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## "Varieties whose finitely generated members are free", Keith Kearnes, Emil Kiss, Agnes Szendrei, 2016

*Reviewed March 23, 2024*

*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...