The n-Category Café

I wrote a little book about entropy; here’s the current draft:

• What is Entropy?

If you see typos and other mistakes, or have trouble understanding things, please let me know!

An alternative title would be 92 Tweets on Entropy, but people convinced me that title wouldn’t age well: in decade or two few people may remember what ‘tweets’ were.

Here is the foreword, which explains the basic idea.

guest post by Wilf Offord

One of the earliest and most well-studied definitions in “higher” category theory is that of a monoidal category. These have found ubiquitous applications in pure mathematics, physics, and computer science; from type theory to topological quantum field theory. The machine making them tick is MacLane’s coherence theorem: if anything deserves to be called “the fundamental theorem” of monoidal categories, it is this. As such, numerous other proofs have sprung up in recent years, complementing MacLane’s original one. One strategy with a particularly operational flavour uses rewriting systems: the morphisms of a free monoidal category are identified with normal forms for some rewriting system, which can take the form of a logical system as in (UVZ20,Oli23), or a diagrammatic calculus as in (WGZ22). In this post, we turn to skew-monoidal categories, which no longer satisfy a coherence theorem, but nonetheless can be better understood using rewriting methods.

Guest post by Matt Kukla and Tanjona Ralaivaosaona

Double limits capture the notion of limits in double categories. In ordinary category theory, a limit is the best way to construct new objects from a given collection of objects related in a certain way. Double limits, extend this idea to the richer structure of double categories. For each of the limits we can think of in an ordinary category, we can ask ourselves: how do these limits look in double categories?

guest post by Leonardo Luis Torres Villegas and Guillaume Sabbagh

Introduction

String diagrams are ubiquitous in applied category theory. They originate as a graphical notation for representing terms in monoidal categories and since their origins, they have been used not just as a tool for researchers to make reasoning easier but also to formalize and give algebraic semantics to previous graphical formalisms.

On the other hand, it is well known the relationship between simply typed lambda calculus and Cartesian Closed Categories(CCC) throughout Curry-Howard-Lambeck isomorphism. By adding the necessary notation for the extra structure of CCC, we could also represent terms of Cartesian Closed Categories using string diagrams. By mixing these two ideas, it is not crazy to think that if we represent terms of CCC with string diagrams, we should be able to represent computation using string diagrams. This is the goal of this blog, we will use string diagrams to represent simply-typed lambda calculus terms, and computation will be modeled by the idea of a sequence of rewriting steps of string diagrams (i.e. an operational semantics!).

guest post by Laura González-Bravo and Luis López

In this blog post for the Applied Category Theory Adjoint School 2024, we discuss some of the limitations that the measure-theoretic probability framework has in handling uncertainty and present some other formal approaches to modelling it. With this blog post, we would like to initiate ourselves into the study of imprecise probabilities from a mathematical perspective.

My brilliant and wonderful PhD student Adrián Doña Mateo and I have just arXived a new paper:

Adrián Doña Mateo and Tom Leinster. Magnitude homology equivalence of Euclidean sets. arXiv:2406.11722, 2024.

I’ve given talks on this work before, but I’m delighted it’s now in print.

Our paper tackles the question:

When do two metric spaces have the same magnitude homology?

We give an explicit, concrete, geometric answer for closed subsets of $\mathbb{R}^N$:

Exactly when their cores are isometric.

What’s a “core”? Let me explain…

A milestone! By my count, there are now 100 papers on magnitude, including several theses, by a total of 73 authors. You can find them all at the magnitude bibliography.

Here I’ll quickly do two things: tell you about some of the hotspots of current activity, then — more importantly — describe several regions of magnitude-world that haven’t received the attention they could have, and where there might even be some low-hanging fruit.

On Mathstodon, Paul Schwahn raised a fascinating question connected to the octonions. Can we explicitly describe an irreducible representation of $SO(3)$ on 7d space that preserves the 7d cross product?

I explained this question here:

• 3d Rotations and the 7d cross product (Part 1).

This led to an intense conversation involving Layra Idarani, Greg Egan, and Paul Schwahn himself. The result was a shocking new formula for the 7d cross product in terms of the 3d cross product.

Let me summarize.

The lanthanides are the 14 elements defined by the fact that their electrons fill up, one by one, the 14 orbitals in the so-called f subshell. Here they are:

lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium.

They are also called ‘rare earths’, but that term is often also applied to 3 more elements. Why? That’s a fascinating puzzle in its own right. But what matters to me now is something else: an apparent connection between the lanthanides and the exceptional Lie group G₂!

Alas, this connection remains profoundly mysterious to me, so I’m pleading for your help.

In the real world, the rope in a knot has some nonzero thickness. In math, knots are made of infinitely thin stuff. This allows mathematical knots to be tied in infinitely complicated ways — ways that are impossible for knots with nonzero thickness! These are called ‘wild’ knots.

Check out the wild knot in this video by Henry Segerman. There’s just one point where it needs to have zero thickness. So we say it’s wild at just one point. But some knots are wild at many points.

There are even knots that are wild at every point! To build these you need to recursively put in wildness at more and more places, forever. I would like to see a good picture of such an everywhere wild knot. I haven’t seen one.

Wild knots are extremely hard to classify. This is not just a feeling — it’s a theorem. Vadim Kulikov showed that wild knots are harder to classify than any sort of countable structure that you can describe using first-order classical logic with just countably many symbols!

Very roughly speaking, this means wild knots are so complicated that we can’t classify them using anything we can write down. This makes them very different from ‘tame’ knots: knots that aren’t wild. Yeah, tame knots are hard to classify, but nowhere near that hard.

There’s a dot product and cross product of vectors in 3 dimensions. But there’s also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There’s nothing really like this in other dimensions.

The following stuff is well-known: the group of linear transformations of $\mathbb{R}^n$ preserving the dot and cross product is called $SO(3)$. It consists of rotations. We say $SO(3)$ has an ‘irreducible representation’ on $\mathbb{R}^3$ because there’s no linear subspace of $\mathbb{R}^3$ that’s mapped to itself by every transformation in $SO(3)$, except for $\{0\}$ and the whole space.

Ho hum. But here’s something more surprising: it seems that $SO(3)$ also has an irreducible representation on $\mathbb{R}^7$ where every transformation preserves the dot product and cross product in 7 dimensions!

That’s right—no typo there. There is not an irreducible representation of $SO(7)$ on $\mathbb{R}^7$ that preserves the dot product and cross product. Preserving the dot product is easy. But the cross product in 7 dimensions is a strange thing that breaks rotation symmetry.

There is, apparently, an irreducible representation of the much smaller group $SO(3)$ on $\mathbb{R}^7$ that preserves the dot and cross product. But I only know this because people say Dynkin proved it! More technically, it seems Dynkin said there’s an $SO(3)$ subgroup of $G_2$ for which the irreducible representation of $\mathrm{G}_2$ on $\mathbb{R}^7$ remains irreducible when restricted to this subgroup. I want to see one explicitly.

The Eisenstein integers $\mathbb{E}$ are the complex numbers of the form $a + b \omega$ where $a$ and $b$ are integers and $\omega = \exp(2 \pi i/3)$. They form a subring of the complex numbers and also a lattice:

Last time I explained how the space $\mathfrak{h}_2(\mathbb{C})$ of $2 \times 2$ hermitian matrices is secretly 4-dimensional Minkowski spacetime, while the subset

$$\mathcal{H} = \left\{A \in \h_2(\mathbb{C}) \, \vert \, \det A = 1, \, \mathrm{tr}(A) \gt 0 \right\}$$

is 3-dimensional hyperbolic space. Thus, the set $\mathfrak{h}_2(\mathbb{E})$ of $2 \times 2$ hermitian matrices with Eisenstein integer entries forms a lattice in Minkowski spacetime, and I conjectured that $\mathfrak{h}_2(\mathbb{E}) \cap \mathcal{H}$ consists exactly of the hexagon centers in the hexagonal tiling honeycomb — a highly symmetrical structure in hyperbolic space, discovered by Coxeter, which looks like this:

Now Greg Egan and I will prove that conjecture.

Last time I introduced a 2-dimensional complex variety called the Eisenstein surface

$$E = \mathbb{C}/\mathbb{E} \times \mathbb{C}/\mathbb{E}$$

where $\mathbb{E} \subset \mathbb{C}$ is the lattice of Eisenstein integers. We worked out the Néron–Severi group $\mathrm{NS}(E)$ of this surface: that is, the group of equivalence classes of holomorphic line bundles on this surface, where we count two as equivalent if they’re isomorphic as topological line bundles. And we got a nice answer:

$$\mathrm{NS}(E) \cong \mathfrak{h}_2(\mathbb{E})$$

where $\mathfrak{h}_2(\mathbb{E})$ consists of $2 \times 2$ hermitian matrices with Eisenstein integers as entries.

Now we’ll see how this is related to the ‘hexagonal tiling honeycomb’:

We’ll see an explicit bijection between so-called ‘principal polarizations’ of the Eisenstein surface and the centers of hexagons in this picture! We won’t prove it works — I hope to do that later. But we’ll get everything set up.

You thought this series was dead. But it was only dormant!

In Part 1, I explained how the classification of holomorphic line bundles on a complex torus $X$ breaks into two parts:

• the ‘discrete part’: their underlying topological line bundles are classified by elements of a free abelian group called the Néron–Severi group $\mathrm{NS}(X)$.

• the ‘continuous part’: the holomorphic line bundles with a given underlying topological line bundle are classified by elements of a complex torus called the Jacobian $\mathrm{Jac}(X)$.

In Part 2, I explained duality for complex tori, which is a spinoff of duality for complex vector spaces. I used this to give several concrete descriptions of the Néron–Severi group $NS(X)$.

But the fun for me lies in the examples. Today let’s actually compute a Néron–Severi group and begin seeing how it leads to this remarkable picture by Roice Nelson:

This is joint work with James Dolan.

Was it really seventeen years ago that John broke the news on this blog that I had finally landed a permanent academic job? That was a long wait – I’d had twelve years of temporary contracts after receiving my PhD.

And now it has been decided that I am to move on from the University of Kent. The University is struggling financially and has decreed that a number of programs, including Philosophy, are to be cut. Whatever the wisdom of their plan, my time here comes to an end this July.

What next? It’s a little early for me to retire. If anyone has suggestions, I’d be happy to hear them.

We started this blog just one year before I started at Kent. To help think things over, in the coming weeks I thought I’d revisit some themes developed here over the years to see how they panned out:

1. Higher geometry: categorifying the Erlanger program
2. Category theory meets machine learning
3. Duality
4. Categorifying logic
5. Category theory applied to philosophy
6. Rationality of (mathematical and scientific) theory change as understood through historical development