nLab monoidal monad

Contents

Context

Higher algebra

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher 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.

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

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

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

cCjoin c :(c)(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 \mathcal{E}

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

    c,cCμ c,c :(c)(c)(cc) 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 ret^{\mathcal{E}} and join join^{\mathcal{E}} are monoidal transformations in that together with the lax monoidal structure ϵ \epsilon^{\mathcal{E}} and μ \mu^{\mathcal{E}} they make the following diagrams commute:

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

(1)

which already implies the unit diagram for the join operation:

and then the two main conditions:

and

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.

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

τ A,B:AT(B)T(AB) \tau_{A, B} \;\colon\; A \otimes T(B) \to T(A \otimes B)
\,
σ A,B:T(A)BT(AB), \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) opCat \tilde{V} \colon B V \times (B V)^{op} \to Cat

where

Remark

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

AT(B) τ A,B T(AB) c T(c) T(B)A σ B,A T(BA) \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 cc‘s are instances of the symmetry isomorphism.

Definition

There is a category of strong functors VVV \to V, whose morphisms are natural transformations λ:ST\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:VV,m:TTT,u:1T)(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 α=β\alpha = \beta where

  • α\alpha is the composite

    TATBσ A,TBT(ATB)T(τ A,B)TT(AB)m(AB)T(AB).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

    TATBτ TA,BT(TAB)T(σ A,B)TT(AB)m(AB)T(AB).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:VV,u:idT,m:TTT)(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

α A,B:T(A)T(B)T(AB),ι=u(I):IT(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(AT(B)u AidT(A)T(B)α A,BT(AB)), \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)Bidu BT(A)T(B)α A,BT(AB)). \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) \,.

Proposition

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

Proof

In fact, the two composites

TATBσ A,TBT(ATB)T(τ A,B)TT(AB)m(AB)T(AB)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)
\,
TATBτ TA,BT(TAB)T(σ A,B)TT(AB)m(AB)T(AB)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 α A,B\alpha_{A, B}. We show this for the second composite; the proof is similar for the first. If α T\alpha_T denotes the monoidal constraint for TT and α TT\alpha_{T T} the constraint for the composite TTT T, then by definition α TT\alpha_{T T} is the composite given by

TTXTTYα TTT(TXTY)Tα TTT(XY)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

TTXTY α T T(TXY) u1 1Tu T(1u) TXTY uTu TTXTTY α TT T(TXTY) 1 mm α TT Tα T TXTY TT(XY) α T m T(XY)\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 SS and TT on a monoidal category CC, a morphism of monads α:ST\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


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.

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:XTAk:X\to TA and h:YTBh:Y\to TB, their product is the map XYT(AB)X\otimes Y \to T(A\otimes B) obtained by the composition

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

References

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

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:

Further discussion:

Discussion in the context of monads in computer science:

A statement in the above text is from

Last revised on February 19, 2024 at 00:50:13. See the history of this page for a list of all contributions to it.