The n-Category Café

I was wondering when Lamarck would be proved right and John’s (academic) children would show his acquired characteristic of wanting to broadcast over the Web. Now Jeff Morton has begun to blog at Theoretical Atlas.

So we’ve seen, in this thread that, even when we work with nice basic finite sets, the probability monad doesn’t get along too well with logical structure, there being none of that pleasant adjointness between categories of predicates over sets.

However, the probability monad, $P$, does come along with other structure of its own. In particular $Hom_{Kleisli(P)}(1, Y)$ is a Riemannian manifold, with the Fisher information metric, which is crucially invariant under reparameterization. Recall that this space is composed of maps $1 \to P Y$, i.e., probability distributions over $Y$. The Fisher information metric (see, e.g., section 4 of Guy Lebanon’s thesis) takes on a simple form for finite $Y$, (p. 21).

You know a conference is a good one when there’s no time to report from it. Categories in Geometries and Physics is of this kind.

But before I forget it, I want to record a couple of things worth mentioning and remembering.

I had very long and interesting discussions with Nils Baas. Let me share the following question he poses:

Suppose you have a lot of silver rings. You join them to form a necklace. Then, given more such silver rings, you form more such necklaces. Then, from all these necklaces, you build, by similarly joining them, yet another necklace: a necklace of necklaces. A second order necklace. And so on.

Now suppose on a table sits a huge pile of silver rings. How do you decide if they form an order $n$ necklace?

Shouldn’t there be something like a higher order knot/link invariant which detects knots of knots and links of links?

Further topics today: a comparison with Enrico Vitale’s work on weak categorical cokernels and mapping cones, as well as a speculation on weak Lie $n$-algebras triggered by discussion with Pavol Ševera.

Squark emailed me an interesting question about spin groups, which I take the liberty of reproducing here:

Dear John,

I would appreciate to hear your thoughts on the following matter.

In dimensions $\le 6$ we have the exceptional isomorphisms. In complex form:

$$Spin(2) \cong GL(1)$$ $$Spin(3) \cong SL(2)$$ $$Spin(4) \cong SL(2) \times SL(2)$$ $$Spin(5) \cong Sp(2)$$ $$Spin(6) \cong SL(4)$$

In dimension 8 the triality isomorphisms play the same role, in a way:

$$Spin(8) \cong Spin(8)$$

What about dimension 7? Does anything of the sort happen there?

Best regards,

Squark

I mentioned that I was aked to give a public lecture in the context of Categories in Geometry and Physics next Friday, to high school kids and journalists. I am imagining the lecture might proceed along the following lines.

Since this is still in progress, I would enjoy receiving comments and suggestions. The following already incorporates advice by John Baez and Tom Leinster.

An Invitation to Higher Dimensional Mathematics and Physics

In which sense is summing two numbers a 2-dimensional process? Everybody who knows that $2+3$ is the same as $3+2$ will be lead in this talk to a simple but profound result in a branch of mathematics known as $n$-category theory. This simple insight in higher dimensional mathematics alone will be sufficient to make understandable some fancy ideas in modern theoretical high energy physics.

My research time at present largely consists of the time it takes me to travel to Canterbury and back. Here’s what I came up with on the subject of progic on yesterday’s trip down.

A probabilistic predicate, $P$, I said back here, is a map from $X$ to $[0, 1]$. But another way to look at it is as a conditional probability distribution on the set $2$ given $X$. This is a map in the Kleisli category of the probability monad. In fact, our probabilistic predicate is a map $P: X \to 2$. Now we can look at the composition of a distribution over $X$ and this $P$.

Let’s take $X$ a set of dogs, $Q(x)$ is a probability distibution over $X$, perhaps recording my degree of belief in which dog I saw just now as it dashed past. Then if $P(x)$ is the predicate recording my belief as to whether $x$ is a poodle, then $\sum_x P(x) \times Q(x)$ is my degree of belief that I just glimpsed a poodle.

This is one of Mumford’s random constants, i.e., probabilistic predicates over {*}, or maps in the Kleisli category $1 \to 2$.

• The Catsters, 5 lectures on String diagrams.

Last week I visited Yale, where I talked with Hisham Sati about supergravity, Chern-Simons 3-bundles and other higher structures in String theory, with Mikhail Kapranov about parallel surface transport, with Todd Trimble about Hecke algebras, groupoidification and the sociology of quantum gravity, and with Anton Zeitlin about BV formulation of Yang-Mills theory. With a short stop at home I went from there straight to Croatia, where I am currently attending Categories in Geometry and Physics.

The first day here was quite remarkable, with an astonishing amount of Croatian media attention for us followed by a couple of rather impressive talks. After Pavol Ševera’s talk I had the chance to chat with with him about differentiation and integration of Lie $n$-algebras, much to my delight.

Last night at about 3 am I learned that I am supposed to give a public talk to high school kids and journalists next Friday. And sure enough, later towards the pm part of the day I found myself being interviewed by some TV team on how I plan to convey the ideas of Categories in Geometry and Physics to the broad public.

Trying to think of something which is both interesting and relevant, as well as easy to understand and expressible in terms of lots of pictures, I thought I’d give a talk revolving around the Eckman-Hilton argument, illustrated by two party-balloons joined at their tip and labeled “A” and “B”.

(The very moment that I am posting this, a journalist sends me the following two questions:

Q1. Your field of interest in mathematics is the relation between categories and quantum physics. During the Split categories conference you will be giving a popular lecture on this topic to the high school students, so can you briefly summarize that lecture for our readers?

Q2. You are also one of the admins at $n$-category cafe blog. As it has become an interesting and influential internet based communication and discussion tool for resarchers in category theory, can you tell me a bit more about it - how did the idea start, how it developed, and what is its influence on the category maths community professional work?

)

Hence there’d be many fun things to write about here. And I should. But right now I will instead talk more about two of the issues I had mentioned recently in Obstructions for $n$-bundle lifts and related.

A lot of us old-timers feel sad about the decline of serious popular science magazines like Scientific American and New Scientist. Perhaps with the rise of science blogging we don’t really need such magazines anymore. But I’m not so sure…

Lately I’ve been browsing through this magazine at my local bookstore:

• American Scientist.

It makes me feel the way I used to feel when reading Scientific American. They’ve got articles on all sorts of topics, and tons of book reviews in each issue. They’re well-written, clear, and — best of all, when compared to some of the competition — moderately demanding, not dumbed-down or drawn like a moth to the candle flame of lowest-common-denominator sensationalism.

Why are they so good? Maybe it’s because they’re run by a scientific society, Sigma Xi. The way multimedia conglomerates demand their newspapers, magazines and journals keep boosting their profits may eventually kill certain writerly crafts… but magazines put out by professional organizations may do better. Let’s hope so! A lot of the best things in life are created just for the love of it.

Who knew category theory could be so popular? The Catster’s videos on Monads have taken Youtube by storm! As of this moment, the first episode has garnered 3758 viewings, prompting one disgruntled teenager to suggest that the Catsters are cheating in the battle for high ratings:

> Its really funny that your most of
> your subscribers have no videos, are
> subscribed to only you, and have only your
> videos as favorites. Do I sense a youtube
> cheater?

Of course, we Catsters fans know it ain’t so — they’re just satisfying a hitherto unnoticed craving for math lectures on YouTube!

And the Catsters are on a roll. Now they’ve tackled adjoint functors, also known as ‘adjunctions’:

• The Catsters, 7 lectures on Adjunctions.

Their edgy, low-budget production makes for gripping cinema. This is the real stuff.

There were many amazing mathematicians in the 20th century — people with world-transforming powers, like Gödel, Mac Lane and Grothendieck. But surely, no matter how short your list of greats, John von Neumann would have to be on it. From topics so abstract as the foundations of set theory and quantum mechanics, to topics so practical as game theory, the Manhattan project and the first computers, he seemed to be everywhere… right at the cutting edge.

Soon there will be a symposium honoring the 75th anniversary of von Neumann’s book The Mathematical Foundations of Quantum Mechanics:

• Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, Princeton University, October 3-4, 2007. Organized by Hans Halvorson.

Hans Halvorson is a philosopher at Princeton. I really like his idea that new mathematics may be needed to make quantum mechanics more intelligible, and that philosophers should get involved. I think it’s true.

My colleague here in Canterbury Jon Williamson is part of an international research group, progicnet, whose aim is to find a good integration of probability theory and first-order logic. For one reason or another, some technical projects get counted as philosophy, while others don’t. Where I’m sure I’d have a hard time getting funding for 2-geometry, progicnet’s goals are philosophically pukka.

Now, as Jon says,

We see then that there are a plethora of combinations of probability and logic, and that these approaches are being investigated in some detail.

But is there a ‘natural’ way of doing it? In the quest to have philosophy take notice of category theory, all we need do is use it to blend probability theory and logic powerfully. Does anyone know of any progress along these lines? Perhaps involving the probability monad?

You can see all sorts of silly things on YouTube these days — from guys flying in wingsuits to math professors making mistakes converting miles to kilometers. But what about serious stuff? Can you watch math lectures on YouTube?

Can you learn category theory?

Can you, say, learn about monads?

Yes! Now you can!

Just try these videos from the new sensation out of Sheffield — the group that has all the British press raving — the Catsters!

• Monads 1: Definition of monad. Example: the monad for monoids.
• Monads 2: Examples, continued. The monad for categories.
• Monads 3: Algebras for monads.
• Monads 3a: Answers to some questions.
• Monads 4: The category of algebras of a monad.

I was asked to forward the announcement of a conference called

Groupoidfest 07

to the $n$-Café. No webpage or similar official site for this seems to exist at this moment. What does exist are two email announcements circulated by Paul S. Muhly. So for the time being I simply reproduce these:

From: Paul S. Muhly
Subject: Groupoidfest 07, First Announcement
Date:September 12, 2007

Dear Friends,

I am writing to announce that Groupoidfest 07 will be held this Fall in Iowa City, at the University of Iowa. The dates are November 3 and 4. Most participants will arrive the 2nd and leave on the 4th in the afternoon. Talks will be scheduled for Saturday and Sunday morning.

As has been the tradition since the beginnings of our meetings, we will split the conference budget of $0 evenly among all participants. Nevertheless, I hope you can come.

I will send out a more detailed announcement later, but for those of you who may want to plan to fly to the meeting, thought you would like to know that the closest airport with major air carrier service is the Cedar Rapids Airport. It is also known as the Eastern Iowa Airport. It’s “identification” code is CID.

I will look forward to seeing you all in November. In the meantime, as always,

Best wishes,

Paul

I am in the process of preparing some slides which are supposed to eventually provide an overview of the current state of the art of the second edge, with an emphasis on understanding String and Chern-Simons Lie $n$-algebras, the Lie $n$-algebra cohomology and the corresponding theory of bundles with Lie $n$-algebra connection in terms of the differential Lie analog of parallel transport $n$-functors, like String 2-bundles and Chern-Simons 3-bundles.

Here a Chern-Simons 3-bundle associated to a $G$-bundle should be the obstruction to lifting the $G$-bundle to a $\mathrm{String}(G)$ 2-bundle.

I know how to say this in a pedestrian way. But now I want to say it in the most elegant possible way. I want to understand how it really™ works.

I am too tired to do the topic justice. But all the more grateful I’d be for comments on the following considerations.

Another Fields Medallist has started a blog – Gowers’s Weblog.

In his first post, you can find out all about the Princeton Companion to Mathematics, which he is editing. You can sign in to read a sample of the articles. Next, in the second post, he starts a discussion on how to set up a wiki for exposition.

Branislav Jurčo asks me to draw attention here to the following notes which he provides on his website:

a) Associated 2-vector bundle construction using Baez-Crans 2-vector spaces

b) 2-crossed module bundle 2-gerbes

c) Differential geometry of 2-crossed module bundle 2-gerbes

These are generalizations building on his work on nonabelian bundle gerbes as presented in

P. Aschieri, L. Cantini B. Jurčo
Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
hep-th/0312154

and

Branislav Jurčo
Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry
math/0510078.

The first one discusses the bundle gerbe analog of how a vector bundle may be associated to a principal bundle. The second and third generalize bundle (1-)gerbes to bundle 2-gerbes. This are then gadgets that are to 3-bundles with some structure 3-group like transitions functions are to ordinary principal bundles.

(A crossed module is the realization of a strict 2-group in the world of complexes of groups. A “crossed module bundle gerbe” is to an abelian bundle gerbe like a 2-bundle for a general strict 2-group is to a 2-bundle for the 2-group “shifted $U(1)$”. A 2-crossed module is a realization of a suitably well behaved 3-group in the world of complexes of groups.)

See also the entry

Jurčo on Gerbes and Stringy Applications

This here is supposed to move the discussion triggered by John’s “nice problem” in Higher Gauge Theory and Elliptic Cohomology into a new thread. If we already had that Wiki I’d develop everything there and just drop you all a link here. Since we are not at this point yet, here is another entry.

Question

i) What is a characteristic class of an $n$-bundle with structure Lie $n$-algebra $g_{(n)}$?

ii) How are these characteristic classes described in terms of deRham cohomology, starting from an arbitrary connection on the $n$-bundle?

ii) In particular, how do the characteristic classes of 2-bundles with $g_{(n)} := g_{\langle \cdot, [\cdot, \cdot\rangle]}$ the String group Lie 2-algebra relate to those of ordinary $g$-bundles?

Here I’ll describe what looks like a nice answer to this nice problem. It is obtained by combining the $n$-groupoid realization of universal $n$-bundles in terms of tangent categories and inner automorphisms with the Lie $n$-algebra technoloy described in Chern Lie $(2n+1)$ and String and Chern-Simons Lie 3-algebras.

More details on the Lie $n$-algebraic aspects are in

Lie $n$-algebra cohomology

The tangent-categorical background useful to put this into perspective is discussed in

Tangent Categories (html).

The higher morphisms of morphisms of Lie $n$-algebras which play a role are discussed in

Higher morphisms of Lie $n$-algebras (html)

I was asked recently by someone for my opinion on the possibility that category theory might prove useful in machine learning. First of all, I wouldn’t want to give the impression that there are signs of any imminent breakthrough. But let me open a thread by suggesting one or two places where categories may come into play.

For other areas of computer science the task would be easier. Category theory features prominently in theoretical computer science as described in books such as Barr and Wells’ Category Theory for Computing Science. Then there’s Johnson and Rosebrugh’s work on databases.

As for machine learning itself, perhaps one of the most promising channels is through probability theory. One advantage of working with the Bayesian approach to machine learning is that it brings with it what I take to be more beautiful mathematics. Take a look at this paper on statistical learning theory. It belongs to the side of the cultural divide where category theory doesn’t flourish. If, on the other hand, we encounter mathematics of the culture I prefer, it is not unlikely that category theory will find its uses.

With everybody growing more comfortable with the idea of tangent categories, in good part due to the various cross-relations which we found in More on Tangent Categories, the obvious open question became more urgent:

What about cotangent categories?

(asked by David Corfield here and by Jim Stasheff here).

Defining a cotangent means first of all agreeing on a codomain in which it takes values. That’s the reason why one gets a headache when trying to define cotangent categories in the same manner as tangent categories.

I think a good idea is to consider the arrow-theoretic analogue of vector valued differential forms. This should include scalar-valued differential forms then as an obvious special case. So we should be after morphisms of tangent categories, or better: morphisms of flows on categories.

The n-Category Café has been around for just over a year now!

Congratulations! Thanks to everyone for making it a success!

Perhaps the time has come for a wiki.

So, let’s talk about it.