nLab monoidal monad



Higher algebra

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

2-Category theory




A monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations.


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.

Tensorial strengths and commutative monads

As a preliminary, 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

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

(BVB V is the one-object 2-category associated with a monoidal category VV, and (BV) op(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 V˜V˜\tilde{V} \to \tilde{V}.


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}}{\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 cc‘s are instances of the symmetry isomorphism.

There is a category of strong functors VVV \to V, where the morphisms are 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.


Monoids in this monoidal category are called strong monads.


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 monads to commutative monads

Let (T:VV,u:1T,m:TTT)(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

α A,B:T(A)T(B)T(AB),ι=uI: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)uA1T(A)T(B)α A,BT(AB)),\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)),
σ A,B=(T(A)B1uBT(A)T(B)α A,BT(AB)).\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)).

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


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.

Functoriality of the correspondence

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


Given monoidal monads SS and TT on a monoidal category CC, a morphism of monads α:ST\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.

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.


See examples of commutative monads.

See also


  • 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

Last revised on November 30, 2021 at 16:40:58. See the history of this page for a list of all contributions to it.