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 a suitable general notion of commutative monad (see def. 3 below), as discussed at commutative algebraic theory. 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. 2) 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 monads to commutative monads

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 first composite; the proof is similar for the second. 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.

References

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