Contents

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Definition

Let $C$ and $D$ be monoidal categories, and $F \,\colon\, C \rightleftarrows D \,\colon\, G$ a comonoidal adjunction, i.e. an adjunction $F\dashv G$ in the 2-category of colax monoidal functors. (By doctrinal adjunction, this is actually equivalent to requiring that $G$ is a strong monoidal functor.) This adjunction is a Hopf adjunction if the canonical morphisms

$F(x \otimes G y) \longrightarrow F x \otimes y$
$F(G y \otimes x) \longrightarrow y \otimes F x$

are isomorphisms for any $x\in C$ and $y\in D$.

Of course, if $C$, $D$, $F$, and $G$ are symmetric, then it suffices to ask for one of these. If $C$ and $D$ are moreover cartesian monoidal, then any adjunction is comonoidal, and the condition is also (mis?)named Frobenius reciprocity.

## Properties

• If $C$ and $D$ are closed, then by the calculus of mates, saying that $F\dashv G$ is Hopf is equivalent to asking that $G$ be a closed monoidal functor, i.e. preserve internal-homs up to isomorphism.

• If $F\dashv G$ is a Hopf adjunction, then its induced monad $G F$ is a Hopf monad. Conversely, the Eilenberg-Moore adjunction of a Hopf monad is a Hopf adjunction.