On Hopf adjunctions, Hopf monads and Frobenius-type properties

TL;DR

This paper explores the relationships between Hopf and coHopf adjunctions, monads, and Frobenius properties in monoidal categories, establishing conditions for dualities and Frobenius structures in these contexts.

Contribution

It provides new characterizations of Hopf and coHopf adjunctions, conditions for Frobenius monoidal functors, and extends these results to Hopf monads under certain exactness and dualizability assumptions.

Findings

Characterization of linearly distributive adjunctions via Hopf and coHopf adjunctions

Conditions under which a strong monoidal functor has a Frobenius monoid as its image of the unit

Extension of Frobenius properties to Hopf monads with dualizable objects

Abstract

Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)\dashv (R,L)$ is a linearly distributive adjunction, if and only if $L\dashv U$ is a Hopf adjunction and $U\dashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $L\dashv U$, to have as right adjoint a twisted version of the left adjoint $L$. In particular, the resulting adjunction will be (left) coHopf. One step further, we prove that if $L$ is precomonadic and $L\mathbf I$ is a Frobenius monoid (where $\mathbf I$ denotes the unit object of the monoidal category), then $L\dashv U\dashv L$ is an ambidextrous adjunction, and $L$ is a Frobenius monoidal functor. We transfer these results to Hopf monads: we show that under suitable exactness assumptions, a Hopf monad $T$ on a monoidal category has a right adjoint which is also a Hopf comonad, if the object $T\mathbf I$ is dualizable as a free $T$-algebra. In particular, if $T\mathbf I$ is a Frobenius monoid in the monoidal category of $T$-algebras and $T$ is of descent type, then $T$ is a Frobenius monad and a Frobenius monoidal functor.