The n-Category Café

In week272 of This Week’s Finds, see what the Cassini probe saw as it shot through the icy geysers of Enceladus:

See what happens when the Io flux tube hits Jupiter. Learn two new properties of the number 6. And discover the wonders of standard Borel spaces, commutative von Neumann algebras, and Polish groups!

There were a lot of interesting talks at last weekend’s Groupoidfest here at Riverside. However, due to a lack of organization and shortage of time on my part, I can give only a brief report.

After a bit of discussion about going beyond the blog now it is starting to exist: a Wiki accompanying the $n$-Café:

$n$Lab

Having come across Peter Freyd’s coalgebraic characterisation of the reals, I’ve been spending some time of late trying to understand coalgebras. Where an algebra for an endofunctor $F: C \to C$ is an object $A$ of $C$ together with an arrow $\alpha : F(A) \to A$, a coalgebra for $F$ requires an object $A$ with an arrow $\beta : A \to F(A)$.

I’ve traced out what seemed to me an interesting path. First I stumbled upon Bart Jacob’s book Introduction to Coalgebra: Towards Mathematics of States and Observations. This I’d thoroughly recommend. Let me give you some nuggets from it:

The duality with algebras forms a source of inspiration and of opposition: there is a “hate-love” relationship between algebra and coalgebra. (p. v)

As already mentioned, ultimately, stripped to its bare minimum, a programming language involves both a coalgebra and an algebra. A program is an element of the algebra that arises (as so-called initial algebra) from the programming language that is being used. Each language construct corresponds to certain dynamics, captured via a coalgebra. The program’s behaviour is thus described by a coalgebra acting on the state space of the computer. (p. v)

For some purpose I was asked to formulate what motivates for me interdisciplinary research at the border of mathematics, physics and philosophy. Here is something that I have come up with today:

Motivation für interdisziplinäre Arbei im Grenzbereich von
Mathematik, Physik und Philosophie
(pdf, 5 pages)

Sorry that it’s in German. After I have slept over this and become more convinced by the form of the text I might produce a translation.

After the workshop in Göttingen yesterday I visited Zoran Škoda at the MPI in Bonn, where we worked on differential nonabelian cohomology. In the evening there was a talk by Christian Bär over at the university:

Christian Bär
Fiber integration and Cheeger-Simons characters
(my pdf notes taken during the talk)

With the publication of the first year of Terry Tao’s blog posts, appearing with the AMS, and the prospect of a second book on Poincaré’s legacies, I’m left wondering whether we are doing enough to make accessible the wisdom contained in this blog.

It’s quite common for me to be googling about for some concept and find that a forgotten conversation we had here many months ago is just what I’m looking for. The trouble is that our conversations are dispersed, and many individuals may be involved, so the effort to extract them for a Wiki, were we to have one, might be off-putting.

A low cost solution would be to form an index pointing to particularly good pieces of exposition. For example, there’s some great material on classifying toposes beginning here, with Tom, Todd, Robin and John contributing.

This workshop looks interesting:

• Categorification and Geometrisation from Representation Theory, April 13–18, 2009, Department of Mathematics, University of Glasgow, organized by Ken Brown, Iain Gordon, Catharina Stroppel, Nicolai Reshetikhin and Raphael Rouquier.

In his paper The Ontology of Complex Systems William Wimsatt explains how he chooses to approach the issue of scientific realism with the concept of robustness.

Things are robust if they are accessible (detectable, measureable, derivable, defineable, produceable, or the like) in a variety of independent ways.

The robustness of Jupiter’s moons was precisely up for debate when Galileo let leading astronomers of his day look towards the planet through his telescope. Even if telescopes had proved their worth on Earth, allowing merchants to tell which ship was heading towards port beyond the range of the naked eye, this did not completely guarantee its accuracy as an astronomical instrument. How do we know that light travels and interacts with matter in the same way in the superlunary realm as down here on Earth? How could we trust this device when what appeared to the eye to be a single source of light was split in two in the telescope’s image?

Now we have robustness for the moons established, and can send probes close to their surfaces to report back on phenomena such as the Masubi Plume on Io. And we have an array of means to tell us that many stars are binary, so know that Galileo’s telescope was reliable.

Does anything like robustness happen in mathematics?

There is a mini-workshop extending the Born-Hilbert seminar in Göttingen, organized by Chenchang Zhu.

CRCG Miniworkshop
Higher and graded geometric structures
Monday, Nov. 24
room: “Sitzungszimmer” of the Mathematical Institute Göttingen.

The schedule is:

Ping Xu (Penn State)
Groupoid extensions and Non-ablian gerbes
9:30-10:30

Matthieu Stiénon (Penn State)
2-Group bundles and characteristic classes
11:00-12:00

Christian Blohmann (Regensburg)
Groupoid symmetry of general relativity
13:15-14:15

Rajan Metha (Washington University)
On models for the adjoint representation of a Lie algebroid
14:15-15:15

Urs Schreiber (Hamburg)
Differential non-abelian cohomology
16:00-17:00

Sharpen your brain after the New Year break at the fourth workshop on Categories, Logic and Physics, held at Imperial College on Wednesday, 7th January 2009, 11:00 - 19:00, Lecture Theatre 2, Blackett Laboratory.

The focus will “mainly be on n-categories and related fields”.

Before getting back to Bruce’s message about BV-quantization of Chern-Simons theory below I need to post some musings and references on the relevance – or not – of codescent for those co-presheaves known as local nets of of quantum observables. I was led to reconsider this after being pointed to the relevance of the split property for wedges of such nets and from some discussion with Jacques Distler.

guest post by Bruce Bartlett

A nice paper appeared on the arXiv today:

Alberto Cattaneo and Pavel Mnëv, Remarks on Chern-Simons invariants.

Does the algebra $\Omega(M)$ of differentiable forms on a smooth manifold $M$ form a Frobenius algebra? Help me understand and straighten out the definitions!

Igor Baković – whose work I had mentioned here – has finished his thesis:

Igor Bakovič
Bigroupoid 2-torsors
(pdf)

He defines and classifies $G$-principal bundles for $G$ a bigroupoid (weak 2-groupoid), as bigroupoids $P \to X$ over base space $X$ – internal to some exact category (a topos, in particular, say of generalized smooth spaces).

Igor relates bigroupoid 2-torsors to the simplicial definition of $n$-torsors for any $n$ by Duskin and Glenn. Glenn essentially defined actions of $\infty$-groupoids (modeled as Kan complexes) in terms of the corresponding action $\infty$-groupoids (recalled as definitions 15.4, 15.5 in Igor’s thesis). Igor realizes this action-$n$-groupoid interpretation (section 13), for $n=2$, by first introducing the concept of an action bigroupoid (theorem 13.2) encoding the weak quotient of a bigroupoid action on some category.

Then he shows, theorem 15.3, that the nerve of the action bigroupoid encoding the action of the bigroupoid $G$ on a $G$-principal bundle is indeed a 2-bundle (2-torsor) in the sense of Duskin-Glenn.

Along the way he generalizes the tangent 2-categories from [Roberts-S.] (p. 114) and identifies our inner automorphism 3-group $INN(G) = \mathbf{E}G$ as, indeed, the action bigroupoid of the 2-group $G$ acting on itself.

I am about to give a talk at the Hamburg Seminar for Quantum Field Theory and Mathematical Physics #

Local nets from parallel transport 2-functors
(pdf notes 6 pages + diagram proof)

Abstract. For every 2-functor on the 2-category of paths in a Lorentzian space we can define its endomorphism co-presheaf. We show that this copresheaf is automatically a local net of monoids satisfying the time slice axiom. For suitable codomains of the 2-functor it is a local net of $C^*$-algebras. It is covariant if the 2-functor is equivariant. One can interpret this as the passage from the Schrödinger to the Heisenberg picture in QM raised to 2-dimensional field theory.

This is based on AQFT from $n$-functorial QFT. The talk is aimed at an audience well familiar with QFT and in particular with AQFT.

This is a followup to my post on Pontryagin duality. In the comments to that, Yves de Cornulier said:

It’s well known (and not trivial) that any locally compact abelian group $A$ has a compact subgroup $K$ such that $A/K$ is a Lie group.

This sent me back to the books. I learned some stuff I should have known a long time ago. Let me tell you about it.

I’m slaving away on a paper about infinite-dimensional representations of 2-groups, which needs to be done in time for the thesis defense of Aristide Baratin — a coauthor who is due to finish grad school very soon.

This paper has forced me to brush up on my analysis — a nostalgic if slightly painful experience. I’ll probably explain some of what I’ve learned in This Week’s Finds. But to finish it, we still need the answers to some questions related to Pontryagin duality.

Help!

Guest post by Tom Leinster

What happened last time?

Ecologically, I explained some ways to measure the diversity of an ecosystem — but only taking into account abundance (how many of each species there are), not similarity (how much the various species have in common). Mathematically, I explained how to assign a number to a finite probability space, measuring how uniform the distribution is.

What’s happening now?

Ecologically, we’ll see how to measure diversity in a way that takes into account both abundance and similarity. So, for instance, an ecosystem consisting of three species of eel in equal numbers will count as less diverse than one consisting of equal numbers of eels, eagles and earwigs. Mathematically, we’ll see how to assign a number to any finite ‘probability metric space’ (a set equipped with both a probability distribution and a metric).

By thinking about diversity, we’ll be led to the concept of the cardinality of a metric space. (Some ecologists got there years ago.) And by building on progress made at this blog, we’ll solve some open problems posed in the ecology literature.

A conversation over conference # dinner I just had revolved around the observation that even though since the 1990s, due to work by Kontsevich, Zwiebach and others (see for instance Kajiura and Stasheff), it has been clear that string theory is fundamentally governed by $A_\infty$ and $L_\infty$-algebraic structures, this insight is not reflected in some of the work done in the community where one would expect it to be relevant.

From that point of view a recent phenomenon may be noteworthy, which those involved modestly address as the membrane mini revolution (a recent impression by Thomas Klose is here). It started with an article by Bagger and Lambert (see Jacques Distler’s useful review) in which the authors managed to construct an $N=2$ supersymmetic version of the worldvolume theory of the M-theory membrane. In their description they use a trinary skew (or partly skew) linear bracket. The authors addressed this bracket as a 3-algebra [sic].

A Lie $n$-algebra is an $L_\infty$-algebra concentrated in the lowest $n$ degrees. An $n$-Lie algebra is a vector space with an $n$-ary skew bracket on it satisfying a Jacobi-like condition. Up to a potential issue of grading (see below), $n$-Lie algebras are special cases of $L_\infty$-algebras, as proven in

A. S. Dzhumadil’daev
Wronskians as $n$-Lie multiplications
(arXiv).

I am grateful to Calin Lazaroiu for this and the following reference.

The published evidence for the relevance of the homtopy-theoretic interpretation of the trinary Lambert-Bagger bracket remains somwehwat inconclusive (see Jacques Distler’s useful second review). In

de Medeiros, Figueroa-O’Farill, Méndez-Escobar, Ritter
On the Lie-algebraic origin of metric 3-algebras
(arXiv)

it says on p. 3 about this:

All this prompts one to question whether the 3-algebras appearing in the constructions [1-3,10,11] play a fundamental role in M-theory or, at least insofar as the effective field theory is concerned, are largely superfluous.

The authors then go on to discuss all these trinary brackets entirely in terms of pairs consisting of an ordinary Lie algebra and a representation.

Apart from usefulness issues of the 3-algebraic perspective, it is noteworthy that the Bagger-Lambert trinary bracket is in general not consistent with $L_\infty$-algebra grading conventions (as for instance in Lada-Stasheff p. 7), for no grading one puts on the underlying vector space $V$ – at least not unless one assumes that there are secretly two differently graded copies of $V$ in the game. One can consider the definition of $L_\infty$-algebras without the grading, in particular if there is just a single arity of brackets involved, as in the above articles. In this ungraded sense then the Bagger-Lambert trinary bracket, at least for the case that it is totally skew, really is an example of a Lie 3-algebra. On the other hand, it makes me wonder that this trinary bracket is in general taken to be skew only in the first two arguments, as described down on p. 2 of de Medeiros et al. One could potentially accomodate for this by an $L_\infty$-algebra proper (with grading, that is) by having two copies of the underlying vector space, one of them shifted in degree down by one (which would mean we’d end up with an $L_\infty$-algebroid).

I probably ought to be much better prepared before I start this venture, but let’s give it a go anyway. A 2-topos $(\mathcal{K}, (-)^{\circ}, \tau)$ is a finitely complete cartesian closed 2-category $\mathcal{K}$ equipped with a duality involution $(-)^{\circ}$ and a classifying discrete opfibration $\tau: \Omega_{\bullet} \to \Omega$. We can worry about what these terms mean as we go along.

So we ‘just’ have to imitate Lambek and Scott’s Introduction to higher order categorical logic. On p. 143 they tell us that the internal language of a topos $\mathcal{T}$ has as types the objects of $\mathcal{T}$, including the special types $1$, the terminal object, $\Omega$, the subobject classifier, and $N$, the natural numbers object, if $\mathcal{T}$ has one.

While we eagerly await Tom’s second post on Entropy, Diversity and Cardinality, here’s a small puzzle for you to ponder. If all of the members of that family of entropies he told us about are so interesting, why is it that so many of our best loved distributions are maximum entropy distributions (under various constraints) for Shannon entropy?

For example, the normal distribution, $N(\mu, \sigma^2)$, is the maximum Shannon entropy distribution for distributions over the reals with mean $\mu$ and variance $\sigma^2$. And there are others, including exponential and uniform (here) and Poisson and Binomial (here). So why no famous distributions maximising Tsallis or Renyi entropy? (People do look at maximising these, e.g., here.)

I am about to catch a night train to Lausanne, to attend this workshop:

Higher Structures in Mathematics and Physics 2008
at Bernoulli Center (EPFL) Lausanne, Switzerland
November 3-7, 2008
(program)

Among the other participants, I am looking forward to meeting friend and $n$-Café regular Bruce Bartlett, who recently handed in his PhD thesis and now has time to travel. Hopefully the two of us will manage to report on some of the highlights (and secret highlights) to the Café.