# Contents

## Idea

One can consider various compatibilities between a (co)monad on a monoidal category and the underlying monoidal product; they are variants of the idea of a distributive law. Similarly, one can look at a compatibility between an action of a monoidal category and a (co)monad on the same category. A Hopf monad satisfies conditions analogous to that of a Hopf monoid (Bruguières 06).

## Definitions

Warning: more than one notion comes up with the names of bimonad and Hopf monad; there are also related notions of monoidal monad and opmonoidal monad as well of a strong monad.

A monoidal monad on a monoidal category is a monad whose underlying endofunctor is a lax monoidal functor and such that the unit and multiplication are monoidal natural transformations. Consequently, the Kleisli category of a monoidal monad has a canonical monoidal structure such that the forgetful functor is strict monoidal.

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### Opmonoidal monads with invertible fusion

An opmonoidal monad on a monoidal category is a monad whose underlying endofunctor is a colax monoidal functor, and such that the unit and multiplication are monoidal natural transformations. An opmonoidal monad is also called a bimonad.

For an opmonoidal monad $T$, one can define the left fusion operator to be the natural transformation

$T(X\otimes T Y) \to T X \otimes T^2 Y \to T X \otimes T Y$

of the opmonoidal constraint of $T$ and its monad multiplication. Similarly, we have the right fusion operator

$T(T X\otimes Y) \to T^2 X \otimes T Y \to T X \otimes T Y$

The fusion operators satisfy certain axioms. In fact, an opmonoidal monad structure on $T$ is uniquely determined by a fusion operator (of either sort) along with the monad unit $Id\to T$ and the opmonoidal unit constraint $T I \to I$, satisfying appropriate axioms.

An opmonoidal monad is called a Hopf monad if both of its fusion operators are invertible. More generally we have left and right Hopf monads where only one fusion operator is invertible. This definition is in Bruguières-Lack-Virelizier.

Alternatively, one might be tempted to define a Hopf monad to be a Hopf monoid in the monoidal category of endofunctors, but that monoidal category is not symmetric or even braided or duoidal, so this doesn’t make sense. However, we can instead ask for a “local” braiding in the form of a mixed distributive law.

Thus, we might define a bimonad to be an endofunctor $H$ equipped with the structure of both a monad and a comonad, along with a distributive law $\lambda: H H \to H H$ satisfying suitable axioms analogous to those of a bimonoid. A bimonad in this sense is a Hopf monad if it has an antipode $s:H\to H$ making the same diagrams commute as for a Hopf monoid. This definition is in Mesablishvili-Wisbauer.

## Examples

• If $H$ is a bimonoid in a braided monoidal category, then the monad $T_H X =H\otimes X$ (whose algebras are $H$-modules) is opmonoidal, with constraints induced by the comultiplication and counit of $H$. If $H$ is moreover a Hopf monoid, then $T_H$ is a (Bruguières-Lack-Virelizier) Hopf monad.

• In fact, if $H$ is a bimonoid as before, then $T_H$ is also a bimonad in the sense of Mesablishvili-Wisbauer, with monad and comonad structures induced by the monoid and comonoid structures of $H$. Moreover, if $H$ is a Hopf monoid, then $T_H$ is a Hopf monad in their sense as well.

## Properties

If $T$ is a (Bruguières-Lack-Virelizier) Hopf monad on a closed monoidal category, then its category of algebras is also closed and the monadic forgetful functor preserves internal-homs.