Hopf polyads

TL;DR

This paper introduces Hopf polyads as a unifying framework for Hopf monads and group actions on monoidal categories, providing new insights into their structure and relationships.

Contribution

It defines Hopf polyads, explores their properties, and shows how they generalize existing concepts like Hopf monads and Hopf categories, including a new description of centers of graded tensor categories.

Findings

Lift of a Hopf polyad yields a simpler, equivalent structure.

Hopf polyads can be 'wrapped' into Hopf monads under certain conditions.

Generalizes the description of centers of graded fusion categories.

Abstract

We introduce Hopf polyads in order to unify Hopf monads and group actions on monoidal categories. A polyad is a lax functor from a small category (its source) to the bicategory of categories, and a Hopf polyad is a comonoidal polyad whose fusion operators are invertible. The main result states that the lift of a Hopf polyad is a strong (co)monoidal action-type polyad (or strong monoidal pseudofunctor). The lift of a polyad is a new polyad having simpler structure but the same category of modules. We show that, under certain assumptions, a Hopf polyad can be `wrapped up' into a Hopf monad. This generalizes the fact that finite group actions on tensor categories can be seen as Hopf monads. Hopf categories in the sense of Batista, Caenepeel and Vercruysse can be viewed as Hopf polyads in a braided setting via the notion of Hopf polyalgebras. As a special case of the main theorem, we generalize a description of the center of graded fusion category due to Turaev and Virelizier to tensor categories: if $C$ is a $G$-graded (locally bounded) tensor category, then $G$ acts on the relative center of $C$ with respect to the degree one part $C_1$, and the equivariantization of this action is the center of $C$.