Surface Diagrams for Frobenius Algebras and Frobenius-Schur Indicators in Grothendieck-Verdier Categories

Abstract

Grothendieck-Verdier categories (also known as $\ast$-autonomous categories) generalize rigid monoidal categories, with notable representation-theoretic examples including categories of bimodules, modules over Hopf algebroids, and modules over vertex operator algebras. In this paper, we develop a surface-diagrammatic calculus for Grothendieck-Verdier categories, extending the string-diagrammatic calculus of Joyal and Street for rigid monoidal categories into a third dimension. This extension naturally arises from the non-invertibility of coherence data in Grothendieck-Verdier categories. We show that key properties of Frobenius algebras in rigid monoidal categories carry over to the Grothendieck-Verdier setting. Moreover, we introduce higher Frobenius-Schur indicators for suitably finite $k$-linear pivotal Grothendieck-Verdier categories and prove their invariance under pivotal Frobenius linearly distributive equivalences. The proofs are carried out using the surface-diagrammatic calculus. To facilitate the verification of some of our results, we provide auxiliary files for the graphical proof assistant homotopy$.$io.