\array{ & \nearrow\searrow^d \\ A &\Downarrow^f& B \\ & \searrow\nearrow_{d'} }$ such that $d \stackrel{f}{\to} d'$ is a duality.

I am pretty sure that this is what John had in mind, when he wrote the above.

Posted by: urs on March 27, 2007 3:26 PM | Permalink | Reply to this

### Re: Duality of dualities

I think the comments about double-duality arise from the comment that duality is related to symmetry via $\mathbb{Z}_2$ by “applying duality twice gets you back where you started”. This turns out to be subtle: I knew that double dualities can lead to isomorphic objects rather than identical ones, and I’ve discovered from other comments that some infinite dimensional spaces lead to double-duals which are strictly larger. So the definition of symmetry needs to be formulated in terms of this; indeed part of why I mentioned it is that this “not equal but isomorphic/containing” relationship under double duality might correspond to some nice categorical notion (which is beyond me). (Incidentally, IIRC Greg Egan’s science fiction novel Diaspora has the non-identity of double duality as one of plot its devices towards the end. Won’t spoil it any more.)

All this is a different thought to the “duality of dualities” comment.

Posted by: dave tweed on March 27, 2007 4:08 PM | Permalink | Reply to this

### Re: Duality of dualities

I think the comments about double-duality arise from the comment that duality is related to symmetry via Z/2 by “applying duality twice gets you back where you started”.

That is indeed what I had in mind. The notion of duality of dualities certainly sounds 2-categorical, and I’m happier staying at the level of the category of Banach spaces and thinking about just the one functor. (Although to be honest I rarely think about anything using the words “category” or “functor”.)

I guess the point is that duality of Banach spaces is in general related not to the group Z/2 but to the semigroup N.

Posted by: Mark on March 28, 2007 12:30 AM | Permalink | Reply to this

### Re: Duality of dualities

We were chatting about duality back here and here. I was wondering (somewhere) what kind of entity that dualizing object in the second post, $V$, could be, e.g., whether it could be a category such as Sets, or Connes’ cyclic category. No doubt in Nirvana we’ll find $\omega$-categories in that dualizing role.

Posted by: David Corfield on March 29, 2007 8:45 AM | Permalink | Reply to this

### Re: Duality of dualities

The second link above is broken, so unfortunately I’m missing the context of most of David C.’s comment.

But looking at the first thread linked above, right away I see “Fourier duality” which is closely related to duality for Hilbert spaces, which is the best behaved special case of duality for Banach spaces. (This is not exactly true, because of complex conjugation issues, but is morally true.)

With that in mind, I wonder how common it is that nice Z/2 dualities (double-dual of X is X again) turn out to be special cases of duality in some more general context in which duality isn’t so well-behaved (double-dual of X may properly contain X). Going back to John A.’s remarks, if my rather dim memory of Galois theory is accurate, the duality there is better behaved if one makes some topological assumptions. And within classicial logic not not P is equivalent to P, so I guess that example fits in here too.

Is all of this part of some larger story? Without attempting to put things in category-theoretic terms (I’d only embarrass myself) I’m thinking along the following lines: one defines a notion of duality which satisfies certain axioms, for which the double-dual of X is the same type of thing as X, and is canonically comparable to X (for example, if X is a set, it has a canonical injection into its double-dual). Then those X which are “equal to” their double-dual form a class worth study on their own.

In the Banach spaces example this leads to looking at reflexive spaces, not just Hilbert spaces. Hilbert spaces are (again ignoring pesky complex conjugation) canonically their own duals. But it’s unclear whether some analogous phenomenon could arise for other types of duality, as for example in Galois theory, for which X and the dual of X are objects in different categories.

Posted by: Mark on March 29, 2007 2:54 PM | Permalink | Reply to this

### Re: Duality of dualities

Posted by: David Corfield on March 29, 2007 3:18 PM | Permalink | Reply to this

### Re: Duality of dualities

In the Banach spaces example this leads to looking at reflexive spaces, not just Hilbert spaces. Hilbert spaces are (again ignoring pesky complex conjugation) canonically their own duals. But it’s unclear whether some analogous phenomenon could arise for other types of duality, as for example in Galois theory, for which X and the dual of X are objects in different categories.

Oh, but duals of Banach spaces do live in a different category: the opposite category! Remember that duality isn’t just an assignment of another space, it’s a functor. The fact that the functor turns linear transformations backwards is very important here.

The same thing does happen in Galois theory, and in all other “Galois connections”. The fact that the two dualizing functors are adjoint gives the canonical maps you’re talking about, called the “unit” and “counit” of the adjunction. The objects that are (isomorphic to) their own double-duals are generally called “closed”, and yes the closed subgroups and closed intermediate fields are very important in Galois theory.

Now, before someone comes along to cut me off at the knees, yes I know that the duality for Banach spaces is not, strictly speaking, a Galois connection. However it’s not that far off (imho), and the important thing is that they’re both described by adjoints.

Posted by: John Armstrong on March 29, 2007 4:57 PM | Permalink | Reply to this

### Re: Duality of dualities

Oh, but duals of Banach spaces do live in a different category: the opposite category! Remember that duality isn’t just an assignment of another space, it’s a functor.

So I take it the self-duality of Hilbert spaces (let’s say real Hilbert spaces so we can forget about complex conjugation altogether) should be formalized using the functor from the category of Hilbert spaces to its opposite category which simply reverses all the arrows. (I further assume this functor has a standard name.) Then the extra-special niceness of duality for Hilbert spaces can be summarized as something like:

“The composition of the duality functor with the arrow-reversing functor is, up to a natural transformation, the identity functor on the category of Hilbert spaces.”

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

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Concerning the idea of groupoidification, which you say involves replacing vector spaces by groupoids and linear maps by spans of groupoids:

spans in some category (like $\mathrm{Set}$) form themselves a weak 2-category.

Spans in some 2-category (like $\mathrm{Grpd}$) should form themselves even a weak 3-category.

For instance, monads in spans in groupoids are nothing but double groupoids, I think.

So it would seem that replacing $\mathrm{Vect}$ by $\mathrm{Span}(\mathrm{Grpd})$ is actually an act of categorifying twice – unless you want to divide out isomorphisms.

Actually, I am not sure yet that I understand what the special role of groupoids is in this program: why not consider replacing linear maps by spans in arbitrary (finite, probably) categories?

You motivate groupoids as the categorification of non-negative rational numbers, like sets categorify natural numbers.

But we have learned from Tom Leinster that general finite categories generalize this even to possibly negative rational numbers. So if I wanted to categorify a matrix with arbitrary rational entries, I would think of considering spans in finite categories, instead of just in groupoids.

Or not? Why not?

By the way, what I find very inspiring here is this:

over at our discussion of the canonical 1-particle I complained about how it is not clear to me precisely what 2-category should replace $\mathrm{Vect}$ when we think of the 1-particle as coupled to a vector bundle with connection, but keeping in mind that the corresponding parallel transport should be a pseudofunctor, such that its sections are functors, such that they are subject to Tom’s theorem.

I was thinking about replacing $\mathrm{Vect}_\mathbb{C} = \mathbb{C}-\mathrm{Mod}$ with the 2-category $\mathbb{C}\mathrm{Set}-\mathrm{Mod}$. But maybe I should think of replacing it just with $\mathrm{Span}(\mathbb{C}\mathrm{Set})$.

That would actually nicely harmonize with the fact that the quantum theory obtained from that also involves spans for expressing its linear maps.

Posted by: urs on March 26, 2007 6:50 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

This also fits with the tangle hypothesis. I mostly work down with 1-tangles, but the ideas should generalize.

One of my major points is that tangles are really cospans of (3,1)-manifolds (3-manifolds with embedded 1-manifolds), and that the best knot invariants are really restrictions of cospans of “topological” functors that preserve pushouts.

For example, we can get the group of a link by applying the cospan of the fundamental group functor to the complement of a tangle in the cube to get a cospan of groups. If the tangle is just a link, then the two “wings” of the cospan are trivial and the only nontrivial information is the link group in the middle.

I’d like to see more categorifications via (co)spans. In particular, I’d love to see a (co)span categorification of a bracket-extending functor (equivalent to Khovanov or not) whose topological functor might give a better idea of what the topological content of the bracket is.

Posted by: John Armstrong on March 26, 2007 7:18 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Urs writes:

Actually, I am not sure yet that I understand what the special role of groupoids is in this program: why not consider replacing linear maps by spans in arbitrary (finite, probably) categories?

This is a very natural question: why not generalize from groupoids to categories, or $n$-groupoids, or $n$-categories? The best answer is one that will seem more convincing after I’ve explained more about the program. Namely: while generalizing further is tempting, a vast amount can be done just by going up to groupoids! So much, in fact, that it’s going to take a long time to explain it all.

Posted by: John Baez on March 26, 2007 4:53 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

while generalizing further is tempting, a vast amount can be done just by going up to groupoids!

Okay, sure.

Maybe one aspect of my question was: in which sense is groupoidification the right term for what you have in mind? Instead of, say, span-ification?

Form the hints you have given so far, it seems to the essential step is that to spans, whereas, as you say, the restriction to groupoids is more a convenience.

Posted by: urs on March 26, 2007 6:35 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

A big and beautiful story can be told where we take large portions of mathematics that seems to involve linear operators between vector spaces, and reveal that it’s really about spans of groupoids. This story really doesn’t seem to demand further generalizations of the span idea. So, James Dolan calls ‘groupoidification’, and that’s what I’m calling it too. But, neither of us thinks this term is perfect — ‘spanification’ might equally good (or bad). Probably an even better name is ‘Hecke theory’, or ‘geometric representation theory’.

But, the best way I can explain what I mean is to write the next issue of This Week’s Finds — and that’s what I’ll start doing right now!

Posted by: John Baez on March 27, 2007 12:34 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Reading your word “spanification”: I’d like to add the short description of possible categorification of rationals (positive+negative). This is similar to your groupoidification, but using directed graphs between two elements (or maybe simply two-objects categories), 0 and 1.

Let’s write N-arrows set from 0 to 1 as
(0 [+N] 1),
and K-arrows set in opposite direction similarly:
(0 [-K] 1).
Whole set of arrows:
(0 [+N][-K] 1).
Now we only need the way to make isomorphic following constructions:
(0 [n] 1) and (0 [N][n][-N] 1).
After that we may only add groupoidal structure on the set of arrows to obtain rationals.

Posted by: osman on April 2, 2007 6:51 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

Tony Smith found a nice picture created by Günter Ziegler of the D4 root system (that is, the 24-cell) viewed from the Coxeter plane. The 6-fold symmetry is evident:

Posted by: John Baez on March 27, 2007 12:41 AM | Permalink | Reply to this

### Uncountable axiom of choice?

Thanks for the article.

I am afraid if the following question is out of context: if one has to assume the uncountable axiom of choice, then we can prove that the Penrose set of quasi-crystals have the same cardinality of the continum. But how do go about constructing an effective bijection between them? Is it, in general, possible to construct such a bijection?

Posted by: Nagu on March 31, 2007 6:39 AM | Permalink | Reply to this

### Re: Uncountable axiom of choice?

Nagu writes:

[Given the axiom of choice] we can prove that the Penrose set of quasi-crystals have the same cardinality of the continum. But how do go about constructing an effective bijection between them? Is it, in general, possible to construct such a bijection?

Hmm. What sort of set whose cardinality is that of the continuum are you wanting to put into bijective correspondence with the set of Penrose tilings in an effective way? Just something like $2^\mathbb{N}$, I suppose? I guess if I think hard I do know what an effectively computable map from or to this space means

A brief web search of this topic reminded me of something I wrote in week175:

• Alain Connes, Andre Lichnerowicz and Marcel Paul Schutzenberger, A Triangle of Thoughts, AMS, Providence, 2000.

There is only one mistake in this book that I would like to complain about. Following Roger Penrose, Connes takes quasicrystals as evidence for some mysterious uncomputability in the laws of nature. The idea is that since there’s no algorithm for deciding when a patch of Penrose tiles can be extended to a tiling of the whole plane, nature must do something uncomputable to produce quasicrystals of this symmetry. The flaw in this reasoning seems obvious: when nature gets stuck, it feels free to insert a defect in the quasicrystal. Quasicrystals do not need to be perfect to produce the characteristic diffraction patterns by which we recognize them.

But that’s a minor nitpick: the book is wonderful! Read it!

Now I’m wondering, exactly what theorem about uncomputability and Penrose tilings are Connes and Penrose using? If it’s strong enough it might give a negative answer to your question.

Posted by: John Baez on April 9, 2007 9:44 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

This is very cool stuff but it should have been in week 248 since I wouldn’t be so late to the discussion and E8 likes 248 better anyways. I never thought before of thinking of the A-D-E root lattice projections as having N-fold symmetry before. That’s interesting. I noticed the ones I have (see my web page) are all 4-fold symmetry (or two sets of two , left-right and top-bottom). This makes my D3 the only one in a Coxeter plane. It’s nice seeing the general idea that the plane you project into can be important. I ended up with 4-fold symmetry I think cause I was trying to see Triality with the 4-fold coming from how Tony Smith gets a Quaternion out of Triality by adding the Adjoint (bosons) to the Vector (spacetime) and two half Spinors (fermions) of D4. For root lattices the adjoint plus vector looks like a spacetime with two levels of time (past-future) if that makes any sense.

Posted by: John G on April 5, 2007 5:24 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 247)

David Richter has built a Zome model of a 3d projection of the E8 root system.

John Stembridge has created pictures of many more root systems, projected onto their Coxeter planes.

Posted by: John Baez on April 14, 2007 5:02 AM | Permalink | Reply to this
Read the post Calculations Inside Semisimple Categories
Weblog: The n-Category Café
Excerpt: Bruce Bartlett on computations in semisimple categories.
Tracked: May 18, 2007 12:26 PM
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 11:41 AM
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 11:41 AM
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 2:08 PM

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