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Owen Lynch (Dec 29 2021 at 19:36):The canonical PROP with a supply of Frobenius algebras is .
However, has in fact two Frobenius algebras on it. One is given by equality, where the generators are the "clone" operation and the "forget" operation. The other is given by summation, where the generators are addition and zero.
Is there a combinatorial PROP with both of these Frobenius structures on it that has a functor into ?
Owen Lynch (Dec 29 2021 at 19:37):I suppose I would have to also think about exactly how the two Frobenius algebras interact.
John Baez (Dec 29 2021 at 20:09):I think the answer you want is in paper with Jason Erbele, Categories in control, and also in various papers by Zanasi and collaborators, including Interacting Hopf algebras by Bonchi, Sobocinski and Zanasi, and Zanasi's thesis Interacting Hopf Algebras: the theory of linear systems.
John Baez (Dec 29 2021 at 20:11):Part of the point is that if you take the multiplication of one of the Frobenius algebras you're talking about and the comultiplication of the other, you get a Hopf algebra! So part of the answer to your question "how do the two Frobenius algebras interact?" is that they are also a pair of interacting Hopf algebras.
John Baez (Dec 29 2021 at 20:11):It's an amazingly rich structure, and Sobocinski set up a whole blog to talk about it.
Nathaniel Virgo (Dec 30 2021 at 03:42):the blog in question: https://graphicallinearalgebra.net