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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.

Posted at 10:23 PM UTC | Permalink | Followups (19)

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).

Posted at 4:02 PM UTC | Permalink | Post a Comment

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.

Posted at 12:06 PM UTC | Permalink | Followups (7)

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 XX is a finite set, and for each x,yXx,y \in X we have a number Z(x,y)=Z(y,x)[0,1]Z(x,y) = Z(y,x) \in [0,1] that specifies how “similar” xx and yy are. (Typically we also assume Z(x,x)=1Z(x,x) = 1.) Fix a parameter q[0,]q \in [0,\infty]. If pp is a probability distribution on XX, then the quantity D q Z(p)=( xsupp(p)( ysupp(p)Z(x,y)p(y)) q1p(x)) 1/(1q) D_q^Z(p) = \left(\sum_{x\in supp(p)} \left( \sum_{y\in supp(p)} Z(x,y) p(y)\right)^{q-1} p(x)\right)^{1/(1-q)} (with the cases q=1,q=1,\infty defined by taking limits) can be interpreted as the “effective number of points” in XX, taking into account both the similarities between points as quantified by ZZ and the weights specified by pp. Its logarithm logD q Z(p)\log D_q^Z(p) is a refinement of the qq-Rényi entropy of pp. The main motivating example is when XX is a set of species of organisms present in an ecosystem, and D q Z(p)D_q^Z(p) quantifies the “effective number of species” in XX, 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.

Posted at 4:34 PM UTC | Permalink | Followups (4)

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.

Posted at 12:22 PM UTC | Permalink | Followups (2)

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 12(p 2+q 2)\frac{1}{2}(p^2 + q^2), while quantum mechanically it gets some extra ‘ground state energy’ making the Hamiltonian

H=12(p 2+q 2+1) H = \frac{1}{2}(p^2 + q^2 + 1)

I’m wondering if there’s any way to see the extra +12+ \frac{1}{2} here as arising from the Duflo isomorphism. I’m stuck because this would seem to require thinking of HH 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.

Posted at 1:48 PM UTC | Permalink | Followups (6)

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.

Posted at 11:18 AM UTC | Permalink | Followups (3)

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:

Posted at 2:58 PM UTC | Permalink | Followups (6)

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?

Posted at 12:48 PM UTC | Permalink | Followups (15)

July 7, 2025

How to Count n-Ary Trees

Posted by John Baez

How do you count rooted planar nn-ary trees with some number of leaves? For n=2n = 2 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 n>2n \gt 2?

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’.

Posted at 8:24 AM UTC | Permalink | Followups (8)

June 26, 2025

Counting with Categories (Part 3)

Posted by John Baez

Here’s my third and final set of lecture notes for a 412\frac{1}{2}-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.

Posted at 1:03 PM UTC | Permalink | Followups (7)

Counting with Categories (Part 2)

Posted by John Baez

Here’s my second set of lecture notes for a 412\frac{1}{2}-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.

Posted at 1:00 PM UTC | Permalink | Followups (3)

June 22, 2025

Counting with Categories (Part 1)

Posted by John Baez

These are some lecture notes for a 412\frac{1}{2}-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 nn-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.

Posted at 2:59 PM UTC | Permalink | Followups (5)

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:

We tackle something even more classical than the classical groups: the monoid of n×nn \times n matrices, with matrix multiplication as its monoid operation.

Posted at 4:25 PM UTC | Permalink | Followups (15)

April 16, 2025

Position in Stellenbosch

Posted by John Baez

guest post by Bruce Bartlett

Stellenbosch University is hiring!

Posted at 12:21 AM UTC | Permalink