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September 13, 2025
A Shadow of Triality?
Posted by John Baez
It’s well known that you can construct the octonions using triality. One statement of triality is that has nontrivial outer automorphisms of order 3. On the other hand, the octonions have nontrivial inner automorphisms of order 3. My question: can we deduce one of these facts from the other?
August 29, 2025
Equivalence via Surjections
Posted by Tom Leinster
Pick a type of categorical structure: say bicategories, or monoidal categories, or whatever you like. Some of the functors between structures are equivalences, in whatever the appropriate sense might be. And some of those equivalences have one or both of these two properties:
They’re not just essentially surjective in every dimension — they’re actually surjective in every dimension.
They don’t just preserve the structure up to isomorphism or equivalence — they strictly preserve it.
Call an equivalence with both these properties a strict surjective equivalence. So a strict surjective equivalence is an equivalence of a very special and easy kind.
General principle: the standard notion of equivalence between structures is generated by just these very special ones. For example, two bicategories are biequivalent if and only if they can be linked up by a zigzag of strict surjective equivalences.
Why should we care? Because there are some types of structure where the right notion of equivalence isn’t clear, and this principle guides us to it. For example, it tells us the right notion of equivalence for double categories.
All this is done in my new paper:
Tom Leinster, Equivalence via surjections. arXiv:2508.20555, 2025.
August 28, 2025
Burrito Monads, Arrow Kitchens, and Freyd Category Recipes
Posted by Tom Leinster
Guest post by Khyathi Komalan and Andrew Krenz
From Lawvere’s Hegelian taco to Baez’s layer cake analogy to Eugenia Cheng’s How to Bake Pi, categorists have cultivated a rich tradition of culinary metaphors and similes. A well-known example in the world of computation is Mark Dominus’s “monads are like burritos” — where a tortilla (computational context) wraps diverse ingredients (values) to create a cohesive entity (effectful value) whose burrito structure is maintained as the meal moves down the assembly line (undergoes computations).
August 14, 2025
Safeguarded AI Meeting
Posted by John Baez
This week, 50 category theorists and software engineers working on “safeguarded AI” are meeting in Bristol. They’re being funded by £59 million from ARIA, the UK’s Advanced Research and Invention Agency.
The basic idea is to develop a mathematical box that can contain a powerful genie. More precisely:
By combining scientific world models and mathematical proofs we will aim to construct a ‘gatekeeper’, an AI system tasked with understanding and reducing the risks of other AI agents. In doing so we’ll develop quantitative safety guarantees for AI in the way we have come to expect for nuclear power and passenger aviation.
August 5, 2025
(BT) Diversity from (LC) Diversity
Posted by Tom Leinster
Guest post by Mark Meckes
Around 2010, in papers that both appeared in print in 2012, two different mathematical notions were introduced and given the name “diversity”.
One, introduced by Tom Leinster and Christina Cobbold, is already familiar to regular readers of this blog. Say is a finite set, and for each we have a number that specifies how “similar” and are. (Typically we also assume .) Fix a parameter . If is a probability distribution on , then the quantity (with the cases defined by taking limits) can be interpreted as the “effective number of points” in , taking into account both the similarities between points as quantified by and the weights specified by . Its logarithm is a refinement of the -Rényi entropy of . The main motivating example is when is a set of species of organisms present in an ecosystem, and quantifies the “effective number of species” in , accounting for both similarities between species and their relative abundances. This family of quantities turns out to subsume many of the diversity measures previously introduced in the theoretical ecology literature, and they are now often referred to as Leinster–Cobbold diversities.
August 2, 2025
Jack Morava
Posted by John Baez
Today I heard from David Benson that Jack Morava died yesterday. This comes as such a huge shock that I can’t help but hope Benson was somehow misinformed. Morava has been posting comments to the n-Café and sending emails to me even very recently.
This is all I know, now.
July 28, 2025
The Duflo Isomorphism and the Harmonic Oscillator Hamiltonian
Posted by John Baez
Quick question. Classically the harmonic oscillator Hamiltonian is often written , while quantum mechanically it gets some extra ‘ground state energy’ making the Hamiltonian
I’m wondering if there’s any way to see the extra here as arising from the Duflo isomorphism. I’m stuck because this would seem to require thinking of as lying in the center of the universal enveloping algebra of some Lie algebra, and while it is in the center of the universal enveloping algebra of the Heisenberg algebra, that Lie algebra is nilpotent, so it seems the Duflo isomorphism doesn’t give any corrections.
July 25, 2025
The Clowder Project
Posted by John Baez
guest post by Emily de Oliveira Santos
I’d like to share here a personal project which might be of interest to the readers of this blog: the Clowder Project.
Clowder is a wiki and reference work for category theory built using the same general infrastructure and tag system of the Stacks Project, Gerby. The intention is for it to eventually become for category theory what the Stacks Project is for algebraic geometry.
July 24, 2025
2-Rig Conjectures Proved?
Posted by John Baez
Kevin Coulembier has come out with a paper claiming to prove some conjectures that Todd Trimble, Joe Moeller and I made in 2-Rig extensions and the splitting principle:
- Kevin Coulembier, Invertible exterior powers.
Lawvere’s Work on Arms Control
Posted by John Baez
Did you know that Lawvere did classified work on arms control in the 1960s, back when he was writing his thesis? Did you know that the French government offered him a job in military intelligence?
July 7, 2025
How to Count n-Ary Trees
Posted by John Baez
How do you count rooted planar -ary trees with some number of leaves? For this puzzle leads to the Catalan numbers. These are so fascinating that the combinatorist Richard Stanley wrote a whole book about them. But what about ?
I’ll sketch one way to solve this puzzle using generating functions. This will give me an excuse to talk a bit about something called ‘Lagrange inversion’.
June 26, 2025
Counting with Categories (Part 3)
Posted by John Baez
Here’s my third and final set of lecture notes for a 4-hour minicourse at the Summer School on Algebra at the Zografou campus of the National Technical University of Athens.
Part 1 is here, and Part 2 is here.
Counting with Categories (Part 2)
Posted by John Baez
Here’s my second set of lecture notes for a 4-hour minicourse at the Summer School on Algebra at the Zografou campus of the National Technical University of Athens.
Part 1 is here, and Part 3 is here.
June 22, 2025
Counting with Categories (Part 1)
Posted by John Baez
These are some lecture notes for a 4-hour minicourse I’m teaching at the Summer School on Algebra at the Zografou campus of the National Technical University of Athens. To save time, I am omitting the pictures I’ll draw on the whiteboard, along with various jokes and profoundly insightful remarks. This is just the structure of the talk, with all the notation and calculations.
Long-time readers of the -Category Café may find little new in this post. I’ve been meaning to write a sprawling book on combinatorics using categories, but here I’m trying to explain the use of species and illustrate them with a nontrivial example in less than 1.5 hours. That leaves three hours to go deeper.
Part 2 is here, and Part 3 is here.
June 1, 2025
Tannaka Reconstruction and the Monoid of Matrices
Posted by John Baez
You can classify representations of simple Lie groups using Dynkin diagrams, but you can also classify representations of ‘classical’ Lie groups using Young diagrams. Hermann Weyl wrote a whole book on this, The Classical Groups.
This approach is often treated as a bit outdated, since it doesn’t apply to all the simple Lie groups: it leaves out the so-called ‘exceptional’ groups. But what makes a group ‘classical’?
There’s no precise definition, but a classical group always has an obvious representation, you can get other representations by doing obvious things to this obvious one, and it turns out you can get all the representations this way.
For a long time I’ve been hoping to bring these ideas up to date using category theory. I had a bunch of conjectures, but I wasn’t able to prove any of them. Now Todd Trimble and I have made progress:
- John Baez and Todd Trimble, Tannaka reconstruction and the monoid of matrices.
We tackle something even more classical than the classical groups: the monoid of matrices, with matrix multiplication as its monoid operation.