nLab monoidal monad

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

We discuss how monoidal monads functorially give rise to 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

• Cat dentes the 2-category of categories, functors and natural transformations,

• $B V$ denotes the delooping one-object 2-category of the monoidal category $V$,

• $(B V)^{op}$ denotes its 1-cell dual, hence the same 2-category except with 1-morphism composition (here: tensor product) in reverse order),

• 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

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

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:

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

$$\,$$

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

Proposition

The strong monad structures obtained from monoidal monads via Def. are commutative monads (Def. ).

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:

1. a commutative strong monad

2. a symmetric monoidal monad.

Tensor product of algebras and multimorphisms

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

See examples of commutative monads.

See also

• strong monad, tensorial strength
• monoidal functor
• commutative monad, commutative algebraic theory

References

• Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970) 1-10 [doi:10.1007/BF01220868]

• Anders Kock, Closed categories generated by commutative monads, Journal of the Australian Mathematical Society 12 4 (1971) 405-424 [doi:10.1017/S1446788700010272, pdf]

• Anders Kock, Strong functors and monoidal monads, Arch. Math 23 (1972) 113–120 [doi:10.1007/BF01304852, 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

• Tobias Fritz, Paolo Perrone and Sharwin Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, 2019 (arXiv:1910.03752)