Interacting Frobenius Algebras are Hopf - Duncan, Ross; Dunne, Kevin (Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, 2016)
Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi [9] have shown that, given a suitable distribution law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise [9] by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model, and recover the system of Bonchi et al as a subtheory in the prime power dimensional case. However the more general theory does not arise from a distributive law.