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July 7, 2009

A Prehistory of n-Categorical Physics II

Posted by John Baez

My previous attempt to finish this paper did not succeed:

I got blindsided by final exams, my arrival in Paris, my work with Paul-André Melliès here, and the need to work on other papers.

But I’m back at it again… and today, tour of the Catacombs instilled me with a new sense of urgency.

I hadn’t really understood how many people were down there until today. Someone said 6 million, but the impressive part is not the number, but actually hiking for almost an hour down dim-lit corridors lined with bones, skulls, and plaques in French and Latin. I have little sense of how much of the whole labyrinth we saw — many passageways were blocked off, probably to keep people from getting lost. Some regions were wet underfoot, dripping from the ceiling. A few large chambers had been formed by cave-ins.



The moral: eat, drink, and publish, for tomorrow we die!

In the second draft version of the paper you’ll see a few new things:

  • A longer survey of various definitions of nn-category in the section on Grothendieck’s 1983 letter called Pursuing Stacks. I also added some elementary remarks about the problem of comparing definitions.
  • A much longer explanation of Freyd and Yetter’s 1986 paper on categories of tangles, leading up to a description of Shum’s theorem. I still need to explain how this led Jim Dolan and me to ponder ‘braided monoidal categories with duals’.
  • More about quantum groups in the section on Reshetikhin and Turaev’s 1989 paper on tangle invariants from quantum groups.
  • A brief sketch of loop quantum gravity and spin foams in the section on Rovelli and Smolin’s 1990 paper introducing loop quantum gravity. This is very short but I’m afraid that’s how it will have to stay.
  • A section on Kashiwara and Lusztig’s work on crystal bases starting around 1990. I need to polish this a bit more.

There’s still a lot to be done — I’ll get going right now!



Posted at July 7, 2009 1:53 PM UTC

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Re: A Prehistory of n-Categorical Physics II

Typo – Frank Wilczek’s name is spelled wrong on page 50.
Kevin

Posted by: Kevin on July 7, 2009 5:56 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

JB wrote: ” I still need to explain how this (^)Jim Dolan and I to ponder ‘braided monoidal categories with duals’.”


(^) = I think you need to insert a verb such as “led” or [“brought”].

Posted by: Stephen Harris on July 7, 2009 7:45 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

… in which case it should be ‘Jim Dolan and me’ (like how you would write it if it were only you).

Posted by: Toby Bartels on July 7, 2009 7:50 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Fixed. But find typos in the paper, not the blog entry!

Posted by: John Baez on July 8, 2009 3:07 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

I think the paper is well-written. Also that your
defense made to Urs that his objections are covered
by your very thorough treatment of these issues in
the Introduction is logical and literate.

———————————————

rothendieck’s dreams involving oo-categories and
“recent”? attempts to realize this dream. [page 1 Intro]

This is a minor point but if you cut the history
off at 2000 is the word “recent” appropriate?
Maybe it is. Maybe “emerging” attempts?

———————————————
The paper reads now,
subject will become important. We believe it will—but so far, all we have is a
‘prehistory’. [page 2]

[rather I think the “a” should be on the next line]

We believe it will—but so far, all we have is
‘a prehistory’.
I think you should override the LaTeX formatting
and move the “a” to the second line and then
enclose it with ’ so that it establishes a better
link to your paper title. That would avoid having
to explain the differnce between “a” and “the” in
the preface as you contemplated with jim stasheff.

Or you could just put a space before and after the -
in “will-but” so that it reads, will - but and this
will likely move the “a” to the next TeXt line so that
you could use ‘a prehistory’ visually standing alone.

———————————————–

I will read more of the paper later. I like it from the view of one who benefits from it.

Posted by: Stephen Harris on July 9, 2009 7:37 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

I’m glad you like the paper, Stephen. I’ve rewritten the first paragraphs to clarify some of the issues people have been talking about:

This paper is a highly subjective chronology describing how physicists have begun to use ideas from nn-category theory in their work, often without making this explicit. Somewhat arbitrarily, we start around the discovery of relativity and quantum mechanics, and lead up to conformal field theory and topological field theory. In parallel, we trace a bit of the history of nn-categories, from Eilenberg and Mac Lane’s introduction of categories, to later work on monoidal and braided monoidal categories, to Grothendieck’s dreams involving \infty-categories, and subsequent attempts to realize this dream. Our chronology ends at the end of the 20th century; after then, developments have been coming so thick and fast that we have not had time to put them in proper perspective.

We call this paper a prehistory because nn-categories, and their applications to physics, are still in their infancy. We call it a prehistory because it represents just one view of a multi-faceted subject: many other such stories can and should be told. Ross Street’s Conspectus of Australian Category Theory is a good example: it overlaps with ours, but only slightly. There are many aspects of nn-categorical physics that our chronology fails to mention, or touches on very briefly; other stories could redress these deficiencies. It would also be good to have a story of nn-categories that focused on algebraic topology, one that focused on algebraic geometry, and one that focused on logic. For nn-categories in computer science, we have John Power’s Why Tricategories?, which while not focused on history at least explains some of the issues at stake.

What is the goal of this prehistory? We are scientists rather than historians of science, so we are trying to make a specific scientific point, rather than accurately describe every twist and turn in a complex sequence of events. We want to show how categories and even nn-categories have slowly come to be seen as a good way to formalize physical theories in which ‘processes’ can be drawn as diagrams—for example Feynman diagrams—but interpreted algebraically—for example as linear operators. To minimize the prerequisites, we include a gentle introduction to nn-categories (in fact, mainly just categories and 2-categories). We also include a review of some revelant aspects of 20th-century physics.

The most obvious roads to nn-category theory start from issues internal to pure mathematics. Applications to physics only became visible much later, starting around the 1980s. So far, these applications mainly arise around theories of quantum gravity, especially string theory and `spin foam models’ of loop quantum gravity. These theories are speculative and still under development, not ready for experimental tests. They may or may not succeed. So, it is too early to write a real history of nn-categorical physics, or even to know if this subject will become important. We believe it will—but so far, all we have is a ‘prehistory’.

Posted by: John Baez on July 10, 2009 9:48 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Fixed! Thanks!

Posted by: John Baez on July 7, 2009 8:22 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Other typos: I think the final tensor product in the top diagram on p21 should be y(xz) and not y(zx), and there’s a random “l” after the words “3 dimensions” on page 35.

Reads very nicely so far!

Posted by: anon on July 7, 2009 6:19 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Thanks very much! There turned out to be even worse mistakes in those hexagon identities on page 21, so thanks for spotting the tip of that iceberg.

I think the diagrams are correct now.

Posted by: John Baez on July 9, 2009 1:41 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Hi John - that’s a great article! thanks for sharing it. Are you still taking suggestions for papers? If so I would suggest Kontsevich’s 1994 ICM, one of the seminal papers of the 1990s IMHO (and I think mentioned by Urs in the previous discussion). This paper invented the categorical side of mirror symmetry (homological mirror symmetry), discovered D-branes (before physicists realized their role - and directly inspiring many physicists) and the fact that they naturally form (dg - or A_oo-) categories, and thus led to a deluge of papers involving category theory and higher category theory in close relation with mathematical physics.

In particular the Moore-Segal paper you discuss can (should?) be seen in the light of this development. On a similar note, roughly contemporary with Moore-Segal (which is arXiv:math/0609042 though developed earlier) are the works of Costello (arXiv:math/0412149) and Kontsevich-Soibelman (arXiv:math/0606241 — some of the results were lectured on in various places by Kontsevich in 2003 and in particular helped inspire Costello) proving a much stronger result, which is the TCFT (or equivalently differential graded / A_oo-) version of the classification of open-closed 2d TFTs. These papers were directly motivated by homological mirror symmetry and topological string theory, and have greatly influenced work in areas such as string topology which you mention and the cobordism hypothesis (Hopkins-Lurie started from Costello’s paper and abstracted the argument, before the general argument in n-dimensions came around). I realize this is beyond your timeframe but I feel it’s appropriate since you do discuss Moore-Segal.

Posted by: David Ben-Zvi on July 7, 2009 6:51 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

I agree with David that with so many pages devoted to knot theory/quantum groups motivated categorical structures, the k-linear higher categorical structures like A-infinity, dg and triangulated categories massively appearing in homological mirror symmetry, deformation quantization, LG-models, BV formalism etc. (Costello, Kontsevich, Cataneo, Tamarkin…) deserve comparable attention; surely every author has a taste, background and inclinations of his own, but still the title-word like “history” is self-contradictory if balancing is not seriously attempted.

Posted by: Zoran Skoda on July 7, 2009 8:43 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Completely agree with David and Zoran about the role of linear higher categorical structures. I would also say a little bit more about Boardman-Vogt book and appearance of operads. Many late definitions of n-categories are inspired by operadic approache. My own work on A_{\infty}-categories (“Homotopy coherent category theory and $A_{\infty}$-structures in monoidal categories”, “Journal of Pure and Appl. Algebra” 123 (1998), 67-103) in 1994 directly led me to my higher operads and my definition of weak \omega-categories. I would say that operads (in many forms) is an important part of higher category theory now.

One more reference concerning comparison of different definitions and history. May be you can mention my paper “On Penon method of weakening algebraic structures”, “ Journal of Pure and Appl. Algebra” 172(1) (2002), 1–23. In this paper I introduced the “correct” Penon’s definition and compare it with my own by constructing a contractible operad for it. Even more, I showed that Penon’s definition (corrected) is strictly stronger in the sense that any Penon’s n-category is also n-category in my sense but not vice versa (although I have a conjecture that they are weakly equivalent definitions).

It was several years before Minnesota and Cheng-Makkai paper. Eugenia knew about my paper. What Makkai observed is that the original Penon’s definition is too strict (my hypothesis was that the “correct” one is still equivalent in a weak sense to the original) and I remember I explained it to Eugenia in Minnesota. I think this counterexample was published later by her and Makkai. I must admit I did not read their paper (it is not in the archive I guess).

Posted by: Michael Batanin on July 8, 2009 2:14 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

“I did not read their paper (it is not in the archive I guess). ”

There are too many papers being cited as preprints - not even on the arXiv much less in print or even submitted for publication!

A word to the wise - especially the young’uns without tenure.

Posted by: jim stasheff on July 9, 2009 4:02 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Michael wrote:

I would also say a little bit more about Boardman-Vogt book and appearance of operads.

I’ve added a bit about operads in my discussion of Lawvere’s 1963 paper on algebraic theories, and I’ll probably add a bit more—but of course not enough.

May be you can mention my paper “On Penon method of weakening algebraic structures”, “ Journal of Pure and Appl. Algebra” 172(1) (2002), 1–23. I

Good point. This portion of my ‘prehistory’ was taken from the preface of my book with Peter May. I’ve corrected that preface, too.

You’re right, the Cheng–Makkai paper is not on the arXiv. I have been unable to find it on the internet. Tsk tsk!

Posted by: John Baez on July 16, 2009 7:59 PM | Permalink | Reply to this

Not on the arXiv NOR the internet

It’s worse than Tsk!Tsk! More and more papers appear with References given as `preprint’ with no further location. Even some are listed as arXiv years after they appeared there. As an editor of a few journals, PLEASE submit paprs for publication before the paper or electronic bits disintegrate!

Posted by: jim stasheff on July 17, 2009 2:24 PM | Permalink | Reply to this

Re: Not on the arXiv NOR the internet

Jim, I take your point, but sometimes things are out of our hands. For example, there is a paper I put on the arXiv and submitted SIX YEARS ago and it still hasn’t appeared in the journal. One year of that was my fault, but five years of that was the journal’s fault. In fact, I’ve just discovered that they apparently never received the final version I sent one year ago, so I’ve just had to send it again. And yes, the lag time seems to be worse for more “prestigious” journals which we “young people” are supposed to try and publish in.

This then leads to another problem which is that our papers get cited while they are only preprints, and this does not get counted by eg MathSciNet. Even when the preprint turns into a published article, MathSciNet does not count it as a citation.

I’m just glad I don’t have to go through the American tenure process.

Posted by: Eugenia Cheng on July 23, 2009 12:27 AM | Permalink | Reply to this

Re: Not on the arXiv NOR the internet

Eugenia wrote:

This then leads to another problem which is that our papers get cited while they are only preprints, and this does not get counted by eg MathSciNet. Even when the preprint turns into a published article, MathSciNet does not count it as a citation.

Actually MathSciNet seems to be trying to rectify this. I’ve seen a number of citation lists on MathSciNet which include a preprint, followed by somethings like “cf. [MR number]” referring to the published version. They seem to be more likely to fail to do this if the title changes in publication.

In any case, MathSciNet has other much more serious failings as a citation tracker, like the fact that many works’ citations aren’t recorded, and that citations to math papers from papers in other fields don’t appear.

Posted by: Mark Meckes on July 23, 2009 1:34 AM | Permalink | Reply to this

Re: Not on the arXiv NOR the internet

Mark Meckes said:

They seem to be more likely to fail to do this if the title changes in publication.

Be that as it may, none of my papers has changed title in publication, and all of them have suffered this silly fate. There may well be more serious failings in MathSciNet as a citation tracker, but none more pertinent to the issue of advanced Arxiv posting, or to “young” mathematicians whose careers to date are bound to be dominated by years of articles in preprint status.

Posted by: Eugenia Cheng on August 1, 2009 12:19 AM | Permalink | Reply to this

Re: Not on the arXiv NOR the internet

I hope all such suggestions are sent to Mathsci net
and not just aired here.

Posted by: jim stasheff on August 2, 2009 3:54 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Apologies - that paper of mine with Makkai had a long (5 year) and slightly complicated gestation, which is why it’s not on the Arxiv yet. But I have been duly spurred into action and it will go up ASAP.

The main point of the paper is that we prove Penon’s original definition is too strict, in that it cannot produce braided monoidal categories. Makkai made the key observation in Minnesota which I did indeed then hear about from Michael Batanin later that day (or so); turning it into a rigorous proof took some more work.

We also look at the non-reflexive version (as introduced by Batanin in the paper he mentions above), and show that this eliminates the problem. Thus the non-reflexive version is not equivalent to the reflexive original.

Posted by: Eugenia Cheng on July 22, 2009 11:53 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

John made it clear last time that he doesn’t feel like adding more aspects to the article, so I presume we ought not try to push him to do so.

What we should rather do is

- provide our feeling that without these aspects the current title of the article would be somewhat misleading to the non-experts and somewhat unfair to the experts, and suggest maybe that it could be changed to something that expresses the angle a bit more, like One road to higher categorical physics or Higher categorical physics via tangles, knots and and quantum groups, or the like.

- add the aspects that we feel are missing in a comprehensive “History of higher categorical physics” and that we feel ought to be exposed to some kind of public ourselves.

Some stub for that was created already at [[nn-categorical physics]].

So far this is barely a back-of-the-envelope sketch, but we can make it grow. A minute ago I have taken the liberty of pasting David BZs message above into it. Have a look.

Notice that a different but related entry was created a few days ago, by migrating and expanding Wikipedia material:

[[Timeline of category theory and related mathematics]].

That would probably be the place to fill in the less physical and more purely categorical material that Michael Batanin mentions above.

I mean, I guess many of us would enjoy seeing a truly comprehensive survey of higher categorical physics, while probably no single one of us has the expertise, energy and leisure to write a truly comprehensive account. (A general problem in science: experts are too busy with pushing the theory than with surveying and exposing it.) But when some of us each spend ten minutes a day pouring a bit of material into a wiki entry, given that experience shows that there is always somebody around investing time in polishing formatting and coherence, eventually we’d get something of value by swarm intelligence.

Posted by: Urs Schreiber on July 8, 2009 10:53 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

David B-Z wrote:

Are you still taking suggestions for papers? If so I would suggest Kontsevich’s 1994 ICM, one of the seminal papers of the 1990s IMHO…

I’m not eager to expand my prehistory, since it’s almost 100 pages long and overdue. But since both you and Zoran recommended mentioning this paper, and made a persuasive case, I’ll probably do it.

Urs wrote:

What we should rather do is

- provide our feeling that without these aspects the current title of the article would be somewhat misleading to the non-experts and somewhat unfair to the experts, and suggest maybe that it could be changed to something that expresses the angle a bit more, like One road to higher categorical physics or Higher categorical physics via tangles, knots and and quantum groups, or the like.

I think my current title — A Prehistory of nn-Categorical Physics — expresses quite well the idea that this is one account, namely my own, reflecting my own biases. And don’t forget that the very first words of the article are these:

This paper is a highly subjective chronology describing how physicists have begun to use ideas from nn-category theory in their work, often without making this explicit. Somewhat arbitrarily, we start around the discovery of relativity and quantum mechanics, and lead up to conformal field theory and topological field theory. In parallel, we trace a bit of the history of nn-categories, from Eilenberg and Mac Lane’s introduction of categories, to later work on monoidal and braided monoidal categories, to Grothendieck’s dreams involving \infty-categories and recent attempts to realize this dream.

Many different histories of nn-categories can and should be told. Ross Street’s Conspectus of Australian Category Theory is a good example: it overlaps with our tale, but only slightly. There are many aspects of nn-categorical physics that our tale fails to mention; other histories could redress these deficiencies. It would also be good to have a history of nn-categories that focused on algebraic topology, one that focused on algebraic geometry, and one that focused on logic. For higher categories in computer science, we have John Power’s Why Tricategories? which while not a history at least explains some of the issues at stake.

What is the goal of this history — or rather, prehistory? We are scientists rather than historians of science, so we are trying to make a specific scientific point, rather than accurately describe every twist and turn in a complex sequence of events. We want to show how categories and even nn-categories have slowly come to be seen as a good way to formalize physical theories in which ‘processes’ can be drawn as diagrams—for example Feynman diagrams—but interpreted algebraically—for example as linear operators.

I also added, at your suggestion, several paragraphs explaining another major line of work — involving gauge theory, gerbes, stacks and the like — which I decided not to discuss at all.

This should be sufficient to prevent readers from thinking that this is meant to be some sort of definitive treatise on the history of nn-categories and physics. I’m not aiming for completeness; I’m trying to tell a story with a clear plot — and tell it in a way that’s comprehensible to people who only know a little bit of math and physics.

I’m glad the timeline of category theory and related mathematics on the nnLab is growing. This can provide a longer, more sprawling story.

Posted by: John Baez on July 8, 2009 1:07 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

A Prehistory?
remember many of your readers are fairly fluent in English but still have trouble with A vs The vs \null

Posted by: jim stasheff on July 9, 2009 4:06 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Okay, in the preface I’ll explain what I mean by the word “a”.

Posted by: John Baez on July 9, 2009 4:39 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

David B-Z wrote:

… I would suggest Kontsevich’s 1994 ICM, one of the seminal papers of the 1990s IMHO (and I think mentioned by Urs in the previous discussion). This paper invented the categorical side of mirror symmetry (homological mirror symmetry), discovered D-branes (before physicists realized their role - and directly inspiring many physicists) and the fact that they naturally form (dg - or $A_\infty$-) categories, and thus led to a deluge of papers involving category theory and higher category theory in close relation with mathematical physics.

Did he call them D-branes??? and what were they for him?

Posted by: jim stasheff on July 9, 2009 3:59 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

David B-Z wrote:

I realize this is beyond your timeframe but I feel it’s appropriate since you do discuss Moore-Segal.

By the way, I’ll end the chronology at 2000, so in the final version there won’t be an entry for Moore–Segal. But I’ll figure out some way to mention that stuff, and a few other ideas that came along after 2000 but help round out the story I’m trying to tell.

Posted by: John Baez on July 8, 2009 3:03 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

A postscript: Shades of things to come
would do nicely

Posted by: jim stasheff on July 9, 2009 4:08 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

NO!

I had to truncate my history at the year 2000 to avoid making it vastly longer. That’s roughly when Urs Schreiber, David Ben-Zvi, Jacob Lurie, Stolz and Teichner, Khovanov and Lauda, Bruce Bartlett, Jamie Vicary, and many others got into the act. That’s when ‘prehistory’ started becoming ‘history’.

I started writing this paper 5 years ago, in the summer of 2004. The whole subject is very different now. But that’s okay: I want to explain the past, not the present. And there’s no way in the world I’m going to include the future.

Posted by: John Baez on July 9, 2009 4:52 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Page 6:

Maxwell (1876)

In his book Matter and Motion, Maxwell [4] wrote:

Readers less familiar with the history of physics may be surprised to see these words, written when Einstein was 3 years old.

Since Einstein was born in 1879, something is wrong with your chronology here.

Also, minor typo on p. 3: “hints at the two way of” should be “… ways …”

Posted by: Mark Meckes on July 8, 2009 3:10 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Mark wrote:

Maxwell (1876)

In his book Matter and Motion, Maxwell [4] wrote:

Our whole progress up to this point may be described as a gradual development of the doctrine of relativity of all physical phenomena. Position we must evidently acknowledge to be relative, for we cannot describe the position of a body in any terms which do not express relation. The ordinary language about motion and rest does not so completely exclude the notion of their being measured absolutely, but the reason of this is, that in our ordinary language we tacitly assume that the earth is at rest…. There are no landmarks in space; one portion of space is exactly like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, sounding, wind or tide, and we cannot tell in what direction we are going. We have no log which we can case out to take a dead reckoning by; we may compute our rate of motion with respect to the neighboring bodies, but we do not know how these bodies may be moving in space.

Readers less familiar with the history of physics may be surprised to see these words, written when Einstein was 3 years old.

Since Einstein was born in 1879, something is wrong with your chronology here.

Ahem.

I guess I meant to say negative three years old!

Thanks for helping me avoid looking like an idiot for the rest of eternity. (Except for here.)

Also, minor typo on p. 3: “hints at the two way of” should be “… ways …”

Thanks! Fixed!

Posted by: John Baez on July 9, 2009 10:11 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Minor typo on p.53, para 2, line 4: the Russian school were especially successful (not expecially so).
Posted by: Peter on July 9, 2009 12:33 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Thanks — fixed!

I’m impressed by how you managed to create a thinner than usual space above and below the words in your comment. Not that I think it looks good: I’m just impressed that you could do it!

Posted by: John Baez on July 9, 2009 10:33 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Cvitanovic’s book [35] (Group Theory - Birdtracks, Lie’s, and Exceptional Groups) has a new printing 2008 and a new web-book address:

http://birdtracks.eu/

It has a nice section 4.9.: A brief history of birdtracks.

Posted by: Florifulgurator, PhD dropout on July 9, 2009 12:16 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Thanks for the info! I like that book. And thanks for reminding me: I should cite his brief history of birdtracks.

Posted by: John Baez on July 9, 2009 4:43 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Okay — in the spare moments when I wasn’t busy arguing with people who want me to put more stuff into this paper, I actually got a bit of writing done.

Namely: I polished and expanded the section on Fukuma, Hosono and Kawai’s 1992 paper on 2d TQFTs. The reason I put in so much detail is that their construction is easy to follow, and pretty… and it categorifies to give the Turaev–Viro construction of 3d TQFTs. It also illustrates in a simple way how ‘centers’ arise in topological quantum field theory. Later I’ll talk about the ‘center’ as a trick for moving one notch down the Periodic Table.

Posted by: John Baez on July 9, 2009 6:01 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Excellent paper…but why stop at year 2000 ? Go ahead please!

Posted by: proaonuiq on July 9, 2009 9:15 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

No! I’ve been working on this paper intermittently for 5 years, it’s almost 100 pages long, I’m tired of writing it, the editor of the book it will be published in wants it now, and I have a lot more to write even without going beyond the year 2000! Also, I would need to learn and think a lot more to do a good job of covering the material after the year 2000.

In my current plan, I still need to explain:

  • The work of Barrett–Westbury on getting 3d TQFTs from spherical categories.
  • The work of Witten–Reshtekhin–Turaev on getting 3d TQFTs from modular tensor categories.
  • The work of Ooguri–Crane–Yetter on getting 4d TQFTs from modular tensor categories.
  • Lawrence’s work on extended TQFTs.
  • The paper by Crane and Frenkel on categorified quantum groups.
  • Freed’s paper on higher algebraic structures and quantization.
  • My paper with Jim Dolan on higher-dimensional algebra and TQFTs.
  • Kontsevich’s paper on open-closed strings and A A_\infty categories.
  • Mackaay’s paper on 4d TQFTs from spherical 2-categories.
  • Khovanov’s paper on categorified quantum groups.

Some of these will be quick to explain, but others not so quick. So, I’ll have plenty of work just finishing these.

Posted by: John Baez on July 9, 2009 9:37 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

I apologize to everyone for getting a bit snappy lately. I just got a funny email, which I’ll take the liberty of quoting anonymously. It cheered me up.

Over at the cafe, you wrote a couple of times:

No!

and gave some very good reasons why not.

But surely once you get this paper off your plate you’ll be ready to set aside your research, teaching, university service, personal life, and hobbies in order to write that complete, up-to-the-minute history of all of mathematics and physics – including their relationships to computer science, biology, finance, meteorology, and backgammon strategy, and a preview of the major trends of the next fifty years – that we’re all dying to read. :P

Posted by: John Baez on July 10, 2009 9:15 AM | Permalink | Reply to this

minor typo

On page 5, first full paragraph, last sentence has ‘sending’ twice.

Posted by: ron on July 10, 2009 5:10 PM | Permalink | Reply to this

Re: minor typo

Thanks! Fixed fixed.

Posted by: John Baez on July 11, 2009 8:09 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Okay, here’s a new version. So far today I’m just finishing sections that were mostly written, but incomplete in various ways:

  • I added a more detailed description of dagger categories and unitarity to my discussion of Atiyah’s 1988 paper on TQFTs.
  • I added a pretty precise statement of the result “2Cob is the free symmetric monoidal category on a commutative Frobenius algebra” to my discussion of Dijkgraaf’s 1989 paper on 2d TQFTs. This is the third theorem I mention that gives a purely algebraic description of a category showing up in topology.
  • I gave a precise definition of symmetric monoidal categories with ‘duals for morphisms’ (aka ‘symmetric dagger categories’) and ‘duals for objects and morphisms’ (aka ‘symmetric dagger compact categories’) to my discussion of Doplicher and Roberts’ 1990 paper on group representation theory. This is all warmup for explaining the Cobordism Hypothesis and Tangle Hypothesis. I also mentioned how Abramsky, Coecke and Selinger used these notions in their work on quantum computation: this is a peek beyond the 2000 cutoff of the paper, but an almost necessary one, since while these concepts were lurking around earlier, their currently popular names weren’t!

So, I think the paper is approximately done up to page 80, where I need to sketch Barrett and Westbury’s construction of 3d TQFTs.

Here’s a question: the timeline of nn-categories over on the nnLab gives Bruce Bartlett credit for introducing the phrase “the primacy of the point” in 2008. I guess Bruce said that in here. I also think I’ve seen Lurie say it, perhaps roughly the same time. Who said it first?

Posted by: John Baez on July 10, 2009 5:37 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

I probably should be marching on, but I couldn’t resist adding a bit more to the section on Freyd and Yetter’s 1986 paper. I explained how the framed tangle that looks like a ribbon with a 360 degree twist is related to the spin-statistics theorem in particle physics. I think this is a great example of how the mysteries of ‘nn-categories with duals’ impinge on real-world physics. And, as I explain, there are still some mysteries connected to this topic.

Posted by: John Baez on July 12, 2009 3:32 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

if i understood your writings then it seems that either on p. 50 there should be: …check that the Kauffman bracket satisfies q 34- q^{-\frac{3}{4}} (i.e. a minus sign in the exponent) or that on p. 47 there should be …related to A by q=A 4q= A^{-4}
Posted by: bnonymous on July 13, 2009 8:40 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

You’re right! Thanks for catching this mistake.

The reason it’s taken me so long to reply is that I hate these normalization issues: different people use different formulas relating AA to qq, and none of them seems to make everything pretty.

Posted by: John Baez on August 2, 2009 7:44 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Nice paper! A few minor things:

On page 5, it would be nice to give P_1(M) and Trans(E) descriptive names in addition to their symbolic names.

On page 10, near the top, you’re missing an “if” in the “if-then” statement defining strong continuity.

On page 16, at the end of the subsection on Feynman, “Joyal and Street” should be “Joyal and Street’s” or perhaps “Joyal’s and Street’s”.

Posted by: John Huerta on July 16, 2009 12:24 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Thanks! Fixed!

Posted by: John Baez on July 16, 2009 7:49 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Okay, here’s another big round of changes:

  • Aaron Lauda wrote a nice description of Kashiwara and Lusztig’s 1990 papers constructing canonical or crystal bases for quantum groups, and
  • I wrote a brief entry on Reshetikhin and Turaev’s 1991 paper on 3d TQFTs from braided monoidal categories. I’ll put more in a discussion of Turaev’s ‘shadow world’ construction of 4d TQFTs from roughly the same data — because as Bruce Westbury pointed out here, the Reshetikhin-Turaev 3d TQFTs are really just the tip of a 4d iceberg.
  • I wrote a brief entry on Turaev and Viro’s 1992 paper on 3d TQFTs from representations of quantum SU(2)SU(2). My emphasis is on how this serves as a version of the Ponzano–Regge model of 3d quantum gravity in which the infrared divergences have been removed. I treat more mathematical aspects in a later entry.
  • I polished up the long entry on Fukuma Hosono and Kawai’s 1992 paper constructing 2d TQFTs from semisimple algebras. More nice pictures from Aaron!
  • I wrote a long new entry on Barrett and Westbury’s 1992 paper constructing 3d TQFTs from spherical categories. My treatment emphasizes how this is a categorification of the Fukuma–Hosono–Kawai construction, so it’s a bit anachronistic, but I think that’s okay.

The paper now stands at 106 pages. Whew!

Posted by: John Baez on July 23, 2009 2:42 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

I just wrote a section on Turaev’s 1992 paper constructing 4d TQFTs from nice braided monoidal categories. I took this opportunity to explain why braided monoidal categories are really monoidal 2-categories with just one object, and mention Mackaay’s work getting 4d TQFTs from nice monoidal 2-categories. This makes the pattern nice:

Fukuma–Hosono–Kawai get 2d TQFTs from nice monoids.

Turaev–Viro–Bartlett–Westbury get 3d TQFTs from nice monoidal categories.

Turaev–Crane–Yetter–Mackaay get 4d TQFTs from nice monoidal 2-categories.

Posted by: John Baez on July 24, 2009 4:18 PM | Permalink | Reply to this

Mendeleev; Re: A Prehistory of n-Categorical Physics II

Very TQFT-n-Category Periodic Table-like. What is “eka silicon” analogue? That is, how should this pattern be extrapolated to 5d TQFTs and beyond?

Posted by: Jonathan Vos Post on July 24, 2009 5:14 PM | Permalink | Reply to this

Re: Mendeleev; Re: A Prehistory of n-Categorical Physics II

Anyone who can add one to all the numbers in sight can say the right words here. But who can walk the walk?

Posted by: John Baez on July 24, 2009 5:28 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

so a group is REALLY just a groupoid with one object?

Posted by: jim stasheff on July 24, 2009 5:57 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Jim wrote:

so a group is REALLY just a groupoid with one object?

I’m not sure what that gibe is in reply to — but just to confirm your worst suspicions, I’ll say YES, it’s REALLY just a groupoid with one object!

Posted by: John Baez on July 24, 2009 6:28 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

so a group is REALLY just a groupoid with one object?

No, it's REALLY a pointed connected groupoid. Counting the number of objects of a groupoid is evil.

I'm just kidding, of course. There's nothing evil about a groupoid with one object if you say it right; after all, it's really just a group!

Posted by: Toby Bartels on July 24, 2009 8:14 PM | Permalink | Reply to this

deloop deloop

John Baez and Jim Stasheff are discussing the distinction or not between groups and their delooping.

Jim Stasheff famously did work on the fact that in the general higher categorical/homotopical context not everything that looks deloopable actually is deloopable. This is discussed at loop space.

John Baez with Mike Shulman famously have an article with an appendix where they point out one general subtlety with identifying monoidal objects and their deloopings.

For the very simple case of an ordinary group the general subtlety is present, too. This is discussed in detail at group.

To see one aspect of the subtlety in full beauty, consider group objects in (,1)(\infty,1)-categories. There is no reason for them to be deloopable at all in general. It is a big theorem and not a triviality that part of the characterization of (,1)(\infty,1)-toposes is that there they all are.

Posted by: Urs Schreiber on July 24, 2009 9:12 PM | Permalink | Reply to this

Re: deloop deloop

To quote from page 50:

It’s well-known that the hypothesis “a kk-monoidal nn-category is a kk-degenerate (n+k)category(n+k)-category” is false, even in low dimensions, if you interpret ‘is’…

For the purposes of groups k=1k=1 and n=0n=0 and the above says

It’s well-known that the hypothesis “a group is a one-object groupoid” is false, if you interpret ‘is’…

Posted by: Urs Schreiber on July 24, 2009 9:18 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Urs wrote:

John Baez and Jim Stasheff are discussing the distinction or not between groups and their delooping.

“Discussing”? I thought we were just goofing around. At least that’s what I was doing — while taking a break from finishing off this darned paper.

But yes, the subject certainly does merit serious thought, and I apologize if Jim (whose comment seemed a bolt from the blue) expected a serious reply.

Tomorrow I have a better excuse for taking a break from the paper: Tim Silverman is visiting!

Posted by: John Baez on July 24, 2009 11:30 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Okay, perhaps this is all for today: a section on Crane and Frenkel’s 1994 paper called ‘Four dimensional topological quantum field theory, Hopf categories, and the canonical bases’.

The main remaining tasks:

  • A short section on Crane’s 1993 paper ‘Categorical physics’.
  • A section on Lawrence’s paper on extended TQFTs.
  • A section on Kontsevich’s 1994 paper on homological mirror symmetry.
  • A section on Freed’s 1994 paper on higher algebraic structures and quantization.
  • A section on my 1995 paper with Jim Dolan.
  • A section on Khovanov’s 1999 paper on categorifying the Jones polynomial.

A light glimmers at the end of the tunnel.

Posted by: John Baez on July 24, 2009 10:26 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Tim Silverman visited me on Saturday. It was very nice to meet him, and nice to hear that he took at look at this Prehistory and found it surprisingly easy to follow. It was even nicer to hear that he’s hoping to do some guest posts on modular forms here at the nn-Café!

Then I got back to work. The latest version has:

  • An introduction to cobordisms in the section on Atiyah’s 1988 paper on conformal field theory. I realized I hadn’t explained the term ‘cobordism’! My explanation is still sketchy, not technically accurate… but that’s probably what’s called for here. Does anyone know a reference on how to make 2d cobordisms with conformal structure into the morphisms of a category?
  • A tour of once extended TQFTs in a section on Lawrence’s 1993 paper on extended TQFTs. I keep thinking I must be missing more papers that seek to treat ‘once extended TQFTs’ as functors between symmetric monoidal bicategories.
  • An expanded section on Crane and Frenkel’s 1994 paper seeking to build 4d TQFTs related to Donaldson theory from Hopf categories arising as categorified quantum groups.
  • A pathetically short section on Dan Freed’s 1994 paper ‘Higher algebraic structures on quantization’. I had to leave out all the fun gerby stuff and focus on the 2-Hilbert space stuff — and even that is incredibly sketchy.

So, the main things left to discuss are:

  • Kontsevich’s 1994 paper on open-closed strings, A A_\infty categories and homological mirror symmetry.
  • My 1995 paper with Jim Dolan.
  • Khovanov’s 1999 paper on categorifying the Jones polynomial. Aaron will write this section.

Almost done!

Posted by: John Baez on July 27, 2009 11:41 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

how to make 2d cobordisms with conformal structure into the morphisms of a category?

isn’t this where things get tricky due to the lack of identity elements in the naive approach? There may be something about collars, but I’ll leave it to the experts to provide details

Posted by: David Roberts on July 27, 2009 2:36 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

how to make 2d cobordisms with conformal structure into the morphisms of a category?

At least for genus 0 morphisms one way to do this is in the work of Huang et al. The theorem there is that holomorphic representations of the corresponding cobordism categories are precisely vertex operator algebras. The references are collected at vertex operator algebra.

Later Huang’s student Liang Kong generalized this from holomorphic aka chiral CFT to “full field CFT” (reference also there) so I’d think he must give the full CFT bordisms category, but I am not sure.

In the FFRS work the full structure appears in principle, only that these authors stick with Atiyah’s original point of view and don’t exactly exhibit a category, but “just” a sewing prescription. Of course that removes the problem with the identities.

isn’t this where things get tricky due to the lack of identity elements in the naive approach? There may be something about collars, but I’ll leave it to the experts to provide details

Yes, the problem is with the algebraic definition of higher category. That makes it hard to define categories of cobordisms (with extra structure).

The definitions for categories of higher cobordisms become vastly easier using geometric definition of higher category.

That’s why Lurie’s definition of the (,n)(\infty,n)-category of cobordisms is effectively just a 2-liner:

it defines the space of nn-tuples of composable morphisms essentially simply as the space of manifolds with (n1)(n-1) markings on it that indicate where we think of the manifold as being glued from submanifolds.

I haven’t tried to think about this carefully, but I wouldn’t be surprised if you could just literally take Lurie’s definition of Bord nBord_n and substitute everywhere where it says “manifold” instead “conformal manifold” or “manifold with S-structure” for some other notion S.

Posted by: Urs Schreiber on July 27, 2009 9:33 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

John wrote:

It was very nice to meet him

It was nice to meet you too!

and nice to hear that he took at look at this Prehistory

I was actually feeling quite guilty that I hadn’t looked at it earlier.

and found it surprisingly easy to follow

That it was easy to follow wasn’t the surprise—based on past performance, I expect no less from you! The surprise was the extent to which it took disparate ideas which I sort of knew about, and caused them to fall into place in a coherent whole. Of course, this was partly because of things I’d learned since first hearing about them, but it was good to see this stuff laid out clearly.

nicer to hear that he’s hoping to do some guest posts on modular forms

Mind you, I’ve been hoping to do this for more than a year, and, although the delay has involved a lot of progress rather than just mucking around, I’m still not quite there yet, and I’m busy with a bunch of unrelated things. So I may be some time. But still hoping.

Posted by: Tim Silverman on July 30, 2009 7:32 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Whew! I think the paper is really almost done.

Here’s the new stuff:

  • I added an explanation of the Reidemeister moves to my discussion of Jones’ 1985 paper. No big deal, but it needed to be done.
  • I polished the section on Lawrence’s 1993 paper on extended TQFTs, which segues seamlessly — I hope! — to a quick sketch of Moore and Segal’s work on open-closed topological string theories.
  • By popular demand, I added a section on Kontsevich’s 1994 lecture on homological mirror symmetry. I used this as excuse to explain lots of more elementary stuff: how chain complexes are secretly \infty-groupoids, the rough idea of an A A_\infty category, how objects in an A A_\infty category model DD-branes, and how Costello thinks of a ‘Calabi–Yau’ A A_\infty category as describing an open-closed topological string theory. Since this stuff comes after my description of the Moore–Segal approach to open-closed topological strings, the reader can see how it digs deeper.
  • I added a much-needed section on Gordon, Power and Street’s 1995 book on tricategories.
  • I wrote a section about Baez and Dolan’s 1995 paper on higher-dimensional algebra and TQFTs, leading up to advertisements for Batanin’s explanation of the periodic table, and Lurie’s work on the cobordism hypothesis.
  • Finally, Aaron and I wrote a section on Khovanov’s 1999 paper, which concludes with some fun stuff about Khovanov homology and exotic 4\mathbb{R}^4s.

I would be very happy if experts could scan the section on Kontsevich’s 1994 lecture and see what they think. I don’t want to cover more ground — the paper is 126 pages long and it’s past time for me to quit. In particular, I don’t want to say more about mirror symmetry or homological algebra! But I hope the stuff I say is correct, modulo a bit of oversimplification here and there.

Is there some canonical reference to Fukaya that I should add to my explanation of A A_\infty categories?

Posted by: John Baez on August 2, 2009 8:26 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

In the Gordon–Power–Street section, you vacillate unnervingly between “tricategory” and “3-category”. Could you mention both at the start, but then just stick with one? It reads a bit strange otherwise.

Posted by: Tim Silverman on August 2, 2009 11:32 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Ah, this issue is the bane of my life. I wasn’t really vacillating, just treading very carefully through a minefield.

I’m talking about 3 different concepts whose standard names used to be ‘3-category’, ‘Gray-category’, and ‘tricategory’.

My own preferred names for these 3 concepts are different and I believe more logical: ‘strict category’, ‘semistrict 3-category’, and ‘3-category’.

I hope my way of talking makes it clearer that these are listed in order of increasing generality.

I think I’ve convinced some of the younger, more flexible folks to adopt this new system. But since using ‘3-category’ to mean something very different from the older standard usage can be confusing, we sometimes condescend to say ‘weak 3-category’. Then the trio becomes ‘strict 3-category’, ‘semistrict 3-category’, ‘weak 3-category’. Again the logical progression is clear; the bad thing is that the most important concept is now saddled with a depressing name: ‘weak 3-category’.

Anyway, in this chronology I am torn between imposing my terminology and talking about what people actually said, in their own words. When talking about Gordon Power and Street, I’d feel funny saying something like “Among other things, they defined a monoidal 2-category to be a 3-category with one object”. In their terminology it was already completely well-known that a monoidal 2-category was a 3-category with one object! If you read what they say, they were defining a monoidal bicategory to be a tricategory with one object!

Sigh…

Perhaps I should only introduce my streamlined terminology when we get to the section on my paper with Jim, where we actually introduced that terminology. For example, you’ll see in that section that braided monoidal 6-categories play a significant role. At that point it would start sounding a bit silly to talk about ‘braided monoidal hexacategories’.

And more importantly, it sounds silly to talk about $n$-categories but then turn Greek whenever you substitute a particular low value for $n$.

Posted by: John Baez on August 3, 2009 6:42 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

a much-needed section on Gordon, Power and Street’s 1995 book on tricategories

glad to see it

in re: terminology, in talking about their paper, use their terminology
afterwards you can try to introduce your own
but recall what happened to omology

Posted by: jim stasheff on August 3, 2009 1:45 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Or contrahomology?

For those of you similarly mystified as I was as to what omology is, see

Bourgin, D. G.,
Modern algebraic topology
The Macmillan Co. 1963

This is the only result I got on MathSciNet.

Posted by: David Roberts on August 4, 2009 2:31 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

yes, also contrahomology as in Hilton and Wylie

Posted by: jim stasheff on August 4, 2009 1:04 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

I was sort of aware of this (though I can see now I was confused about semistrict categories) so I’m looking at that section again to try to work out what exactly was bothering me. I think it’s the second paragraph, where the first sentence uses “tricategory” (from the old terminology) and the second uses “strict 3-category” (from the new terminology). So there really is a jump there.

I agree it’s a really difficult problem. I think if it was me writing this, then in the most confusing places, like here, I’d probably use the old terminology, possibly in quotes, followed by the new terminology in parentheses. But, although this would be consistent, it would interrupt the flow of the prose and could get quite annoying. I don’t know what to suggest. All I can say is that this particular switch in the 2nd para jumped out at me, even though I wasn’t particularly paying attention to this aspect.

Posted by: Tim Silverman on August 3, 2009 2:00 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

adopt a technique I learned from Frank Adams:

up front, explaining the problem/conflict clearly and BOLDLY

then: Henceforth in this article/book

algebra shall mean hemi-demi-semi-algebra with chain condition or whatever

Posted by: jim stasheff on August 3, 2009 9:27 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Yes, although it might also be worth pointing out which system is in use in the book/paper under discussion at each point. And I’m sympathetic to John’s point that using new standard terminology can feel like a betrayal of the historical reality (to paraphrase him with a fairly broad brush, hopefully not to the point of inaccuracy).

But yes, consistency does make for clarity, at least about what the concepts actually are, if not what people called them at the time (or how they ranked them in importance/novelty/etc, which I think is maybe somewhat implicit in people’s choice of how close a new piece of terminology is to existing terminology. But perhaps that’s asking for too much in a paper like this.)

Posted by: Tim Silverman on August 3, 2009 9:49 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Tim write:

I think it’s the second paragraph, where the first sentence uses “tricategory” (from the old terminology) and the second uses “strict 3-category” (from the new terminology). So there really is a jump there.

I can certainly understand what you mean. But if it makes any difference: I was in Australia at the time the book came out, and to the best of my recollection, the terms ‘strict 3-category’ and ‘weak 3-category’ were starting to be used by people there, especially as it was becoming clear that strict 3-categories were destined to take a back seat in importance to Gray-categories or tricategories, and various speculations on weak n-categories (called as such) were much in the air.

Various deployments of the adjectives ‘strict’, ‘pseudo’, and ‘lax’ had of course been around a long time in the theory of 2-dimensional categorical algebra, and a certain emphasis on ‘strict’ as the exceptional case would not, if memory serves, have seemed too much out of place at the time. Thus the wording in John and Aaron’s draft doesn’t strike me as all that anachronistic. (But I’ll freely admit that my memory of those days could be faulty.)

Posted by: Todd Trimble on August 4, 2009 12:44 AM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

A couple of simple typos:

Section Kashiwara and Lusztig (1990)” third paragraph “where the constants mz
ij…” the indicies on m appear to be wrong.

Section Barrett-Westbury (1992): first paragraph last sentence should probably end “… we shall present it from that viewpoint.”

Posted by: Mark Biggar on August 6, 2009 5:54 PM | Permalink | Reply to this

Re: A Prehistory of n-Categorical Physics II

Thanks! Fixed.

Posted by: John Baez on August 6, 2009 6:46 PM | Permalink | Reply to this

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