## February 15, 2018

### Physics and 2-Groups

#### Posted by David Corfield

The founding vision of this blog, higher gauge theory, progresses. Three physicists have just brought out Exploring 2-Group Global Symmetries . Two of the authors have worked with Nathan Seiberg, a highly influential physicist, who, along with three others, had proposed in 2014 a program to study higher form field symmetries in QFT (Generalized Global Symmetries), without apparently being aware that this idea was considered before.

Eric Sharpe then published an article the following year, Notes on generalized global symmetries in QFT, explaining how much work had already been done along these lines:

The recent paper [1] proposed a more general class of symmetries that should be studied in quantum field theories: in addition to actions of ordinary groups, it proposed that we should also consider ‘groups’ of gauge fields and higher-form analogues. For example, Wilson lines can act as charged objects under such symmetries. By using various defects, the paper [1] described new characterizations of gauge theory phases.

Now, one can ask to what extent it is natural for n-forms as above to form a group. In particular, because of gauge symmetries, the group multiplication will not be associative in general, unless perhaps one restricts to suitable equivalence classes, which does not seem natural in general. A more natural understanding of such symmetries is in terms of weaker structures known as 2-groups and higher groups, in which associativity is weakened to hold only up to isomorphisms.

There are more 2-groups and higher groups than merely, ‘groups’ of gauge fields and higher-form tensor potentials (connections on bundles and gerbes), and in this paper we will give examples of actions of such more general higher groups in quantum field theory and string theory. We will also propose an understanding of certain anomalies as transmutations of symmetry groups of classical theories into higher group actions on quantum theories.

## February 14, 2018

### Gradual typing

#### Posted by Mike Shulman

*(Guest post by Max New)*

Dan Licata and I have just put up a paper on the arxiv
with a syntax and semantics for a *gradually typed* programming
language, which is a kind of synthesis of statically typed and
dynamically typed programming styles.
The central insight of the paper is to show that the dynamic type
checking used in gradual typing has the structure of a proarrow
equipment.
Using this we can show that some traditional *definitions* of dynamic
type checks can be proven to be in fact *unique solutions* to the
specifications provided by the structure of an equipment.
It’s a classic application of category theory: finding a universal
property to better understand what was previously an ad-hoc
construction.

The paper is written for computer scientists, so I’ll try to provide a more category-theorist-accessible intro here.

## February 8, 2018

### Homotopy Type Theory Electronic Seminar

#### Posted by John Baez

What a great idea! A seminar on homotopy type theory, with talks by top experts, available to everyone with internet connection!

## February 6, 2018

### Linguistics Using Category Theory

#### Posted by John Baez

*guest post by Cory Griffith and Jade Master*

Most recently, the Applied Category Theory Seminar took a step into linguistics by discussing the 2010 paper Mathematical Foundations for a Compositional Distributional Model of Meaning, by Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark.

Here is a summary and discussion of that paper.

## February 5, 2018

*m*Lab

#### Posted by John Baez

Since nothing get parodied until it’s sufficiently well-known to make it worth the effort, this proves the $n$Lab is a success:

• $m$Lab.

Click on the links!

## February 2, 2018

### A Problem on Pushouts and Pullbacks

#### Posted by John Baez

I have a problem involving pullbacks and pushouts. This problem arose in work with Kenny Courser on an application of category theory. But you don’t need to understand anything about that application to understand — and I hope solve! —our problem.

If you can solve it, we will credit you in our paper.

## January 29, 2018

### The Stable Homotopy Hypothesis and Categorified Abelian Groups

#### Posted by Tom Leinster

*guest post by Nick Gurski*

In December, Niles Johnson, Angélica Osorno, and I posted a proof of
the stable homotopy hypothesis in dimension two on the arxiv. This is
the fourth paper we have written together on this project (one also
joint with Marc Stephan), and since proving the SHH in dimension two
was the *first step* in a long line of research which you might
call “an introduction to twice-categorified abelian groups,” I
thought now was a good time to try to explain what we have been up to
and why.

## January 25, 2018

### Announcing the 2018 Talbot Workshop: Model-Independent Theory of Infinity-Categories

#### Posted by Emily Riehl

The Talbot Workshop is a 1-week learning workshop for roughly 35 graduate students and a few postdocs. Most of the talks will be given by participants, and will be expository in nature. This year’s workshop will be held from May 28 - June 2, 2018 at Government Camp, Oregon USA and Dominic Verity and I will be the mentors. The topic is on our joint work developing a theory of $\infty$-categories from first principles in a model-independent fashion, that is, using a common axiomatic framework that is satisfied by a variety of models. The goal is to demonstrate that theorems proven using the combinatorics of a particular model transfer across specified “change of model” functors. More details about the program, including a preliminary list of talks and references, can be found here.

Applications are now open, and close on Wednesday, February 28 at 11:59PM. You can apply online here.

Talbot is meant to encourage collaboration among young researchers, with an emphasis on graduate students. We also aim to gather participants with a diverse array of knowledge and interests, so applicants need not be an expert in the field– in particular, students at all levels of graduate education are encouraged to apply. As we are committed to promoting diversity in mathematics, we also especially encourage women and minorities to apply.

The Talbot Workshop grant will cover all local expenses including lodging and food and also offer partial funding for participants’ travel costs.

I’m happy to answer questions about the scientific content of this workshop on this blog - please comment below. If you have logistical questions, please do not hesitate to e-mail the organizers at talbotworkshop (at) gmail.com.

On behalf of the organizers: Eva Belmont, Calista Bernard, Inbar Klang, Morgan Opie, Sean Pohorence

### More Secrets of the Associahedra

#### Posted by John Baez

The associahedra are wonderful things discovered by Jim Stasheff around 1963 but even earlier by Dov Tamari in his thesis. They hold the keys to understanding ‘associativity up to coherent homotopy’ in exquisite combinatorial detail.

But do they still hold more secrets? I think so!

## January 23, 2018

### Statebox: A Universal Language of Distributed Systems

#### Posted by John Baez

We’re getting a lot of great posts here this week, but I also want to point out this, by one grad students:

- Christian William, Statebox: a universal language of distributed systems,
*Azimuth*, January 22, 2018.

A brief teaser follows, in case you’re wondering what this is about.

### FreeTikZ

#### Posted by Tom Leinster

*guest post by Chris Heunen*

I don’t have to tell you, dear $n$-Category Café reader, that string diagrams are extremely useful. They speed up computations massively, reveal what makes a proof tick without an impenetrable forest of details, and suggest properties that you might not even have thought about in algebraic notation. They also make your paper friendlier to read.

However, string diagrams are also a pain to typeset. First, there is the entrance fee of learning a modern LaTeX drawing package, like TikZ. Then, there is the learning period of setting up macros tailored to string diagrams and internalizing them in your muscle memory. But even after all that investment, it still takes a lot of time. And unlike that glorious moment when you realise that you have cycled about twice the circumference of the Earth in your life, this time is mostly wasted. I estimate I’ve wasted over 2000 string diagrams’ worth of time by now.

Wouldn’t it be great if you could simply draw your diagram by hand, and have it magically converted into TikZ? Now you can!

## January 22, 2018

### A Categorical Semantics for Causal Structure

#### Posted by John Baez

*guest post by Joseph Moeller and Dmitry Vagner*

We begin the Applied Category Theory Seminar by discussing the paper A categorical semantics for causal structure by Aleks Kissinger and Sander Uijlen.

Compact closed categories have been used in categorical quantum mechanics to give a structure for talking about quantum processes. However, they prove insufficient to handle higher order processes, in other words, processes of processes. This paper offers a construction for a $\ast$-autonomous extension of a given compact closed category which allows one to reason about higher order processes in a non-trivial way.

We would like to thank Brendan Fong, Nina Otter, Joseph Hirsh and Tobias Heindel as well as the other participants for the discussions and feedback.

## January 10, 2018

### On the Magnitude Function of Domains in Euclidean Space, I

#### Posted by Simon Willerton

*guest post by Heiko Gimperlein and Magnus Goffeng.*

The magnitude of a metric space was born, nearly ten years ago, on this blog, although it went by the name of cardinality back then. There has been much development since (for instance, see Tom Leinster and Mark Meckes’ survey of what was known in 2016). Basic geometric questions about magnitude, however, remain open even for compact subsets of $\mathbb{R}^n$: Tom Leinster and Simon Willerton suggested that magnitude could be computed from intrinsic volumes, and the algebraic origin of magnitude created hopes for an inclusion-exclusion principle.

In this sequence of three posts we would like to discuss our recent article, which is about asymptotic geometric content in the magnitude function and also how it relates to scattering theory.

For “nice” compact domains in $\mathbb{R}^n$ we prove an asymptotic variant of Leinster and Willerton’s conjecture, as well as an asymptotic inclusion-exclusion principle. Starting from ideas by Juan Antonio Barceló and Tony Carbery, our approach connects the magnitude function with ideas from spectral geometry, heat kernels and the Atiyah-Singer index theorem.

We will also address the location of the poles in the complex plane of the magnitude function. For example, here is a plot of the poles and zeros of the magnitude function of the $21$-dimensional ball.

We thank Simon for inviting us to write this post and also for his paper on the magnitude of odd balls as the computations in it rescued us from some tedious combinatorics.

The plan for the three café posts is as follows:

State the recent results on the asymptotic behaviour as a metric space is scaled up and on the meromorphic extension of the magnitude function.

Discuss the proof in the toy case of a compact domain $X\subseteq \mathbb{R}$ and indicate how it generalizes to arbitrary odd dimension.

Consider the relationship of the methods to geometric analysis and potential ramifications; also state some open problems that could be interesting.

## December 23, 2017

### An M5-Brane Model

#### Posted by John Baez

When you try to quantize 10-dimensional supergravity theories, you are led to some theories involving strings. These are fairly well understood, because the worldsheet of a string is 2-dimensional, so string theories can be studied using 2-dimensional conformal quantum field theories, which are mathematically tractable.

When you try to quantize 11-dimensional supergravity, you are led to a theory involving 2-branes and 5-branes. People call it M-theory, because while it seems to have magical properties, our understanding of it is still murky — because it involves these higher-dimensional membranes. They have 3- and 6-dimensional worldsheets, respectively. So, precisely formulating M-theory seems to require understanding certain quantum field theories in 3 and 6 dimensions. These are bound to be tougher than 2d quantum field theories… tougher to make mathematically rigorous, for example… but even worse, until recently people didn’t know what either of these theories *were!*

In 2008, Aharony, Bergman, Jafferis and Maldacena figured out the 3-dimensional theory: it’s a supersymmetric Chern–Simons theory coupled to matter in a way that makes it no longer a topological quantum field theory, but still conformally invariant. It’s now called the ABJM theory. This discovery led to the ‘M2-brane mini-revolution’, as various puzzles about M-theory got solved.

The 6-dimensional theory has been much more elusive. It’s called the (0,2) theory. It should be a 6-dimensional conformal quantum field theory. But its curious properties got people thinking that it *couldn’t arise from any Lagrangian* — a serious roadblock, given how physicists normally like to study quantum field theories. But people have continued avidly seeking it, and not just for its role in a potential ‘theory of everything’. Witten and others have shown that if it existed, it would shed new light on Khovanov duality and geometric Langlands correspondence! The best introduction is here:

- Edward Witten, Geometric Langlands from six dimensions, 2009.

## December 21, 2017

### Arithmetic Gauge Theory

#### Posted by David Corfield

Around 2008-9 we had several exchanges with Minhyong Kim here at the Café, in particular on his views of approaching number theory from a homotopic perspective, in particular in the post Kim on Fundamental Groups in Number Theory. (See also the threads Afternoon Fishing and The Elusive Proteus.)

I even recall proposing a polymath project based on his ideas in Galois Theory in Two Variables. Something physics-like was in the air, and this seemed a good location with two mathematical physicists as hosts, John having extensively written on number theory in This Week’s Finds.

Nothing came of that, but it’s interesting to see Minhyong is very much in the news these days, including in a popular article in Quanta magazine, Secret Link Uncovered Between Pure Math and Physics.

The Quanta article has Minhyong saying:

“I was hiding it because for many years I was somewhat embarrassed by the physics connection,” he said. “Number theorists are a pretty tough-minded group of people, and influences from physics sometimes make them more skeptical of the mathematics.”

Café readers had an earlier alert from an interview I conducted with Minhyong, reported in Minhyong Kim in The Reasoner. There he was prepared to announce

The work that occupies me most right now, arithmetic homotopy theory, concerns itself very much with arithmetic moduli spaces that are similar in nature and construction to moduli spaces of solutions to the Yang-Mills equation.

Now his articles are appearing bearing explicit names such as ‘Arithmetic Chern-Simons theory’ (I and II), and today, we have Arithmetic Gauge Theory: A Brief Introduction.