Interacting Hopf monoids and Graphical Linear Algebra

          @misc{ pirsa_PIRSA:17080007,
            doi = {10.48660/17080007},
            url = {https://pirsa.org/17080007},
            author = {Sobocinski, Pawel},
            keywords = {Quantum Foundations, Quantum Information},
            language = {en},
            title = {Interacting Hopf monoids and Graphical Linear Algebra},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {aug},
            note = {PIRSA:17080007 see, \url{https://pirsa.org}}
          }
          

Abstract

The interaction of Hopf monoids and Frobenius monoids is the productive nucleus of the ZX calculus, where famously each Frobenius monoid-comonoid pair corresponds to a complementary basis and the Hopf structure describes the interaction between the bases. The theory of Interacting Hopf monoids (IH), introduced by Bonchi, Sobocinski and Zanasi, features essentially the same Hopf-Frobenius interaction pattern. The free symmetric monoidal category generated by IH is isomorphic to the category of linear relations over the field of rationals: thus the string diagrams of IH are an alternative graphical language for elementary concepts of linear algebra. IH has a modular construction via distibutive laws of props, and has been applied as a compositional language of signal flow graphs. In this talk I will outline the equational theory, its construction and applications, as well as report on ongoing and future work.