A Prehistory of n-Categorical Physics

May 18, 2009

A Prehistory of n-Categorical Physics

Posted by John Baez

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I’m valiantly struggling to finish this paper:

John Baez and Aaron Lauda, A prehistory of n-categorical physics (draft version).

Perhaps blogging about it will help…

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… or at least help me feel like I’m not procrastinating.

We’re calling it a ‘prehistory’ because the history hasn’t quite started yet. Nobody yet has used explicitly $n$ -categorical ideas to formulate a successful theory of physics. Nonetheless, many lines of thought are converging in this direction, so it seems worthwhile explaining what people are doing, even if it never amounts to anything substantial.

Here’s one problem with writing a prehistory when the history hasn’t started yet: it’s hard to know where to stop.

Right now the subject of $n$ -categories is really exploding. But I don’t want to talk about the cool papers that came out last week… so I’m arbitrarily drawing a cutoff at the year 2000, and I’ll only talk about work after that it completes a story that was well underway before this date. For example, the cobordism hypothesis was formulated before 2000, but it’s impossible to discuss this hypothesis without at least mentioning Lurie’s recent work on it… so I’ll do that. But I need to be ruthlessly selective in my choice of topics. Someday I may a book on this stuff, but this is not that book.

So: I’m trying to hint that if you suggest topics for me to write about, you shouldn’t feel bad if I say “no” or don’t even reply at all. I’m trying to finish this paper by June 1st, so I even need to say “no” to most of my own great ideas about what to add to it.

But, I thought I might post a few comments here about the subjects I’m trying to tackle — just to build up the psychic energy needed to write the paper. And maybe you can cheer me on, and cheer me up, and post corrections, and chat about the ideas, and suggest enormous extra subjects to write about that I won’t have time for.

Right now I’m on the entry Segal (1988). I have to explain the idea of conformal field theory in very simple terms: especially how it arose from string theory, how it led to Atiyah’s axioms for a topological quantum field theory, and how it connects to higher categories. I suddenly realized that I need to set the stage with an entry String Theory (1980s).

So, let me try that…

Posted at May 18, 2009 8:58 PM UTC

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67 Comments & 1 Trackback

Re: A Prehistory of n-Categorical Physics

Nice paper. I am happy to cheer you on. Being pretty ignorant of the subject, I will not suggest any topics…

Posted by: Eugene Lerman on May 19, 2009 4:05 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Excellent! But maybe you can compensate for all the topics people are suggesting below by suggesting a topic I should remove — something you don’t want to know about.

Posted by: John Baez on May 20, 2009 4:02 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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something you don’t want to know about

You list some standard physics milestones such as those in the development of relativity and quantum mechanics. In some cases it didn’t quite become clear to me how you want your readers to think of these as instances of $n$ -categorical physics.

For instance the list right at the beginning, Maxwell through Einstein. That seems to be about the development of the idea of relativity. If you want to claim that this somehow leads to $n$ -categorical notions you might want to do so more explicitly. I for one am not sure what you have in mind here. (But then, I haven’t read the article in total. But then in turn, maybe I did a few years back, if I am not mistaken.)

Or, otherwise, why not just remove the relativity bit! :-)

(But more exciting I’d find it if you left it in and explained your way of thinking of relativity as something of $n$ -categorical flavor.)

Posted by: Urs Schreiber on May 20, 2009 8:04 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Urs wrote:

You list some standard physics milestones such as those in the development of relativity and quantum mechanics. In some cases it didn’t quite become clear to me how you want your readers to think of these as instances of $n$ -categorical physics.

It’s supposed to become clear eventually, but maybe it’s not clear yet.

For example, I explain how quantum mechanics became the study of unitary group representations so I can explain how Feynman diagrams are string diagrams in a category of unitary group representations so I can explain how string diagrams can be used to compute in any symmetric monoidal category with duals so I can explain how this relies on the tangle hypothesis for 1-dimensional tangles in codimension 3, so I can explain the tangle hypothesis and why it’s important for physics. I’ve done most of this — only the final stages, after 1990, are missing.

Also, I explain general relativity and the fundamental group so I can explain gauge theory and path groupoids so I can explain higher gauge theory and path $n$ -groupoids. I’ve just barely gotten started on this.

In a sense these are the two main themes: higher-dimensional ‘Feynman diagrams’ representing physical processes, and higher gauge theory. But, the first theme is much more developed, and the second theme will take so much more work to develop properly that I almost want to leave it out — but I can’t, because it’s much too important.

In each case, I should sketch the basic idea in the ‘Roadmap’ and then let the story gradually unfold in the ‘Chronology’. The ‘Roadmap’ should let the reader know what’s happening before it happens.

I’ll need to finish this paper long before it’s perfected. I console myself with the thought that it’ll eventually become part of a book, or books, where I explain everything in more detail.

An $n$ Lab entry would help a lot! But right now I need to spend my time on this paper, which is the last of the expository papers I need to write by a deadline.

Posted by: John Baez on May 20, 2009 4:53 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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In a sense these are the two main themes: higher-dimensional ‘Feynman diagrams’ representing physical processes, and higher gauge theory.

Yes. And I think there is a reason: these are the first two steps in the $n$ -categorical “tale of successive quantization”.

Here is how I think second quantization, and hence Feynman diagrams as well as string perturbation theory etc, fits into the general picture of Sigma-model extended QFT.

I’ll use some kind of “n-categorical pseudo code” in the following. It won’t run when put on a machine as is, but every knowlegeable human will know how to turn it into code that does.

So here goes:

The story of $n$ -fold $n$ -categorical quantization

We start with “classical” data: a target space $X$ (some $\infty$ -categorical object, as all of the following) equipped with a background field $\nabla$ given by its parallel transport morphism

$\nabla : X \to V \,.$

First quantization will turn this into a morphism

$\exp(\nabla) : \exp(X) \to \exp(V) \,,$

where

- $\exp(X)$ is some $\infty$ -category of cobordisms from which we may have maps to $X$

- $\exp(V)$ is similarly some kind of spans in $V$

- $\exp(\nabla)$ is the morphism obtained by homming a cobordism into the space $\nabla^* E V$ classified by $\nabla$ and then pull-pushing through the resulting span, for instance as in section 7 here (or similar, see the remark at the end).

The pull push yields the path integral for first quantization. It describes propagation of a single $n$ -particle through $X$ charged under $\nabla$ . The push-forward in the pull-push involves a sum over all possible maps from a given cobordism $\Sigma$ to $X$ , all paths in $X$

Second quantization

The simple observation now is: all we needed to start doing first quantizatoin was a morphism $\nabla : X \to V$ . But the output of first quantization is itself such a morphism

$\exp(\nabla) : Cob \to \exp(V) \,.$

So we may simply iterate the procedure!

Now what used to be “target space” is the $\infty$ -category $Cob$ .

This means that

- a single point now is a collection of $k$ $n$ -particles;

- a path now is a Feynman diagram (or string diagram, or higher) on which the $k$ $n$ -particles move and react to produce an output of $l$ $n$ -particles: another point in $Cob$ .

So we do pull-push quantization again, now with $Cob$ as our “target space”.

This means first of all that we need to pick once again some category of cobordisms. Let me take $1Cob$ for that, lest we blow our brains right away with second quantizoidation.

So , we now hom the interval into target space $Cob$ and do pull-push quantization. The push-forward now tells us to sum over all “paths” in $Cob$ between given in and out point. But that just means to sum over all Feynman diagrams! Each weighted with the value $\exp(\nabla)$ from first quantization, i.e. with its corresponding first quantized correlator.

This is exactly what we expect. (I am not sure if this is clear to everybody, that the Feynman amplitudes are just the 1-quantized corrlators of the single particles: if not, follow the links to “worldline formalism” that are linked to from the entry [[string theory]]).

third quantization

And so on.

Remarks

I am hoping to work this out in detail eventually. But I am also posting it in case somebody reads it who would do it quicker than I can. Recently I had the pleasureful experience that a young guy unknown to me before contact me by email to inform me that he had read, formalized and then proven an aspect of the $n$ -Café Quantum conjecture, which is the basic building block here. That made me think that it is maybe not completely pointless to have such kind of discussion on a blog after all…

Posted by: Urs Schreiber on May 21, 2009 5:56 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Urs wrote:

First quantization… Second quantization (“target space” is now the $\infty$ category Cob)

Wo, that’s freaking my mind out Urs. But I guess you’re right. Never thought about it that way (actually I haven’t thought about this chain of going from first quantization to second quantization and so on for a long while).

Posted by: Bruce Bartlett on May 21, 2009 6:45 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Never thought about it that way

Oh, I seemed to remember that we had talked about this before. Maybe not. At least I mentioned this idea before here.

It’s the kind of idea that feels “obviously right” but needs some energy to work out in detail. Eventually we’ll get there…

Posted by: Urs Schreiber on May 21, 2009 10:45 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Urs wrote:

Recently I had the pleasureful experience that a young guy unknown to me before contact me by email to inform me that he had read, formalized and then proven an aspect of the n-Café Quantum conjecture, which is the basic building block here.

Is there some reason to keep the identity of this young guy secret? Shouldn’t we instead do a blog post in his honor, or collect money to buy him a present?

Posted by: John Baez on May 21, 2009 7:38 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Is there some reason to keep the identity of this young guy secret?

No, I think there’ll be a chance to post something, soon.

Shouldn’t we instead do a blog post in his honor,

Will soon.

or collect money

Already did that, actually.

buy him a present?

Yup, a trip to Hamburg, I thought.

Posted by: Urs Schreiber on May 21, 2009 10:38 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

This reminded me of your

Exercise in Groupoidification

I really hope to get back to that one day. Maybe after the first week in June, life will begin to resemble mere normal levels of chaos as opposed to the heightened levels of chaos of the past year.

Anyway, one of my goals since the nLab was created was to convert that thread into a clean article because I think it is fairly simple to understand (I ALMOST understood it) and I think it brings in many important concepts that are important generally.

It’s long overdue, but I finally got around to creating a page for it on the nLab:

An Exercise in Groupoidification

Right now, it is just ported over, but eventually, I hope we can edit it a bit.

Posted by: Eric on May 22, 2009 3:12 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Thanks, Eric. i should try to find some time to work more on this. You should know that the “exercise in groupoidification” has been “expanded”, if you wish, to these notes, which are closely related, I think, to the construction appearing in “geometric $\infty$ -function theory” and also, methinks, to the construction sketched in the latest Freed-Hopkins-Lurie-Teleman.

So there is definmitely reason for me to try to find time to work more on this. But time is a rare good.

Posted by: Urs Schreiber on May 22, 2009 5:37 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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I will have a look. Thanks. I’m afraid that my brain can grok at most a finitary version of all this, which is why I liked that Exercise so much.

By the way, by accident, I discovered that you can include symbols in wiki links by pasting the symbol rather than using itex. I fixed a typo on your page and changed [[sigma-model]] to [[ $\sigma$ -model]].

I hope the “pasting” trick is not browser sensitive. It works in Firefox.

Posted by: Eric on May 22, 2009 5:54 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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The pasting trick works even when you create pages:

$\infty$ -groupoid

If this works on all browsers, I might do some reconstruction on the nLab (if no objections). For example, I might copy [[infinity-groupoid]] to [[ $\infty$ -groupoid]], etc.

Posted by: Eric on May 22, 2009 6:11 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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If this works on all browsers, I might do some reconstruction on the nLab (if no objections).

We originally decided not to do this (or I originally decided it and everybody else went along, I'm not sure which) on the grounds that it's harder for many people to type article names that way. That is, if somebody writes ‘infinity-groupoid’ in text since it's not easy for them to do it otherwise, then somebody else can change it later to ‘ $\infty$ -groupoid’ without any trouble, not to mention that iTeX makes it fairly easy to do it right the first time. But if somebody creates [[infinity-groupoid]], then it's more of a job to move that to [[∞-groupoid]] later (although you are willing to do it). What's worse, if somebody creates [[∞-groupoid]], then it may be difficult for someone to figure out how to even link to it! They'll have to link to [[infinity-groupoid]], hope that somebody has already made a redirect page there, and we'll still want someone to come in later and make the link direct.

It would not be as big a deal if we had Wikipedia-style redirects, since then it doesn't matter if links are direct or go to a redirect page. All the same, anyone creating a page like [[∞-groupoid]] would have to take care to also create a redirect from [[infinity-groupoid]] to ensure that people with limited computer expertise can link to it easily. (Or am I wrong about how savvy people are? Can everybody figure out how to type ‘∞’ without using iTeX or & entities, which don't work in links, and also without doing anything inconvenient like looking up Unicode or copy and paste?) So the naming convention is to use only ASCII symbols in page names (except for user pages, the one so far being [[Zoran Škoda]]).

If Instiki parsed typographical symbols in page names, then this would be another matter; we could do [[$\infty$-groupoid]], [[Chevalley--Eilenberg algebra]], etc. But I don't think that this would work very easily with Jacque's code (happy to be proved wrong, of course!).

Anyway, Eric, we should probably start a thread at the Forum or the discussion post if you want to talk about this more.

Posted by: Toby Bartels on May 22, 2009 8:22 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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I experimented with a bunch of pages, but have rolled them all back now.

As you noted, I may still change some of the link displays, but without changing the links themselves, e.g. now and then, I may change a

[[infinity-category]]

[[infinity-category| $\infty$ -category]]

so that it looks better in the page without changing the structure.

If some day you do decide to make a wholesale change, I’d be glad to volunteer to do a lot of the grunt work. Many of the pages I created still exist, but with nothing more than a redirect. If I can’t contribute much to the mathematical content, maybe I could at least contribute to the aesthetics :)

If the day comes when we do get a real redirect feature, it may make sense to reconsider the idea.

Posted by: Eric on May 22, 2009 9:18 PM | Permalink | Reply to this

ASCII page titles

Note to all: There is indeed discussion of this at an appropriate place here.

Posted by: Toby Bartels on May 22, 2009 8:58 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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John wrote:

In a sense these are the two main themes: higher-dimensional ‘Feynman diagrams’ representing physical processes, and higher gauge theory.

Urs replied:

Yes. And I think there is a reason: these are the first two steps in the $n$ -categorical “tale of successive quantization”.

I’ve tried to allude to this — in an extremely vague and allusive way — at the end of the road map to my paper, where I explain two lines of thought leading to $n$ -categorical physics, and apologize for only talking about the first.

Don’t bother reading this, Urs — you won’t learn anything new. But some other people will!

Posted by: John Baez on May 22, 2009 8:37 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Urs wrote:

Or, otherwise, why not just remove the relativity bit! :-)

I think that’s what I’ll do. Telling the story of general relativity, gauge theory and then higher gauge theory is too much for me to tackle in the next 10 days — I’ll be plenty busy finishing the other aspects. Also, a lot of interesting work on higher gauge theory started after the 2000 AD cutoff point for this history.

I just added a longish apology for this at the end of the ‘Roadmap’, and deleted some stuff about general relativity and gauge theory.

Posted by: John Baez on May 21, 2009 7:31 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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An $n$ Lab entry would help a lot! But right now I need to spend my time on this paper

[…]

Telling the story of general relativity, gauge theory and then higher gauge theory is too much for me to tackle in the next 10 days

I understand that probably in this case it is not an option for you, but I feel like at least mentoning the meme that developing such a task on the $n$ Lab can in general mean alleviation instead of additional work.

I think most regular contributors to the $n$ Lab had had this nice experience: you post an entry to the best of your abilities. Next day or two you come back to the Lab and find that a few people not only corrected a bunch of typos and trivial mistakes, but also expanded on aspects one hadn’t oneself heard of before, added links one didn’t think of adding and all in all produced something quite beyond what one was able to produce oneself in the given time frame.

Once you feel like you have time (not that I have much time!) we (by which I mean the whole $n$ -Café gang) should start working on [[ $n$ -categorical physics]].

If you are all right with it, we could start by filling in the material from your article and then continue expanding it indefinitely. All the stuff you need to omit now we should eventually ad. Piece by piece, little step by little step.

Posted by: Urs Schreiber on May 21, 2009 10:56 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Urs wrote:

I understand that probably in this case it is not an option for you, but I feel like at least mentioning the meme that developing such a task on the $n$ Lab can in general mean alleviation instead of additional work.

Right. It doesn’t work when you foolishly need to deliver a fully finished product in 10 days. But for more reasonable projects I bet it could help a lot.

I haven’t been contributing to the $n$ Lab much because I’ve been busy finishing off papers that have deadlines. But luckily, THIS IS THE LAST ONE!

I have a bunch of other projects lined up:

Higher-dimensional algebra VII: Groupoidification (with Alex Hoffnung and Christopher Walker).
Higher-dimensional algebra VIII: Hecke algebras (with Alex Hoffnung).
Higher-dimensional algebra IX: Hall algebras (with Christopher Walker).
Division algebras and supersymmetric Yang–Mills theory (with John Huerta).
My favorite numbers.

but I don’t feel so stressed about these, for some reason.

I expect my next big project will be a book — an introductory textbook on diagrammatic methods in $n$ -category theory, with applications to all sorts of subjects. My ‘Rosetta’ paper and this ‘Prehistory’ should become part of that.

For this project I may try to use the $n$ Lab… but I have a lot of questions about how. I’m not sure how much people will want to help me write a book — or how much I want to cede control over its final form. Should it really be a book, or just a perpetually evolving bunch of webpages? I’m afraid I still have the urge to write a nicely worded, perfected book. It seems old-fashioned, I know…

Anyway, I don’t need to decide now.

If you are all right with it, we could start by filling in the material from your article and then continue expanding it indefinitely. All the stuff you need to omit now we should eventually add. Piece by piece, little step by little step.

That’s fine with me. I can donate the article when it’s done — it probably makes sense to wait until it’s done; that shouldn’t take too long.

Posted by: John Baez on May 22, 2009 4:39 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

How about describing the CFT/TFT correspondence through its connection to boundary CFT? I haven’t yet come across a very clear explanation of the physics behind this connection (but would love to see one). The usual story says that if we have a topological field theory on a manifold with boundary, then massless chiral excitations can exist on the boundary, which are described by a related CFT. AFAIK, a general theory has not been worked out, but lots of examples are known.

People began exploring this stuff back in the 80’s, shortly after the basics of CFT were getting hammered out - certainly appropriate for your prehistory, but maybe this is what you were planning anyway…

Posted by: Jon on May 19, 2009 6:19 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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How about describing the CFT/TFT correspondence through its connection to boundary CFT?

Good point. By the way for the case 3dTFT/2d CFT this has a beautiful comprehensive and categorical (on both sides of the duality) description in terms of the FFRS-model #.

Concerning the case for $n$ -categorical physics made by this:

I had argued # that holography in QFT relating an $n$ -dimensional QFT to a boundary $(n-1)$ -dimensional theory is about having transformations

$\eta : 1 \to Z_n$

from the trivial $n$ -dimensional theory into the $n$ -dimensional one $Z_n$ , both regarded as functors

$1, Z_n : n Cob \to somewhere \,.$

The components of the transformation itself form a functor on $(n-1)$ -dimensional cobordisms

$\eta : (n-1)Cob \to somewhere$

which is the lower-dimensional QFT.

Later this was picked up by Peter Teichner and Stefan Stolz, who then called this $\eta$ a “twisted QFT”.

The idea also appears in the latest Freed-Hopkins-Lurie-Teleman, where it is used to describe 3d Chern-Simons theory as being twisted by a 4d theory!

This is not yet the full realization of the holography Chern-Simons/Wess-Zumino-Witten by natural transformations of QFT functors that I think # should exist, but it seems to be getting close.

Posted by: Urs Schreiber on May 19, 2009 7:02 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Last night I was reading a bit of Chris Schommer-Pries’s thesis, and in the introduction he mentions that one of the reasons extended TQFT is attractive is because by going to lower and lower dimension you are able to go to simpler building blocks for your manifold, making the invaiants inherently more computable.

The thought suddenly struck me that this is precisely what Witten did in his original paper, “Quantum field theory and the Jones polynomial”. Recall that in that paper he attacks the problem on two different fronts. In the beginning he proceeds perturbatively, expanding around the classical solutions, and obtains nice topological invariants, which hints that the full “nonperturbative” theory actually exists.

When he actually computes the invariants though, such as

(1)

Z(S^3) = \frac{2}{k+2} \sin( \frac{\pi}{k+2}),

he does it by establishing the link between Chern-Simons theory in 3d and the spaces of conformal blocks in the 2-dimensional WZW model. We know that the latter has to do with representations of loop groups, and nowadays we “know” that these are the things which make up the category assigned to the circle. I write “know” in inverted commas because my impression is that there are still quite a few t’s to be crossed and i’s to be dotted on this point (at least I hope so because I hope to be involved in that!).

So, Witten was secretly using the category assigned to the circle in order to calculate the 3-manifold invariants! He was the first extended TQFT theorist, though he never knew it. Unless I am mistaken here, this offers a distinct selling point of higher-categorical physics: extended TQFT apparantly enables you to calculate certain quantities in topological field theories nonperturbatively. Am I misguided here?

Posted by: Bruce Bartlett on May 22, 2009 12:46 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

By the way, that comment of mine was meant to be read as: “I agree with what Urs is saying here, here are some extra arguments to weigh in”

Posted by: Bruce Bartlett on May 22, 2009 12:49 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Nice read.

Here are suggestions that came to my mind:

John Roberts might deserve being mentioned not only for the Doplicher-Roberts theorem, but also for emphasizing the description of QFT as a co-presheaf of observables, and for motivating from that the notion of $\omega$ -catgegories and of nonabelian cohomology with coefficients in them. I find this quite remarkable. Roberts’ role in this development is described in some detail in Street’s “conspectus” that you mention, and in his “aspects of descent”.

That was

John E. Roberts, Mathematical Aspects of Local Cohomology, talk at Colloqium on Operator Algebras and their Applications to Mathematical Physics, Marseille 20-24 June, 1977

(On the other hand, I wonder how clear it was to people that the theory of descent for $\omega$ -category valued presheaves being born here from quantum field theory was to a large extent at least overlapping with if not included as a special case of the model for $\infty$ -stacks and nonabelian sheaf cohomology presented by Kenneth Brown already 1973 (here))that later grew into to Brown-Joyal-Jardine model for $\infty$ -stacks that is now seen to be the model for $\infty$ -stacks.)

Kontsevich deserves to be mentioned notably for understanding that homological mirror symmetry is about a duality of derived categories, an insight that deeply injected categorical methods of homological algebra into formal high energy physics and maybe more so, made it come to its full glory there.

If you stop up to and including the year 2000, I would say also Freed’s work on charges in terms of differential cohomology in physics deserves to be mentioned. (winter 2000). It shows not only that already Maxwell’s work is actually concerned not just with line bundles as Dirac noticed (you don’t mention Dirac!, do you?) but also with line bundle gerbes, and then goes on to identify and study a plethora of higher bundles with connection in higher supergravity theories coupled to higher gauge theory.

This may make one also want to mention Cheeger-Simons, whose differential characters really first captured the idea of higher dimensional parallel transport.

Jim Stasheff famously developed and promoted the idea of homotopical physics, which is really nothing but another word for $(\infty,1)$ -categorical physics.

There is so much $\infty$ -categorical physics out there which many of its very practitioners tend not to recognize as such. All of BV-BRST formalism is really a way to generalize the idea of quantization from spaces of configurations to Lie $\infty$ -algebroids of configurations. Lots of differential-graded algebra techniques in physics reveal under Dold-Kan that large bits of physics have been inherently and strongly $\infty$ -categorical long before this term has even been made up.

Notably it seems that much of string field theory (see Stasheff’s version, again) is deeply $\infty$ -categorical, if only waiting to be realized manifestly so. That was the big thing throughout the 90s, now having ceased for some reason.

Posted by: Urs Schreiber on May 19, 2009 6:46 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

In response to:

It shows not only that already Maxwell’s work is actually concerned not just with line bundles as Dirac noticed

explicitly?

Jim Stasheff famously developed and promoted the idea of homotopical physics, which is really nothing but another word for (infty,1)-categorical physics.

Thanks for pointing that out -TO ME!

There is so much infty-categorical physics out there which many of its very practitioners tend not to recognize as such. All of BV-BRST formalism is really a way to generalize the idea of quantization from spaces of configurations to Lie infty-algebroids of configurations.

Or JUST as homological algebra!

Notably it seems that much of string field theory (see Stasheff’s version, again) is deeply infty-categorical, if only waiting to be realized manifestly so. That was the big thing throughout the 90s, now having ceased for some reason.

Not quite ceased - see my work with Kajiura on OCHA and work of Hoefel - all on the arXiv.

Posted by: jim stasheff on May 19, 2009 2:33 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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explicitly?

Oh, I thought I was being clear, I am referring to the discussion of charges as here.

Jim Stasheff famously developed and promoted the idea of homotopical physics, which is really nothing but another word for $(infty,1)$ -categorical physics.

Thanks for pointing that out -TO ME!

You are wecome. :-)

Notably it seems that much of string field theory (see Stasheff’s version, again) is deeply $\infty$ -categorical, if only waiting to be realized manifestly so. That was the big thing throughout the 90s, now having ceased for some reason.

Not quite ceased - see my work with Kajiura on OCHA and work of Hoefel - all on the arXiv.

Ah, I hadn’t seen Hoefel’s work. You mean

Eduardo Hoefel, OCHA and the swiss-cheese operad

I suppose. Interesting.

So maybe I shouldn’t have said “ceased”. But it seems to be evident that interest in structural aspects of string field theory used to be more wide-spread? No? Maybe it never was very wide-spread.

But whatever, the point here is that this is a fillet piece of $\infty$ -categorical physics, if only somewhat implicitly so.

Posted by: Urs Schreiber on May 19, 2009 2:55 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Cohomological Physics in the XXth Century: A Survey

Never finished, though a semi-final draft is available. Could even post to the blog as tex or pdf if someone will tell me how.

I first referred to cohomological physics in the context of
anomalies in gauge theory, cf. my work with Bonora, Cotta-Ramusino and
Rinaldi \cite{BCRSI,BCRSII}, but it all began
with Gauss in 1833.
The cohomology referred to in Gauss was that of differential forms,
div, grad, curl and especially Stokes Theorem (the de Rham complex).
This survey is limited to the years before 2001 since there has been an explosion
of cohomological applications in theoretical physics (including K-theory) in
the new century.

Indeed, this is a rather idiosyncratic snapshot of these developments as I saw them at the turn of the century, only slightly updated. It is being posted now for fear it would otherwise be consigned to an electronic bottom drawer and never see the light of day. Additions, corrections, suggested would be most welcome.

This survey is limited to the years before 2001 since there has been an explosion
of cohomological applications in theoretical physics (even of K-theory) in the new century.
Since 1931 but especially toward the end of the XXth century, there has been increased
use of cohomological and more recently homotopy theoretical
techniques in mathematical physics. The chart on the next page
will give you some
indication, though I’m sure it is not complete and would
appreciate any additions.

In this survey intended for a mixed sudience of mathematically inclined physicists
and physically inclined mathematicians, I’ll paint with a very broad brush, hoping
to provide an overview and guide to the literature.
I will emphasize the comparatively recent development of two
aspects, the use of configuration and moduli spaces (cf. operads)
and the use of homological algebra, where
I’ve been actively involved, at least in spreading the gospel.
Each section begins with a brief synopsis.

Posted by: jim stasheff on May 19, 2009 2:46 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Oh! I would love to see a copy of this.

Posted by: Eric on May 19, 2009 3:52 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Here is a link to the paper on your web page:

Survey of Cohomological Physics

Posted by: Eric on May 19, 2009 3:59 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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We should create an $n$ -Lab entry [[ $n$ -categorical physics]], pour all our thoughts into it, revise it viciously, add plenty of links to other resources and to $n$ Lab entries with further details.

That’s really what we should do.

Posted by: Urs Schreiber on May 20, 2009 11:05 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

At one point in your paper you say: “This is another occurrence of the seemingly strange occurrence of needing a 2d object (a monoidal category) to get an invariant in 3d topology.”

Is this related in any way to the “Holographic hypothesis”?

Posted by: Mark Biggar on May 20, 2009 10:28 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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At one point in your paper you say: “This is another occurrence of the seemingly strange

occurrence of needing a 2d object (a monoidal category) to get an invariant in 3d topology.”

Is this related in any way to the “Holographic hypothesis”?

Indirectly it has to do with holography, as the modular tensor category controls both the 3d TFT as well as the 2dCFT holographically related to it.

But more directly, I’d think it is more illuminating to think of the monoidal category that controls 3d invariants as the thing assigned by an extended 3d TFT to the circle. This alone implies that it carries a monoidal structure and that patching the 3d invariant together involves putting objects in the monoidal category on pieces of surface.

Then when using the 3d TFT to define a 2d CFT hologhaphically, that data rearranges itself and then also controls the 2d theory, of course.

(Apart from that, I had a comment on the $n$ -formal meaning of holography above.)

Posted by: Urs Schreiber on May 21, 2009 5:27 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Mark wrote:

At one point in your paper you say: “This is another occurrence of the seemingly strange occurrence of needing a 2d object (a monoidal category) to get an invariant in 3d topology.”

Is this related in any way to the “Holographic hypothesis”?

It must be somehow, but there’s something pathetically simple going on here and that’s all I really want to explain.

3d topology should be related to 3-categories: that’s what simnple dimension-counting suggests. But the most basic example of a 3-category is 2Cat, the 3-category of all 2-categories. So, we should not be surprised that 3-dimensional topology also has a lot to do with 2-categories!

I need to flesh this out in the paper. It’s an example of a certain ‘level slip’ that I found very confusing when I first started thinking about $n$ -categorical physics, which turns out to be pathetically simple.

Posted by: John Baez on May 21, 2009 8:00 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Slow work today:

I finished writing the section on the Ponzano–Regge model of 3d quantum gravity, giving a few viewpoints on the $6j$ symbols. The viewpoint I left out was the one I call ‘spin foams as categorified linear algebra’. This is the one that really explains why a 2-category (and in particular a monoidal category) is the right sort of thing to build a 3d theory. I’ll explain this later.
I wrote a pathetically terse section on string theory, just enough to set the stage for Segal’s work on conformal field theory.
I described 2-vector spaces in the section on Kapranov and Voevodsky’s 1991 paper. This will help lay the groundwork for my explanation of ‘spin foams as categorified linear algebra’.

I also made lots of minor corrections and changes on the pictures… it’s amazing how quickly the day goes by when one is struggling with this sort of stuff, and how little seems to get done!

Posted by: John Baez on May 23, 2009 5:28 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Is this the place to report typos?
Under String Theory:
conformal structure, which is an equivalence classes of Riemannian

Posted by: Charlie C on May 23, 2009 4:25 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Yes, this is the place to report typos! I’ll give anyone here one doughnut per typo — shipping and handling not included until you reach a dozen.

conformal structure, which is an equivalence classes of Riemannian

Thanks — fixed.

Posted by: John Baez on May 24, 2009 2:17 AM | Permalink | Reply to this

Pants; Re: A Prehistory of n-Categorical Physics

Not a typo, but in the `string worldsheets’ illustration isn’t it worth saying whether a “pair of pants” surface is sewn from sheets or the like? Or which Feynman diagrams correspond to the torus (closed string) or the annulus (disc with a puncture) (open string) worldsheet (in the oriented case) or even the Klein bottle and the Moebius strip (in the unoriented case)?

Posted by: Jonathan Vos Post on May 24, 2009 5:00 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Page 19:

The category $Rep(G)$ becomes [a] symmetric monoidal category …

On page 27, the diagram of the tetrahedron on the RHS of the equation starting $tr(Ta_{i,j,k}S) =$ is vertically displaced and intrudes on the text below; this problem occurs whether I look at the PDF on a Mac or under Windows.

Posted by: Greg Egan on May 24, 2009 11:48 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Page 4

… when [we?] generalize from point particles to strings and higher-dimensional objects we meet …

Page 8

Some subtleties appear when we take some findings from particle physics into account.

What follows seems to be saying that time reversal and parity (which both have determinant -1) are, individually, elements of SO(3,1).

Posted by: Greg Egan on May 24, 2009 4:51 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Thanks for catching those blunders! They’re fixed now!

I had earlier defined the Lorentz group to be all linear transformations of $\mathbb{R}^4$ preserving the Minkowski metric… so when I wrote $SO(3,1)$ , I should have written $\mathrm{O}(3,1)$ .

I’ll introduce this notation when the Lorentz group first appears, in the section on Minkowski.

Posted by: John Baez on May 25, 2009 1:16 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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In the first diagram on page 10, I think the arrows between the categories $C$ and $D$ and joined by the natural transformation $\alpha$ should be labelled uppercase $F$ and $G$ rather than lowercase $f$ and $g$ .

Page 19, near the bottom:

where $\rho(g)$ is the unitary operator by which $g$ acts on $H$ , and $\rho'(g)$ is the one by which $g$ acts on $H$ [should be $H'$ ].

Posted by: Greg Egan on May 24, 2009 11:13 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Fixed! Thanks!

Posted by: John Baez on May 25, 2009 1:20 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Page 28:

In 1995, Baez and Dolan initiated another [approach?] to $n$ -categories …

Page 39:

We have mentioned how Jones [should be Jones’ or Jones’s] discovery in 1985 …

Also page 39:

finite-dimnensional [dimensional] representations of a quantum group

Page 42: There are two lower-case $\delta s$ – in the last diagram on the page, and in the equation following it – that I think should be capital $\Delta s$ .

Page 30 and page 44, braidings are given as $B:x\otimes x\to x\otimes x$ that I guess should be $B:x\otimes y\to y\otimes x$ .

Posted by: Greg Egan on May 24, 2009 12:59 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Thanks again!

The section with those screwed-up $\Delta$ ’s — my explanation of bialgebras — is still not finished. So, please forgive it.

On the pages you mention, the right thing is really

$B_{x,x}: x \otimes x \to x \otimes x$

The idea is that a solution of the Yang–Baxter equation is really a way of braiding an object past itself. In Yang’s original work, it was an operator that describes what happens when two quantum particles of the same type pass through each other in 2d spacetime. This was later seen as a special case of a braiding

$B_{x,y}: x \otimes y \to y \otimes x$

Posted by: John Baez on May 25, 2009 1:52 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Page 45:

Such matrices indeed act to give functors from $T:Vect^n\to Vect^m$ .

[Maybe should be “functors from $Vect^n$ to $Vect^m$ ” or “functors $T:Vect^n\to Vect^m$ ”?]

Page 48:

In ohter words …

In the references, there are multiple occurrences of:

Also available as Available as

Sorry for the drizzle of short comments instead of putting everything in one, but I wasn’t sure if I’d have time to read the whole paper today so it seemed better not to wait until I got to the end.

I’ll forego the doughnut reward, as I don’t think they’d be edible by the time they reached Perth!

Posted by: Greg Egan on May 24, 2009 1:34 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Thanks for all these corrections, Greg. I’ve fixed all these mistakes.

I’ll forego the doughnut reward, as I don’t think they’d be edible by the time they reached Perth!

Good point. As a consolation prize, I’ll email you a new track from the long version of my forthcoming album, Treq Lila. It’s called ‘Swirl’. The mp3 file is only 4 megabytes; that shouldn’t bust your inbox. It may not be as good as a box of doughnuts, but it’s probably better than a box of stale doughnuts.

Posted by: John Baez on May 25, 2009 6:06 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

I’ll forego the doughnut reward, as I don’t think they’d be edible by the time they reached Perth!

Try http://www.krispykreme.com.au/order/. (This is probably not the only company that offers this; I see that delivery on a dozen costs nearly as much as the dozen.)

But perhaps you like the consolation prize better.

Posted by: Toby Bartels on May 25, 2009 11:28 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Toby wrote:

But perhaps you like the consolation prize better.

Definitely. And it’s also healthier.

Posted by: Greg Egan on May 26, 2009 2:38 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Oddly, when I google the phrase order doughnuts online, I see the Australian Krispy Kreme website that Toby pointed out, but no comparable American website. So in fact Greg may have been uniquely well-placed for that doughnut prize.

In America I see donuts2go.com, but when I search for a participating doughnut shop within 250 miles of my ZIP code — 92507 — it says there are no locations near me. Since all of Los Angeles is within this radius, that’s ridiculous.

In Canada you can order Tim Hortons coffee online, but not doughnuts. So, ‘coffee for theorems’ is a good idea in Canada, but not ‘doughnuts for typos’.

So: I will offer rewards other than doughnuts to non-Australians. But Alex Hoffnung, who emailed me a long list of typos, will receive hand-delivered doughnuts if he likes.

Posted by: John Baez on May 26, 2009 4:05 AM | Permalink | Reply to this

Where are the surface knots, braids, tangles?; Re: A Prehistory of n-Categorical Physics

Probably a stupid question, already answered in your Periodic Table discussions. But, by analogy, what I expected to see somewhere near the end of the paper was 3-d or 4-d TQFTs that draw from, by analogy, not the theories of knots and tangles of flexible lines in 3-space, but of flexible sheets in 4-space. That is, surface knots and surface braids and surface tangles.

It is well-known that you can’t tie a rope in 4-dimensional space (say, an infinitely long flexible line) – the rope just goes around itself and unknots. But you can tie a surface (say, an infinitely extended flexible plane).

I expected discussion of:
arXiv:0905.3488
Title: Widths of surface knots
Authors: Yasushi Takeda
(Submitted on 21 May 2009)

“Abstract: We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of points in the inverse image of a point in each of the regions. We determine the widths of certain surface knots and characterize those surface knots with small total widths. Relation to the surface braid index is also studied.”

Not a proper scholarly citation (I’ve misplaced my Knot Theory text and Caltech’s Library has charged me for it) but Wikipedia “Knot Theory” currently has:

“In four dimensions, any closed loop of one-dimensional string is equivalent to an unknot. This necessary deformation can be achieved in two steps. The first step is to “push” the loop into a three-dimensional subspace, which is always possible, though technical to explain. The second step is changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. An analogy for the plane would be lifting a string up off the surface.”

“Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two dimensional sphere embedded in a four dimensional sphere. Such an embedding is unknotted if there is a homeomorphism of the 4-sphere onto itself taking the 2-sphere to a standard ‘round’ 2-sphere…”

Also, as Dr. Tim Poston pointed out to me: in 4-space, although two linked rings can trivially unlink, and two spherical surfaces don’t quite link to each other, one can have a chain of alternating sphere and rings. 4-dimensional sailors take note.

I expected to see discussion of (still drawing from that wikipedia) the technique called “general position” which implies that for a given n-sphere in the m-sphere, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n-spheres form knots only in (n+2)-space [Zeeman, E. C. (1963), “Unknotting combinatorial balls”, Annals of Mathematics (2) 78: 501–526], although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted 4k-1-spheres in 6k-space, e.g. there is a smoothly knotted 3-sphere in the 6-sphere [Haefliger, Andre’ (1962), “Knotted (4k − 1)-spheres in 6k-space”, Annals of Mathematics (2) 75: 452–466][Levine, Jerome (1965), “A classification of differentiable knots”, Annals of Mathematics (2) 1982: 15–50].”

“Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k-sphere in an n-sphere with 2n-3k-3 > 0 is unknotted.”

Again, I’m sure that you or Witten or Kauffman have written all about this. I probably missed it. But I expected to see this. Am I merely confused? Was there a reason that this was shown irrelevant?

Posted by: Jonathan Vos Post on May 24, 2009 8:07 AM | Permalink | Reply to this

Re: Where are the surface knots, braids, tangles?; Re: A Prehistory of n-Categorical Physics

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Jonathan wrote:

Probably a stupid question, already answered in your Periodic Table discussions. But, by analogy, what I expected to see somewhere near the end of the paper was 3-d or 4-d TQFTs that draw from, by analogy, not the theories of knots and tangles of flexible lines in 3-space, but of flexible sheets in 4-space. That is, surface knots and surface braids and surface tangles.

Yes, these are often called ‘2-knots’, ‘2-braids’ and ‘2-tangles’. I will definitely talk about them, since they’ve been an obsession of mine for years. Scott Carter, who posts comments here often, practically earns his living from studying them.

I expect that just as tangles are connected to 3d TQFTs, 2-tangles will be connected 4d TQFTs. But this connection remains murky.

The good news is that people have almost succeeded in categorifying quantum groups to the point of showing that categorified quantum groups give 2-tangle invariants, just as quantum groups give tangle invariants. I hope Aaron Lauda writes up an explanation of this, since he’s one of the key players in this subject.

Posted by: John Baez on May 25, 2009 1:58 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Nice paper, I’m also cheering you guys on naturally.

I really like the 6j symbols, I find them intriguing. It’s also really cool how the string diagram calculus actually makes it explicit how a half-spin representation picks up a minus sign when you move it round in a loop.

In addition to the 6j symbols, which encode the associator information, there are also things which could be called the “3j symbols” which encode the duality information. I’ve been trying to understand these things for a while, though I haven’t thought as much about it as I should have in the last few months.

In a fusion category (semisimple linear category where each object has a dual, like $Rep(G)$ ), the nature of duals for objects in the category can be encoded in a ‘principal $\mathbb{C}^\times$ -bundle’ over the simple objects. Firstly, go ahead and choose an arbitrary dual object $V_i^*$ for every simple object $V_i$ . Write

(1)

Dual (V_i \dashv V_i^*) = set of ways V_i^* can be expressed as a right dual of V_i

Notice that $Dual(V_i \dashv V_i^*)$ is a $\mathbb{C}^\times$ -torsor (you can multiply the unit by a nonzero complex number, following which you are forced to multiply the counit by the reciprocal of this number if you want to obey the snake equations).

In a fusion category, if $V_i^*$ is a right dual of $V_i$ , then it is also a left dual of $V_i$ , though not in a canonical way. This gives us our line bundle $L$ over the simple objects — the fiber above $V_i$ is

(2)

L_{V_i} = \{isomorphisms Dual(V_i \dashv V_i^*) \rightarrow Dual(V_i^* \dashv V_i)\}.

A ‘pivotal’ or ‘even-handed structure’ is a section of this bundle satisfying some naturality properties. In other words, it is a consistent way to turn right duals into left duals. The most stringent condition is that the invariant vectors in a tensor product of three particles must either be totally ‘bosonic’ or totally ‘fermionic’ when you rotate them through a full rotation. That is, the endomorphism

(3)

T^i_{jk} : Hom(1, V_i \otimes V_j \otimes V_k) \rightarrow Hom(1, V_i \otimes V_j \otimes V_k)

(which by the way, can be proved to be an involution) given by sending

must be equal to $\pm \id$ . Assuming this is satisfied (if not, there is no consistent way to turn right duals into left duals), we can call this collection of signs the ‘3j symbols’ or the ‘pivotal symbols’ $\epsilon^i_{jk}$ of the fusion category:

(4)

T^i_{jk} = \epsilon^i_{jk} \id.

To repeat: the 3j symbols are the signs arising from choosing a trivialization of the ‘duality bundle’ and analyzing how 3-particle states transform when they go through a full revolution.

It turns out that one can express the concept of a ‘pivotal structure’ directly in terms of the 3j symbols $\epsilon^i_{jk}$ . Namely, a pivotal structure on the category amounts to a $\epsilon$ -twisted monoidal natural tranformation of the identity on the category. That is to say, a collection of nonzero complex numbers $t_{ijk}$ satisfying

(5)

t_j t_k = \epsilon^i_{jk} t_i \quad whenever V_i appears in V_j \otimes V_k

If all the 3j symbols were equal to 1, this would just be an ordinary monoidal natural transformation of the identity. Perhaps one might be able to get a new angle on this phenomenon using the language of planar algebras. Started nLab entry.

Posted by: Bruce Bartlett on May 24, 2009 2:55 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Nice graphics, Bruce! And good idea to upload it to the $n$ Lab.

With more and more graphics used that way, we may need top think about a way to systematize this. So that people who look for a graphic to include in a comment or elsewhere can find.

I mean, first of all of course I am hoping you’ll maybe find the time to create a stub entry [[fusion category]] where you could include this particular figure.

But maybe we could even have a page that lists all graphics we have listed suitably, e.g. alphabetically and/or by subject.

That also serves to remind us that we may need to be careful with naming of graphics. turn.png is convenient, but who knows how many different notions of “turns” we may want to illustrate graophically in the future. Maybe it would be good to have longer, more descriptive file names for graphics. But that’s maybe not so important. More important is that the link to the file is included in a context that makes it searchable and, more importantly, findable.

Posted by: Urs Schreiber on May 24, 2009 4:44 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

That also serves to remind us that we may need to be careful with naming of graphics

Yes I know this is a problem. I think one would like to include graphics or external files on the page “Monoidal category” in a folder like way, that is as [[Monoidal category/associahedron.png]]. But when I tried to do that, Instiki seemed to barf at the slash symbol “/”.

Posted by: Bruce Bartlett on May 25, 2009 11:21 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

But when I tried to do that, Instiki seemed to barf at the slash symbol “/”.

I wonder why! Perhaps because if you allow slashes all over the place then someone might just try to include that fantastic picture: ../../../etc/passwd

(Yes, I know.)

Perhaps if you put something a little less controversial as a separator, say an underscore or a hyphen, then it may work.

By the way, I know Bruce knows but others may not. There’s a fair bit of discussion as to how best to do graphics over on the n-Forum.

Posted by: Andrew Stacey on May 25, 2009 3:29 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

There’s a fair bit of discussion as to how best to do graphics over on the n-Forum.

That's mostly about avoiding having to upload files, of course. We could move the discussion here to this entry.

Posted by: Toby Bartels on May 26, 2009 1:56 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Well, it’s meant to be extendible to any aspect of getting graphics on the n-lab. More generally, the n-Forum (need I say this again?) is for discussion of things relating to the n-lab. This blog entry is about John and Aaron’s paper not graphics on the ‘lab! I know that our esteemed hosts have no objection to discussion wandering from topic to topic (indeed, actively encourage it), but it must be a bit frustrating to see that there’s a new comment on this thread, run over to see what the latest comment on the paper is, and find out that it’s some technical thing on not being able to use forward slashes in titles on the ‘lab!

<rant>

I’m not trying to drum up traffic to the ‘forum for vanity’s sake. It’s extremely irritating and time-wasting to have to follow conversations jumping from the ‘lab to the ‘cafe to the ‘forum and back again. I’ve said before that if the ‘forum isn’t working then I’m happy to take it down to remove one area of confusion, but I do believe that it is the best of the three for actual discussion of this kind of thing. I found the various discussions on graphics over there extremely useful and relatively easy to follow.

</rant>

Posted by: Andrew Stacey on May 26, 2009 9:08 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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I do believe that it is the best of the three for actual discussion of this kind of thing

Yes.

We just need to keep pointing patiently to moving appropriate discussion there.

Above the slash-discussion got started from a comment of mine replying to Bruce who made a technical comment on John and Aaron’s article. Somehow I had to say this here. After all, I can’t reply on the $n$ -forum to a comment that sits here.

I could of course have said here: Bruce, I have a reply to that, but since it’s off-topic here you can find it over at the $n$ -forum.

I’ll do that next time.

Posted by: Urs Schreiber on May 27, 2009 9:47 AM | Permalink | Reply to this

Forum (Was: A Prehistory of n-Categorical Physics)

More generally, the n-Forum (need I say this again?) is for discussion of things relating to the n-lab.

I agree; that's why I created a Forum discussion for it (since I didn't think that the existing ones were the right place) and linked to it above. Since you created your own Forum discussion for these matters, I'll link to it too: http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=25.

Posted by: Toby Bartels on May 27, 2009 12:46 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

Instiki seemed to barf at the slash symbol “/”.

If you create a link (or inclusion) in a page, the Instiki interprets / as %2F. But I can't get anything to work in a picture name.

Of course, if a picture might be reused, then it really shouldn't have a filename built in this way. I did this for the SVGs on the grounds that we really want some generating code (like TikZ) directly in the page instead, and even then I was unsure.

Posted by: Toby Bartels on May 25, 2009 11:19 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

That also serves to remind us that we may need to be careful with naming of graphics

Yes I know, this is a problem. I think one would like to include graphics or external files on the page “Monoidal category” in a folder like way, that is as [[Monoidal category/associahedron.png:pic]]. But when I tried to do that, Instiki seemed to barf at the slash symbol “/”.

Posted by: Bruce Bartlett on May 25, 2009 11:21 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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Today’s progress:

I moved the definition of ‘braided monoidal functor’ to the discussion of Joyal and Street’s 1985 paper on braided monoidal categories. I also added the definition of ‘braided monoidal natural transformation’. This allowed me to give a precise statement of their result that braids form the free braided monoidal category on one object. Since this is one of the first theorems demonstrating ‘the primacy of the point’, I explained it rather carefully.
I almost completed the section on Reshetikhin and Turaev’s 1989 paper on getting tangle invariants from quantum groups. There’s a picture missing.

The section on Freyd and Yetter’s paper is a complete mess, but I want to fix it soon since it gives a second easy example of ‘the primacy of the point’.

Posted by: John Baez on May 26, 2009 4:20 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

The discussion on this thread has gotten so off-topic that when I first read this I thought that you were talking about entries on the n-Lab! I’d completely forgotten what it was originally about.

Posted by: Andrew Stacey on May 26, 2009 8:48 AM | Permalink | Reply to this

topic: off-topic topics

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Hi Andrew,

you write

The discussion on this thread has gotten so off-topic that when I first read this I thought that you were talking about entries on the n-Lab!

Is that a fair characterization?

I count seven comments of 61 on pure $n$ Lab administrative matters.

In the light of 4 on worldwide doughnut availability that’s not yet being rudely off-topic, I’d think, in particular since the $n$ Lab entries were in response to direct comment on the article John is writing.

Whose content, I am hoping, will eventually make it into [[ $n$ -categorical physics]] in one way or other.

But I take your point: discussion of administrative $n$ Lab matters should be taken to the $n$ -Forum, yes!

Posted by: Urs Schreiber on May 26, 2009 9:13 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

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For what it’s worth, I don’t mind people talking about the $n$ Lab on this blog entry. I think it’s actually good for you guys working down in the $n$ Lab to come up to the Café now and then, to get some fresh air and let ordinary folks see you still exist. I think there are more people browsing the Café than the Lab. (I could be wrong.)

Posted by: John Baez on May 26, 2009 9:41 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics

I hope the treatment of (derived and) A-infinity categories in homological mirror symmetry will appear later. This is early (Kontsevich 1994) and very influential employment of higher category theory in mathematical physics, which now seems completely missing in this draft. Nowdays physicists at the conferences seem more interested in mirror symmetry than quantum groups which are treated in many steps in this interesting ‘history’ paper.

Posted by: Zoran Skoda on May 26, 2009 10:20 PM | Permalink | Reply to this

Read the post A Prehistory of n-Categorical Physics II
Weblog: The n-Category Café
Excerpt: I'm back to writing that Prehistory of n-Categorical Physics. I want to finish it before I die.
Tracked: July 7, 2009 2:56 PM

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May 18, 2009