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November 22, 2022

Inner Automorphisms of the Octonions

Posted by John Baez

What are the inner automorphisms of the octonions?

Posted at 5:03 PM UTC | Permalink | Followups (7)

November 16, 2022

The Icosidodecahedron

Posted by John Baez

The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron. It has 30 vertices, one at the center of each edge of the icosahedron—or equivalently, one at the center of each edge of a dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a shadow of a more symmetrical shape with twice as many vertices, which lives in a space with twice as many dimensions! Namely, it is a projection down to 3d space of a 6-dimensional polytope with 60 vertices.

Even better, it is also a slice of a more symmetrical 4d polytope with 120 vertices, which in turn is the projection down to 4d space of an even more symmetrical 8-dimensional polytope with 240 vertices: the so-called ‘E8 root polytope’.

Note how the numbers keep doubling: 30, 60, 120 and 240.

Posted at 5:24 PM UTC | Permalink | Followups (8)

November 1, 2022

Categories and Epidemiology

Posted by John Baez

I gave a talk about my work using category theory to help design software for epidemic modeling:

• Category theory and epidemiology, African Mathematics Seminar, Wednesday November 2, 2022, 3 pm Nairobi time or noon UTC. Organized by Layla Sorkatti and Jared Ongaro.

This talk is a lot less technical than previous ones I’ve given on this subject, which were aimed mainly at category theorists. You can watch it on YouTube.

Posted at 11:05 PM UTC | Permalink | Followups (4)

October 25, 2022

Booleans, Natural Numbers, Young Diagrams, Schur Functors

Posted by John Baez

There’s an adjunction between commutative monoids and pointed sets, which gives a comonad. Then:

Take the booleans, apply the comonad and get the natural numbers.

Take the natural numbers, apply the comonad and get Young diagrams.

Take the Young diagrams, apply the comonad and get Schur functors.

Let me explain how this works!

Posted at 3:07 PM UTC | Permalink | Followups (15)

October 16, 2022

Partition Function as Cardinality

Posted by John Baez

In classical statistical mechanics we often think about sets where each point has a number called its ‘energy’. Then the ‘partition function’ counts the set’s points — but points with large energy count for less! And the amount each point gets counted depends on the temperature.

So, the partition function is a generalization of the cardinality |X||X| that works for sets XX equipped with a function E:XE\colon X \to \mathbb{R}. I’ve been talking with Tom Leinster about this lately, so let me say a bit more about how it works.

Posted at 4:12 PM UTC | Permalink | Followups (26)

October 11, 2022

Two Talks on Measuring Diversity

Posted by Tom Leinster

Later this month, I’ll give a pair of lectures at Riken, the major research institute in Japan, at the kind invitation of Ryosuke Iratani.

Both lectures will be on measuring diversity, and the aim is to touch on some biological and information theory aspects as well as the mathematics. Titles, abstracts and dates follow. You can attend online (and I’ll be speaking online too), but Ryo asks that you fill in the registration form.

Posted at 9:20 AM UTC | Permalink | Followups (2)

October 7, 2022

The Eventual Image, Eventually

Posted by Tom Leinster

More than ten years ago, I wrote a series of posts (1, 2, 3) about what happens when you iterate a process for an infinite amount of time. I know just how that feels: it’s taken me until now to finish writing it up. But finally, it’s done:

Tom Leinster, The eventual image. arXiv:2210.00302, 2022.

What is the eventual image? I’ll tell you in nine ways.

Posted at 6:03 PM UTC | Permalink | Followups (5)

October 5, 2022

CWRU Is Hiring

Posted by Tom Leinster

Guest post by Nick Gurski and Mark Meckes

The Department of Mathematics, Applied Mathematics, and Statistics at Case Western Reserve University is hiring a tenure-track assistant professor in pure math, to start in fall 2023. Preference will be given to candidates in algebra, topology, and related areas, although excellent candidates in all areas of pure mathematics will be considered.

At CWRU, patrons of the nn-Category Café may be familiar with not only our names, Nick Gurski and Mark Meckes, but also our colleagues Juan Orendain (Math, Applied Math and Statistics) and Colin McLarty (Philosophy).

Applications are through MathJobs, where you can see the full ad. Questions can be emailed to Nick (, who is chairing the search committee.

Posted at 9:51 PM UTC | Permalink | Post a Comment

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 (9)

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 | Followups (1)

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)