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February 26, 2025
Potential Functions and the Magnitude of Functors 2
Posted by Tom Leinster
Despite the “2” in the title, you can follow this post without having read part 1. The whole point is to sneak up on the metricky, analysisy stuff about potential functions from a categorical angle, by considering constructions that are categorically reasonable in their own right. Let’s go!
February 23, 2025
Potential Functions and the Magnitude of Functors 1
Posted by Tom Leinster
Next: Part 2
In the beginning, there were hardly any spaces whose magnitude we knew. Line segments were about the best we could do. Then Mark Meckes introduced the technique of potential functions for calculating magnitude, which was shown to be very powerful. For instance, Juan Antonio Barceló and Tony Carbery used it to compute the magnitude of odd-dimensional Euclidean balls, which turn out to be rational functions of the radius. Using potential functions allows you to tap into the vast repository of knowledge of PDEs.
In this post and the next, I’ll explain this technique from a categorical viewpoint, saying almost nothing about the analytic details. This is category theory as an organizational tool, used to help us understand how the various ideas fit together. Specifically, I’ll explain potential functions in terms of the magnitude of functors, which I wrote about here a few weeks ago.
February 17, 2025
Category Theorists in AI
Posted by John Baez
Applied category theorists are flocking to AI, because that’s where the money is. I avoid working on it, both because I have an instinctive dislike of ‘hot topics’, and because at present AI is mainly being used to make rich and powerful people richer and more powerful.
However, I like to pay some attention to how category theorists are getting jobs connected to AI, and what they’re doing. Many of these people are my friends, so I wonder what they will do for AI, and the world at large — and what working on AI will do to them.
Let me list a bit of what’s going on. I’ll start with a cautionary tale, and then turn to the most important program linking AI and category theory today.
February 4, 2025
A Characterization of Standard Borel Spaces
Posted by John Baez
People in measure theory find it best to work with, not arbitrary measurable spaces, but certain nice ones called standard Borel spaces. I’ve used them myself.
The usual definition of these looks kind of clunky: a standard Borel space is a set equipped with a -algebra for which there exists a complete metric on such that is the -algebra of Borel sets. But the results are good. For example, every standard Borel space is isomorphic to one of these:
- a finite or countably infinite set with its -algebra of all subsets,
- the real line with its sigma-algebra of Borel subsets.
So standard Borel spaces are a good candidate for Tom Leinster’s program of revealing the mathematical inevitability of certain traditionally popular concepts. I forget exactly how he put it, but it’s a great program and I remember some examples: he found nice category-theoretic characterizations of Lebesgue integration, entropy, and the nerve of a category.
Now someone has done this for standard Borel spaces!
February 2, 2025
Backing Up US Federal Databases
Posted by John Baez
I hope you’ve read the news:
- Ethan Singer, Thousands of U.S. government web pages have been taken down since Friday, New York Times, 2 Feburary 2025.
Many of the pages taken down mention DEI, but they also include research papers on optics, chemistry, medicine and much more. They may reappear, but at this time nobody knows.
If you want to help save US federal web pages and databases, here are some things to do.
January 29, 2025
Comagnitude 2
Posted by Tom Leinster
Previously: Part 1
Last time, I talked about the magnitude of a set-valued functor. Today, I’ll introduce the comagnitude of a set-valued functor.
I don’t know how much there is to the comagnitude idea. Let’s see! I’ll tell you all the interesting things I know about it.
Along the way, I’ll also ask an elementary question about group actions that I hope someone knows how to answer.
January 22, 2025
Comagnitude 1
Posted by Tom Leinster
Next: Part 2
In this post and the next, I want to try out a new idea and see where it leads. It goes back to where magnitude began, which was the desire to unify elementary counting formulas like the inclusion-exclusion principle and the simple formula for the number of orbits in a free action of a group on a finite set.
To prepare the ground for comagnitude, I need to present magnitude itself in a slightly different way from usual. I won’t assume you know anything about magnitude, but if you do, watch out for something new: a connection between magnitude and entropy (ordinary, relative and conditional) that I don’t think has quite been articulated before.
January 15, 2025
The Dual Concept of Injection
Posted by Tom Leinster
We’re brought up to say that the dual concept of injection is surjection, and of course there’s a perfectly good reason for this. The monics in the category of sets are the injections, the epics are the surjections, and monics and epics are dual concepts in the usual categorical sense.
But there’s another way of looking at things, which gives a different answer to the question “what is the dual concept of injection?”
December 15, 2024
Random Permutations (Part 14)
Posted by John Baez
I want to go back over something from Part 11, but in a more systematic and self-contained way.
Namely, I want to prove a wonderful known fact about random permutations, the Cycle Length Lemma, using a bit of category theory. The idea here is that the number of -cycles in a random permutation of things is a random variable. Then comes a surprise: in the limit as , this random variable approaches a Poisson distribution with mean . And even better, for different choices of these random variables become independent in the limit.
I’m stating these facts roughly now, to not get bogged down. But I’ll state them precisely, prove them, and categorify them. That is, I’ll state equations involving random variables — but I’ll prove that these equations come from equivalences of groupoids!
December 10, 2024
Martianus Capella
Posted by John Baez
I’ve been blogging a bit about medieval math, physics and astronomy over on Azimuth. I’ve been writing about medieval attempts to improve Aristotle’s theory that velocity is proportional to force, understand objects moving at constant acceleration, and predict the conjunctions of Jupiter and Saturn. A lot of interesting stuff was happening back then!
As a digression from our usual fare on the -Café, here’s one of my favorites, about an early theory of the Solar System, neither geocentric nor heliocentric, that became popular thanks to a quirk of history around the time of Charlemagne. The more I researched this, the more I wanted to know.
December 4, 2024
ACT 2025
Posted by John Baez
The Eighth International Conference on Applied Category Theory (https://easychair.org/cfp/ACT2025) will take place at the University of Florida on June 2-6, 2025. The conference will be preceded by the Adjoint School on May 26-30, 2025.
This conference follows previous events at Oxford (2024, 2019), University of Maryland (2023), Strathclyde (2022), Cambridge (2021), MIT (2020), and Leiden (2019).
Applied category theory is important to a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social science, systems, linguistics and other subjects using category-theoretic tools. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, strengthen the applied category theory community, disseminate the latest results, and facilitate further development of the field.
If you want to give a talk, read on!
November 22, 2024
Axiomatic Set Theory 10: Cardinal Arithmetic
Posted by Tom Leinster
Previously: Part 9.
The course is over! The grand finale was the theorem that
for all infinite sets and . Proving this required most of the concepts and results from the second half of the course: well ordered sets, the Cantor–Bernstein theorem, the Hartogs theorem, Zorn’s lemma, and so on.
I gave the merest hints of the world of cardinal arithmetic that lies beyond. If I’d had more time, I would have got into large sets (a.k.a. large cardinals), but the course was plenty long enough already.
Thanks very much to everyone who’s commented here so far, but thank you most of all to my students, who really taught me an enormous amount.
November 15, 2024
Axiomatic Set Theory 9: The Axiom of Choice
Posted by Tom Leinster
Previously: Part 8. Next: Part 10.
It’s the penultimate week of the course, and up until now we’ve abstained from using the axiom of choice. But this week we gorged on it.
We proved that all the usual things are equivalent to the axiom of choice: Zorn’s lemma, the well ordering principle, cardinal comparability (given two sets, one must inject into the other), and the souped-up version of cardinal comparability that compares not just two sets but an arbitrary collection of them: for any nonempty family of sets , there is some that injects into all the others.
The section I most enjoyed writing and teaching was the last one, on unnecessary uses of the axiom of choice. I’m grateful to Todd Trimble for explaining to me years ago how to systematically remove dependence on choice from arguments in basic general topology. (For some reason, it’s very tempting in that subject to use choice unnecessarily.) I talk about this at the very end of the chapter.
November 8, 2024
Axiomatic Set Theory 8: Well Ordered Sets
Posted by Tom Leinster
Previously: Part 7. Next: Part 9.
By this point in the course, we’ve finished the delicate work of assembling all the customary set-theoretic apparatus from the axioms, and we’ve started proving major theorems. This week, we met well ordered sets and developed all the theory we’ll need. The main results were:
every family of well ordered sets has a least member — informally, “the well ordered sets are well ordered”;
the Hartogs theorem: for every set , there’s some well ordered set that doesn’t admit an injection into ;
a very close relative of Zorn’s lemma that, nevertheless, doesn’t require the axiom of choice: for every ordered set and function assigning an upper bound to each chain in , there’s some chain such that .
I also included an optional chatty section on the use of transfinite recursion to strip the isolated points from any subset of . Am I right in understanding that this is what got Cantor started on set theory in the first place?
November 5, 2024
The Icosahedron as a Thurston Polyhedron
Posted by John Baez
Thurston gave a concrete procedure to construct triangulations of the 2-sphere where 5 or 6 triangles meet at each vertex. How can you get the icosahedron using this procedure?
Gerard Westendorp has a real knack for geometry, and here is his answer.