monoidal adjunction


Monoidal categories

2-Category theory



A monoidal adjunction is an adjunction between monoidal categories which respects the monoidal structure.

Since there are several types of monoidal functors (lax, colax, and strong) there are several types of “adjunctions between monoidal categories which respect the monoidal structure.” Namely, we could have:

  • An adjunction in the 2-category MonCatMonCat of monoidal categories and strong monoidal functors. In this case both the left and right adjoint are strong.

    Rstrong monoidalLstrong monoidal \array{ \underoverset {\underset{R \, \text{strong monoidal}}{\longrightarrow}} {\overset{L \, \text{strong monoidal}}{\longleftarrow}} {} }

    We call this a strong monoidal adjunction.

  • An adjunction in the 2-category MonCat MonCat_\ell of monoidal categories and lax monoidal functors. In this case the right adjoint is lax, while the left adjoint is necessarily strong (by doctrinal adjunction; see here).

    Rlax monoidalLstrong monoidal \array{ \underoverset {\underset{R \, \text{lax monoidal}}{\longrightarrow}} {\overset{L \, \text{strong monoidal}}{\longleftarrow}} {} }

    In fact, since the right adjoint of an oplax monoidal functor is necessarily a lax monoidal functor (this prop.), it is sufficient to demand that LL be strong monoidal.

    This version, which is one of the most frequently occurring, is often called simply a monoidal adjunction.

  • The dual: an adjunction in the 2-category MonCat cMonCat_c of monoidal categories and colax monoidal functors, in which case the left adjoint is colax and the right adjoint is strong. One might call this an opmonoidal adjunction.

  • A mixed situation, in which the left adjoint is colax, the right adjoint is lax, and the lax and colax structure maps are mates under the adjunction. This is a conjunction in the double category of monoidal categories and lax and colax monoidal functors, so we may call it a monoidal conjunction or a lax/colax monoidal adjunction. By doctrinal adjunction, given any adjunction between monoidal categories, if the right adjoint is lax monoidal, then the left adjoint automatically acquires a colax monoidal structure making the adjunction into a monoidal conjunction, and dually.


As mentioned above, the nature of monoidal adjunctions follows as a special case from generalities of doctrinal adjunctions. For the record, here is an explicit discussion:



𝒞RL𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}

be a pair of adjoint functors between monoidal categories, such that the left adjoint LL is a strong monoidal functor by natural isomorphisms

μ L(X,Y):L(X)L(Y)L(XY) \mu_L(X,Y) \;\colon\; L(X) \otimes L(Y) \overset{\simeq}{\longrightarrow} L(X \otimes Y)


e L:1L(1). e_L \;\colon\; 1 \overset{\simeq}{\longrightarrow} L(1) \,.


  1. the right adjoint RR becomes a lax monoidal functor via natural morphisms

    μ R(X,Y):R(X)R(Y)η(R(X)R(Y))RL(R(X)R(Y))R(μ L 1(R(X),R(Y)))R(LR(X)LR(Y))R(ϵ(X)ϵ(Y))R(XY) \mu_R(X,Y) \;\colon\; R (X) \otimes R(Y) \overset{\eta(R(X) \otimes R(Y))}{\longrightarrow} R L (R(X) \otimes R(Y)) \underoverset{}{ R( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow} R ( L R(X) \otimes L R (Y) ) \overset{R( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow} R(X \otimes Y)


    e R:1η(1)RL(1)R(e L 1)R(1), e_R \;\colon\; 1 \overset{\eta(1)}{\longrightarrow} R L(1) \overset{R(e_L^{-1})}{\longrightarrow} R(1) \,,

    where η\eta denotes the adjunction unit and ϵ\epsilon denotes the adjunction counit, as usual.

  2. For any object A𝒟A \in \mathcal{D} carrying the structure of a monoid object (A,μ A,e A)(A, \mu_A, e_A), then

    1. the unit of the adjunction η(A):ARL(A)\eta(A) \;\colon\; A \longrightarrow R L(A) is a monoid homomorphism with respect to the canonically induced monoid structure on RL(A)R L(A) (this prop.) given by

      μ RL(A):RL(A)RL(A)μ R(L(A))R(L(A)L(A))R(μ L(A))RL(AA)RL(μ A)RL(A) \mu_{R L(A)} \;\colon\; R L(A) \otimes R L(A) \overset{\mu_R(L(A))}{\longrightarrow} R( L(A) \otimes L(A)) \overset{R(\mu_L(A))}{\longrightarrow} R L(A \otimes A) \overset{R L(\mu_A)}{\longrightarrow} R L(A)


      e RL(A):1e RR(1)R(e L)RL(1)RL(e A)RL(A) e_{R L(A)} \;\colon\; 1 \overset{e_R}{\longrightarrow} R(1) \overset{R(e_L)}{\longrightarrow} R L(1) \overset{R L(e_A)}{\longrightarrow} R L(A)
    2. similarly for the counit of the adjunction.


The first statement is discussed at oplax monoidal functor.

For the second statement, we need first need to check that the following square commutes:

AA η(A)η(A) RL(A)RL(A) μ A μ RL(A) A η(A) RL(A) \array{ A \otimes A &\overset{\eta(A) \otimes \eta(A)}{\longrightarrow}& R L (A) \otimes R L (A) \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_{R L(A)}}} \\ A &\underset{\eta(A)}{\longrightarrow}& R L(A) }

Now by definition, the composite of the top and right morphism here is the total diagonal composite in the following diagram:

AA η(AA) RL(AA) R(μ L 1) R(L(A)L(A)) η(A)η(A) RL(η(A)η(A)) R(L(η(A))L(η(A))) R(idid) μ R(L(A)) : RL(A)RL(A) η(RL(A)RL(A)) RL(RL(A)RL(A)) R(μ L 1(RL(A))) R(LRL(A)LRL(A)) R(ϵ(L(A))ϵ(L(A))) R(L(A)L(A)) R(μ L(A)) RL(AA) RL(μ A) RL(A) \array{ && A \otimes A &\overset{\eta(A \otimes A)}{\longrightarrow}& R L(A \otimes A) &\overset{R(\mu_L^{-1})}{\longrightarrow}& R( L(A) \otimes L(A) ) \\ && {}^{\mathllap{\eta(A) \otimes \eta(A)}}\downarrow && {}^{\mathllap{ R L( \eta(A) \otimes \eta(A) ) }}\downarrow && {}^{\mathllap{R( L(\eta(A)) \otimes L(\eta(A)) )}}\downarrow & \searrow^{\mathrlap{R(id \otimes id)}} \\ \mu_R(L(A)) &\colon& R L (A) \otimes R L ( A) &\overset{\eta(R L (A) \otimes R L (A))}{\longrightarrow}& R L (R L ( A) \otimes R L (A)) &\underoverset{}{ R( {\mu_L^{-1}(R L(A))} ) }{\longrightarrow}& R ( L R L (A) \otimes L R L (A) ) &\overset{R( \epsilon(L(A)) \otimes \epsilon(L(A)) )}{\longrightarrow}& R(L(A) \otimes L(A)) \\ && && && && \downarrow^{\mathrlap{ R( \mu_L(A)) }} \\ && && && && R L( A \otimes A ) \\ && && && && \downarrow^{\mathrlap{ R L (\mu_A) }} \\ && && && && R L (A) }

Here the top sqaures commute by naturality of η\eta and μ L\mu_L, and top right diagonal morphism is the identity morphism, as shown, by the zig-zag identity for the adjunction (LR)(L \dashv R). Therfore R(μ L 1)R(\mu_L^{-1}) cancels agains R(μ L)R(\mu_L). so that the composite morphism in question becomes just AAη(AA)RL(AA)RL(μ A)RL(A)A \otimes A \overset{\eta(A \otimes A)}{\longrightarrow} R L(A \otimes A) \overset{ R L(\mu_A) }{\longrightarrow} R L(A). Again by the naturality of the adjunction unit η\eta

AA η(AA) RL(AA) μ A RL(μ A) A η(A) RL(A) \array{ A \otimes A &\overset{\eta(A \otimes A)}{\longrightarrow}& R L (A \otimes A) \\ {}^{\mathllap{ \mu_A }}\downarrow && \downarrow^{\mathrlap{R L(\mu_A)}} \\ A &\underset{\eta(A)}{\longrightarrow}& R L (A) }

this equals η(A)μ A\eta(A) \circ \mu_A, as required.

Finally we need to check that the following diagram commutes:

1 e A e RL(A) A η(A) RL(A) \array{ && 1 \\ & {}^{\mathllap{e_A}}\swarrow && \searrow^{\mathrlap{ e_{R L (A)} }} \\ A && \underset{\eta(A)}{\longrightarrow} && R L(A) }

Now unwinding the above definitions of e RL(A)e_{R L}(A) in terms of the definition of e Re_R we find that

e RL(A):1η(1)RL(1)R(e L 1)R(1)R(e L)RL(1)RL(e a)RL(A). e_{R L (A)} \;\colon\; 1 \overset{\eta(1)}{\longrightarrow} R L(1) \overset{R(e_L^{-1})}{\longrightarrow} R(1) \overset{R(e_L)}{\longrightarrow} R L(1) \overset{R L(e_a)}{\longrightarrow} R L(A) \,.

Here the two morphisms in the middle cancel, so that we are left just with

e RL(A):1η(1)RL(1)RL(e A)RL(A). e_{R L(A)} \;\colon\; 1 \overset{\eta(1)}{\longrightarrow} R L(1) \overset{R L(e_A)}{\longrightarrow} R L (A) \,.

That this equals η(A)e A\eta(A)\circ e_A, as required, follows by the naturality of η\eta:

1 η(1) RL(1) e A RL(e A) A η(A) RL(A). \array{ 1 &\overset{\eta(1)}{\longrightarrow}& R L(1) \\ {}^{\mathllap{e_A}}\downarrow && \downarrow^{\mathrlap{R L(e_A)}} \\ A &\underset{\eta(A)}{\longrightarrow}& R L(A) } \,.

The argument for the homomorphism property of the counit should be formally dual to the above.



(stabilization in stable homotopy theory)

The stabilization adjunction

Ho(Spectra)Ω Σ ()Ho(Spaces */)() +Ho(Spaces) Ho(Spectra) \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty(-)}{\longleftarrow}} {\bot} Ho(Spaces^{\ast/}) \underoverset {\underset{}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {} Ho(Spaces)

between the classical homotopy categories Ho(Spaces)Ho(Spaces) and Ho(Spaces */)Ho(Spaces^{\ast/}) of (pointed) topological spaces and the stable homotopy category Ho(Spectra)Ho(Spectra) is a monoidal adjunction, since the left adjoint Σ () +\Sigma^\infty(-)_+ (forming the suspension spectrum of a space after freely adjoining a basepoint) is strong monoidal with respect to forming product topological spaces and forming smash product of spectra, respectively.

In fact this is the derived functors of what is even a monoidal Quillen adjunction between the classical model structure on topological spaces and the stable model structure on orthogonal spectra (this cor.), which implies (strong) monoidality of the derived functors on homotopy categories (this prop.).

In detail, let

(LR):𝕊 orthMod stableΩ orth Σ orth Top Quillen */() +Top Quillen (L \dashv R) \;\colon\; \mathbb{S}_{orth}Mod_{stable} \underoverset {\underset{\Omega^\infty_{orth}}{\longrightarrow}} {\overset{\Sigma^\infty_{orth}}{\longleftarrow}} {\bot} Top^{\ast/}_{Quillen} \underoverset {\underset{}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} Top_{Quillen}

be the Quillen adjunction on orthogonal spectra (here). The left adjoint LL is a strong monoidal functor, and hence so is its derived functor Σ () +:Ho(Top)Ho(Spectra)\Sigma^\infty(-)_+ \colon Ho(Top) \to Ho(Spectra) (by this prop.).

We want to see that the structure of a lax monoidal functor which is induced on the derived right adjoint Ω ():Ho(Top)Ho(Spectra)\Omega^\infty(-) \colon Ho(Top) \to Ho(Spectra) via prop. 1 is the expected one, given on Omega-spectra XX and YY by

Ω (X)Ω (X)=X 0Y 0(XY) 0=Ω (XY). \Omega^\infty(X) \wedge \Omega^\infty(X) = X_0 \wedge Y_0 \overset{}{\to} (X \wedge Y)_0 = \Omega^\infty( X \wedge Y ) \,.

To see this, observe that if XX and YY are CW-Omega-spectra and hence cofibrant and fibrant in 𝕊 orthMod stable\mathbb{S}_{orth}Mod_{stable} then the derived lax monoidal structure is given by the total bottom composite in the following diagram

R(X)R(Y) η(R(X)R(Y)) RL(R(X)R(Y)) R(μ L 1(R(X),R(Y))) R(LR(X)LR(Y)) R(ϵ(X)ϵ(Y)) R(XY) derived adjunction unit RjL(R(X)R(Y)) Rj(LR(X)LR(Y)) Rj(XY) RPL(R(X)R(Y)) RQ(μ L 1(R(X),R(Y))) RP(LR(X)LR(Y)) RQ(ϵ(X)ϵ(Y)) RP(XY), \array{ R (X) \otimes R(Y) &\overset{\eta(R(X) \otimes R(Y))}{\longrightarrow}& R L (R(X) \otimes R(Y)) &\underoverset{}{ R( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}& R ( L R(X) \otimes L R (Y) ) &\overset{R( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}& R(X \otimes Y) \\ &{}_{\mathllap{ \text{derived } \atop {\text{adjunction unit}} }}\searrow& \downarrow^{\mathrlap{R j L (R(X) \otimes R(Y))}} && \downarrow^{\mathrlap{R j ( L R (X) \otimes L R(Y) )}} && \downarrow^{\mathrlap{ R j( X \otimes Y ) }} \\ && R P L (R(X) \otimes R(Y)) &\underoverset{}{ R Q( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}& R P ( L R(X) \otimes L R (Y) ) &\overset{R Q( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}& R P (X \otimes Y) } \,,

where we write for brevity (LR)(Σ orth Ω orth )(L \dashv R) \coloneqq (\Sigma^\infty_{orth} \dashv \Omega^\infty_{orth}), and where j:idPj \colon id \longrightarrow P denotes functorial fibrant replacement (which exists since the small object argument applies in 𝕊 orthMod stable\mathbb{S}_{orth}Mod_{stable}). By functoriality of the replacement, all the squares commute, so that the derived lax monoidal structure on CW-Omega spectra is seen to be equivalently the underived one.

But that underived top horizontal composite is manifestly just the canonical isomorphism X 0Y 0(XY) 0X_0 \wedge Y_0 \simeq (X \wedge Y)_0 (since RR simply picks the component space in degree-0 and LL preserves the component space in degree 0).


(exponential modality in linear type theory)

In linear type theory (see there for more) the exponential modality !! may have categorical semantics as the comonad induced by a monoidal adjunction.

Revised on June 25, 2017 09:19:45 by Urs Schreiber (