We know that Frobenius objects in a monoidal category obey a diagrammatic string calculus. We also know that trees are polynomial functors (Kock  Polynomial functors and trees). The string calculus for Frobenius Algebras is very much like trees, except for the fact that since the monoidal product in the base category is symmetric, the strings can be crossed and uncrossed with no meaning as to which line is on top of another line. I am wondering if the string calculus itself can be modelled as a polynomial monad? Also, we know that internal Frobenius algebras in a monoidal category are generated by Frobenius monads on the base category (Heunen and Karvonen  Monads on dagger categories). What is the relationship between the Frobenius monad and the polynomial monad for the Frobenius string calculus?
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$\begingroup$ Could you clarify what you mean by "the string calculus for Frobenius Algebras is very much like trees, except [...] strings can be crossed and uncrossed"? The combinators of a Frobenius object can be used to build up arbitrary unitrivalent graphs (i.e., they can contain cycles, and are not necessarily trees). Do you mean something else? $\endgroup$ – Noam Zeilberger May 2 '18 at 17:47