Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

September 16, 2022

Young Diagrams and Classical Groups

Posted by John Baez

Young diagrams can be used to classify an enormous number of things. My first one or two This Week’s Finds seminars will be on Young diagrams and classical groups. Here are some lecture notes:

Young diagrams and classical groups.

I probably won’t cover all this material in the seminar. The most important part is the stuff up to and including the classification of irreducible representations of the “classical monoid” End( n)\mathrm{End}(\mathbb{C}^n). (People don’t talk about classical monoids, but they should.)

Just as a reminder: my talks will be on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building at the University of Edinburgh. The first will be on September 22nd, and the last on December 1st.

If you’re actually in town, there’s a tea on the fifth floor that starts 15 minutes before my talk. If you’re not, you can attend on Zoom:
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

Posted at 5:49 PM UTC | Permalink | Followups (7)

September 12, 2022

The Algebra of Grand Unified Theories

Posted by John Baez

Fans of the nn-Category Café might like The Cartesian Café, where Timothy Nguyen has long, detailed conversations with mathematicians. We recently talked about the fascinating mathematical patterns in the Standard Model that led people to invent grand unified theories:

For more details, go here:

Posted at 11:27 PM UTC | Permalink | Post a Comment

September 11, 2022

Seminar on This Week’s Finds

Posted by John Baez

Here’s something new: I’m living in Edinburgh until January! I’ll be working with Tom Leinster at the University of Edinburgh, supported by a Leverhulme Fellowship.

One fun thing I’ll be doing is running seminars on some topics from my column This Week’s Finds. They’ll take place on Thursdays at 3:00 pm UK time in Room 6206 of James Clerk Maxwell Building, home of the Department of Mathematics. The first will be on September 22nd, and the last on December 1st.

We’re planning to

1) make the talks hybrid on Zoom so that people can participate online:
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

2) record them and eventually make them publicly available on my YouTube channel.

3) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars: it’s here.

More details soon!

I have the topics planned out….

Posted at 10:09 AM UTC | Permalink | Followups (19)

August 24, 2022

Joint Mathematics Meetings 2023

Posted by John Baez

This is the biggest annual meeting of mathematicians:

  • Joint Mathematical Meetings 2023, Wednesday January 4 - Saturday January 7, 2023, John B. Hynes Veterans Memorial Convention Center, Boston Marriott Hotel, and Boston Sheraton Hotel, Boston, Massachusetts.

As part of this huge meeting, the American Mathematical Society is having a special session on Applied Category Theory on Thursday January 5th.

I hear there will be talks by Eugenia Cheng and Olivia Caramello!

Posted at 11:45 PM UTC | Permalink | Followups (6)

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.

Posted at 9:35 PM UTC | Permalink | Post a Comment

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:

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.

Posted at 2:57 PM UTC | Permalink | Followups (1)

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 QQ (state space),

  • an initial state q 0Xq_0\in X and a set FQF\subseteq Q of accepting states,

  • a finite set AA of input symbols and a transition map τ a:QQ\tau_a:Q\to Q for every aAa\in A.

Posted at 6:24 PM UTC | Permalink | Followups (2)

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 vv a real number x vx_v that represents the agreement of vv to respect a certain topic. For instance, fix the topic “category theory is cool”. In a social network of nn mathematicians, we will have some high positive x vx_v \in \mathbb{R} representing a strong agreement with this assertion, some high negative x vx_{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

(1)dxdt=αLxforsomeα>0 \frac{d x}{d t} = -\alpha L x \; for \; some \; \alpha \gt 0

where x=(x v 1,...,v v n) nx = (x_{v_1},...,v_{v_n})\in \mathbb{R}^n and LL is the graph Laplacian, a matrix that represents a graph defined by the difference L=DAL = D - A, with DD the degree matrix and AA 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.

Posted at 4:22 PM UTC | Permalink | Followups (4)

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:

  1. Things equal to the same thing are also equal to one another.
  2. And if equal things are added to equal things then the wholes are equal.
  3. And if equal things are subtracted from equal things then the remainders are equal.
  4. And things coinciding with one another are equal to one another.
  5. 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

Posted at 5:51 PM UTC | Permalink | Followups (5)

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.

Posted at 9:54 PM UTC | Permalink | Followups (4)

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 Categories 38 (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.

Posted at 9:47 AM UTC | Permalink | Followups (8)

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 nn-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.

Posted at 10:06 PM UTC | Permalink | Followups (1)

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….

Posted at 3:18 PM UTC | Permalink | Post a Comment

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.

Posted at 6:47 PM UTC | Permalink | Post a Comment

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.

Posted at 5:32 PM UTC | Permalink | Followups (2)