## November 26, 2007

### Categories, Logic and Physics in London

#### Posted by John Baez

Category theory and logic seem to be finding more connections with physics. Bob Coecke and Andreas Döring have decided to run a series of workshops on these connections, starting with this:

It’s too bad I can’t make it! I hope someone here does, and tells us what happened. Maybe Jamie Vicary? Maybe even David?

More details follow…

Here’s an email I got:

Dear all,

We hereby wish to invite you to participate in a workshop series which aims at nourishing research in the area of “Categories, Logic and Foundations of Physics”. There seems to be a substantial number of people in the Loxbridge area and beyond with interest in this field to sustain such a series. We also welcome activity from other research strands aiming to gain structural insights into foundational physical theories, for example by means of toy models, operational methodologies for general physical theories, structures for dynamics and space-time etc. Besides the workshops we plan to have an online presence in terms of a (moderated) interactive website on which open problems, news, discussions, tutorials, recordings of talks and pointers to literature will be exhibited. The scheduled date for the first meeting, which will take place at Imperial College in London, is

Wednesday, January 9, 2008.

While we do not want to stick to an a priori fixed format for the workshop, we propose the following schedule for the first meeting, to get the ball rolling:

12.00-13.00 Survey talk (Chris Isham on the topos apporach)
13.00-13.30 Research talk
13.30-14.15 Buffet lunch session
14.15-15.15 Survey talk (Samson Abramsky on the symmetric monoidal approach)
15.15-15.45 Research talk
15.45-16.30 General discussion session aiming at:

• stating open problems
• requests for presentation of certain topics
• plans for the future, e.g. proposals for collaboration

16.30-16.50 Coffee break
16.50-17.30 Work session
17.30-18.00 Research talk (?)
18.15-19.00 Pub session

We invite you to put forward suggestions for whom you would like to see give a (survey or) research talk, and please let us know about others who would be interested.

For planning reasons, please do let us know if you will participate in the January 9 workshop!

We are looking forward to seeing you at “Categories, Logic and Foundations of Physics”.

Best regards,

Bob Coecke (Oxford) and Andreas Doering (Imperial)

Posted at November 26, 2007 12:58 AM UTC

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### Re: Categories, Logic and Physics in London

Maybe even David?

I’m very tempted. Perhaps I’ll propose a category theoretic study of Charles Peirce’s representation of modal logic via tinctured gamma existential graphs.

Posted by: David Corfield on November 26, 2007 9:55 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

David wrote:

Perhaps I’ll propose a category theoretic study of Charles Peirce’s representation of modal logic via tinctured gamma existential graphs.

If so, I sure hope you talk to Todd Trimble about this, and read all the stuff he and Gerry Brady wrote about Peirce, including the stuff that’s really hard to find. For starters, look under ‘Research’ on Gerry Brady’s webpage.

Posted by: John Baez on November 26, 2007 9:29 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Yes, I’d be very interested to hear what you have to say on this, David; you may be one of the few people with both the philosophical and mathematical creds to give a proper updated account of Peirce’s Gamma.

When Gerry and I were talking about this, we found (buried deep in archives, not in Peirce’s Collected Papers) some material which suggested, at least to us, other possibilities for interpreting what he was up to with Gamma. In particular, we found some passages with existential graphs which, taken in conjunction with what we knew about Beta, are interpretable as the crucial axioms for power allegories (as in Freyd and Scedrov’s book), an alternative approach to topos theory and higher-order logic!!

Now, the archival material we found is probably (by itself) too thin to adequately support the idea that Peirce was an inventor of higher-order logic in this particular way, but I do have a sneaking suspicion that there may be a little more to Gamma than what found its way into his Collected Papers (eds. Charles Hartshorne and Paul Weiss, Harvard University Press). Peirce left behind voluminous piles of unpublished writings and scraps of paper, which (if I understand correctly) remain in the care of the Philosophy Department of Harvard. While we owe scholars such as C.I. Lewis a great debt for the herculean effort required to sort through and assemble and archive these mountains of material, it could be that some of these scholars were operating through philosophical filters which preclude alternative readings. It would be wonderful if scholars equipped with modern-day mathematical insights were to take another look!

Posted by: Todd Trimble on November 27, 2007 2:24 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Todd, I still have a stack of photocopies of Peircean jottings you once gave me, which I confess I never did much with. Perhaps it is time.

Posted by: David Corfield on November 27, 2007 9:07 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

David — yes, it’s going to be held on Wednesday 9th January 2008. I tried to reply directly to your message, but “an error occurred” with “movable type”! I hope you can come.

The talks will all be videod and uploaded, as is the fashion these days, but harder to video will be the “discussion sessions” in whatever form these will take. I’m especially looking forward to these: hearing mathematicians and physicists argue amongst themselves is a great spectator sport!

Posted by: Jamie Vicary on November 26, 2007 9:57 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

I decided to correct the date in John’s post and then remove the message pointing it out, which might have caused the error message.

Looks good. I’ll go.

Posted by: David Corfield on November 26, 2007 11:25 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

John, thanks for publicising, and David, good that you’re coming. Anybody else reading this blog who wishes to come (either now, or wishes to be included in future announcements), just write us:
* a [dot] doering [at] imperial [dot] ac [dot] uk
* coecke [at] comlab [dot] ox [dot] ac [dot] uk
As Jamie mentioned above, he and some others will be maintaining a site archiving and reporting on the event.

Posted by: bob on November 26, 2007 2:20 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

I’m glad you appreciate the publicity, Bob!

Good luck! Despite my previous paragraph, I’m sure it’ll be a really interesting workshop.

Posted by: John Baez on November 26, 2007 8:52 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Can any crackpot with theories about categories, logic and the foundations of physics who reads this blog please reveal himself here and now.

You are not welcome at our event!

(hope this does the job)

Posted by: bob on November 26, 2007 9:16 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Then I’ll have to present my work here, I guess. A new proof of the Curry-Howard isomorphism:

Posted by: Mike Stay on November 27, 2007 5:28 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Could you transform Ron Howard in a plate of vindaloo too?

Posted by: bob on November 27, 2007 11:25 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

You are not welcome at our event!

Oh, well. The airfare is too expensive, anyway.

Posted by: Kea on November 29, 2007 7:08 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

A slight update: Louis Crane will be coming and give one of the 30 min research talks.

Posted by: bob on November 29, 2007 8:40 PM | Permalink | Reply to this
Read the post The Principle of General Tovariance
Weblog: The n-Category Café
Excerpt: Landsmann proposes that physical laws should be formulated such that they may be internalized into any topos.
Tracked: December 5, 2007 7:10 PM

### Re: Categories, Logic and Physics in London

We now have a webpage/wiki/blog dedicated to this new seminar series on Categories, Logic and Physics.

The page contains a list of participants. In addition, there is a related facebook group.

The January 9th workshop is listed as a facebook event here.

Posted by: jake on December 15, 2007 11:02 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

I had thought: “cool workshop, wish I could attend it, wish it weren’t in the middle of the semester, wish it would refund my travel expenses and wish, generally, that travelling wouldn’t take time”.

Now my friend Igor Baković calls me and asks me where I will be accomodated since, he tells me to my surprise, I am listed as a participant.

Heh, that’s fun. I take it as a sign that I should actually go.

I’ll try to take a night train. That’ll take me right through the tunnel to St. Pancras International.

Can anyone provide any help on what’s the best way to get from there right to the workshop? And how long might it take?

Posted by: Urs Schreiber on January 7, 2008 9:11 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Cool!

If I were just going to Imperial College to visit Isham, I’d take the underground from St. Pancras to South Kensington, and then walk north. Check out a map of the underground for the best route. It’d take somewhere between .5 and 1 hours. But I’m not sure where the conference is, so I’m not sure South Kensington is the stop you want.

It’s cool that Louis Crane will be there — he’s the guy who got me interested in 2-categories in the first place.

Posted by: John Baez on January 7, 2008 10:40 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Hi Urs,

I’m going :-) Like Igor, I also thought you were going!

I’m not a london local, but I would think the easiest public transport way from St Pancras to the South Kensington campus is to take the tube. Victoria line from Kings Cross to Green Park, and then the Piccadilly line to South Kensington, and then you walk to the workshop (5 min according to website) from there. I guess this will take about half an hour.

The only difference between this and the Transport for London recommended route is that they would have you take the bus from Green Park. Tube’s probably easier.

If you don’t have one already, you might want to buy a pay-as-you-go Oyster card, costs 3 pounds but will save you money ultimately. Otherwise just buy as you go, or buy a day pass.

St Pancras is da bomb; that’s where the Sheffield trains come in too. I’m taking the early morning bus though, arriving 10.30 at Victoria.

It will be great if you can make it!

Posted by: Bruce Bartlett on January 7, 2008 10:49 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Why not take the Piccadilly line direct from King’s Cross to South Kensington?

Posted by: David Corfield on January 8, 2008 8:24 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Why not take the Piccadilly line direct from King’s Cross to South Kensington?

Woops, sorry Urs, you should never trust me on directions ! Some idiot turned the Southern Cross upside-down in these parts, it must be that which has has screwed up my navigational sensors

Posted by: Bruce Bartlett on January 8, 2008 9:24 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Thanks for all your kind replies!

Bought a train ticket today. Will arrive

Jan 9th, 7:55 am

at St. Pancras and will leave again from there

Jan 9th, 19:34 .

A rather quick trip to London. And with an impressive cost/time ratio, but, hey, it’ll be an unforgettable workshop, right? ;-)

Will any of you be around early at South Kensington Campus? Otherwise I’ll search a public WLAN and spend the morning bombarding the $n$-Café with messages…

Posted by: Urs Schreiber on January 8, 2008 4:44 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Cool! I won’t be there real early… but I hope to be standing outside the Blackett laboratory at 10.50am, if you want to meet up.

Posted by: Bruce Bartlett on January 8, 2008 5:34 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Cool! I won’t be there real early… but I hope to be standing outside the Blackett laboratory at 10.50am, if you want to meet up.

Okay, great. I’ll have to leave in a minute and might not have much internet access until tomorrow morning. I’ll try to be at the Blacket laboratory ar 10:50.

Or let me say, since it’s topos theory we’ll be doing:

I’ll be there

or not

or something else.

Posted by: Urs Schreiber on January 8, 2008 6:34 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

I’ll just barely make it before noon, so will hve to wait until the first break to meet up.

Posted by: David Corfield on January 8, 2008 7:21 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

One of the remarkable aspects of the workshop had been the discussion session.

Two points were emphasized there:

1) The organizers expressed their great positive surprise at the resonance that their workshop had found. Originally intended as a small local meeting, it turned out to attract lots of people, partly from far away.

It was generally felt that signs are good that a certain critical momentum of research interest in the kinds of topics discussed at the workshop has built up, and the need was felt not to let that chance pass without making use of it.

The hope was expressed several times that the wiki which accompanied the workshop, Wiki ” Categories, Logic and the Foundations of Physics”, could serve, with the blog and other interactive features it contains, to help create more of an active community of people working in “the field”, or maybe help that field ripen as such in the first place, I guess.

2) Closely related the second main point, various contributors to the discussion addressed the question: why has the application of category theoretic methods in physics not, so far, created the kind of enthusiasm among physicists which those propagating it think it deserves?

Louis Crane gave an impressive statement in which he emphasized how quantum physics exhibits categorical phenomena all over the place (he in particular amplified the perturbative path integral and its renormalization), only that very few researchers in the field explicitly make these manifest.

The general feeling was expressed, I think Eugenia Cheng made that point explicitly, that if the identification of categorical aspects in modern physics could be improved, pure category theorists would have an easier time joining in the fun and helping out with the powerful methods that already exist, preventing physicists from reinventing wheels and thus propelling progress.

I very much agree with this point. I think, and tried to say so, that part of the scepticism which categorical methods are met with among physicists results from the fact that there have been in the past quite a few very interesting but very speculative proposals for how category theory might help resolve the problems of quantum gravity, while a comparative effort to first sort out the categorical basics of just ordinary quantum field theory has been missing.

But maybe, that’s what this workshop possibly has shown best, this is precisely the point that is changing now.

My humble opinion, in conclusion:

if only we manage to see the unifying frame for all “three approaches” to categorical physics discussed at this workshop (topos theoretic, $C^*$-algebraic (AQFT) and dagger-compact closed monoidal theory, and on top of that functorial quantum field theory) as one single categorical conception of quantum physics (which will have lots of dual points of view on it), then we’re in business.

The rest will then be a corollary. :-)

Posted by: Urs Schreiber on January 10, 2008 2:11 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

why so reluctant?
remember their reaction to the gruppenpest

Posted by: jim stasheff on January 10, 2008 8:43 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

remember their reaction to the gruppenpest

That’s precisely the example I mentioned in the discussion session.

I said, once upon a time, some mathematicians tried to point out that physics depends all over the place on one-object categories with invertible morphisms.

There was lots of opposition to this idea, among physicists, back then. Things have changed since then. If today you have a question about $E_8$, chances are that you find more answers in your physics theory group than in your math department.

What is it that made the difference?

That mathematicians published papers which conjectured that one-object categories with invertible morphsims (aka groups) might solve some far-out fancy almost unthinkable problems one might imagine on the horizon of physical research?

No. The difference came when it became clear that ignoring one-object categories with invertible morphisms in your physics research meant that you would be left behind soon, because for your those of your colleagues who already did accept it, the dust suddenly settled and they could see the light and progress.

The same will be true for categories with more than one object and potentitally non-invertible morphisms, eventually.

But it takes time to read a book on group theory. While physicist A was locked away reading books on group theory, physicist B kept cranking out papers on physics. At the end of the year physicist A was beginning to have a vision of the theory of the next century (eventually called quantum mechanics) while phyicist B had a vision of tenure.

While that period lasts, it’s hard. But towards the end of that period, suddenly things change.

Posted by: Urs Schreiber on January 10, 2008 8:57 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

I am back in Hamburg now.

On my way back I had a stop in Brussels, where I wrote the entry The Concept of a Space of States, and the Space of States of the Charged $n$-Particle which describes a couple of observations, remarks and ideas I had while and after listening to Chris Isham’s review of his work with Andreas Döring.

Very unfortunately, I could attend only the first ten minutes of Louis Crane’s talk, which was the very last one. I really would have liked to listen to that, given our discussion of categorified Dijkgraaf-Witten theory here and here.

But Jamie Vicary videotaped the entire workshop, so I am confident that I will eventually be able to see this talk and talk about it here.

There were also a coupld of talks about what they call the “monoidal approach” to quantum mechanics. First I was confused about what “monoidal approach” is supposed to mean.

It refers to the work by Bob Coecke, Samson Abramsky and other. This is based on the crucial fact the right codomain for a transport functor which describes quantum mechanical propagation is a dagger-compact (or whatevery you call it) monoidal category (essentially: a monoidal category where each morphism has an “adjoint”, modeled on the archetypical excample of the category of Hilbert spaces).

For a great list of information on that see John’s old entry Quantum computation and symmetric monoidal categories.

On the other hand, none of the speakers mentioned that this is in fact what these monoidal categories are: codomains for the propagation transport.

It was Bruce Bartlett who raised his voice in the discussion session to remark that the “functorial approach” to quantum mechanics had been missing from all the talks.

In fact, some speakers had mentioned the “different approaches” to categorical quantum mechcanics:

- topos theoretic

- C-star algebraic

- monoidal

I found that slightly irritating. These are, or at least should be, different aspects of one and the same categorical formalism.

After I said something to this extent, I was later asked for more details. Let me use this opportunity here to point to my notes Local nets form 2-transport, which are supposed to indicate how the C-star algebraic approach should connect to the “monoidal approach” and to the functorial approach.

And all of this should embed in some topos theoretic context, as I try to indicate in The Concept of a Space of States, and the Space of States of the Charged $n$-Particle.

Have to run now. More later.

Posted by: Urs Schreiber on January 10, 2008 11:31 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

I think you’ll have to wait for that video to find out what Louis Crane said next, unless Bruce, Simon, Eugenia or someone else can reconstruct it.

In 30 minutes we got to see many things, including the importance of singularity theory for gravitational lensing.

Posted by: David Corfield on January 11, 2008 9:40 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

none of the speakers mentioned … what these monoidal categories are: codomains for the propagation transport.

Isn’t there a sense in which the monoidal approach could be considered as more general than this? After all, taking the transport functor as fundamental assumes a lot of things — like the nature of space, ‘particle’ as a fundamental notion, that space and quantum mechanics have equal status — which we might want to assume, but we might not. But we can study monoidal categorical generalisations of Hilb directly without having to make firm commitments on these issues.

In fact, some speakers had mentioned the “different approaches” to categorical quantum mechcanics: topos theoretic, C-star algebraic, and monoidal. I found that slightly irritating.

I would have preferred to have talked about this during the discussion session, rather than revisit the peristalithic ‘why does nobody give enough respect to category theory’ debate!! :)

Although it’s tempting to hope that the approaches are related, I don’t see why they should be any more related to each other than to other approaches that don’t directly rely on category theory. After all, at the end of the day, given $n$ approaches to fundamental physics, $n$ of them are likely to be wrong — but that’s okay, because hopefully they will each bring something new to the table. But why should any two necessarily be deeply related? I would say that using the same broad mathematical framework is neither necessary nor sufficient for any deep relationship to exist.

And seems to me that the philosophical content of the three category theoretic approaches you list is not really all that similar. Of course, since they’re all trying to be theories of quantum mechanics, we can probably use any one of them to describe another if we really want to — but why should we, unless something really beautiful comes out that’s more than the sum of its parts?

Posted by: Jamie Vicary on January 12, 2008 11:45 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Hi Jamie,

it is good that you voice your disagreement with my assertions here, because that shows that there is need for more discussion and it opens the possibility to sort things out by actually starting to discuss.

I am a little shorter of time than I would hope (we might have to invent something like a blog-throttle, there is too much going on here) but let me try to say a couple of things:

After all, taking the transport functor as fundamental assumes a lot of things — like the nature of space, ‘particle’ as a fundamental notion, that space and quantum mechanics have equal status

You are probably thinking that by that “transport functor” I meant a “classical transport” that assigns phases to paths in space time. But this is not so. I am talking about the “quantum propagation transport”. I am calling that a transport, too, because I think it’s cool to do so, but here it just means the plain propagation functor

$QM : 1Cob_{Riem} \to Hilb$

which takes any 1-dimensional manifold with length $t$, sends its endpoints to some Hilbert space and the cobordism itself to the propagator over time interval $t$:

$QM : (x \stackrel{t}{\to} y ) \mapsto (H \stackrel{\exp(i t \Delta)}{\to} H) \,.$

Here “$\Delta$” denotes the Hamiltonian operator.

You see what I mean now? The point is that such a functor might be obtained by quantizing the kind of classical transport which you apparently thought I mentioned, but it need not. In particular, no notion of space is invoked here explicitly (but might be extractible from $\Delta$ by the usual spectral Yoga).

Here $Hilb$ could be replaced by other categories. But $1Cob_{Riem}$ is a monoidal category, and $QM$ is a monoidal functor on that. So whatever we generalize $Hilb$ to, we want it to be monoidal.

That’s why monoidal categories show up in quantum mechanics.

Generally, quantum field theory is about representations of cobordism categories – or $n$-representations of $n$-cobordism categories if you refine things, and the monoidal structure in the game is that coming from these cobordisms.

And the little argument above is just the tip of a beautiful vast iceberg. John Baez and James Dolan once tried to sketch this iceberg in

Quantum field theory is all about $n$-categories which are monoidal in so many ways, that making them more monoidal doesn’t change anything: they are stably monoidal.

(There is something really deep going on here with this monoidalness, which must be related to the sphere spectrum. We talked about that here, but I certainly don’t fully understand it.)

So the fact that quantum mechanics is about monoidal categories is not a coincidence, and not something one just makes up or doesn’t. It has to do with the fact that quantum field theory is about representations of cobordisms categories.

In my idiosyncratic ways, I like to think of such representations as “quantum transports”. I have a reason for doing so, but it’s not important right now.

I would have preferred to have talked about this during the discussion session, rather than revisit the peristalithic ‘why does nobody give enough respect to category theory’ debate!! :)

I very much agree with that. In particular since I had so little time to spend there. But that’s the way it goes. If we are good, we’ll have this discussion here now, instead!

After all, at the end of the day, given $n$ approaches to fundamental physics, $n$ of them are likely to be wrong

Do you really think so? Of course any one random approach will be wrong. But here we are talking about “approaches” that a bunch of people have been following a considerable amount of time, growing fond of them while doing so. This cannot work unless a large fraction of these approaches is to a large degree detecting some parts of an existent elephant. That elephant we need to sketch.

But why should any two necessarily be deeply related?

currently, what is called “quantum field theory” is a mess. In large parts it’s a mess because people don’t know what it really is. In other parts it’s a mess because people decided what it really is, but cannot agree on their definitions. Some work here, some there.

But in 100 years from now we will known “what quantum field theory is”. Just like we know today what classical mechanics is. We know that all these different approaches that have been considered over the centuries (Hamiltonian, Lagrangian, virtual displacements, Liouville flows, symplectic geometry, whatnot) are just different aspects of one singe unique nice thing.

It didn’t use to be that way. But it certainly had to work out this way after a sufficient amount of effort. There is just one concept “classical mechanics” our there in the space of concepts. And just one concept “quantum field theory”. And we need to find the true formulation of it.

Posted by: Urs Schreiber on January 14, 2008 9:34 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Let me add my two cents worth from left field, and say that I learnt some interesting stuff from the merry men (one of whom was Jamie Vicary) at the recent conference. For instance, this idea that a Frobenius algebra in in one of these quantum categories is profitably to be thought of as a “classical object”. I learnt that from Bob Coecke’s talk when he came to Sheffield… but then I forgot it. It seems one can get a lot of mileage out of that idea. I think Jamie also has some secret results along these lines.

Posted by: Bruce Bartlett on January 14, 2008 11:31 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Hi Urs!

I did in fact know what you meant by “transport functor”. And of course, this is an extremely elegant approach. But I do think that it makes unwarranted assumptions which — from the point of view of quantum gravity — lack sufficient hard evidence.

If ‘all we want to do’ is understand quantum field theory, then this is all quite different. But, to the extent that what I’m doing has a slogan, it’s “come up with new versions of quantum mechanics that might be good for constructing a theory of quantum gravity” — not very catchy, perhaps, but there you go. Everything that I am saying is from this perspective.

So, I will repeat my list of assumptions from my previous post. The propogation functor assumes something about the nature of space — there’s just no way that Riemannian manifolds are at all fundamental, and I appreciate that perhaps some better category could be used for the domain instead, but I have yet to see one. Secondly, I suppose it is not fair to say that the notion of ‘particle’ is assumed, since presumably you would really be interested in a category like 3Cob${}_{\mathrm{Riem}}$ where the ‘particles’ are pretty well-hidden.

But this links with the most important point: for a transport functor to somehow underlie quantum gravity implies that spacetime (the domain) and quantum mechanics (the codomain) somehow exist at the same ontological level. To my mind, this is not compatible with the hopes of many that spacetime will ultimately prove to be a ‘quantum process’.

You say:

“the fact that quantum mechanics is about monoidal categories is not a coincidence … It has to do with the fact that quantum field theory is about representations of cobordisms categories.”

To summarise my position: perhaps I would be tempted to argue this way from the point of view of quantum field theory, but I would strongly reject it from the point of view of quantum gravity. I think that it is uncontroversial to say that the mathematical basis of quantum gravity is incredibly uncertain, and to accept at this stage the statement that “these monoidal categories are … codomains for the propagation transport”, denying any greater role for them in the job of constructing a quantum theory of gravity, would be a serious mistake.

Posted by: Jamie Vicary on January 16, 2008 1:04 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Jamie Vicarie mentioned

quantum gravity

Ah, now I see where you are coming from.

I am slightly taken aback by how you want to jump from the harmonic oscillator right to quantum gravity, with a certain disdain for “ordinary” quantum field theory, but I have seen that happen before… ;-)

But okay, let’s discuss this, this will be getting interesting.

Right this moment I don’t at all have time for a detailed response, but two quick remarks, with more details later:

a) yes, the “quantum propagation $n$-transport” for $n$-dimensional gravity will be a functor not on $1Cob_{Riem}$ but on plain $n\mathrm{Cob}$. Or rather on an $n$-categorical refinement of that.

b) second quantization will be a procedure that reads in quantum $n$-transport of $n$-particles ($n=1$: points, $n=2$: string, $n=3$: membrane, etc.) propagating on $m$-dimensional target space, and spits out the quantom, $m$-transport of the corresponding second quantized field theory on target space ($n=1$: ordinary field theory, $n=2$: string field theory, etc.)

c) $1Cob_{Riem}$ is just a toy model to get started (well, and to make contact with standard non-relativistic textbook quantum mechanics).

More later.

Posted by: Urs Schreiber on January 16, 2008 2:30 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Jamie Vicarie mentioned

y! y! y! y! y!

Ah, now I see where you are coming from.

Sorry about that — I should probably have mentioned ‘quantum gravity’ earlier…

I am slightly taken aback by how you want to jump from the harmonic oscillator right to quantum gravity, with a certain disdain for “ordinary” quantum field theory

This is clearly a very important point! As far as I can see, there is no obvious reason why we have to sort out the problems with flat-space quantum field theory before we move on to quantum gravity. Perhaps you are saying this because you would like the categorifying process to operate like a steamroller, slowly moving through physics in a linear fashion, straightening out problems in fundamental physics one after the other in the order that they historically arose… but surely, it doesn’t have to be like that!

Perhaps you are confused because you assume that my research is proceeding in a sensible and orderly way, moving from the quantum harmonic oscillator to quantum field theory… I wish :). I am more mercenary than that!

In fact, there were three distinct camps present at the Imperial College meeting: those interested in the foundations of quantum field theory, those interested in the foundations of quantum gravity, and those interested in the high-level semantics of quantum computation. Of course, many people will be in more than one camp — but only the quantum field theory camp might agree that Hilb exists solely to be the codomain of a propagation functor. And I suspect that the quantum field theory camp was perhaps the least well-represented at the meeting, which surely explains why “none of the speakers mentioned … what these monoidal categories are: codomains for the propagation transport.” Hopefully this can be rectified at future workshops!

I should point out that the reason these meetings are a good idea is that these three camps often have a lot to say to each other. An excellent example: I came across my results on the quantum harmonic oscillator at almost exactly the same time as Marcelo Fiore came across almost exactly the same stuff, but from the computer-science point of view of categorical models of linear logic. I think this is pretty amazing!

Posted by: Jamie Vicary on January 16, 2008 3:30 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Jamie suggested:

you assume that my research is proceeding in a sensible and orderly way, moving from the quantum harmonic oscillator to quantum field theory… I wish :)

I am not assuming that. While research is probably bound to be erratic, what I am assuming is that we want to try to get a clear picture in our heads of what is going on.

To have any chance at all with quantum gravity, we need to have some things straightened out. Currently the discussion is at a point here, where the two of us are disagreeing on what the Hamiltonian in non-relativistic 1-particle quantum mechanics really is.

With that state of affairs, I am a little reluctant to delve into lots necessarily vague discussion of what the “ontological level of spacetime is”, to slightly paraphrase you.

Can you see what I mean?

But I am looking forward to discussing this at the length it requires, just not right at the moment.

Until I am done with what I am doing and can continue discussing this here, let me quickly throw the ball back to you:

are you aware that it’s the very idea that whatever quantum gravity is, it should give us something like a functor $n\mathrm{Cob} \to Hilb \,,$ namely a “topological QFT”, which made, historically, the functorial perspective very popoular in the first place?

If not, please do. It’s not long.

Posted by: Urs Schreiber on January 16, 2008 4:43 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

By the way, Jamie: one thing I wanted to remark after your talk in London, but didn’t get a chance before the question session was over:

to me it seems that it might be worthwhile not advertizing your work as being about the harmonic oscillator. My impression is that what you really do is that you look at the familiar old second quantizaton functor and show how to discuss it in full beauty internal to any suitable monoidal category, using just abstract tools.

I really like that work of yours a lot, but I know in the beginning the title confused me a bit, when we discussed this last time. (But not just me, maybe. That last discussion was under the headline of “categorified QM”. I don’t think you actually consider categorifying QM in your work, in the sense of raising categorical dimensions.)

You should please correct me if necessary. Maybe I am still confused. But when listening to your talk I thought to myself: he should call and advertize his work as second quantization in its most general context.

You see, it is kind of a coincidence (a deep one, though) that the harmonic oscillator itself has some of the features of a second quantized system. I think the best explanation of this phenomenon is the one John tells in The Story of $n$-th Quantization.

So what you do does apply to the harmonic oscillator. But it is way more general. It is second quantization. Fully abstractly. That’s very nice. Maybe it is worthwhile pointing that out more explicitly to your audience.

But let me know what you think.

Posted by: Urs Schreiber on January 14, 2008 11:10 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

My impression is that what you really do is that you look at the familiar old second quantizaton functor and show how to discuss it in full beauty internal to any suitable monoidal category, using just abstract tools.

Indeed, that makes sense to me. Somehow you are saying that we can apparantly make sense of quantum field theory (or many-body quantum mechanics) in any category that looks sufficiently like $Hilb$. We can even say what the “coherent states” are and everything. Cool.

Posted by: Bruce Bartlett on January 14, 2008 11:45 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

We can even say what the “coherent states” are and everything. Cool.

I think the coolest thing is being able to define exponentials fully abstractly: starting with some endomorphism $f:A \rightarrow A$, and constructing an endomorphism $e ^f: A \rightarrow A$. This all satisfies $e ^0 = \mathrm{id} _{A}$ and $e ^f e^g = e ^{f+g}$ if $f\circ g = g \circ f$, just like you’d expect.

When I worked this out it was really a “who ordered this?” moment. I still don’t really know why they’re there…

Posted by: Jamie Vicary on January 15, 2008 11:07 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Dear Urs,

Thank you for your comments. In a nutshell, I agree, and I should not give a talk on this subject without mentioning the phrase “second quantisation”, as was the case last Wednesday. But I can explain why I described it that way originally, and in the process mention some of the ‘loose ends’ that never quite came together the way that I was hoping.

Initially I used the phrase “categorical quantum harmonic oscillator” for two main reasons: firstly, my motivation was to find a new way to approach single-particle quantum mechanics; and secondly, everybody else (i.e. Jeffrey Morton, John Baez) starts saying “quantum harmonic oscillator” whenever they get a whiff of the CCRs, so I didn’t see why I shouldn’t be able to! Also, I was hopeful for a long time that I would be able to pull off this trick for other constrained single-particle systems, but that did not come to fruition.

There’s a beautiful fact about constrained single-particle quantum mechanics that gave me hope that this would work. It has been known at least since the Blute, Panangaden and Seely paper of 1994 that the state space of the quantum harmonic oscillator has the structure of a commutative monoid, with the lowest energy state being the unit for the monoid. But in fact, quite generally — certainly for constrained potentials in finite dimensions — the ground state is always nondegenerate! Wow. So we can always find a ‘unit’ state, only ambiguous up to a phase factor. So perhaps we can also always find some sort of monoid structure, that corresponds to ‘combining’ two states? In general, for other systems than the quantum harmonic oscillator, given two eigenstates of the Hamiltonian with energies $E_1$ and $E_2$ there will fail to be an eigenstate with energy $E_1 + E_2$ — which is a shame, but doesn’t completely kill the idea.

One of the most serious ways in which the work so far falls short of truly describing the quantum harmonic oscillator is that it is difficult to motivate the definition of the Hamiltonian, $\sum_i a_i a_i^\dagger$. Sure, you can construct it, you just get the right morphisms and compose them — but category-theoretically, why is this special? I couldn’t find anything very impressive to say along these lines. (Does anybody have any ideas?)

But anyway, this is also a problem for the free quantum field theory case, because that too comes with the same Hamiltonian. So I didn’t see that calling it ‘second quantisation’, and associating it directly with quantum field theory, was necessarily an improvement. So given my ambitions in single-particle quantum mechanics, as outlined above, I didn’t go down this route.

There’s also the fact that I want mathematicians and computer scientists to be interested in my paper, and ‘second quantisation’ perhaps sounds a bit more esoteric and technical than ‘the quantum harmonic oscillator’. But this is a small point.

Finally, I should point out that I have never used the word ‘categorification’ other than to indicate an increase in $n$-categorical dimension!

Posted by: Jamie Vicary on January 15, 2008 11:01 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Hi Jamie,

everybody else (i.e. Jeffrey Morton, John Baez) starts saying “quantum harmonic oscillator” whenever they get a whiff of the CCRs

To be precise, one should not really talk about the “oscillator” unless and until one has something satisfying the CCRs and then “equips” that something with the corresponding number operator (the “Hamiltonian”).

The point is that the space of states of the particle on the line in any potential is obtained by acting with raising operators on the state that would have been the vacuum state for the particle in the square potential. All these inf. dimensional Hilbert spaces are isomorphic, after all!

So it’s the Hamiltonian that makes a difference. And this relates to that other point I made above, by the way: most of what was said at the London conference about QM was purely kinematical. The dynamics was missing.

The dynamics gets into the game once we realize that the monoidal categories we are talking about (as well as their $C^*$-like algebras of (bounded) endomorphisms of their objects) are just the codomain of a propagation functor. On objects, that functor encodes the kinematics. On morphisms, it encodes the dynamics.

So, if you ask me: what is “the harmonic oscillator”, abstractly? When can I rightly say that my abstract categorical stuff is about generalized harmonic oscillators?

precisely if you are looking at a functor $QM : 1dCob_{Riem} \to C \,,$ where $C$ is your monoidal category of choice such that it supports CCR algebra on its objects in the sense you nicely describe in your work, and where the morphism in $C$ associated by the functor $QM$ to the 1-dimensional Riemannian manifold of length $t$ is the endomorphism $\exp( - t N ) \,,$ where $N$ is the number operator on the object in $C$ which $QM$ assigns to the siungle 0-dimensional Riemannian manifold (the point) in your context.

$QM_{\mathrm{oscillator}} : (\bullet \stackrel{t}{\to} \bullet) \mapsto H \stackrel{\exp(-t N)}{\to} H$

(Strictly speaking, that would be the “euclidean” harmonic oscillator, unless you can somehow manage to make sense in your categorical framework of what it would mean to put a factor that behaves like the square root of -1 into the above exponent. But euclidean is fine for many purposes. Even though one might express the hope that a deep abstract understanding of QM will in the end also help explain what that business with complex numbers in QM really is about, and maybe make it arise automatically from some deeper reasoning. We have talked about that a couple of times here on the blog.)

a beautiful fact about constrained single-particle quantum mechanics

I think you mean to say: bound instead of constrained. Right? Systems on the line whose potential goes to $+ \infty$ to both sides?

(Constrained systems are those where we have a Hilbert space of states such that not every vector in that Hilbert space has an interpretation as a physical state of the system. They, and their abstract categorical formilation, are extremely important, but I don’t think that’s what you are looking at.)

difficult to motivate the definition of the Hamiltonian

[…]

(Does anybody have any ideas?)

This, I think, cannot (and has no right to) be solved by just looking at the internalized CCR structure.

You are asking the question: “Why does that particular quantum propagation functor come to us?”

Pretty much generally the answer will be: “Because it arises as the quantization of some classical transport functor.”

(Remove the term “transport functor” from these two statements if you don’t like them, and read “dynamics” in both cases.)

So then the question is: what’s the thing about classical transport/dynamics that makes the harmonic oscillator appear so prominently?

I guess I could give something like an answer to that. But it won’t be really short. It comes down to understanding why free field theory Lagrangians like to be of the form $\phi \mapsto (\nabla \phi)^2 - m^2 \phi^2$ for “fields” $\phi : X \to \mathbb{R}$ of “mass” $m$.

It’s that $\phi^2$ term here which makes the oscillator appear.

Why is that square term singled out, though?

The best answer we have to that (which I think is actually the final answer already) is

a) fermions

b) Higgs mechanism.

(I really do think we need to go all that way, but I can sense readers thinking: “now he is overdoing it”.)

Together this says: we see square potentials all around us in field theory, because they come from bilinear fermion pairings $\langle \psi , \Phi \psi\rangle \,,$ where the endomorphism-valued field $\Phi$ (the Higgs field) is assumed to be constant.

If that sounds far-fetched from the internal category-theoretical point from which we started out looking at the world around us here, it should be emphasized that there is, in turn, a beautiful abstract way to explain this: that’s Alain Connes’ spectral action way of understanding things.

With that in hand, we get full circle back to the transport functor point of view:

we learn: quantum propagation is not just a functor on 1-dimensional Riemannian cobordisms, it is really thought of as a functor on 1-dimensional super-Riemannian cobordisms:

$QM : 1dCob_{superRiem} \to C$

If that functor is smooth (as it should be), then such functors are no longer characterized just by a Hamiltonian, but by a Dirac operator, a spectral triple in fact.

(Have a look at the first part of Stolz-Teichner’s What is an elliptic object? the here relevant aspects of which I review towards the end of Seminar on 2-Vector Bundles and Elliptic Cohomology, V).

Once we accept that 1-particles should be taken to be superparticles (which for 1-particles just means that they are spinning particles, which indeed our particles are) then we see that Connes’ spectral action is the natural action for the second quantized version, and there we see that it’s the fact that fermions want to pair in bilinears which gives rise to the Higgs effect and the appearance of quadratic potentials all over the place.

This sketches, I think, one big aspect of the full story which eventually needs to be told in detail.

There’s also the fact that I want mathematicians and computer scientists to be interested in my paper, and ‘second quantisation’ perhaps sounds a bit more esoteric and technical than ‘the quantum harmonic oscillator’. But this is a small point.

I don’t know about computer scientists, but I imagine that mathematicians will be rather pleased to learn (if they hadn’t heard about it before) that “second quantization is a functor”, and that this functor even arises as naturally and abstractly as you show it does.

Posted by: Urs Schreiber on January 16, 2008 10:25 AM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Jamie wrote:

…everybody else (i.e. Jeffrey Morton, John Baez) starts saying “quantum harmonic oscillator” whenever they get a whiff of the CCRs

Just to be clear: Jeffrey Morton has given a specific prescription for categorifying not only the canonical commutation relations and the harmonic oscillator Hamiltonian, but also a large class of perturbed harmonic oscillator Hamiltonians and the time evolution operators these give rise to. He expresses these time evolution operators as a sum of Feynman diagrams in the standard way, but categorified.

So, he’s gotten quite a bit more when a ‘whiff’ of the quantum harmonic oscillator. Indeed, he’s categorified enough of the usual techniques of perturbative quantum field theory that I think the way is clear to march down this road and do much more. And, there’s a nice relation to the ‘generating functions’ familiar in combinatorics, since the categorified version of the Fock space for an oscillator with $n$ degrees of freedom is the groupoid of $n$-tuples of finite sets.

Right now in the seminar we’re starting to:

1. review this material,
2. integrate it with our new improved understanding of groupoidification,
3. $q$-deform a bunch of this stuff, replacing the groupoid of finite sets by the groupoid of finite-dimensional vector spaces over $F_q$.
4. show how the resulting math is related to the Hecke and Hall algebras associated to the $A_n$ Dynkin diagrams.

(I know you were trying to defend your use of the term ‘harmonic oscillator’, not attack ours. But, you did it in a self-and-other-deprecatory British style that Americans don’t understand. Also, you should know that a good thesis advisor defends its students as a mother bear defends her cubs!)

Posted by: John Baez on January 16, 2008 6:37 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Aarg! Sorry Jeffrey: I think your work is fantastic. I really, really never meant to criticise anything…

I was under the impression that there is one pertinent similarity between your work and mine: in both cases, although the Hamiltonian can be written down, there seems to be no magical, categorical reason why it is the way it is. Apart from the fact that you can do lots of cool stuff with it, of course. Is this something you would agree with?

Posted by: Jamie Vicary on January 16, 2008 7:44 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Jamie wrote:

Aarg! Sorry Jeffrey: I think your work is fantastic. I really, really never meant to criticise anything…

I didn’t actually think so — and Jeffrey is probably even more laid-back, if he’s reading this at all. I just couldn’t keep myself from explaining why he obtained much more than a ‘whiff’ of the harmonic oscillator. Anyway:

although the Hamiltonian can be written down, there seems to be no magical, categorical reason why it is the way it is.

I’ll take this as a puzzle about the harmonic oscillator itself rather than its categorified version.

I think the answer goes like this. Let $HILB$ be the groupoid of Hilbert spaces and unitary operators. Then second quantization is a functor

$K : HILB \to HILB$

In particular, if $H$ is a Hilbert space and $K(H)$ the corresponding Fock space, we get a group homomorphism

$K: Aut(H) \to Aut(K(H))$

where (just to make things painfully explicit) $Aut(H)$ is the group of unitary operators on $H$, and similarly for $Aut(K(H))$.

In short: every symmetry of $H$ gives a symmetry of its Fock space.

We can differentiate this group homomorphism and get a Lie algebra homomorphism sending self-adjoint operators on $H$ to self-adjoint operators on $K(H)$:

$d K : u(H) \to u(K(H))$

Here we follow physicists and think of the Lie algebra $u(H)$ of the group $U(H)$ as the space of self-adjoint operators on $H$; mathematicians would prefer skew-adjoint operators.

In short: every Hamiltonian on $H$ gives a Hamiltonian on its Fock space.

If we apply this to the case where our Hilbert space $H$ is $\mathbb{C}$ and our self-adjoint operator on $H$ is just $1$, we get the harmonic oscillator Hamiltonian. If we take $H$ to be a big fat Hilbert space and still use $1$ as our self-adjoint operator, $d K(1)$ is called the ‘number operator’.

This is about as nice as one could want.

Posted by: John Baez on January 17, 2008 5:46 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

If we apply this to the case where our Hilbert space $H$ is $\mathcal{C}$ and our self-adjoint operator on $H$ is just 1, we get the harmonic oscillator Hamiltonian.

Oh!

I wasn’t aware of that. But now that you say it, it’s clear. That’s cool.

This construction Jamie should be able to incorporate easily in his generalized second quantization functor discussion, I guess!

So I was overdoing it in my previous comment. ;-)

Posted by: Urs Schreiber on January 17, 2008 8:44 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

I regret missing this historic meeting. I hope we can have some blog entries about it — for example posts by David and Urs, and guest posts by Bruce Bartlett and Jamie Vicary, and maybe also Bob Coecke and Andreas Döring and other people who’ve been known to hang out here!

(To write a guest post, just act like you’re going to post a comment, and then quit at the last minute, and instead email the text to me. Be sure to say which ‘text filter’ you used.)

If a bunch of people write short posts on only what they found most interesting about the conference, that’ll be more fun and less work than if one person tries to summarize everything.

Posted by: John Baez on January 8, 2008 9:39 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Greetings from somewhere between Hamburg and Brussels, on a night train, happen to have WLAN access for a moment, since we are in a station.

I’ll try to blog about the meeting as far as time permits. And as far as I am not absorbed with thinking about other things than topoi, which actually I am.

Posted by: Urs Schreiber on January 8, 2008 11:11 PM | Permalink | Reply to this

### Re: Categories, Logic and Physics in London

Thanks for the invitation John, I’ll surely take that one up. I wasn’t aware that this discussion is taking place until Jamie just knocked on my office door and pointed me to it. He seems to be moving to Oxford, great!

A short note on Urs’ discomfort with the use of *approaches*. Computer scientists have learned for already quite a while that abstraction is a tool to make particular features explicit, transparent and easier to use.

Now, if one looks at the foundations of quantum mechanics literature, *compositionality* is something one never finds there. The symmetric monoidal structure is exactly what makes this explicit, precisely capturing parallel and sequential composition of processes; and as we know, an amazing amount of quantum reasoning either emerges from it or can be adjoined to it e.g. adding *Bell-states* gives all the power of dagger compactness, and the classical objects mentioned above even allow you to reason about classical-quantum interaction in sophisticated quantum informatic protocols. As another example, referring to John’s discussion here, Rob Spekkens’ toy model can be axiomatized in a few lines as an SMC with classical objects. I am currently writing this down with my students Bill Edwards and Ben Jackson. This category is then characteristic for the particular quantum features Spekkens’ is trying to capture in his discrete model (in fact a sub-dagger-SMC of Rel; but one which doesn’t have biproducts i.e. is not semi-additive).

So the word approach basically boils down to a particular pair of glasses by means of which one tries to understand an important feature of nature. On the other hand, Isham and Doering discovered that to understand the propositional content of quantum observables, mainly in relation to physical space, one better puts on his topos-glasses.

These are just two different ways of looking at the same thing, namely nature, but having different perspectives on one thing is sometimes more useful than trying to see everything at once. Admittedly, of course, each *camp* thinks that their ansats is the better one to adjoin more and more features to it, hoping to obtain a more comprehensive theory, but not necessarily of everything.

Posted by: bob on January 17, 2008 3:38 PM | Permalink | Reply to this
Read the post The Concept of a Space of States, and the Space of States of the Charged n-Particle
Weblog: The n-Category Café
Excerpt: On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
Tracked: January 9, 2008 10:26 PM

### Re: Categories, Logic and Physics in London

It was a great meeting! For me, the best thing was to meet lots of people who I had only known by reputation, like Bruce Bartlett, Eugenia Cheng and Simon Willerton.

Unfortunately, I never got a chance to talk properly to David or Urs because I was having to spend so much time playing with the video camera. For the same reason, I couldn’t take notes from the talks, so I look forward to getting the chance to look back over the recordings — which all of you will hopefully be able to do within a week or so.

Something that Bob Coecke was originally hoping to do was formally set aside some time for people to work on specific problems in small groups. In the end this wasn’t included in the formal schedule, as too many people had registered to come! But I’d be interested to know if people have experienced this sort of thing at workshops before, and whether or not they think it’s worthwhile.

Posted by: Jamie Vicary on January 12, 2008 8:57 PM | Permalink | Reply to this

### Videos are online

I am happy to announce that the videos of the talks are now online, at http://categorieslogicphysics.wikidot.com/video. Sorry it took a while to get them ready, but there are so many technical problems that crop up when doing this for the first time.

Unfortunately, because of the microphone we used, you can’t hear the questions asked by the audience. If you gave a talk or asked a question, or if you think you can remember what any of the questions were, it would be great if you could email me at jamievicary@gmail.com so that I can add them as subtitles.

Posted by: Jamie Vicary on February 3, 2008 6:41 PM | Permalink | Reply to this
Read the post Categories, Logic and Physics in London, II
Weblog: The n-Category Café
Excerpt: Second workshop on "Categories, Logic and Foundations of Physics" in London.
Tracked: February 19, 2008 12:17 AM
Read the post 6th Workshop on Categories, Logic and Foundations of Physics
Weblog: The n-Category Café
Excerpt: The 6th Workshop in a series organized by Bob Coecke and Andreas Döring.
Tracked: January 25, 2010 7:34 PM

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