The n-Category Café

You may have heard of the Mathematics Genealogy Project. This is a wonderful database that lets you look up the Ph.D. advisor and students of almost any mathematician. This is how I traced back my genealogy to Gauss back in week166.

I was feeling pretty proud of myself, too — until I found someone who had two Ph.D. students before he was even born!

At HIM this week there is a Noncommutative Geometry Conference.

Just heard Thomas Schücker talk about The noncommutative standard model and its post- and predictions, which, as it turns out, closely followed his entry for the Encyclopedia of Mathematical Physics: Noncommutative geometry and the standard model

The question of the relationship between category theory and model theory emerged in this thread. So I was interested to read some things David Kazhdan had to say about this relationship in his Lecture notes in Motivic Integration.

In spite of it successes, the Model theory did not enter into a “tool box” of mathematicians and even many of mathematicians working on “Motivic integrations” are content to use the results of logicians without understanding the details of the proofs.

I don’t know any mathematician who did not start as a logician and for whom it was “easy and natural” to learn the Model theory. Often the experience of learning of the Model theory is similar to the one of learning of Physics: for a [short] while everything is so simple and so easily reformulated in familiar terms that “there is nothing to learn” but suddenly one find himself in a place when Model theoreticians “jump from a tussock to a hummock” while we mathematicians don’t see where to “put a foot” and are at a complete loss.

Vaughan Pratt asked me some questions about the physics FAQ on light mills. I’ve become quite puzzled. So, it’s time to revive the long-dormant thread on ‘gnarly issues in physics’.

guest post by Gavin Wraith

Excuse me pestering you with a query about an article in the Scientific American — ‘The Self-Organizing Quantum Universe’ by Ambjörn, Jurkiewicz and Loll. I found the article interesting but frustrating. It gives hints but no definite description of the mathematics involved. The references given were evidently written for a readership of physicists not mathematicians.

In rapid-fire succession we’ve seen two conferences on this subject in London… now another, organized by the same team, at Oxford!

• Categories, Logic and Foundations of Physics, Oxford University, August 23-24, 2008, organized by Bob Coecke and Andreas Döring

Over here, we’re looking at the differences between two famous infinitely large fields:

$\mathbb{C}$ is uncountably categorical, that is, it is uniquely described in a language of first order logic among the fields of the same cardinality.

In case of $\mathbb{R}$, its elementary theory, that is, the set of all closed first order formulae that are true in $\mathbb{R}$, has infinitely many models of cardinality continuum $2^{\aleph_0}$.

In naive terms, $\mathbb{C}$ is rigid, while $\mathbb{R}$ is soft and spongy and shape-shifting. However, $\mathbb{R}$ has only trivial automorphisms (an easy exercise), while $\mathbb{C}$ has huge automorphism group, of cardinality $2^{2^{\aleph_0}}$ (this also follows with relative ease from basic properties of algebraically closed fields). In naive terms, this means that there is only one way to look at $\mathbb{R}$, while $\mathbb{C}$ can be viewed from an incomprehensible variety of different point of view, most of them absolutely transcendental. Actually, there are just two comprehensible automorphisms of $\mathbb{C}$: the identity automorphism and complex conjugation. It looks like construction of all other automorphisms involves the Axiom of Choice. When one looks at what happens at model-theoretic level, it appears that “uniqueness” and “canonicity” of a uncountable structure is directly linked to its multifacetedness.

Apparently, there is something Galoisianly model theoretic going on.

Now, I see Jamie Vicary has a new paper – Categorical properties of the complex numbers, and I am wondering whether bridges can be built.

In week267 of This Week’s Finds see the tile patterns of the Alhambra:

Then learn about the 17 wallpaper groups, their corresponding 2d orbifolds, the role of 2-groups as symmetries of orbifolds, the work of Carrasco and Cegarra on hypercrossed complexes, and the work of João Faria Martins on the fundamental 2-group of a 2-knot.

What do people make of Jean-Yves Girard’s entry for CATEGORY in his paper Locus Solum?

Category-theory played an important role in the disclosure of the deep structure of logic : for instance coherent spaces form a categorical model for linear logic, and by the way came from this model, not the other way around. The pregnancy of categories in our area made me style the period 1970-2000 as the time of categories, a period which opened with the Curry-Howard isomorphism. Ludics originates in the category-theoretic approach, but eventually took some distance.

The limitations of categories–insofar as we can judge them from the sole logical viewpoint–lies in their spiritualism, their extreme spiritualism : everything is up to isomorphism. In particular categories cannot explain locative logical constructions such as intersection types–or if you prefer, the categorical viewpoint compelled us to consider these artifacts as non-logical. In the same way, category-theory cannot explain the prenex form of ludics, which are based on equalities and which are definitely impossible to explain by means of isomorphisms.

To sum up, category theory only presents a limited aspect of logic; provided we realise this, it remains a very important tool. (Locus Solum, pp. 96-97)

He has later entries on LOCATIVE LOGIC and on PRENEX FORM.

We were wondering back here whether higher categories met other demands of his.

Many entities with which we deal can be said to fit into a hierarchical order:

• sound, vocable, word, utterance, conversation, discourse
• character, word, sentence, paragraph, chapter, book, encyclopaedia/collection
• sound wave, note, chord, phrase, passage, movement, symphony, style (Baroque, etc.)
• particle, molecule, molecular assembly, organelle, cell, cell assembly, organ, …

• …organism, herd/shoal, species

• …person, household, local community, nation state

Now you could claim that mathematics works best at the lower levels of these hierarchies, whereas narrative is the necessary tool to describe the higher levels. For example, we have a good mathematical theory of sound waves, but ask what Beethoven achieved with his Eroica symphony and we start talking about romanticism, Napoleon and the ideals of the French Revolution.

We can set anyone, or better a computer, the task of counting the relative frequency of the letters of the alphabet in a text, a reasonably educated person to count the frequency of metaphors, but we only listen to the subtle critic to learn about War and Peace, who will tell us among other things about the state of Russia at the time Tolstoy wrote it. Mathematics struggles to grasp reality as it climbs the hierarchy leaving us little better off than the numbering of the Eroica as Beethoven’s third symphony, or the count of the chapters of War and Peace.

I have some questions on $\omega$-categorical issues in the context of descent and cohomology to the experts. Some of them are accompanied by the figures collected here.

Over at his Theoretical Atlas blog, Jeffrey Morton reports from Quantum Gravity and Quantum Geometry 2008, briefly indicating the content of a couple of talks. This one here especially caught my attention:

Benjamin Bahr gave another talk dealing with categorical issues - namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold. In fact, he treated various cases under the same framework: flat and non-flat connections, on manifolds and on graphs - and others.

I have posted the following comment, which however is still “awaiting moderation”. While it awaits, I thought I’d share this here:

I’m in Paris now. I’ve spent this week attending a conference on Algebraic Topological Methods in Computer Science. Yesterday during one of the breaks my host, Paul–André Melliès, introduced me to a student I’d seen somewhere before. I didn’t catch the guy’s name, but his eyes glittered strangely as we sipped our coffee and talked, and then I realized: it was Samuel Mimram, winner of the first Theorems Into Coffee challenge! In this paper:

• Samuel Mimram, Presentation of a game semantics for first-order propositional logic, Thm. 14, page 22.

he proved the result I sought, namely:

Theorem: $Mat(\mathbb{N})$ is the PROP for bicommutative bimonoids.

So, I handed him an elegant handmade Korean envelope containing 20 euros — to be spent only on coffee.

Yesterday it was my turn again in our “internal seminar” at HIM: I had been asked to talk about the ideas on AQFT from $n$-functorial QFT (arXiv, blog).

The notes for the talk

AQFT from FQFT and Applications
(pdf)

dwell, after some motivation and a quick tour through the main theorem, a bit more on applications than the currently available article on the arXiv does. In particular, there was some further progress on my part with understanding the questions concerning lattice models. This owes a lot to very helpful discussion I had with Pasquale Zito who educated me more about his thesis work and especially about hard-to-find work (still have to try to track down some of it) by A. Ocneanu on asymptotic inclusion of von Neumann algebra subfactors.

See section 3.5 and 3.6

All experts I talked to so far assure me that there should be nice constructions of AQFT nets from continuum limits of lattice models. But no literature at all seems to exist. Over on his blog, Alain Connes once said, in a closely related context, that

It is not really nicely spelled out anywhere

In our little “internal seminar” at HIM the last two times Alejandro Cabrera gave an introduction to BV-formalism and Poisson reduction. He had some useful slides

Alejandro Cabrera
Homological BV-BRST methods: from QFT to Poisson reduction
(pdf)

on the BV background. Then he summarized Poisson reduction as follows below. The combination of the two is the content of his next talk.

For more on symplectic reduction see for instance

J. Butterfield
On symplectic reduction in classical mechanics
pdf.

guest post by Scott Carter, a commentary on his and Sarah Gelsinger’s sphere eversion (50MB). See also John Armstrong’s commentary.

Consider the $2$-category given as follows. The objects correspond to a (possibly empty) set of dots along a vertical axis. Each dot has an associated sign: $-$ will indicate that a $1$-morphism emanating from the dot should be a left pointing strand; $+$ indicates a right pointing strand. The orientation information will be completely suppressed here. The $1$-morphisms are generated (in the sense of composition and tensor products) by $\subset$, $\supset$, and $X$. Of course a predominantly horizontal arc indicates the identity $1$-morphism. There are two flavors of $\subset$ and $\supset$ when orientations are drawn, and 4 flavors of $X$ in that case.

A $1$-morphism from the empty object to the empty object is a collection of generically immersed circles in the plane.

The set of $2$-morphisms are generated by birth, death, saddle, cusp,(the projection of) type II and type III, and the (projection of) Yetter $\psi$ move in which a double point on one arc near an optimum bounces to the other arc near the optimum. Here, of course, height is measured in a left to right direction. Finally, there are tensorators which allow the interchange of critical levels from right to left.

Urs defined the Schreiber 2-group of linear automorphisms of a skeletal Baez-Crans 2-vector space back here. For the space $\delta = 0: F^n \to F^m$, it has $GL(n, F) \times GL(m, F)$ as objects and $F^{n \cdot m}$ worth of arrows from an object to itself. Arrows are linear maps $F^n \to F^m$, and the group of objects acts on them by a kind of conjugation.

Now, the small thought occurred to me that interchanging the $m$ and $n$ makes little difference. So when I suggested that the Poincaré 2-group was a sub-2-group of the 2-group for $n = 1, m = 4$, I might also have said $n = 4, m = 1$.

But all this is not so surprising, as this area is quite span-ish and bimodule-esque.

The highlight of today’s talks at The manifold geometries of QFT for me was a talk by Walter v. Suijlekom in which he made a connection between the Connes-Kreimer Hopf algebra of Feynman diagrams and the BV-formalism.

Building on the Connes-Kreimer fact that Feynman diagrams form a Hopf algebra, there is a certain Hopf quotient which one can form. The question is what this corresponds to physically. The answer Walter Suijlekom gives is: it corresponds to imposing the BV master equation!

I have taken rather detailed notes, with everything that was on the blackboard, here. For everything except the BV-stuff and relations to it look at his latest very readable article

Walter D. van Suijlekom
Renormalization of gauge fields using Hopf algebras
arXiv:0801.3170