## March 31, 2009

### Categories, Quanta and Concepts at the Perimeter Institute

#### Posted by John Baez

Many customers here at the $n$-Café believe — do I dare say know? — that category theory provides crucial clues about the foundations of quantum theory. This summer there will be a workshop at the Perimeter Institute devoted to pursuing these clues:

Here’s a description of the workshop:

In recent years, more and more mathematical methods and results from the field of category theory have been employed in the foundations of physics. These methods enable us to reconsider conceptual and structural issues in the foundations of quantum theory and possibly beyond, often providing surprising insights and new points of view. The workshop “Categories, Quanta, Concepts” (CQC) will bring together researchers from a variety of backgrounds, physicists, mathematicians, theoretical computer scientists and philosophers, all with a common interest in the foundations of physics. With talks in the morning and workgroups and in-depth sessions in the afternoon, the workshop will provide ample opportunity for interaction between the participants from this emerging community. The workshop also serves as an ideal platform for PhD students to get insight into the field and to discuss their own work.

The invited speakers are:

• Samson Abramsky, Wolfson College, Oxford University
• Harvey Brown, Faculty of Philosophy, Oxford University
• Howard Barnum, Los Alamos National Laboratory
• Richard Blute, Dept. Mathematics and Statistics, Ottawa University
• Keye Martin, U.S. Naval Research Laboratory
• Prakash Panangaden, School of Computer Science, McGill University
• Peter Selinger, Mathematics and Statistics, Dalhousie University
• Mike Stay, University of California at Riverside

Mike may talk about using categories as a Rosetta Stone to understand the links between physics, topology, logic and computation.

Posted at March 31, 2009 4:05 PM UTC

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### Re: Categories, Quanta and Concepts at the Perimeter Institute

Hi John,

Do you know any specific resource
for Category Theory specific for the topology of 4 manifolds?

Posted by: Daniel de França MTd2 on March 31, 2009 7:42 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Do you know any specific resource for Category Theory specific for the topology of 4 manifolds?

There is an $(\infty,4)$-category of 4-dimensional manifolds (brief idea at $n$Lab: $(\infty,n)$-category of cobordisms)

the resource for which is On the classification of TFT.

Posted by: Urs Schreiber on March 31, 2009 8:13 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

If you want to study 4-manifolds topologically, an h-cobordism induces a homeomorphism, so this should be an iso in this category right?
And if you want to study whether two manifolds are diffeomorphic, you need to go beyond cobordism classes, so I would guess that this cobordism category is not the “right” category of smooth 4-manifolds.

This, confusingly, is also studied in physics using topological quantum field theory. I’m always confused by these things, so please correct me if I’m wrong.

On-topic: seems like a nice workshop, I hope some abstracts go up soon, then I can decide to apply.

Posted by: Jan on April 1, 2009 8:53 AM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

“If you want to study 4-manifolds topologically, an h-cobordism induces a homeomorphism, so this should be an iso in this category right?”

Sure. Indeed, that’s the main reason. I really want to know what Category Theory has for these crazy cases of smoothness.

Posted by: Daniel de França MTd2 on April 1, 2009 11:48 AM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

And if you want to study whether two manifolds are diffeomorphic, you need to go beyond cobordism classes,

The morphisms in the 1-category $n$Cob are themselves (diffomorphism classes) of manifolds, not cobordism classes of manifolds.

Similarly the $k \leq n$-morphisms in an $(\infty,n)$-category of cobordisms are represented by manifolds, not by cobordism classes of manifolds.

This, confusingly, is also studied in physics using topological quantum field theory.

A topological field theory would be a representatrion of a category of cobordisms whose morphisms are, in top degree, diffeomorphism classes of manifolds.

Some more motivating remarks on this are for instance in

Eugenia Cheng, Nick Gurski, Towards an $n$-category fo cobordisms.

It’s true probably that technically this should be called in general “Diff-Class field theory” then, I suppose, at least if the representation (the QFT) in question really depends on the diffeomorphism classes and not just on the homeomorphism classes.

Historically, the word “topological” in physics was mainly to emphasize the “does not depend on (pseudo)-Riemannian or conformal structure”.

But then, the axiomatization of TFT and QFT in terms of representations of cobordism categories is something that hasn’t come out of the physics community.

Posted by: Urs Schreiber on April 1, 2009 4:08 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Thanks for the explanation. I still have tons of questions on this subject, but I’ll first entertain mysel by reading some real introductions on TFT before I will embarrass myself further, and have something more useful to add to the discussion.

Posted by: Jan on April 1, 2009 4:39 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

reading some real introductions on TFT

Maybe that helps. If not, you can do me a favor by adding what you think is missing.

Posted by: Urs Schreiber on April 1, 2009 5:23 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

But I can’t find in any of those links how one can see how the issues with exotic smoothness in 4D, the non diffeomorphism of the poicare sphere in 4k dimensions, the non equivalence of top, triangulation and smoothness etc… with computation, logics, etc.

Maybe the rosetta stone idea just works with a tiny tiny part of topology.

Posted by: Daniel de França MTd2 on April 1, 2009 4:57 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

I can’t find in any of those links how one can see how the issues with exotic smoothness

I’ts not clear to me which issue precisely you want to be solved by the category-theory-power-tool™.

From the pure math perspective, what the existence and understanding of an $(\infty,n)$-category of cobordisms buys you is that it provides you with a tool for translating global questions about spaces into local questions.

The Baez-Dolan-Lurie-Hopkins theorem is just the most vivid incarnation of that: for understanding where a big space is sent to under a representation, it is sufficient to know where the point is sent to plus understanding higher dimensional gluing.

I haven’t worked sufficiently with exotic smooth structures to know if such local-to-global tools are of much interest there. Maybe somebody else knows.

Posted by: Urs Schreiber on April 1, 2009 5:30 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Let me put it simple. Does it make sense to think in using category-theory-power-tool™ to find in logics, computing, physics, etc, the equivalent of Rokhlin’s theorem?

Posted by: Daniel de França MTd2 on April 1, 2009 7:59 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Daniel wrote:

But I can’t find in any of those links how one can see how the issues with exotic smoothness in 4D, the non diffeomorphism of the Poincaré sphere in 4k dimensions, the non equivalence of top, triangulation and smoothness etc… with computation, logics, etc.

Maybe the Rosetta Stone idea just works with a tiny tiny part of topology.

I really doubt it’s so limited. The Generalized Tangle Hypothesis relates smooth, piecewise-linear, topological, oriented, spin, etcetera manifolds to various algebraically defined $n$-categories. You can read about this here. Check out all the links, especially the link back to the café discussion where the Generalized Tangle Hypothesis was first publicly promulgated.

There’s a lot of work left to be done, relating various kinds of manifolds to various kinds of $n$-categories and various generalizations of TQFT. But don’t forget: Donaldson theory was what make Witten invent the notion of TQFT in the first place, and this theory detects exotic smooth strutures! A great challenge is to understand Donaldson theory (or now: Seiberg–Witten theory) from a more category-theoretic approach.

Posted by: John Baez on April 2, 2009 7:00 AM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Thank you very much John. I was already aware that SW invariants could be used to detect exotic smothness, but I didn’t know Witten invented TQFT with Donaldson theory in mind. So, It seems then Witten does everything not just with string in mind, but mostly with Topology.

Since we are talking about topologies in 4 dimensions, and you mentioned witten’s topological mind, I rememebered that his most recent mathematical advanture is trying to see Langland dualities from a geometrical point of view. It happens that this happen with a N=4 SYM theory. So, we have topological dualities with N=2,N=4 SYM in 4 dimensions.

Now, is there any way of relating N=2 with N=4 in 4 dimensions through topological invariants? That is, a way to relate to TQFT and Langlands program? It seems there is a lot of category theory Langlands program, so, relating these two, it could help describe Donaldson-SW witten in the the categorical way you like.

Posted by: Daniel de França MTd2 on April 2, 2009 5:12 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Be aware though, that as far as I know, it is unknown whether Witten-type TQFT’s are described by the Atiyah-Segal axioms.

The difference between Witten-type TQFT’s and “normal” TQFT’s are that in the former case, the action and the path integral “measure” do depend on the metric, but these dependencies magically cancel each other out. In the latter, the invariance is more explicit, in that the action and measure are invariant under changes of the metric.

I can’t comment on the N=2 and N=4 relation. I wrote a master’s thesis on Seiberg-Witten invariants, but I still don’t understand why it needs a N=2 theory. My problem probably comes from a lack of background in physics.

Posted by: Jan on April 3, 2009 9:44 AM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

” but I still don’t understand why it needs a N=2 theory”

Neither do I. I just memorized the facts.

In fact, I really do not like the idea of supersymmetry, at least until a few days ago, when Renata Kallosh provided the strongest argument until now of why N=8 SUSY is renormalizable at all orders. Given that this is related to N=4 Susy by KlT relations, I suddenly became interested in SUSY, even though mostly the things I know are just facts.

Posted by: Daniel de França MTd2 on April 3, 2009 1:05 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Jan wrote:

On-topic: seems like a nice workshop, I hope some abstracts go up soon, then I can decide to apply.

Apparently 30 people have already applied to attend. But yes, abstracts would be nice.

Posted by: John Baez on April 2, 2009 4:26 AM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

There is also the stuff by Prakash Panangaden, and Keye Martin on domain theoretic approaches to ‘A domain of spacetime intervals for General
Relativity’ which is on the webpage

http://www.cs.mcgill.ca/~prakash/talks.html

as a Dagstuhl talk from 2006.

Posted by: Tim Porter on April 1, 2009 8:42 AM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

Looks like a great conference! I hope that someone picks up on the categorical implications of the new results about the time variation of dimensionless fundamental constants .

Posted by: stefan on April 2, 2009 4:01 PM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

You can see Jeffrey Morton’s description of his talk at Categories, Quanta and Concepts here.

Posted by: John Baez on June 9, 2009 1:45 AM | Permalink | Reply to this

### Re: Categories, Quanta and Concepts at the Perimeter Institute

And all the talks are available online.

Posted by: Mike Stay on June 10, 2009 12:19 AM | Permalink | Reply to this
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 7:31 PM

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