Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Monoidal categories

monoidal categories

## In higher category theory

#### 2-Category theory

2-category theory

# 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.

## Tensorial strengths and commutative monads

As a preliminary, 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

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

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

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

($B V$ is the one-object 2-category associated with a monoidal category $V$, and $(B V)^{op}$ is the same 2-category but with 1-cell composition (= tensoring) in reverse order), and the two-sided strength means we have a structure of lax natural transformation $\tilde{V} \to \tilde{V}$.

###### 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

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

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

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

###### Definition

Monoids 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

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

Let $(T \colon V \to V, u \colon 1 \to T, m \colon T T \to T)$ be a monoidal monad, with structural constraints 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

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

$(m \colon T T \to T, u \colon 1 \to T)$ is a commutative monad.

###### 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.

## Functoriality of the correspondence

The correspondence between monoidal monads and commutative monads is functorial. More precisely,

###### Proposition

Given monoidal monads $S$ and $T$ on a monoidal category $C$, a morphism of monads $\alpha:S\Rightarrow T$ is a morphism of strong monads if and only if it is a monoidal natural transformation.

For a reference, see FPR ‘19, Proposition C.5.

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 $C$.

## Examples

• Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970), 1–10.

• Anders Kock, Strong functors and monoidal monads, Arhus Universitet, Various Publications Series No. 11 (1970). PDF.

• Anders Kock, Closed categories generated by commutative monads (pdf)

• H. Lindner, Commutative monads in Deuxiéme colloque sur l’algébre des catégories. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

• William Keigher, Symmetric monoidal closed categories generated by commutative adjoint monads, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293 (NUMDAM, pdf)

• Gavin J. Seal, Tensors, monads and actions (arXiv:1205.0101)

• Martin Brandenburg, Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)

A statement in the text appears in Appendix C of