Frobenius monoidal functors from (co)Hopf adjunctions

TL;DR

This paper characterizes when right adjoint functors between certain monoidal categories are Frobenius monoidal, linking their properties to Frobenius algebra structures in the domain category, and applies this to construct Frobenius algebras in the Drinfeld center.

Contribution

It provides necessary and sufficient conditions for Frobenius properties of right adjoint functors in monoidal categories, extending previous work and applying to the Drinfeld center.

Findings

R is separable Frobenius monoidal iff R(1) is a separable Frobenius algebra

R is special Frobenius monoidal iff R(1) is a special Frobenius algebra

Construction of Frobenius algebras in the Drinfeld center

Abstract

Let $U:\mathcal{C}\rightarrow\mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a symmetric Frobenius algebra in $\mathcal{C}$. As an application, we construct Frobenius monoidal functors going into the Drinfeld center $\mathcal{Z}(\mathcal{C})$, thereby producing Frobenius algebras in it.