Frobenius Algebras and Dual Bimodules in Monoidal 2-Categories

Abstract

We explicitly construct dual bimodules in a semistrict monoidal 2-category, using Frobenius algebra structure. The main result shows that a coherent dual of the underlying object can be promoted to a coherent dual of the bimodule, with zigzag 2-isomorphisms additionally require special Frobenius structures. We also prove that every special Frobenius algebra in $2Vect$ is rigid, via a categorified Casimir object argument, and discuss the relationship between the Frobenius, rigid, special Frobenius, and separable algebra hierarchies.

24 pages, 56 string diagrams, 31 commutative diagrams