The n-Category Café

Recently John Huerta asked me about some things I once said in the context of discussion such as in the entry SuGra 3-Connection Reloaded.

I had had only time for a brief reply by email. But since this deserves a comprehensive discussion, I have now started typing what I have to say into an $n$Lab entry:

D’Auria-Fré formulation of supergravity (with an eye towards higher nonabelian gauge theory)

This, too, remains unfinished for the moment, since I have to run now to a seminar. But it already may contain some details that might be of interest.

When I am back and have a minute to spare, I would like to discuss with anyone who feels like replying in more detail the issue of rheonomy (and, for a start, give a review of this concept at the $n$Lab page). But this has to wait a bit.

A while back, Steven Krantz asked me to write an opinion piece about math blogging in the AMS Notices. I asked you what I should say, and we had a great conversation about it.

Today, with the deadline fast approaching, I quickly cranked out a draft of this piece. I’d love to hear your comments. But beware: there’s a word limit of 800 words, and I’ve spent 750 so far! So, I can’t add much unless I take something away.

One thing I will add is a pointer to the online list of math blogs and wikis at the n-Lab, which also contains a link to our previous conversation. Do you know math blogs and wikis that should be on that list, but aren’t? If so, please add them!

In week280 of This Week’s Finds, hear about the courses taught by Ashtekar and Rovelli at the quantum gravity summer school in Corfu. Ashtekar spoke about loop quantum cosmology, and how it could turn the Big Bang into a Big Bounce. Rovelli spoke about spin foam models, and how the new EPRL model cures many problems of the old Barrett-Crane model. The graviton propagator seems to work!

After a visit in 1939 to the Monastery of the Prophet Elijah on the Greek island of Santorini, the Oxford philosopher R. G. Collingwood entered into discussion with his students about the value of monastic life. It appears that the students were a little perplexed to find their prejudice that monks were “at worst idle, self-indulgent, and corrupt; at best selfishly wrapped up in a wrongheaded endeavour to save their souls by forsaking the world and cultivating a fugitive and cloistered virtue” clash with their admiration for

…the atmosphere of earnest and cheerful devotion to a sacred calling, the dignity of the services and beauty of their music, the eager welcome and the loving hospitality, and above all the graces of character and mind which the life either generated in those who had adopted it or at least demanded of aspirants to it and thus focused, as it were, in the place where the life went on. (‘Monks and Morals’, Essays in Political Philosophy, Oxford, 1989)

I’d like to ask Urs some questions about n-curvature. I thought I’d ask them publicly, because it might help other people learn about this subject.

This year the Fall Western Section Meeting of the American Mathematical Society will be held here at UC Riverside. Julie Bergner and I are running a session on homotopy theory, $n$-categories and related topics. If you’re anywhere nearby, I hope you drop by!

• Special session on homotopy theory and higher algebraic structures, UC Riverside, November 7-8, 2009. Organized by Julie Bergner and John Baez.

If you’re interested in our session, you may also like this one:

• Special session on algebraic structures in knot theory, UC Riverside, November 7-8, 2009. Organized by Alissa Crans and Sam Nelson.

It’ll include talks by Louis Kauffman, Mikhail Khovanov, Scott Carter, Masahico Saito, Scott Morrison and other people who live near the interface of topology, categories and physics.

As if that weren’t enough, there’s also another session on knot theory:

• Special session on knotting around dimension three: a special session in memory of Xiao-Song Lin, UC Riverside, November 7-8, 2009. Organized by Martin Scharlemann and Mohammed Ait Nouh.

I’m back at Riverside after a quantum gravity school in Corfu where I learned many wonderful things, which I’m eager to explain…

Contuinuing with my by now well-established practice – see A Seminar on a Survey of Elliptic Cohomology and A Seminar on Gromov-Witten Invariants – of working seminar notes here at MPI Bonn into the $n$Lab, I am now continuing with the main event, the lecture series by Stephan Stolz and Peter Teichner on their work on

geometric models for elliptic cohomology and tmf

See there for a (little) bit of background and context.

Notes on today’s sessions are beginning to find an incarnation as wiki-entries here:

Axiomatic field theories and their definition from topology

(1|1)-dimensional Euclidean field theories and K-theory

(2|1)-dimensional Euclidean field theories and tmf.

So far this sketches the outline of the construction. Concrete details are the topic of the next days. Those reader who haven’t seen aspects of this program before might find the notes so far a bit impenetrable, for instance a basic understanding of supermanifolds is assumed. So I am posting this here for the time being more for the experts who might lend me a hand in polishing this, than for the laymen. For them, hopefully, this will develop into something useful as this proceeds.

In particular, please notice, as before, that the notes behind these links are, for the time being, notes taken real-time during the seminar, with no real post-production yet. So everything is a bit rough at the moment. Help me make it become more smooth!

The sophisticated take on mathematical induction is that the natural numbers form the inital algebra of the endofunctor on $Set$, $F: X \to 1 + X$. As such we have a map $\langle 0, s \rangle: 1 + \mathbb{N} \to \mathbb{N}$, picking out the zero and the successor function. As an initial object in the category of $F$-algebras we know that any monomorphism into it must be an isomorphism. So if I have a property, $P$, of the natural numbers which holds for $0$, and for which $P(n)$ implies $P(s(n))$ for all $n$, then the set, $Y$, of natural numbers satisfying $P$ forms an $F$-algebra, $\langle 0, s \rangle: 1 + Y \to Y$, with an obvious monomorphism into $\mathbb{N}$. Hence $Y$ is isomorphic to $\mathbb{N}$ and $P$ holds for all natural numbers.

Now in the case of coalgebras, coinduction is going to rely on the fact that epimorphisms from terminal coalgebras are isomorphisms, i.e., there’s no quotient coalgebra of a terminal coalgebra. So if I want to establish that two elements of a terminal coalgebra are equal all I need do is find an equivalence relation which relates the elements and which is compatible with the coalgebra structure. Equivalence classes cannot contain more than one element, so I know these two elements are the same.

In a fashion similar to the Seminar on a Survey of Elliptic Cohomology, Kevin Lin and two other grad students here at MPI Bonn are now running a seminar on Gromov-Witten invariants.

And as with the former I am taking notes right into the $n$Lab with this one, too, now.

Today’s notes are here:

basic ideas of moduli stacks of curves and Gromov-Witten invariants

This is a bit rough, as you will see, as it is essentially real-time note-taking. All the post-production I had time and leisure for was spent on one entry about the terminology of fine/coarse moduli spaces/stacks.

But I am thinking that this raw material may be a good basis to base some further editorial effort on. Maybe some lurking $n$Café guest or other feels like creating one of the grayish links spread all over the text.

guest post by Andrew Stacey

We are proposing shifting the place for recording $n$Lab latest changes to the $n$-Forum. This will provide the following advantages:

1. It will be better organised and so easier to see new items, and replies to existing items.

2. There is an RSS feed so you can follow developments without having to check this page every 5 minutes (you know who you are).

3. Updates can be made simultaneously so you don’t have to wait for the person currently editing latest changes to finish.

4. If you create an account on the forum (this is not mandatory), it will keep track of the items that you have read and inform you of new things.

5. It will be easier for you to find out about other possible changes to the infrastructure of the n-lab, and so have some influence on what makes this place run.

The system will be no more complicated than the current one:

1. Make your changes in the nLab.

2. Click on the latest changes link in the contents sidebar of the homepage (we’ll include a link ( not a redirect) from this page for those who’ve bookmarked this one).

3. Click on the ‘start a new discussion’ in the panel (left hand side) of the page on the forum.

4. Fill in the details of the change. I suggest that the title of the discussion makes it clear which pages on the n-lab are referred to.

5. If you have an account, you will be invited to log in (if you are not already logged in). If not, there is a reCaptcha to solve to prove that you are human (or elven). If you do post “anonymously”, may we ask that you put your name at the top of your comment so that we know who you are.

You can subscribe to RSS or ATOM feeds from the forum. This will keep you up to date in your preferred ‘feed reader’ of what’s going on.

The syntax on the forum is ever-so-slightly different to that here. You can use markdown syntax, but it’s not extended markdown. Wikilinks work just as they do here (and point to the correct pages here). Mathematics is, currently, enabled via a Wordpress-style system where it gets converted to pictures. Enclose your mathematics within double dollars to get it to work. (Hopefully, this will change to a MathML system in the near future).

To have a look at the system, go to this page. When we “go live” (which will happen soon), I’ll clean out the discussions so feel free to have a play first. Any comments and suggestions should be recorded here (you’ll have to create an account to comment on this, though).

Jacques Distler has asked us for a list of citations to entries on the $n$-category Café and the $n$Lab.

Since I find it hard to remember them myself, and since it might be of interest in general to keep track of these, I thought I’d create an entry at which to collect them. This is now here:

inbound citations

So far this contains just a handful of citations that I could think of off the top of my head. If you can think of more, please add them.

I’ve known Arnold Neumaier for quite a while, thanks to many discussions on the newsgroup sci.physics.research. Recently he sent me a proposal for a system called FMATHL (Formal Mathematical Language), designed to be:

a formal framework which will allow — when fully implemented in some programming language — the convenient use of and communication of arbitrary mathematics on the computer, in a way close to the actual practice of mathematics, with emphasis on matching this practice closely.

He asked me for comments, and I gave him a few. But I said that some of you have thought about this subject more deeply, so your comments might be more valuable. So he agreed to let me post links to his proposal here.

The German Center for Media and Interactivity is organizing a

Symposium

Title: Communication Formats and their Dynamics in Digital Science Communication (program pdf)

Sept. 9 - 11, 2009

Schloss Rauischholzhausen

on the usage of digital web media among scientists. If you can read German, you can find a detailed description here.

I am being asked to participate in a roundtable discussion tomorrow evening, representing the $n$-Category Café.

guest post by Mike Stay

John Baez has been struggling to understand the pi calculus — a formalism for modelling concurrency in computer science.

I’m going to try to explain it nice and slowly. I’ll use the “polyadic asynchronous” variant of the pi calculus.

Recently I’ve been looking into the coalgebra community’s take on modal logic, which in a phrase states simply that

Modal Logics are Coalgebraic.

The key idea is to start from a dual adjunction between the semantics and syntax of propositional theories, as explained on page 3 of Exemplaric Expressivity of Modal Logics. Typically this is a Stone-style duality between (Lindenbaum algebras of) propositions and spaces of models or valuations for a propositional theory.

Modalities are then introduced on the syntactic side in the shape of an endofunctor, where the new modal propositional theories are algebras for this functor. Meanwhile, on the semantic side, we have a endofunctor whose coalgebras provide the semantics for the modal propositional theories. If all goes well, we end up lifting the Stone adjunction to one between coalgebras on one side and algebras on the other. It would be handy at this point if I’d learned how to draw suitable diagrams. As I haven’t you can see what’s going on on page 2 of Coalgebras and Their Logics.

My position in Hamburg is about to terminate, currently I am based at the MPI Bonn and about to start a postdoc position in Ieke Moerdijk’s group in Utrecht.

In Bonn we are preparing for a program by Stephan Stolz and Peter Teichner on, you know, QFT, cohomology and such things. There are lots of students of them around and they are starting to run a

Seminar on Elliptic Cohomology

This is supposed to essentially go through, guess what, Jacob Lurie’s notes. On that occasion I started a corresponding $n$Lab entry

$n$Lab:A Survey of Elliptic Cohomology

So far this contains just a table of contents, a summary of one of the main ideas and a few links, as far as existent. I am planning to expand this entry as we go along in the Seminar, and hopefully others will join in, too.

In week279 of This Week’s Finds, hear how we can describe vectors and spinors in 3d, 4d, 6d and 10d spacetime using the real numbers, complex numbers, quaternions and octonions.

See the magic identity that makes super-Yang-Mills theory and the classical superstring possible in these dimensions! Read the history of this subject… and then a bit about tuning systems and infinitely groupoidified logic.

guest post by Mike Shulman

Regular readers of the cafe may be familiar with the microcosm principle: certain algebraic structures can be defined in any category equipped with a categorified version of that same structure. Kate Ponto and I are working on a paper about a perhaps less familiar instance of this principle, regarding the horizontal categorification of traces; and we’d appreciate any feedback you have to offer. Here’s the link:

• Kate Ponto and Mike Shulman, Shadows and traces in bicategories

I have been asked for suggested reading to help with a school project on solitons. Partial derivatives aren’t on the syllabus, so even a brief sketch of the balancing of dispersion and steepening of waves gleaned from Palais’s excellent survey is already stretching the bounds of knowledge. Rather than more mathematics then, something along the lines of physical and engineering applications would go down well.

There are some handy websites, e.g., here, which reminded me that a number of years ago there was a project to use solitons to send vast amounts of data rapidly down optical fibres. Erbium-doping was supposed to be the answer to transmission problems. But that was more than a decade ago. Are solitons actually used practically today in this way?

As for natural phenomena, it seems that Jupiter’s Great Red Spot is an autosoliton, and that solitons have been observed emerging from the Straits of Gibraltar.

Any other candidates?

I’ve been in love with the octonions for over a decade now. They seem downright distasteful when you first meet them: an 8-dimensional algebra where division is allowed, but multiplication is nonassociative! Only later do you realize that they’re deeply connected to a wide range of mysterious and exotic things: the icosahedron, the exceptional Lie groups, the exceptional Jordan algebra… and the way superstring theory and super-Yang–Mills theory love to live in 10 dimensions.

I wrote a paper on the octonions back in 2001, where I tried to assemble all the clues I’d found. But I didn’t get far in relating the octonions to supersymmetry. I just explained the basic facts: how octonions are naturally tied to 10-dimensional Minkowski spacetime.

Now I’m lucky to have a brave graduate student who can do calculations like a physicist and proofs like a mathematician: John Huerta. Since March — when we finished our review article on the algebra of grand unified theories — we’ve been digging into the relation between octonions and supersymmetry. As so often the case in this game, the clues are already available, lurking in existing papers. The trick is assembling them into a clear story that people can understand without going on a lengthy and painful quest.

• John Baez and John Huerta, Division algebras and supersymmetry — latest version with all known corrections or arXiv version.

It’s not quite done yet, but it’s close. As always, comments and corrections would be appreciated.

Thanks to Andrew Stacey, it’s finally done: after a bit of testing the $n$Lab has finally successfully migrated to a new server.

That means:

- as far as you go about accessing and using it nothing in the usage has changed: in particular you still access the lab at http://ncatlab.org/nlab

- but using the $n$Lab should be a better experience now: the server reaction is much quicker now, the whole thing should be stable now.

If you are interested in more technical discussion of what happened, see the $n$Forum. But of course, you don’t have to be interested in any behind-the-scenes technical discussion for using the $n$Lab.

Instead, you may want to be the one to create entry number 2000 by writing a piece on your favorite topic that is ignorantly ignored so far. The moment of this writing the nominal count of entries is 1965.

Viewed from one perspective this may look like a large number. But viewed from the right perspective it is a very small number. We are only just getting started, still. But it’s looking promising.

You’ll recall from my earlier blog entry that on July 10th, a team led by the geophysicist Didier Sornette predicted a crash in the Shanghai Stock Exchange Composite Index. They said it would happen between July 17th and 27th, with 60% probability.

Now some say they were right.

together with Orit Davidovich

The following is the third set of notes following the talks of David Ben-Zvi at a workshop on topological field theories, held at Northwestern University in May 2009. This post follows our second post found here. We’ll again give a brief introduction, and then send you over to a PDF file for the full set of notes.