monoidal monad (Rev #36, changes) in nLab

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Definition

\begin{definition} \label{MonoidalMonad} \linebreak A monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations. \end{definition}

This notion originates inside the statement of Kock 1970, Thm. 3.2.

In components, this means (cf. Kock 1970, p. 8, review includes Seal 2012, §1.2):

Monoidal structure on a monad

ℰ:C→C \mathcal{E} \,\colon\, \mathbf{C} \to \mathbf{C}

c∈C⊢ret c ℰ:c→ℰ(c) c \in \mathbf{C} \;\;\;\; \vdash \;\;\;\; ret^{\mathcal{E}}_c \,\colon\, c \to \mathcal{E}(c)

c∈C⊢join c ℰ:ℰ∘ℰ(c)→ℰ(c) c \in \mathbf{C} \;\;\;\; \vdash \;\;\;\; join^{\mathcal{E}}_{c} \,\colon\, \mathcal{E} \circ \mathcal{E}(c) \to \mathcal{E}(c)

acting on a monoidal category (C,⊗,𝟙)(\mathbf{C}, \otimes, \mathbb{1}) is:

1. lax monoidal functor-structure on ℰ\mathcal{E}

   ⊢ϵ ℰ:11→ℰ(11) \vdash \;\;\;\; \epsilon^{\mathcal{E}} \,\colon\, 1\!\!1 \to \mathcal{E}(1\!\!1)

   c,c′∈C⊢μ c,c′ ℰ:ℰ(c)⊗ℰ(c′)→ℰ(c⊗c′) c,c' \in \mathbf{C} \;\;\;\; \vdash \;\;\;\; \mu^{\mathcal{E}}_{c,c'} \,\colon\, \mathcal{E}(c) \otimes \mathcal{E}(c') \to \mathcal{E}(c \otimes c')

2. such that the monad structure transformations ret ℰret^{\mathcal{E}} and join ℰjoin^{\mathcal{E}} are monoidal transformations in that together with the lax monoidal structure ϵ ℰ\epsilon^{\mathcal{E}} and μ ℰ\mu^{\mathcal{E}} they make the following diagrams commute:

First of all, the lax monoidal unit must coincide with the monad unit

(1)

\begin{tikzcd}[% nodes={scale=1.25}, arrows={thick},% sep=large%] & 1!!1 \ar[dl, equals] \ar[dr, { \epsilon^{\mathcal{E}} }] \ 1!!1 \ar[rr, { \mathrm{ret}^{\mathcal{E}}_{1!!1} }{swap}] && \mathcal{E}(1!!1) \end{tikzcd}

which already implies the unit diagram for the join operation:

\begin{tikzcd}[% nodes={scale=1.25}, arrows={thick},% sep=large%] & 1!!1 \ar[dl, { \mathcal{E}(\epsilon^{\mathcal{E}}) \circ \epsilon^{\mathcal{E}} }{description}] \ar[dr, { \epsilon^{\mathcal{E}} }] \ \mathcal{E}\circ\mathcal{E}(1!!1) \ar[rr, { \mathrm{join}^{\mathcal{E}}_{1!!1} }{swap}] && \mathcal{E}(1!!1) \end{tikzcd}

and then the two main conditions:

\begin{tikzcd}[% nodes={scale=1.25}, arrows={thick},% sep=large%] c \otimes c’ \ar[dd, equals] \ar[rr, { \big(\mathrm{ret}^{\mathcal{E}}_c\big) \otimes \big(\mathrm{ret}^{\mathcal{E}}_{c}\big) }] && \mathcal{E}(c) \otimes \mathcal{E}(c’) \ar[dd, { \mu^{\mathcal{E}}_{c,c} }] \ \ c \otimes c’ \ar[rr, { \mathrm{ret}^{\mathcal{E}}_{c \otimes c} }] && \mathcal{E}(c \otimes c’) \end{tikzcd}

and

\begin{tikzcd}[% nodes={scale=1.25}, arrows={thick},% sep=large%] \big(\mathcal{E}\circ\mathcal{E}(c)\big) \otimes \big(\mathcal{E}\circ\mathcal{E}(c’)\big) \ar[d, { \mu^{\mathcal{E}}_{\mathcal{E}(c), \mathcal{E}(c)} }] \ar[rr, { \big( \mathrm{join}^{\mathcal{E}}_c \big) \otimes \big( \mathrm{join}^{\mathcal{E}}_{c} \big) }] && \big(\mathcal{E}(c)\big) \otimes \big( \mathcal{E}(c’) \big) \ar[dd, { \mu^{\mathcal{E}}_{c,c} }] \ \mathcal{E} \Big( \big(\mathcal{E}(c)\big) \otimes \big(\mathcal{E}(c’)\big) \Big) \ar[d, { \mathcal{E}\big( \mu^{\mathcal{E}}_{c,c} \big) }] \ \mathcal{E} \circ \mathcal{E}(c \otimes c’) \ar[rr, { \mathrm{join}^{\mathcal{E}}_{c \otimes c} }{swap}] && \mathcal{E}(c \otimes c’) \end{tikzcd}

Moreover, if C\mathbf{C} is even a symmetric monoidal category with braiding σ\sigma, then a monoidal monad on C\mathbf{C} as above is a symmetric monoidal monad if the underlying monoidal functor is a symmetric monoidal functor.

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

τ A,B:A⊗T(B)→T(A⊗B) \tau_{A, B} \;\colon\; A \otimes T(B) \to T(A \otimes B)

\,

σ A,B:T(A)⊗B→T(A⊗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 TT preserves the two-sided monoidal action of VV on itself, in an appropriate 2-categorical sense. More precisely: the two-sided action of VV on itself is a lax functor of 2-categories

V˜:BV×(BV) op→Cat \tilde{V} \colon B V \times (B V)^{op} \to Cat

where

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

• BVB V denotes the delooping one-object 2-category of the monoidal category VV,

• (BV) op(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 V˜→V˜\tilde{V} \to \tilde{V}.

\begin{definition} \label{StrongMonad} There is a category of strong functors V→VV \to V, whose morphisms are natural transformations λ:S→T\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. \end{definition}

\begin{definition} \label{CommutativeMonad} A strong monad (T:V→V,m:TT→T,u:1→T)(T \colon V \to V, m \colon T T \to T, u: 1 \to T) (def. \ref{StrongMonad}) is a commutative monad if there is an equality of natural transformations α=β\alpha = \beta where

• α\alpha is the composite

  TA⊗TB→σ A,TBT(A⊗TB)→T(τ A,B)TT(A⊗B)→m(A⊗B)T(A⊗B).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).

• β\beta is the composite

  TA⊗TB→τ TA,BT(TA⊗B)→T(σ A,B)TT(A⊗B)→m(A⊗B)T(A⊗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).

\end{definition}

From monoidal to commutative strong monads

\begin{definition} \label{StrengthFromMonoidalness} (strength from monoidalness) \linebreak For (T:V→V,u:id→T,m:TT→T)(T \colon V \to V, u \colon id \to T, m \colon T T \to T) a monoidal monad (Def. \ref{MonoidalMonad}), with the monoidal monad-structure on the underlying functor denoted by

α A,B:T(A)⊗T(B)→T(A⊗B),ι=u(I):I→T(I), \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:

τ A,B≔(A⊗T(B)→u A⊗idT(A)⊗T(B)→α A,BT(A⊗B)), \tau_{A, B} \;\coloneqq\; \big( A \otimes T(B) \overset{u_A \otimes id}{\to} T(A) \otimes T(B) \overset{\alpha_{A, B}}{\to} T(A \otimes B) \big) \,,

\,

σ A,B≔(T(A)⊗B→id⊗u BT(A)⊗T(B)→α A,BT(A⊗B)). \sigma_{A, B} \;\coloneqq\; \big( T(A) \otimes B \overset{id \otimes u_B}{\to} T(A) \otimes T(B) \overset{\alpha_{A, B}}{\to} T(A \otimes B) \big) \,.

\end{definition}

This construction is functorial:

\begin{proposition} \label{cf} Given monoidal monads SS and TT on a monoidal category CC, a morphism of monads α:S⇒T\alpha \colon S\Rightarrow T is a morphism of the induced strong monad structures (Def. \ref{StrengthFromMonoidalness}) if and only if it is a monoidal natural transformation. \end{proposition}

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

This relation has a converse:

\begin{proposition} \label{BijectSymmMonoidalMonadsWithCommStrongMonads} \linebreak For a monad TT on (the underlying category of) a symmetric closed monoidal category, there is a bijection between the structure on TT of:

1. a commutative strong monad

2. a symmetric monoidal monad.

\end{proposition}

This is due to Kock (1972), Thm. 2.3), a detailed review is in GLLN08, §7.3, §A.4.

Tensor product of algebras and multimorphisms

See here.

Monoidal structure on the Kleisli category

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

• On objects, by the tensor product ⊗\otimes of CC;

• On morphisms, given k:X→TAk:X\to TA and h:Y→TBh:Y\to TB, their product is the map X⊗Y→T(A⊗B)X\otimes Y \to T(A\otimes B) obtained by the composition \begin{tikzcd}[% nodes={scale=1.25}, arrows={thick},%] X\otimes Y \ar{r}{f\otimes g} & TA \otimes TB \ar{r}{\nabla} & T(A\otimes B), \end{tikzcd} where ∇\nabla is the monoidal multiplication of TT.

• The associator and unitor are induced by those of CC.

Examples

See examples of commutative monads.

See also

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

References

The definition originally appears inside statement and proof of Thm. 3.2 in:

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

establishing right away the relation to commutative strong monads (in the case that the underlying monoidal category is symmetric monoidal closed) which is further expanded on in:

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

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

Further discussion:

• 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) [[numdam:CTGDC_1975__16_3_217_0](http://www.numdam.org/item/CTGDC_1975__16_3_217_0), pdf]

• William Keigher, Symmetric monoidal closed categories generated by commutative adjoint monads, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978) 269-293 [[numdam:CTGDC_1978__19_3_269_0](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1978__19_3_269_0), pdf]

• Kosta Dosen, Zoran Petric, Coherence for Monoidal Monads and Comonads, Mathematical Structures in Computer Science , 20 4 (2010) 545-561 [[arXiv:0907.2199](https://arxiv.org/abs/0907.2199), doi:10.1017/S0960129510000034]

• Gavin J. Seal, Tensors, monads and actions, Theory and Applications of Categories 28 15 (2013) 403-434. [[arXiv:1205.0101](http://arxiv.org/abs/1205.0101), tac:28-15]

  (on the Eilenberg-Moore categories of monoidal monads)

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

Discussion in the context of monads in computer science:

• Jean Goubault-Larrecq, Slawomir Lasota, David Nowak, Logical Relations for Monadic Types, Mathematical Structures in Computer Science, 18 6 (2008) 1169-1217 [[arXiv:cs/0511006](https://arxiv.org/abs/cs/0511006), doi:10.1017/S0960129508007172]

A statement in the above text is from

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