monoidal monad


Higher algebra

Monoidal categories

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


Revised on February 12, 2014 12:00:48 by Urs Schreiber (