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February 12, 2008

Geometric Representation Theory (Lecture 25)

Posted by John Baez

This time in the Geometric Representation Theory seminar, we showed how to categorify the commutation relations between annihilation and creation operators for the harmonic oscillator:

aa *=a *a+1 a a^* = a^* a + 1

obtaining an isomorphism of spans of groupoids:

AA *A *A+1 A A^* \cong A^* A + 1

This reduces a basic ingredient of quantum field theory to pure combinatorics, not involving the continuum in any form.

Even better, we did an in-class experiment demonstrating these commutation relations!

  • Lecture 25 (Jan. 22) - John Baez on groupoidifying the harmonic oscillator. The annihilation and creation operators as spans of groupoids. The groupoidified commutation relations: AA *A*A+1 A A^* \cong A* A + 1 Demonstration: an actual experiment proving these commutation relations! Weak pullbacks for composing spans of groupoids. Examples of weak pullbacks. Using weak pullbacks to compute AA *A A^* and A *AA^* A.

Posted at February 12, 2008 6:39 PM UTC

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Re: Geometric Representation Theory (Lecture 25)

In the lecture notes it says:

First question: how do we compose spans of groupoids? Second: how do we add them?

The answer to the second question has been postponed to next week, I assume?

Would you mind giving a hint for someone too lazy to go through this and figure it out?

Did you think about groupoidifying the exponentiated creation and annihilation operators exp(A) \exp( A ) and exp(A *) \exp(A^*) ?

Given that you can say what a sum of spans is, can you say what a series Id+A+12AA+16AAA+ \mathrm{Id} + A + \frac{1}{2} A \circ A + \frac{1}{6} A \circ A \circ A + \cdots of spans is?

Posted by: Urs Schreiber on February 12, 2008 7:19 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Urs wrote:

First question: how do we compose spans of groupoids? Second: how do we add them?

The answer to the second question has been postponed to next week, I assume?

Yes. But, I’m not trying to keep it secret! Given spans

XSY X \leftarrow S \to Y

and

XTY X \leftarrow T \to Y

their sum is

XS+TY X \leftarrow S + T \rightarrow Y

where S+TS + T is the ‘disjoint union’ (or ‘coproduct’) of the groupoids SS and TT, and the arrows are defined in the obvious way.

In quantum mechanics, an operator gives a ‘transition amplitude’ to go from some state to some other state. The transition amplitude is a number, but now we’re groupoidifying, so it becomes a groupoid. And, the operator becomes a span of groupoids:

XSY X \leftarrow S \to Y

This gives a groupoid of ways to go from any object of XX to any object of YY.

When we add operators in quantum mechanics, we add transition amplitudes. For us, this amounts to adding groupoids.

Think of the double slit experiment — but consider the possibility that our photon has not just an amplitude to get from the light source to any point on the wall, and not even just a set of ways to get from here to there, but a groupoid of ways. That’s the physical intuition behind the math here.

Posted by: John Baez on February 12, 2008 9:13 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Great, thanks.

That reminds me: my latest thinking # about higher gauge theory has made it clear to me once again (I had talked about it before #):

The set U(1)U(1) is an illusion.

What it really is is the groupoid //\mathbb{R}//\mathbb{Z}.

sitting in the short exact sequence of 2-groups

1//B1. 1 \to \mathbb{R} \to \mathbb{R}//\mathbb{Z} \to \mathbf{B} \mathbb{Z} \to 1 \,.

Posted by: Urs Schreiber on February 12, 2008 9:40 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Even better, we did an in-class experiment demonstrating these commutation relations!

Is there, or can there be, some experiment that can distinguish between categorified and ordinary QM?

Posted by: Thomas Larsson on February 14, 2008 7:03 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

This is a great question! I can’t wait to see what everybody thinks. Surely, nobody thinks that there can never be such an experiment in principle. My own perspective is that quantum mechanics gives us accurate answers to anything that we ask it, except when gravity starts to play an important role — then it doesn’t give us any answers at all. Maybe we’re just not being clever enough, but I think it’s more likely that a new formulation of quantum mechanics is needed, and so it’s in the realm of quantum gravity that I hope experiments would arise that would distinguish between bog-standard quantum mechanics and any more fundamental variant, categorified or whatever it turns out to be.

But I wish I knew exactly what these experiments were, and precisely how any new version of quantum mechanics gives rise to different answers than conventional quantum mechanics! When we have answers to this, then we’ll really be getting somewhere.

Posted by: Jamie Vicary on February 14, 2008 9:10 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Is there, or can there be, some experiment that can distinguish between categorified and ordinary QM?

QM is 1-dimensional QFT.

nn-fold categorified QM should be nn-dimensional QFT.

My opinion. But I have some data to back this up.

(But let’s not mix up nn-fold categorification here with other internalizations, like second quantization).

I’ll voice another opinion, as I have done before:

the perception of category-theoretic reasoning among physicists would be in better shape, had there been

- more efforts in the past to work out how much of what physicists are already familiar with and fond of is secretly already a higher categorical structure, just waiting to be fully realized such as to reveal its full power, and

- less efforts to point out how exotic the generalizations are which can also be reached by turning the higher categorical crank.

In a better world, we would already be able to reply to Thomas’ question:

the nn-categorical description of QM differs from the standard one in that it is less mysterious; in that it explains the huge mysteries that contemporary physicist’s have got used to simply accepting – and then takes us further.

In London, Louis Crane made this point very beautifully using the path integral as an example. And that’s probably the example.

It’s all about understanding renormalization. And the two major tools for this which have emerged are both, more or less secretly (not all that secretly, actually) \infty-categorical:

1) BV-formalism #

2) Connes-Kreimer renormalization method #

The first one is a technique in Lie \infty-algebroids. The last one in operads.

Posted by: Urs Schreiber on February 14, 2008 11:40 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Here’s a late evening kind of a question. Would the success of this categorified mathematical physics tell us something about the way the theory hooks onto the world?

There are several options to choose from, including

  • The theory is simply an expression of how the world is.
  • The theory is an expression of how we interact with the world.
  • The theory is an expression of what we can know about the world. E.g., Fuchs/Caves.

Now does phrasing things category theoretically change anything?

Bob Coecke seems to promote the second option: “monoidal categories constitute the actual algebra of practicing physics”.

Posted by: David Corfield on February 18, 2008 12:16 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

You seem to be conflating ‘categorified mathematical physics’ with ‘phrasing physics category theoretically’. I’m very interested in both, but categorification — taking ideas and pushing them further up the nn-categorical ladder by replacing equations with isomorphisms — is different from merely taking ideas and phrasing them in terms of category theory. The latter is a prerequisite for the former.

If phrasing existing physics in terms of categories catches on, this could just be the logical extension of the Gruppenpest that hit quantum theory many decades ago. We can do physics without talking about categories, just as people did quantum mechanics without talking about groups… but we understand things more clearly and simply when we use symmetry concepts, and categories are a useful generalization of groups.

If categorified physics catches on, something more interesting may be at work. Namely, a dethronement of the concept of ‘equality’, where the static concept of ‘xx is yy’ is completely replaced by the dynamic ‘ff is a process whereby xx becomes yy’.’ And this goes straight to the root of physics. After all, the Greek word physis meant something like ‘becoming’.

Posted by: John Baez on February 18, 2008 5:38 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Yes, the conflation. Although on the other hand I suppose the act of writing in category theoretic terms may reveal a theory one level below which shows your old theory is already categorified.

If Urs is right

n-fold categorified QM should be n-dimensional QFT,

then in your terms that’s quite an iterated becoming.

But maybe category theory is neutral between interpretations. There’s nothing to stop processes acting between states of knowledge, like a Bayesian updating their degrees of belief.

Posted by: David Corfield on February 18, 2008 6:23 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

So, I didn’t actually answer David’s question: “Would the success of this categorified mathematical physics tell us something about the way the theory hooks onto the world?”

I guess I’m more competent at thinking about the world than how our theories hook onto the world.

But maybe I can toss this question back to David: if the long-cherished concept of ‘equality’ turned out to be an oversimplification that we had to bypass to make further progress, what would this say about how our theories hook onto the world?

Posted by: John Baez on February 18, 2008 6:23 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

As I suggest above, I think we can tell becoming, and the becoming of becoming, etc. stories equally well about physical states and knowledge states. We can imagine different paths between a pair of knowledge states, and then paths between these paths.

Perhaps then more subtle notions of sameness won’t help us much, and we’ll have to work out the relation between physical states and maximal knowledge states first anyway.

Posted by: David Corfield on February 19, 2008 6:01 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

You seem to be conflating ‘categorified mathematical physics’ with ‘phrasing physics category theoretically’.

Probably we should all specify more precisely what exactly we have in mind when saying “categorification” here.

One thing I meant, which I think is important, is this:

quantum mechanics is about functors which know about “being” over points and about “becoming” along one dimension.

There is nn-fold categorification of this, and it yields, I think, nn-dimensional quantum field theory.

And this is only partly just a rephrasing of known nn-dimensional QFT. To an, eventually, larger extent, it leads to a refined description of nn-dimensional QFT undreamed of in classical terms.

On the other hand, the kind of categorification of quantum mechanics that you have been talking about a lot is – as I think we said elsewhere in a similar discussion some months ago – possibly to be thought of as a way of realizing that there was a hidden categorical dimension already in ordinary 1-dimensional QFT (= Quantum mechanics) – that ordinary QM is itself the result of starting with something higher categorical and then taking equivalence classes.

Somehow this is a way of “categorifying” which goes in the opposite direction than the one I had in mind. Both categorifications yield higher categorical dimensions which add up: if it is right, for instance, that QM itself is actually secretly already about 2-functors, then the kind of categorification that I am talking about will say that nn-dimensional QFT is secretly about (n+1)(n+1)-functors (as opposed to mere nn-functors).

And in fact, as we also said elsewhere already, there are a couple of indications that this is the case. For me, personally, the strongest one currently being that the most generally necessary notion of “background field” for the nn-particle involves, in fact, (n+1)(n+1)-functors, not nn-functors.

(And I also mentioned elsewhere that I am having that hallucinations that I am seeing hints about how that shift relates to the shift (i.e. groupoidification) which you have in mind.)

We should maybe invent precise terms that are able to distinguish between

- internalizing concept XX in nnCat

- realizing concept XX as the decategorification (result of passing to equivalence classes) of a concept XX'.

The second process is maybe best called recategorification!

Posted by: Urs Schreiber on February 18, 2008 11:32 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Urs wrote:

Probably we should all specify more precisely what exactly we have in mind when saying “categorification” here.

Indeed! I like to run around categorifying everything in sight, and sometimes the conceptual relationship between different ways categorification applies to the same subject isn’t so clear!

In particular, you’re right about how this happens in quantum field theory. There’s the old idea that nn-dimensional quantum field theory is all about nn-functors from a cobordism nn-category to nHilbn Hilb. And then there’s another idea, the theme of this seminar: that even for n=1n = 1, Hilbert spaces are sometimes just decategorified (or degroupoidified) versions of groupoids. This seems to give the whole story ‘one extra dimension’.

A big clue is that this degroupoidification stuff is another way of studying Khovanov homology. Like Khovanov homology, we’re taking familiar algebraic gadgets (finite groups, certain Lie algebras, etc…) and saying that their category of representations is secretly a 2-category.

Khovanov homology strongly suggests that all of Chern–Simons theory admits this ‘one extra dimension’. That’s something you should be in a great position to think about.

Posted by: John Baez on February 19, 2008 6:14 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Given a topos, is there a way to tell if it’s of the form FinSet G\mathbf{FinSet^G} for some finite groupoid G\mathbf{G}? A necessary condition is that every hom-set in the topos is finite, but that’s probably not sufficient. A good condition might be that there are only a finite number of isomorphism classes of objects that lack proper subobjects, but I can’t prove anything formally.

I thought I’d ask this here as it seems like the sort of thing that geometric representation theorists would know about!

Posted by: Jamie Vicary on February 15, 2008 9:12 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

I guess you were around when people tackled some related questions in the comments on lecture 19. Back then, Todd Trimble explained that a category XX is equivalent to a category of the form Set CSet^C iff XX is cocomplete and its full subcategory of tiny objects is essentially small and dense. He added that given this, CC is a groupoid iff XX is a Boolean topos. Given this, Denis-Charles Cisinski then explained how to recover the groupoid CC as the ‘points’ of the topos XX.

This gives an implicit answer to your question. But you rightly want a more explicit answer!

It sounds like you’d be happy if something like this were true: a group is finite iff it has finitely many isomorphism classes of transitive actions. Then we should get a similar result for groupoids, and then we should be able to tell when a groupoid GG is finite by looking at Set GSet^G.

Posted by: John Baez on February 16, 2008 3:03 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

Thanks very much for that summary! So, I suppose we’re trying to prove the following:

Conjecture. A topos is equivalent to Set GSet^{\mathbf{G}} for some finite groupoid G iff it has a finite number of connected objects.

Of course, I’m using ‘number’ to mean ‘number of isomorphism classes of’. A connected object is an object XX such that Hom(X,)(X,-) preserves coproducts, as Todd Trimble explained in his post. He also explained that a projective object is an object XX such that Hom(X,)(X,-) preserves coequalisers, and that a tiny object is an object which is connected and projective.

To prove the conjecture, we therefore need to show that in a topos with a finite number of connected objects, the tiny objects form a dense subcategory.

I’m stuck at the first hurdle, just showing that there must be any tiny objects at all. In a category of presheaves of a finite groupoid G, my intuition (i.e. what I’ve stolen from other people’s posts) is that the connected objects are the transitive G-actions, and the tiny objects are the simply-transitive actions for one of the groups in the groupoid G, which are just the regular representations for these groups. It seems to me that a morphism f:ABf:A \rightarrow B between transitive G-actions must be surjective, and will only be injective if ABA \simeq B as G-actions. So if f:ABf:A \rightarrow B, then we know that |A||B||A| \ge |B|. We can use this to organise our transitive G-actions into a partially-ordered set, with B<AB \lt A iff |Hom(B,A)\mathrm{Hom}(B,A)|=0= 0. The tiny actions are then actually the biggest actions, the ones without any action above them in the poset — so maybe tiny isn’t such a such a good name!

But how can these tiny objects be identified categorically? Perhaps as the colimit of the full subcategory of the connected objects?

I also wanted to say something about spotting the index category C that a presheaf category Set C opSet^{C ^{op}} arises from, because I don’t think anyone’s said it yet: it’s always just the smallest full generating subcategory. This is, in fact, just the Yoneda embedding of the index category into the presheaf category, as far as I remember. This is probably equivalent to what Denis-Charles Cisinski said, but it’s worth saying it explicitly.

Posted by: Jamie Vicary on February 19, 2008 5:30 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

I said:

But how can these tiny objects be identified categorically? Perhaps as the colimit of the full subcategory of the connected objects?

I think it works if you use the limit, not the colimit. In a category of representations of finite groupoids, let ii be the full subcategory of connected objects. Then the limit of ii is tiny. No idea how to prove this in an arbitrary topos, though…

Posted by: Jamie Vicary on February 19, 2008 9:14 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

It sounds like you’d be happy if something like this were true: a group is finite iff it has finitely many isomorphism classes of transitive actions.

I would really like to know if this is true — but luckily, I don’t quite require this! All I need is a way to take a finitary topos T with a finite number of isomorphism classes of connected object, and from it construct a groupoid G such that T\simeqSet GSet^G. If there’s also an infinite groupoid that happens to have the same finite transitive actions, it doesn’t really matter!

Posted by: Jamie Vicary on February 19, 2008 7:23 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

By the way, your question was about FinSet GFinSet^G, but I answered in terms of Set GSet^G. I don’t think this should make a vast amount of difference for a finite groupoid GG, since every action of such a groupoid on a set is a coproduct of actions on finite sets. I just know more about the universal properties of Set CSet^C than FinSet CFinSet^C.

Set CSet^C is the free cocomplete category on CC. What about FinSet CFinSet^C?

Posted by: John Baez on February 18, 2008 5:21 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 25)

I guess I’ll give a partial answer to this. If CC is a finite category, then clearly CC is FinFin-enriched, and there is a Yoneda embedding

CFin C opC \to Fin^{C^{op}}

obtained by transforming the hom C(,):C op×CFinC(-, -): C^{op} \times C \to Fin. The category Fin C opFin^{C^{op}} is finitely complete, since FinFin is.

Proposition: Fin C opFin^{C^{op}} is the free finite-cocompletion of CC. More precisely, y Cy_C is 2-universal among functors G:CUDG: C \to U D, where UU is the forgetful 2-functor (from the 2-category of finitely cocomplete categories DD and finitely cocontinuous functors to CatCat).

Proof. Each F:C opFinF: C^{op} \to Fin can be represented as a finite colimit of representables C(,y)C(-, y): there is an exact sequence

y,zCC(,y)×C(y,z)×F(z) zCC(,z)×F(z)F()\sum_{y, z \in C} C(-, y) \times C(y, z) \times F(z) \stackrel{\to}{\to} \sum_{z \in C} C(-, z) \times F(z) \to F(-)

which expresses FF as a coequalizer of maps between finite sums of representables. (One of the maps involves composition in CC, and the other involves the contravariant action C(y,z)×F(z)F(y)C(y, z) \times F(z) \to F(y) of CC on FF.)

Given a functor G:CDG: C \to D to a finitely cocomplete DD, we define an extension G^:Fin C opD\hat{G}: Fin^{C^{op}} \to D of GG along the Yoneda embedding, in the only way which preserves this colimit: namely G^(F)\hat{G}(F) is the evident coequalizer G CFG \otimes_C F in DD:

y,zCG(y)×C(y,z)×F(z) zCG(z)×F(z)G CF.\sum_{y, z \in C} G(y) \times C(y, z) \times F(z) \stackrel{\to}{\to} \sum_{z \in C} G(z) \times F(z) \to G \otimes_C F.

(The ×\times is a slight misnomer here; if SS is a finite set and dd is an object of DD, then d×Sd \times S just means the coproduct of SS copies of dd.) One easily checks that G^()=G C()\hat{G}(-) = G \otimes_C (-) is finitely cocontinuous. But since finite cocontinuity forced G^()\hat{G}(-) to be defined this way (up to unique isomorphism), it is the unique finitely cocontinuous extension up to unique isomorphism, as desired.

Posted by: Todd Trimble on February 19, 2008 9:24 PM | Permalink | Reply to this

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