## August 1, 2022

### Timing, Span(Graph) and Cospan(Graph)

#### Posted by Emily Riehl

*Guest post by Siddharth Bhat and Pim de Haan. Many thanks to Mario Román for proofreading this blogpost.*

This paper explores modelling automata using the Span/Cospan framework by Sabadini and Walters. The aim of this blogpost is to introduce the key constructions that are used in this paper, and to explain how these categorical constructions allow us to talking about modelling automata and timing in these automata.

## July 29, 2022

### Relational Universal Algebra with String Diagrams

#### Posted by Emily Riehl

*guest post by Phoebe Klett and Ralph Sarkis*

This post continues the series from the Adjoint School of Applied Category Theory 2022. It is a summary of the main ideas introduced in this paper:

- Filippo Bonchi, Dusko Pavlovic and Pawel Sobocinski, Functorial Semantics for Relational Theories.

Just as category theory gives us a bird’s-eye view of all mathematical structures, universal algebra gives a bird’s-eye view of all algebraic structures (groups, rings, modules, etc.) While universal algebra leads to a beautiful theory with many general statements — it also enjoys a categorical formulation introduced in F.W. Lawvere’s thesis which inspired the aforementioned paper’s title — it does not deal with several common structures in mathematics like graphs, orders, categories and metric spaces. Relational universal algebra allows to cover these examples and more. In this post, we present this field of study using a diagrammatic syntax based on cartesian bicategories of relations.

## July 28, 2022

### Compositional Constructions of Automata

#### Posted by Emily Riehl

*guest post by Ruben van Belle and Miguel Lopez*

In this post we will detail a categorical construction of automata following the work of Albasini, Sabadini, and Walters. We first recall the basic definitions of various automata, and then outline the construction of the aformentioned authors as well as generalizations and examples.

A *finite deterministic automaton* consists of

a finite set $Q$ (

*state space*),an

*initial state*$q_0\in X$ and a set $F\subseteq Q$ of*accepting states*,a finite set $A$ of

*input symbols*and a*transition map*$\tau_a:Q\to Q$ for every $a\in A$.

## July 27, 2022

### Learning to Lie with Sheaves

#### Posted by Emily Riehl

*guest post by Sean O’Connor and Ana Luiza Tenorio*

Social networks are frequently represented by graphs: each agent/person is a vertex and the interactions between pairs of individuals are the edges. A starting point to think about the evolution of opinions over time is to associate to each vertex $v$ a real number $x_v$ that represents the agreement of $v$ to respect a certain topic. For instance, fix the topic “category theory is cool”. In a social network of $n$ mathematicians, we will have some high positive $x_v \in \mathbb{R}$ representing a strong agreement with this assertion, some high negative $x_{v'}\in \mathbb{R}$ for a strong disagreement, and some neutral opinions. Those mathematicians interact and may change their original opinion. What is the group’s opinion about category theory after a period of time? Clearly, eventually, everyone will agree that category theory is cool. Jokes (or not) aside, a standard way to try to answer this is to study the dynamical system generated by the heat equation

where $x = (x_{v_1},...,v_{v_n})\in \mathbb{R}^n$ and $L$ is the graph Laplacian, a matrix that represents a graph defined by the difference $L = D - A$, with $D$ the degree matrix and $A$ the adjacency matrix of the graph. In this approach, originally proposed in Towards a mathematical theory of influence and attitude change, we study the evolution of opinion distributions without considering that expressed opinions may be different from personal opinions. In the paper Opinion Dynamics on Discourse Sheaves, Jakob Hansen and Robert Ghrist introduced a functor that addresses this distinction, and leads to a flexible model. We briefly present it here.

## July 26, 2022

### Identity Types in Context

#### Posted by Emily Riehl

*guest post by Shreya Arya and Greta Coraglia*

The relation between mathematicians and the notion of identity has been an interesting one. Even what is arguably the first mathematical text (that we know of) in history, Euclid’s Elements (c. 300 BC), deals with the problem of equality. In Book I, after *Definitions* and *Postulates*, Euclid details five *Common Notions*:

- Things equal to the same thing are also equal to one another.
- And if equal things are added to equal things then the wholes are equal.
- And if equal things are subtracted from equal things then the remainders are equal.
- And things coinciding with one another are equal to one another.
- And the whole [is] greater than the part.

On one hand, such notions are considered “common”, so that they are trivial enough that everyone ought to agree with them; on the other, they are not so trivial that one can avoid writing them down. Moreover, Euclid feels the need to pin-point the effect that equality must have on operations involving such objects: the underlying principle here is that two entities that are equal should share the same properties and behaviours.

*What mathematicians really use is a bit stronger than that…*^{1}

## July 25, 2022

### How to apply category theory to thermodynamics

#### Posted by Emily Riehl

*guest post by Nandan Kulkarni and Chad Harper*

This blog post discusses the paper “Compositional Thermostatics” by John Baez, Owen Lynch, and Joe Moeller. The series of posts on Dr. Baez’s blog gives a more thorough overview of the topics in the paper, and is probably a better primer if you intend to read it. Like the posts on Dr. Baez’s blog, this blog post also explains some aspects of the framework in an introductory manner. However, it takes the approach of emphasizes particular interesting details, and concludes in the treatment of a particular quantum system using ideas from the paper.

## July 19, 2022

### Probability Monads as Codensity Monads

#### Posted by Tom Leinster

My PhD student Ruben Van Belle has just published his first paper!

Ruben Van Belle, Probability monads as codensity monads.

Theory and Applications of Categories38 (2022), 811–842.

It’s a treasure trove of theorems demonstrating how many of the monads loosely referred to as “probability monads” arise as codensity monads in a certain uniform manner, which I’ll tell you about now.

## July 16, 2022

### Conversations on Mathematics

#### Posted by John Baez

Now that I’ve retired, I have more time for pure math. So after a roughly decade-long break, James Dolan and I are talking about math again. Here are our conversations. Some are in email, but mainly these are our weekly 2-hour-long Zoom sessions, which I’ve put on YouTube. They focus on algebraic geometry — especially abelian varieties and motives — but also ‘doctrines’ and their applications to algebraic geometry, group representation theory, combinatorics and other subjects.

They may not be easy to follow, but maybe a few people will get something out of them. I have not corrected all the mistakes, some of which we eventually catch. I’ve added lots of links to papers and Wikipedia articles.

These conversations are continuing, but I won’t keep putting links to them here on $n$-Category Café, so if you want more of them you can either check out my webpage at your leisure, or subscribe to my YouTube channel. I’ll probably fall behind in putting up videos, and then catch up, and then fall behind, etc. — so please don’t expect one to show up each week.

## July 9, 2022

### Symposium on Compositional Structures 9

#### Posted by John Baez

The Symposium on Compositional Structures is a nice informal conference series that happens more than once a year. You can now submit talks for this one:

• Ninth Symposium on Compositional Structures (SYCO 9), Como, Italy, 8-9 September 2022. Deadline to submit a talk: Monday August 1, 2022.

Apparently you can attend online but to give a talk you have to go there. Here are some details….

## June 27, 2022

### Compositional Modeling with Decorated Cospans

#### Posted by John Baez

It’s finally here: software that uses category theory to let you build models of dynamical systems! We’re going to train epidemiologists to use this to model the spread of disease. My first talk on this will be on Wednesday June 29th. You’re invited!

• Compositional modeling with decorated cospans, Graph Transformation Theory and Practice (GReTA) seminar, 19:00 UTC, Wednesday 29 June 2022.

You can attend live on Zoom if you click here. You can also watch it live on YouTube, or later recorded, here.

## June 22, 2022

### Motivating Motives

#### Posted by John Baez

I gave an introductory talk on Grothendieck’s ‘motives’ at the conference Grothendieck’s Approach to Mathematics at Chapman University in late May.

Now the videos of all talks at this conference are on YouTube — including talks by Kevin Buzzard, Colin McLarty, Elaine Landry, Jean-Pierre Marquis, Mike Shulman and other people you’ve heard about on this blog.

## June 20, 2022

### Hoàng Xuân Sính

#### Posted by John Baez

During the Vietnam war, Grothendieck taught math to the Hanoi University mathematics department staff, out in the countryside. Hoàng Xuân Sính took notes and later did a PhD with him — by correspondence! She mailed him her hand-written thesis. She is the woman in this picture:

As you might guess, there’s a very interesting story behind this. I’ve looked into it, but what I found raises even more questions. Hoàng Xuân Sính’s life really deserves a good biography.

## June 18, 2022

### Compositional Thermostatics (Part 4)

#### Posted by John Baez

*guest post by Owen Lynch*

This is the fourth and final part of a blog series on this paper:

• John Baez, Owen Lynch and Joe Moeller, Compositional thermostatics.

In Part 1, we went over our definition of **thermostatic system**: it’s a convex space $X$ of **states** and a concave function $S \colon X \to [-\infty, \infty]$ saying the **entropy** of each state. We also gave examples of thermostatic systems.

In Part 2, we talked about what it means to compose thermostatic systems. It amounts to *constrained maximization of the total entropy*.

In Part 3 we laid down a categorical framework for composing systems when there are choices that have to be made for how the systems are composed. This framework has been around for a long time: operads and operad algebras.

In this post we will bring together all of these parts in a big synthesis to create an operad of all the ways of composing thermostatic systems, along with an operad algebra of thermostatic systems!

## June 15, 2022

### Graded Modalities

#### Posted by David Corfield

I’m hosting a workshop tomorrow (13:00-17:00 UK time (UTC+1), Thursday 16 June 2022) which explores what common ground there may be in the treatment of *graded modalities* by linguistics, computer science and philosophy. It’s a hybrid event (in-person and online) and all are welcome, details here.

You very much ought to come along.

A fine example of a graded modality.

## May 17, 2022

### The Magnitude of Information

#### Posted by Tom Leinster

*Guest post by Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca*

The magnitude of a metric space $(X,d)$ does not require further introduction on this blog. Two of the hosts, Tom Leinster and Simon Willerton, conjectured that the magnitude function $\mathcal{M}_X(R) := \mathrm{Mag}(X,R \cdot \mathrm{d})$ of a convex body $X \subset \mathbb{R}^n$ with Euclidean distance $\mathrm{d}$ captures classical geometric information about $X$:

$\begin{aligned} \mathcal{M}_X(R) =& \frac{1}{n! \omega_n} \mathrm{vol}_n(X)\ R^n + \frac{1}{2(n-1)! \omega_{n-1}} \mathrm{vol}_{n-1}(\partial X)\ R^{n-1} + \cdots + 1 \\ =& \frac{1}{n! \omega_n} \sum_{j=0}^n c_j(X)\ R^{n-j} \end{aligned}$

where $c_j(X) = \gamma_{j,n} V_j(X)$ is proportional to the $j$-th intrinsic volume $V_j$ of $X$ and $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$.

Even more basic geometric questions have remained unknown, including:

- What geometric content is encoded in $\mathcal{M}_X$?
- What can be said about the magnitude function of the unit disk $B_2 \subset \mathbb{R}^2$?

We discuss in this post how these questions led us to possible relations to information geometry. We would love to hear from you:

- Is magnitude an interesting invariant for information geometry?
- Is there a category theoretic motivation, like Lawvere’s view of a metric space as an enriched category?
- Does the magnitude relate to notions studied in information geometry?
- Do you have interesting questions about this invariant?