nLab
monoidal monad

Context

Higher algebra

Monoidal categories

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.

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 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}.

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}}{\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.

Definition

Monoids 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. 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)).
Proposition

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

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 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.

References

Revised on February 12, 2014 12:00:48 by Urs Schreiber (82.113.121.111)