Frobenius algebras, Hopf algebras and 3-categories

          @misc{ pirsa_PIRSA:17080011,
            doi = {10.48660/17080011},
            url = {https://pirsa.org/17080011},
            author = {Reutter, David},
            keywords = {Quantum Foundations, Quantum Information},
            language = {en},
            title = {Frobenius algebras, Hopf algebras and 3-categories},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {aug},
            note = {PIRSA:17080011 see, \url{https://pirsa.org}}
          }
          

Abstract

It is well known that commutative Frobenius algebras can be represented as topological surfaces, using the graphical calculus of dualizable objects in monoidal 2-categories. We build on related ideas to show that the interacting Frobenius algebras of Duncan and Dunne, which have a Hopf algebra structure, arise naturally in a similar way, by requiring a single 3-morphism in a 3-category to be invertible. We show that this gives a purely geometrical proof of Mueger's version of Tannakian reconstruction of Hopf algebras from fusion categories equipped with a fibre functor. We also relate our results to the theory of lattice code surgery.