Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Monoidal categories

monoidal categories

## In higher category theory

#### 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

# Contents

## Definition

###### Definition

A monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations.

###### Remark

The notion of monoidal monad is equivalent to the notion of commutative monad. To put it another way: to give a commutative strength for a monad $T$ is to give a monoidal monad whose underlying monad is $T$. We explore this connection below.

## Properties

### Relation to commutative strong monads

#### Strength

First to recall the notion of a strong monad:

Let $V$ be a monoidal category. We say a functor $T \colon V \to V$ is strong if there are given left and right tensorial strengths, namely natural transformations of the form:

$\tau_{A, B} \;\colon\; A \otimes T(B) \to T(A \otimes B)$
$\,$
$\sigma_{A, B} \;\colon\; T(A) \otimes B \to T(A \otimes B) \,,$

which are suitably compatible with one another: The full set of coherence conditions may be summarized by saying $T$ preserves the two-sided monoidal action of $V$ on itself, in an appropriate 2-categorical sense. More precisely: the two-sided action of $V$ on itself is a lax functor of 2-categories

$\tilde{V} \colon B V \times (B V)^{op} \to Cat$

where

###### Remark

In the setting where $V$ is symmetric monoidal, we will assume that the left and right strengths $\tau$ and $\sigma$ are related by the symmetry in the obvious way, by a commutative square

$\array{ A \otimes T(B) & \stackrel {\tau_{A, B}} {\longrightarrow} & T(A \otimes B) \\ \mathllap{^c} \big\downarrow & & \big\downarrow \mathrlap{^{T(c)}} \\ T(B) \otimes A & \underset{ \sigma_{B, A}} {\longrightarrow} & T(B \otimes A) }$

where the $c$‘s are instances of the symmetry isomorphism.

###### Definition

There is a category of strong functors $V \to V$, whose morphisms are natural transformations $\lambda \colon S \to T$ which are compatible with the strengths in the obvious sense. Under composition, this is a strict monoidal category.

The monoid objects in this monoidal category are called strong monads.

###### Definition

A strong monad $(T \colon V \to V, m \colon T T \to T, u: 1 \to T)$ (def. ) is a commutative monad if there is an equality of natural transformations $\alpha = \beta$ where

• $\alpha$ is the composite

$T A \otimes T B \stackrel{\sigma_{A, T B}}{\to} T(A \otimes T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B).$
• $\beta$ is the composite

$T A \otimes T B \stackrel{\tau_{T A, B}}{\to} T(T A \otimes B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B).$

#### From monoidal to commutative strong monads

###### Definition

(strength from monoidalness)
For $(T \colon V \to V, u \colon id \to T, m \colon T T \to T)$ a monoidal monad (Def. ), with the monoidal monad-structure on the underlying functor denoted by

$\alpha_{A, B} \,\colon\, T(A) \otimes T(B) \to T(A \otimes B), \qquad \iota = u(I) \,:\, I \to T(I) \,,$

Define strengths on both the left and the right by:

$\tau_{A, B} \coloneqq (A \otimes T(B) \stackrel{u A \otimes 1}{\to} T(A) \otimes T(B) \stackrel{\alpha_{A, B}}{\to} T(A \otimes B)),$
$\,$
$\sigma_{A, B} \coloneqq (T(A) \otimes B \stackrel{1 \otimes u B}{\to} T(A) \otimes T(B) \stackrel{\alpha_{A, B}}{\to} T(A \otimes B)) \,.$

###### Proof

In fact, the two composites

$T A \otimes T B \stackrel{\sigma_{A, T B}}{\to} T(A \otimes T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B)$
$\,$
$T A \otimes T B \stackrel{\tau_{T A, B}}{\to} T(T A \otimes B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B)$

are both equal to $\alpha_{A, B}$. We show this for the second composite; the proof is similar for the first. If $\alpha_T$ denotes the monoidal constraint for $T$ and $\alpha_{T T}$ the constraint for the composite $T T$, then by definition $\alpha_{T T}$ is the composite given by

$T T X \otimes T T Y \stackrel{\alpha_T T}{\to} T(T X \otimes T Y) \stackrel{T\alpha_T}{\to} T T(X \otimes Y)$

and so, using the properties of monoidal monads, we have a commutative diagram

$\array{ & & T T X \otimes T Y & \stackrel{\alpha_T}{\to} & T(T X \otimes Y) \\ & ^\mathllap{u \otimes 1} \nearrow & \downarrow^\mathrlap{1 \otimes T u} & & \downarrow^\mathrlap{T(1 \otimes u)} \\ T X \otimes T Y & \stackrel{u \otimes T u}{\to} & T T X \otimes T T Y & \stackrel{\alpha_T T}{\to} & T(T X \otimes T Y) \\ & ^\mathllap{1} \searrow & \downarrow^\mathrlap{m \otimes m} & \searrow^\mathrlap{\alpha_{T T}} & \downarrow^\mathrlap{T\alpha_T} \\ & & T X \otimes T Y & & T T(X \otimes Y) \\ & & & ^\mathllap{\alpha_T} \searrow & \downarrow^\mathrlap{m} \\ & & & & T(X \otimes Y) }$

which completes the proof.

This construction is functorial:

###### Proposition

Given monoidal monads $S$ and $T$ on a monoidal category $C$, a morphism of monads $\alpha \colon S\Rightarrow T$ is a morphism of the induced strong monad structures (Def. ) if and only if it is a monoidal natural transformation.

(e.g. FPR (2019), Prop. C.5)

This relation has a converse:

###### Proposition

(Kock (1972), Thm. 2.3)
For a monad $T$ on (the underlying category of) a symmetric closed monoidal category, there is a bijection between the structure on $T$ of:

See here.

### Monoidal structure on the Kleisli category

The Kleisli category of a monoidal monad $T$ on $C$ inherits the monoidal structure from $C$. In particular, the tensor product is given

• On objects, by the tensor product $\otimes$ of $C$;
• On morphisms, given $k:X\to TA$ and $h:Y\to TB$, their product is the map $X\otimes Y \to T(A\otimes B)$ obtained by the composition

where $\nabla$ is the monoidal multiplication of $T$. * The associator and unitor are induced by those of $C$.