Wow! I just learned an objective reason why sets and vector spaces are special!
Of course we all know math relies heavily on set theory and linear algebra. And if you know category theory, you can say various things about why the categories Set and Vect are particularly convenient frameworks for calculation. But I'd never known a *theorem* that picks out these categories, and a few others.
Briefly: these are categories where all objects are 'free'.
(1/n)
@johncarlosbaez This is a really interesting thread – thank you
For anyone who’s interested, the image in the opening post is part of the first page of the 2016 paper by Keith Kearnes, Emil Kiss and Ágnes Szendrei, ‘Varieties whose finitely generated members are free’
The paper that starts as a jumping-off point for John’s thread is freely available here: https://doi.org/10.48550/arXiv.1508.03807
We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.
John Carlos Baez
Anna Nicholson