Posts with this logo use 
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.
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This approach is often treated as a bit outdated, since it doesn’t apply to all the simple Lie groups: it leaves out the so-called ‘exceptional’ groups.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.