The n-Category Café

Let’s take a break from all this type theory and $\infty$-stuff and do some good old 2-dimensional category theory. Although as usual, I want to convince you that plain old 2-categories aren’t good enough, and we need something different. But as far as I know, the structure I have in mind today doesn’t exist yet! I think I know how to define it, but before I go to the trouble, I want to know what people think about it, and whether anyone has seen anything like it before. So I’ll just describe informally what I think this structure should look like.

Briefly, what I want is a “minimal” extension of a 2-category which can also include extraordinary natural transformations.

Yay! Alex Hoffnung has accepted a postdoc at the University of Ottawa, where he’ll work with Rick Blute (who works on categorical logic and its relation to computation), Alistair Savage (who works on representation theory and quivers) and other folks.

Note This is reposted from the temporary site so that comments can continue here.

I’m currently at MSRI at a knot homology meeting. There are lots of people here with pictures of surfaces, some of these even being categorical, so I thought I would return to the subject of computer manipulation of these. In particular I thought I would make use of our (hopefully) brief sojourn at WordPress and take the opportunity to embed some videos (but then I was told how I could do it at our usual place).

Note: This is being reposted from the temporary site, with existing comments copied over so that discussion can continue here.

Today I’m blogging from Washington D.C., at the Annual Meeting of the Association for Symbolic Logic. The ASL is mostly populated with material set theorists and classical logicians, but this year they had a special session on Categorical Logic, and another one on Logic and the Foundations of Physics (including lots of categorical quantum mechanics)—a promising sign for the recognition of category theory. I was invited to speak at the former session this afternoon, about stack semantics and 2-categorical logic.

And not-entirely-coincidentally, at long last I’ve put online a draft of my (first) paper about the stack semantics and comparing material and structural set theories. You can get it from my nlab page:

• The stack semantics

There are also slides from today’s talk and one from last November.

Thanks once more to the sterling work of Jacques Distler, we’re up and running again. So, please come here, rather than the temporary WordPress site, for all your mathematical, physical and philosophical needs.

The software problems seem to be mostly fixed. One or two people still find themselves unable to leave comments. If this happens to you, please tell me or one of the other hosts. We’ll post the comment for you and see if we can get the problem sorted out. (And if you just can’t wait to get that comment into cyberspace, you can put it at the loose ends entry of the WordPress site.)

Thanks for your patience.

In week294 of This Week’s Finds, hear an account of Gelfand’s famous math seminar in Moscow. Read what Jan Willems thinks about control theory and bond graphs. Learn the proof of Tellegen’s theorem. Meet some categories where the morphisms are circuits, and learn why a category object in Vect is just a 2-term chain complex. Finally, gaze at Saturn’s rings edge-on:

The $n$Category Café has moved to a new address:

• The NEW nCategory Café

Unfortunately, this blog’s software is suffering some problems. Since it seems that the fix will take a while, we have decided to take measures and migrate the $n$Category Café to a Wordpress installation. At least for the time being.

Please follow the above link.

We apologize for the inconvenience.

This paper may be finished now:

• John Baez and John Huerta, An Invitation to Higher Gauge Theory.

Tim van Beek wanted a heads-up when it was done, so he could make another round of corrections. Indeed, I’d love comments from all of you! Please post them here.

The March 2010 edition of the Newsletter of the European Mathematical Society is out, and it includes a short piece by me on realism in mathematics. I was invited to contribute to a series on Platonism by Urs Persson, a series which includes Brian Davies’ ‘Let Platonism Die’ (June 2007), Reuben Hersh’s ‘On Platonism’ and Barry Mazur’s ‘Mathematical Platonism and its Opposites’ (June 2008), and David Mumford’s ‘Why I am a Platonist’ and Philip Davis’ ‘Why I Am A (Moderate) Social Constructivist’ (December 2008). Ulf has himself a riposte to Davies – Let Platonism Live!. I decided to drop the term ‘Platonism’ since I prefer to focus on the issue of the nature of the constraints acting on mathematicians, and don’t wish to tie this wholly to the question of the existence of abstract entities.

As you will have noticed, currently this blog is affected by a bug that generates an empty error message whenever a comment is submitted and thereby prevents any comments to be posted.

This is of course rather unfortunate for a blog, and hopefully it will go away, soon. My understanding is the problem is caused by some software update and investigations into the problem are underway.

Meanwhile, I personally found it frustrating that I couldn’t post comments here, and suspect you may feel the same. Since at the same time we are running the nForum intended for meta-discussion related to the nLab, and since that runs just fine, I thought I’d simply forward, for the moment, the comments that I would otherwise post here to that forum.

Notably, I just posted a reply to John’s latest entry Division Algebras and Supersymmetry II to this nForum thread.

This is not meant to be a good solution to the problem. But it does serve as a workaround for the moment.

John Huerta and I are finishing up another paper:

• John Baez and John Huerta, Division algebras and supersymmetry II.

We’d love comments and corrections! In particular, the material on supergravity theories needs improvement. But the story is already quite cool, since it connects the octonions and higher gauge theory to superstrings, super-2-branes and supergravity.

I would like to start to explain something about the “intrinsic volumes” for Riemannian manifolds, which are some fundamental differential geometric invariants including the volume, the total scalar curvature and the Euler characteristic, and they go by various names including curvature measures, Killing-Lipshitz curvatures, generalized curvatures and quermassintegrale.

The reason I’m currently interested in these is that I have just about finished a paper on the magnitude of some homogeneous Riemannian manifolds and these intrinsic volumes or generalized curvatures seem to be related to the asymptotics of the magnitude.

You may well remember that the magnitude (or cardinality) of a metric space was introduced by Tom Leinster in a post here at the Café as a special case of the Euler characteristic of an enriched category. The fact that the asymptotic behaviour of the magnitude is related to intrinsic volumes for polyconvex sets has already been mentioned here and here. However, the paper I’m finishing has to do with smooth things rather than polyconvex ones. I will mention connections with polyconvex sets at the end of this post.

Here I will restrict myself to the basic case of closed surfaces in $3$-space, although everything generalizes, and use as motivation the celebrated Tube Formula of Weyl, which you might wish to ponder before reading on.

Tube Problem. Suppose $\Sigma$ is a surface in $\mathbb{R}^3$; thicken it up to form the $\epsilon$-neighbourhood $N_\epsilon \Sigma$ consisting of all points within $\epsilon$ of $\Sigma$. Supposing that $\epsilon$ is sufficiently small, how does the volume of $N_\epsilon \Sigma$ depend on $\epsilon$ and $\Sigma$?

Very little knowledge of differential geometry will be assumed.

This Saturday, March 13, at Paris Diderot takes place the next

• Séeminaire Itinérant de Catégories (pdf announcement and schedule)

Among the speakers and talks on higher category theory are

• Denis-Charles Cisinski on (∞,1)-operads

and

• Dimitri Ara, on the definition of weak ∞-groupoid from Grothendieck’s Pursuing Stacks

Grothendieck’s original take at the homotopy hypothesis has recently attracted renewed attention. $n$Café readers ask me to ask you if anyone attending the seminar might report on what Ara has to say on this here on the blog – or better yet fill in some remarks at nLab: ∞-groupoid

These days extended topological quantum field theory in the form of FQFT and AQFT and its variants is studied a lot by pure mathematicians, who tend to embrace the higher category theory origin of these notions. Slowly but surely one expects that $(\infty,n)$-category theory finds its way into mathematical string theory this way, where it is of utmost importance.

Eric Sharpe kindly points out the

• Summer School on Mathematical String Theory 2010
  June 21 - July 2, 2010 at Virginia Tech

Among the speakers there is John Francis, who $n$-Café regulars will know from our discussion of geometric ∞-function theory and his work on Ek-Algebra. He will lecture the $\infty$-categorical refinement of AQFT provided by topological chiral homology (factorization algebras ).

The full list of speakers so far is

• Dima Arinkin (UNC-CH) (homological algebra)
• Arend Bayer (Connecticut) (derived categories, $\pi$-stability)
• John Francis (Northwestern) ($E_n$ algebras, topological chiral homology, topological field theories)
• Josh Guffin (U-Penn) (heterotic string compactifications, (0,2) mirrors)
• Simeon Hellerman (IPMU, Japan) (modular averages, D-instantons)
• Ilarion Melnikov (Max-Planck, Potsdam) (gauged linear sigma models)
• Peter Zograf (Steklov, St. Petersburg) (moduli spaces of curves)

See the school’s website for more details.

A few of us here at the Café decided that it would be good to have a short series of posts in which each of us (at least, each of us who wants to) says something about his overall take on higher category theory. So, this is big-picture stuff, not nitty-gritty. Urs effectively kicked the series off. Here’s my contribution.

I also want to take this opportunity to pay tribute to John’s incredible activities in higher-dimensional category theory. One of the first web pages I ever laid eyes on was one of his. It was just before I started my Ph.D., higher category theory had yet to enter my life, and Richard Thomas was showing me this thing called the ‘World Wide Web’. As I remember it, he typed ‘category theory’ into AltaVista and up came an issue of This Week’s Finds all about $n$-categories.

For 15 years now, John’s been inspiring people to go and work on higher category theory; he’s been patiently explaining the basic ideas over and over again; he’s been making famous Hypotheses that shape the current mathematical landscape; he’s been categorifying everything in sight. Simply, he’s been an enormous influence on the subject. Now he’s moving on to other things. John, we salute you! Give that man a round of applause.

People may have noticed that the two cultures of mathematics idea has a certain grip on me. In one culture there’s all the mathematics we love here at the Café, which by 2050 will be condensed into some beautiful statements about $\infty$-adjunctions between $\infinity$-toposes of space and quantity. Algebraic geometry and homotopy theory will find themselves simple consequences of this grand theory.

But will it make inroads into the other culture? Will it penetrate into combinatorics to cover, say, the Erdös discrepancy problem? You’d think not. So what makes for the difference?

Terry Tao has made some interesting comments on the subject, which I’ve gathered here. The first explains why it is difficult to establish a general theory of nonlinear partial differential equations. The second discusses the relationship between ‘structure’ as understood by the combinatorialist and the theory-builder, and offers the prospect of a degree of convergence. The third concerns the different kinds of condition on the entities dealt with in algebra and analysis. This last comment appeared on Buzz, which, as he tells us at What’s New, Tao uses as “an outlet for various things I wanted to say or share, but which were too insubstantial to merit a mention on this blog”.

I’ll reproduce the third comment here in case people would like to discuss it:

There’s a postdoc position in Lisbon for someone working on categorification or gerbes!

This year’s Oporto Meeting is about categorification! This year it will not take place in Oporto. It will be in the city of Faro, which in is the southernmost region of Portugal, called the Algarve:

• XIXth Oporto Meeting on Geometry, Topology and Physics, July 19 – 23, 2010, Faro, Portugal. Organized by Marco Mackaay, Roger Picken and Pedro Vaz.

A while back I mentioned a very old review article by Duff. If you look at his brane scan you’ll see he lists superstring theories in 3, 4, 6, and 10 dimensions. He also lists 2-brane theories in 4, 5, 7 and 11 dimensions, which give the superstring theories upon dimensional reduction. Now John Huerta and I are wondering: are all of these 2-brane theories associated to theories of supergravity? The 2-brane theory in 11 dimensions is rather famously associated to 11d supergravity. But what about the other cases?

I’ve written before about “structural” foundations for mathematics, which can also be thought of as “typed” foundations, in the sense that every object comes with a type, and typing restrictions prevent us from even asking certain nonsensical questions. For instance, whether the cyclic group of order 17 is equal to the square root of 2 is such a nonsensical question, since the first is a group while the second is a real number, and only things of the same type can be compared for equality.

The structural axiom systems we’ve talked about, such as ETCS and SEAR, are presented as first-order theories whose objects of study are things with names such as “sets,” “functions,” and “relations.” Although such presentations look a bit different (and hopefully less syntactically imposing) than what is usually called type theory, it’s not too hard to write down more traditional-looking “type theories” that capture essentially the same models as ETCS and SEAR. However, these theories are missing an important aspect that is present in many type theories, namely dependent types.

If your introduction to type theory has been through topos theory and categorical logic, as mine was, it may take some time to really appreciate dependent types. At least, it’s taken me until now. The Elephant spends all of one section of one chapter (D4.4) on dependent type theory as the internal logic of locally cartesian closed categories, closing with the remark that the generalization from topoi to such categories “…has been won at the cost of a very considerable increase in the complexity of the calculus itself, and so we shall not pursue it further.” However, while dependent types are not necessary in any theory containing powersets (which, I think, includes all the foundational theories of interest to most mathematicians), I’m coming to believe that they are really a very convenient, natural, and ubiquitous notion.

[ guest post by Thomas Nikolaus ]

A few months ago we had here a discussion about $\infty$-groupoids and the homotopy hypothesis. In this discussion I had the idea to use algebraic Kan complexes as a model for $\infty$-groupoids and to endow the category of algebraic Kan complexes with a model stucture. After a fair amount of work I finally could prove that these things really do work out.

Notes on Algebraic Kan Complexes (pdf)

Abstract. We establish a model category structure on algebraic Kan complexes. In fact, we introduce the notion of an algebraic fibrant object in a general model category (obeying certain technical conditions). Based on this construction we propose algebraic Kan complexes as an algebraic model for ∞-groupoids and algebraic quasicategories as an algebraic model for (∞,1)-categories.

I would be grateful for comments and criticism. Please let me know what you think about this.

In the previous entry John described higher gauge theory with the declared intent of not emphasizing its higher category theory.

The other extreme of this has its own charms: try to describe higher gauge theory entirely using formal category theory on an ambient $\infty$-topos. I have been entertaining myself with searching for this intrinsic $\infty$-topos-theoretic formulation. The present state of my understanding is summarized on this page:

• Structures in an $(\infty,1)$-topos.

This lists structures that are available on purely formal grounds in an $(\infty,1)$-topos: its shape, its cohomology, its homotopy, its rational homotopy, its differential cohomology.

Here differential cohomology in an $(\infty,1)$-topos is just another word for higher gauge theory .

In the last days maybe I was able to fill what used to be a gap in the abstract story that I was trying to tell. It helped to read the remarkable recent article

• David Ben-Zvi, David Nadler, Loop spaces and connections

and its emphasis of the issue discussed in the remarkable

• Bertrand Toën, Champs affine

This provides a nice point of view on rational homotopy theory from $\infty$-topos theory. Some aspects of this I had understood before. For instance the left adjoint $\infty$-functor that produces the global function dg-algebra on an $\infty$-stack discussed there is what I used to call the Chevalley-Eilenberg algebra of an $\infty$-stack. But after reading this again now I obtained a clear simple abstract picture that I did not quite have before.

On the page

• Rational homotopy theory in an $(\infty,1)$topos

there is first a section that describes this abstract picture. Then there is a section that recalls key definitions and results from Bertrand Toën’s article from this perspective.

John Huerta and I are writing an expository paper based on the notes of the course I taught in Corfu last summer:

• John Baez and John Huerta, An invitation to higher gauge theory (DRAFT VERSION).

As with the course, the goal here is to give mathematicians and physicists a little taste of higher gauge theory, just enough to whet their appetite. So, the category-theoretic prerequisites have been reduced to the bare minimum. You don’t even need to know what a category is! You just need to promise not to run out of the room screaming when you see the definition.