Citation: Kearnes, Keith A., Emil W. Kiss, and Ágnes Szendrei. "Varieties whose finitely generated members are free." Algebra universalis 79.1 (2018): 3.
Web: https://arxiv.org/abs/1508.03807
Tags: Mathematical, Monoidal-categories
This paper proves a very interesting theorem. It shows that every variety (in the sense of universal algebra) which has all free modules must be equivalent to either the category of sets, pointed sets, modules over a division ring, or affine spaces over a division ring. Outside of this seemingly uninteresting distinction between pointed/non-pointed, this seems to imply there is something very special about sets and modules over a division ring. When the ground field is R, the only division algebras are R, C, and H by Frobenius' theorem. This theorem seems to be singling out classical phases, bosonic quantum phases, and fermionic quantum phases. It is not immediately clear how to apply this to our understanding of classical/bosonic/fermionic topological phases of matter, though...