John Baez and James Dolan, 2023-10-09

Chat about tuning systems, especially just intonation.
The category of commutative separable algebras has special commutative Frobenius algebras as objects but algebra homomorphisms (not necessarily preserving the coalgebra structure) as morphisms, so 'separability' is being treated as a mere property of an algebra, namely the property that their exists a special Frobenius structure. We set things up this way because the only algebra homomorphisms between Frobenius algebras also preserving the coalgebra structure are isomorphisms.
The opposite of the category of separable commutative monoids in any symmetric monoidal category is cartesian - in fact a Boolean pretopos, by Carboni's work:
Aurelio Carboni, Matrices, relations, and group representations, www.sciencedir...
John Baez, Grothendieck-Galois-Brauer Theory (Part 1), golem.ph.utexa...
The only tensor-invertible object in a cartesian monoidal category is the terminal object. A special commutative Frobenius monoid is 'extraspecial' if the inverse of the unit is the counit. Since the unit is a monoid homomorphism, this shows that the only extraspecial commutative Frobenius monoid is the terminal object. Nonetheless the free symmetric monoidal category on an extraspecial commutative Frobenius monoid, i.e. the prop for extraspecial commutative Frobenius monoids, is interesting: it's the category of corelations between finite sets:
Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids, arxiv.org/abs/...
The free symmetric monoidal category on a special commutative Frobenius monoid, i.e. the prop for special commutative Frobenius monoids, is the category of cospans between finite sets:
Steve Lack, Composing PROPs, www.tac.mta.ca/....
The free symmetric monoidal category on a commutative Frobenius monoid, i.e. the prop for commutative Frobenius monoids, is the category of oriented 2-dimensional cobordisms between oriented 1-dimensional compact manifolds:
Joachim Kock, Frobenius algebras and 2D topological quantum field theories (short version), mat.uab.cat/~k...
For more on this whole series of conversations, go here:
math.ucr.edu/h...