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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.