## March 19, 2018

### Magnitude Homology Reading Seminar, I

#### Posted by Simon Willerton

In Sheffield we have started a reading seminar on the recent paper of Tom Leinster and Mike Shulman Magnitude homology of enriched categories and metric spaces. The plan was to write the talks up as blog posts. Various things, including the massive strike that has been going on in universities in the UK, have meant that I’m somewhat behind with putting the first talk up. The strike also means that we haven’t had many seminars yet!

I gave the first talk which is the one here. It is an introductory talk which just describes the idea of categorification and the paper I wrote with Richard Hepworth on categorifying the magnitude of finite graphs, this is the idea which was generalized by Tom and Mike.

## March 11, 2018

### Stabilization of Derivators

#### Posted by Mike Shulman

*(guest post by Ian Coley)*

I recently published a paper to the arXiv which reconstructs an old paper of Alex Heller. Heller’s Stable homotopy theories and stabilization is one of a few proto-derivator papers that are still oft-cited by those of us studying derivators — a subject absent from this website since the two papers of Mike Shulman and Kate Ponto were published in 2014! Therefore before getting into the paper itself, it’s worth recalling what a derivator is supposed to be and do. For those interested in the long version, check out the nLab article or Moritz Groth’s excellent paper.

## March 10, 2018

### Cognition, Convexity, and Category Theory

#### Posted by John Baez

*guest post by Tai-Danae Bradley and Brad Theilman*

Recently in the Applied Category Theory Seminar our discussions have returned to modeling natural language, this time via *Interacting Conceptual Spaces I* by Joe Bolt, Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Marsden, and Robin Piedeleu. In this paper, convex algebras lie at the heart of a compositional model of cognition based on Peter Gärdenfors’ theory of conceptual spaces. We summarize the ideas in today’s post.

Sincere thanks go to Brendan Fong, Nina Otter, Fabrizio Genovese, Joseph Hirsh, and other participants of the seminar for helpful discussions and feedback.

## March 4, 2018

### Coarse-Graining Open Markov Processes

#### Posted by John Baez

Kenny Courser and I have been working hard on this paper for months:

- John Baez and Kenny Courser, Coarse-graining open Markov processes.

It may be almost done. So, it would be great if you folks could take a look and comment on it! It’s a cool mix of probability theory and double categories.

‘Coarse-graining’ is a standard method of extracting a simple Markov process from a more complicated one by identifying states. We extend coarse-graining to open Markov processes. An ‘open’ Markov process is one where probability can flow in or out of certain states called ‘inputs’ and ‘outputs’. One can build up an ordinary Markov process from smaller open pieces in two basic ways:

- composition, where we identify the outputs of one open Markov process with the inputs of another,

and

- tensoring, where we set two open Markov processes side by side.

A while back, Brendan Fong, Blake Pollard and I showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category:

- A compositional framework for Markov processes, $n$-Category Café, January 12, 2016.

Here Kenny and I go further by constructing a symmetric monoidal *double* category where the 2-morphisms include ways of coarse-graining open Markov processes. We also extend the previously defined ‘black-boxing’ functor from the category of open Markov processes to this double category.

But before you dive into the paper, let me explain all this stuff a bit more….

## March 1, 2018

### Univalence From Scratch

#### Posted by Mike Shulman

Martín Escardó has written “a self-contained, brief and complete formulation of Voevodsky’s Univalence Axiom” in English and Agda:

## February 28, 2018

### Hypergraph Categories of Cospans

#### Posted by John Baez

*guest post by Jonathan Lorand and Fabrizio Genovese*

In the Applied Category Theory Seminar, we most recently read Brendan Fong’s article on decorated cospans. This construction is part of a larger framework, developed in Brendan Fong’s PhD thesis, for studying interconnected, open, network-style systems. A paradigmatic example: systems composed of electric circuits having input and output terminals, allowing for composition of smaller circuits into larger. An aim of Brendan’s framework is to give, for any such kind of system, a unified categorical way to describe both the formal, symbolic language of such systems (their “syntax”) as well as the behavior of the systems that these formal symbols represent (the “semantics”). For circuits: syntax is formal rules for combining circuit diagram nomenclature; semantics is a (mathematical) description of how real-life circuits behave in the presence of voltages, currents, etc.. Decorated cospans are a tool ideal for “syntax”; decorated corelations are designed to handle “semantics” and are flexible enough to model any so-called hypergraph category. We’ll focus on the former, and hint at the latter.

## February 23, 2018

### Applied Category Theory at NIST

#### Posted by John Baez

I think it’s really cool how applied category theory is catching on. My former student Blake Pollard is working at the National Institute of Standards and Technology on applications of category theory to electrical engineering. He’s working with Spencer Breiner… and now Breiner is running a workshop on this stuff:

• Applied Category Theory: Bridging Theory & Practice, March 15–16, 2018, NIST, Gaithersburg, Maryland, USA.

### Higher Algebra and Mathematical Physics

#### Posted by John Baez

You all know about Homotopy Type Theory Electronic Seminar Talks. Here’s another way to cut carbon emissions: a **double conference**. The idea here is to have a conference in two faraway locations connected by live video stream, to reduce the amount of long-distance travel!

Even better, it’s about a great subject:

- Higher algebra and mathematical physics, August 13–17, 2018, Perimeter Institute, Waterloo, Canada, and Max Planck Institute for Mathematics, Bonn, Germany. Organized by David Ayala, Lukas Brantner, Kevin Costello, Owen Gwilliam, Andre Henriques, Theo Johnson-Freyd, Aaron Mazel-Gee, and Peter Teichner.

Higher algebra, lower carbon emissions… what more could you want?

## February 19, 2018

### Cartesian Bicategories

#### Posted by John Baez

*guest post by Daniel Cicala and Jules Hedges*

We continue the Applied Category Theory Seminar with a discussion of Carboni and Walters’ paper *Cartesian Bicategories I*. The star of this paper is the notion of ‘bicategories of relations’. This is an abstraction of relations internal to a category. As such, this paper provides excellent, if technical, examples of internal relations and other internal category theory concepts. In this post, we discuss bicategories of relations while occasionally pausing to enjoy some internal category theory such as relations, adjoints, monads, and the Kleisli construction.

We’d like to thank Brendan Fong and Nina Otter for running such a great seminar. We’d also like to thank Paweł Sobociński and John Baez for helpful discussions.

## 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.