Hopf monads on monoidal categories

TL;DR

This paper extends the concept of Hopf monads to arbitrary monoidal categories, providing a unifying framework that generalizes Hopf algebras and algebroids, with applications to tensor categories and module theory.

Contribution

It introduces Hopf monads on monoidal categories, generalizing Hopf algebras and algebroids, and explores their properties and connections to module categories and adjunctions.

Findings

Any finite tensor category is modules over a Hopf algebroid.

Hopf monads from Hopf algebras in the center are exactly augmented Hopf monads.

Extension of Hopf module decomposition theorem to Hopf monads.

Abstract

We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid. Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford-Majid bosonization of Hopf algebras). We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedler's Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).