The n-Category Café

The Riemann sphere $\mathbb{C}\mathrm{P}^1$ leads a double life in physics. On the one hand it’s the set of states of a complex qubit. In this guise, physicists call it the ‘Bloch sphere’. On the other hand it’s the set of directions in which you can look when you’re an inhabitant of 4d Minkowski spacetime — as we all are. In this guise, it’s called the ‘celestial sphere’ . But these two roles are deeply connected! In 4d spacetime a Weyl spinor is described by a complex qubit: that is, a unit vector in $\mathbb{C}^2$. A state — that is, a unit vector modulo phase — simply says which way the spinor is spinning. Its spin can point in any direction, and these directions are points in the celestial sphere.

Last time I started explaining how to generalize some of these ideas from $\mathbb{C}\mathrm{P}^1$ to what I’m really interested in, $\mathbb{O}\mathrm{P}^2$. Again this has two roles. On the one hand it’s the set of states of an octonionic qutrit. On the other hand it’s the heavenly sphere in a 27-dimensional spacetime modeled on the exceptional Jordan algebra. This is a funny spacetime where the lightcone is described not by the usual sort of quadratic equation like

$$t^2 - x^2 - y^2 - z^2 = 0$$

but instead by a cubic equation.

It’ll be easy for me to get lost in the pleasures of this geometry. But I have a concrete goal in mind. The symmetries of $\mathbb{O}\mathrm{P}^2$ form the group $\mathrm{E}_6$, and I’m trying to use this to understand a fact about $\mathrm{E}_6$. Namely, this Lie group has four Lie subgroups:

• the double cover of the Lorentz group in 10 dimensions
• translations in right-handed spinor directions
• translations in left-handed spinor directions
• ‘dilations’ (rescalings)

and these give Lie subalgebras whose direct sum, as vector spaces, is all of $\mathfrak{e}_6$:

$$\mathfrak{e}_6 \cong \mathfrak{so}(9,1) \oplus \mathbb{O}^2 \oplus (\mathbb{O}^2)^\ast \oplus \mathbb{R}$$

I proved these facts back in Part 7, but now I’m trying to understand them better. The duality between points and lines in projective plane geometry turns out to be the key!

Here are slides for a talk I gave last week to my department on some of the advantages for philosophy of adopting dependent type theory. It revolves around Mike’s very interesting theory of n-theories.

MathOverflow saw a flurry of activity the same day on the advantages for proof assistants.

Category theory has an excellent track record of formalizing intuitive statements of the form “this is the canonical such and such”. It has been especially effective in topology and algebra.

But what does it have to say about canonical measures? On many spaces, there is a choice of probability measure that seems canonical, or at least obvious: the first one that most people think of. For instance:

• On a finite space, the obvious probability measure is the uniform one.

• On a compact metric space whose isometry group acts transitively, the obvious probability measure is Haar measure.

• On a subset of $\mathbb{R}^n$, the obvious probability measure is normalized Lebesgue measure (at least, assuming the subset has finite nonzero volume).

Emily Roff and I found a general recipe for assigning a canonical probability measure to a space, capturing all three examples above: arXiv:1908.11184. We call it the uniform measure. It’s categorically inspired rather than genuinely categorical, but I think it’s a nice story, and I’ll tell it now.

I now have a semiannual column in the Notices of the American Mathematical Society! I’m excited: it gives me a chance to write short explanations of cool math topics and get them read by up to 30,000 mathematicians. It’s sort of like This Week’s Finds on steroids.

Here’s the first one:

• The tenfold way, Notices Amer. Math. Soc. 67 (November 2020), 1599–1601.

Guest post by Bruce Bartlett

The sunny campus of Stellenbosch University in South Africa is hiring!

We’re looking to make a permanent appointment of a mathematician at Lecturer or Senior Lecturer level.

• Job description and application webpage. Deadline 30 November 2020.

We’re a department with faculty having a variety of research interests, with functional analysis, model theory, algebra, mathematical physics, categorical algebra and number theory all being represented. I hope we will receive some applications from $n$-Category Café patrons.

Mike Shulman and I have just arXived the final, journal version of our paper Magnitude homology of enriched categories and metric spaces, to appear in Algebraic & Geometric Topology.

The first arXiv version appeared in 2017. There are now quite a few papers on magnitude homology of metric spaces that build on it (and therefore, ultimately, on Richard Hepworth and Simon Willerton’s very provocative work on magnitude homology of graphs).

However, that first version was more complicated than it really needed to be. We were lucky to get an extremely helpful referee, who prodded us into simplifying it. I think it’s now much more accessible and widely readable.

There’s been loads already on this blog about magnitude homology, so I won’t write more now. Enjoy!

I’ve just arXived a new paper: The categorical origins of Lebesgue integration (arXiv:2011.00412). Longtime Café readers may remember that I blogged about this stuff back in 2014, but I’ve only just written it up.

What’s it all about? There are two main theorems, which loosely are as follows:

Theorem A   The Banach space $L^1[0, 1]$ has a simple universal property. This leads to a unique characterization of integration.

Theorem B   The functor $L^1:$ (finite measure spaces) $\to$ (Banach spaces) has a simple universal property. This leads to a unique characterization of integration on finite measure spaces.

But there’s more! The mist has cleared on some important things since that last post back in 2014. I’ll give you the highlights.

Last time I described the symmetry group $\mathrm{E}_6$ of the exceptional Jordan algebra in terms of 10d Minkowski spacetime. We saw that in some sense it consists of four parts:

• the double cover of the Lorentz group in 10 dimensions,
• translations in left-handed spinor directions,
• translations in right-handed spinor directions, and
• ‘dilations’ (rescalings).

But our treatment was computational: the geometrical meaning of this decomposition was left obscure. As Atiyah said:

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.

Now let’s renounce the devil’s bargain and try to understand the geometry! Jordan algebras are deeply connected to projective geometry, and the exceptional Jordan algebra is all about the octonionic projective plane.

The British philosopher R. G. Collingwood wrote

…whenever anybody states a thought in words, there are a great many more thoughts in his mind than there are expressed in his statement. Among these there are some which stand in a peculiar relation to the thought he has stated; they are not merely its context, they are its presuppositions. (An Essay on Metaphysics, 1940, pp. 21-22)

In my book I proposed that dependent type theory is well-suited to represent such presuppositions, largely via the device of its notion of ‘context’ (pp. 57-58, 92). Ideas for this thesis came in part from discussions at the Café, including that generated by this post, concerning Collingwood’s treatment of the presuppositions of a complex question. Something perhaps worth developing then is a dependent type-theoretic treatment of questions.

Collingwood writes in his Autobiography:

…a logic in which the answers are attended and the questions neglected is a false logic. (Autobiography, 1939, p. 31)

What is ordinarily meant when a proposition is called ‘true’, I thought, was this: (a) the proposition belongs to a question-and-answer complex which as a whole is ‘true’ in the proper sense of the word; (b) within this complex it is an answer to a certain question; (c) the question is what we ordinarily call a sensible or intelligent question, not a silly one, or in my terminology it ‘arises’; (d) the proposition is the ‘right’ answer to that question. (R.G. Collingwood, Autobiography, p. 38)

What do we have then on the logical treatment of questions? Well the Stanford Encylopedia of Philosophy is a natural starting point. And consulting SEP: Questions, we find the briefest mention of a couple of type-theoretic approaches, but nothing on dependent types:

• Ciardelli, I., F. Roelofsen and N. Theiler, 2017. Composing alternatives, Linguistics and Philosophy, 40 (1): 1-36.
• Cooper R., Ginzburg J. 2012. Negative Inquisitiveness and Alternatives-Based Negation. In: Aloni M., Kimmelman V., Roelofsen F., Sassoon G.W., Schulz K., Westera M. (eds) Logic, Language and Meaning. Lecture Notes in Computer Science, vol 7218. Springer, Berlin, Heidelberg. doi

If presuppositions are usefully treated by dependent type theory, then we would expect this logic to account well for questions. This is a task Aarne Ranta begins in his Type-Theoretic Grammar (OUP, 1994, pp. 137-142), which I’ll look to expand somewhat here. If anyone knows of a place where this section of his book is developed, then please let me know.

Last time I explained the connection between the exceptional Jordan algebra and 10-dimensional Minkowski spacetime. Today I want to report on some work that Greg Egan, John Huerta and I did in November 2015. We figured out how to describe the big 78-dimensional symmetry group of the exceptional Jordan algebra in terms of 10d spacetime geometry!

This group is called $\mathrm{E}_6$, and we saw how it’s built up from:

• the double cover of the Lorentz group of 10d spacetime, $\mathrm{Spin}(9,1)$
• the right-handed spinors in 10d spacetime, $S_+$
• the left-handed spinors in 10d spacetime, $S_-$
• scalars, $\mathbb{R}$.

This lets us chop up the Lie algebra of $\mathrm{E}_6$ as follows:

$$\mathfrak{e}_6 \cong \mathfrak{so}(9,1) \oplus S_+ \oplus S_- \oplus \mathbb{R}$$

or counting dimensions,

$$78 = 45 + 16 + 16 + 1$$

One of the most interesting functions on the exceptional Jordan algebra is the determinant. The linear transformations that preserve this form the exceptional Lie group $\mathrm{E}_6$. And today I’ll show you how to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and spinors in 10d Minkowski spacetime.