monoidal Quillen adjunction


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Monoidal categories



The concept of monoidal Quillen adjunction is a lift of the concept of strong monoidal adjunctions (adjoint functors for which the left adjoint is a strong monoidal functor so that the right adjoint is, canonically, a lax monoidal functor) from the context of plain categories to that of model categories.


For CC and DD monoidal model categories, a lax monoidal Quillen adjunction

(LR):CRLD (L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D


  • a Quillen adjunction (LR)(L \dashv R) between the underlying model categories;

  • equipped with the structure of a lax monoidal functor on RR with respect to the underlying monoidal categories

  • such that the induced structure of an oplax monoidal functor on LL satisfies:

    1. for all cofibrant objects x,yDx,y \in D the oplax monoidal transformation

      ˜ x,y:L(xy)L(x)L(y) \tilde\nabla_{x,y} : L(x \otimes y) \to L(x) \otimes L(y)

      is a weak equivalence in CC

    2. for some (hence any) cofibrant resolution q:I^ DI Dq : \hat I_D \stackrel{\simeq}{\to} I_D of the monoidal unit object in DD, the composite

      L(I^ D)L(q)L(I D)e˜I C L(\hat I_D) \stackrel{L(q)}{\to} L(I_D) \stackrel{\tilde e}{\to} I_C

      with the oplax monoidal counit is a weak equivalence in CC.

This is called a strong monoidal Quillen adjunction if LL is a strong monoidal functor. In this case the first condition above on LL is vacuous, and the second becomes vacuous if the unit object of DD is cofibrant.

If a monoidal Quillen adjunction is also a Quillen equivalence it is called a monoidal Quillen equivalence.


Recognition of monoidal Quillen adjunctions


Let (LR):CRLD(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D be a Quillen adjunction between monoidal model categories and let RR be equipped with the strcuture of a lax monoidal functor.

Then the following two conditions are sufficient for (LR)(L \dashv R) to be a lax monoidal Quillen adjunction:

  1. for some (hence any) cofibrant resolution q:I^ DI Dq : \hat I_D \stackrel{\simeq}{\to} I_D of the unit object in DD, the composite morphism

    L(I^ D)L(q)L(I D)i˜I C L(\hat I_D) \stackrel{L(q)}{\to} L(I_D) \stackrel{\tilde i}{\to} I_C

    is a weak equivalence, (wher i˜\tilde i is the adjunct of i:I DR(I C)i : I_D \to R(I_C));

  2. the unit object I DI_D detects weak equivalences in that for every weak equivalence f:XYf : X \to Y between fibrant objects the morphism D Δ op(QI D,f)D^{\Delta^{op}}(Q I_D, f) of hom-objects in the category of simplicial objects in DD is an equivalence of Kan complexes, for QI DQ I_D a cofibrant resolution in the Reedy model structure D Reedy Δ opD^{\Delta^{op}}_{Reedy}.

This is proposition 3.16 in (SchwedeShipley).

Lift to an adjunction on monoids

We discuss how a monoidal Quillen adjunction (LR):CRLD(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D induces, under mild conditions, an adjunction (L monR):Mon(C)RLMon(D)(L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} Mon(D) on the corresponding categories of monoids. In the following secton we discuss how this is itself a Quillen adjunction

The lax monoidal functor R:CDR : C \to D induces (as described there) a functor R:Mon(C)Mon(D)R : Mon(C) \to Mon(D) on monoids (which by slight abuse of notation we denote by the same symbol). Write mabla X,Y:RXRYR(XY)\mabla_{X,Y} : R X \otimes R Y \to R(X \otimes Y) for the lax monoidal structure on RR. This induces canonically the structure of a oplax monoidal functor (as described there) on the left adjoint L:DCL : D \to C. Write ˜:L(XY)LXLY\tilde\nabla : L(X \otimes Y) \to L X \otimes L Y for this oplax structure.

While LL will not extend to a functor on the category of monoids unless RR is a strong monoidal functor there is nevertheless an adjoint L monL^{mon} to R:Mon(C)Mon(D)R : Mon(C) \to Mon(D).

As described at category of monoids, if CC has countable coproducts preserved by the tensor product, then we have a free functor/forgetful functor adjunction

(FU):Mon(C)UFC, (F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C \,,

where F(X)F(X) is the tensor algebra over the object XX in (C,)(C, \otimes).


Let (LR):CRLD(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D be a pair of adjoint functors between monoidal categories where RR is a lax monoidal functor and DD has all small colimits.

Then the functor R:Mon(C)Mon(D)R : Mon(C) \to Mon(D) has a left adjoint

L mon:Mon(D)Mon(C) L^{mon} : Mon(D) \to Mon(C)

given by forming the coequalizers

L mon:Blim (F CLF DBF CLB) L^{mon} : B \mapsto \lim_{\to} (F_C L F_D B \stackrel{\to}{\to} F_C L B)

in Mon(C)Mon(C) of the following two morphisms

  • the first one is the image under F CLF_C \circ L of the adjunction counit F DU DBB F_D U_D B \to B;

  • the second is the unique CC-monoid morphism that restricts to the CC-morphism

    LF DB nL(B n)˜ n(LB) nF CLB L F_D B \simeq \coprod_{n \in \mathbb{N}} L( B^{\otimes n}) \stackrel{\coprod \tilde \nabla}{\to} \coprod_{n \in \mathbb{N}} (L B)^{\otimes n} \simeq F_C L B

    which is componentwise given by the oplax monoidal structure on LL induced by the lax monoidal structure on RR.

This is considered on p. 305 of (SchwedeShipley)


To see that (L monR)(L^{mon} \dashv R) first notice that a morphism of monoids

L monXY L^{mon} X \to Y

is by the definition of coequalizer a morphism of monoids f:F CLXYf : F_C L X \to Y satisfying a condition. By the free property of F CLXF_C L X this in turn is a morphism f 1:LXYf_1 : L X \to Y in CC which by (LR)(L \dashv R) is a morphism f˜ 1:XRY\tilde f_1 : X \to R Y in CC. So we need to show that the condition satisfied by ff is precisely the condition that makes f˜ 1\tilde f_1 a morphism of monoids in that

XX f˜ 1f˜ 1 RYRY R(YY) X f˜ 1 RY \array{ X \otimes X &\stackrel{\tilde f_1 \otimes \tilde f_1}{\to}& R Y \otimes R Y \\ \downarrow && \downarrow \\ && R ( Y \otimes Y) \\ \downarrow && \downarrow \\ X &\stackrel{\tilde f_1}{\to}& R Y }

commutes. We insert the definition of the adjunct f˜ 1\tilde f_1 and the lax naturality square of RR to get

XX RLXRLX Rf 1Rf 1 RYRY = R(LXLY) Rf 1 R(YY) X f˜ 1 RY. \array{ X \otimes X &\to& R L X \otimes R L X &\stackrel{R f_1 \otimes R f_1}{\to}& R Y \otimes R Y \\ \downarrow && \downarrow &=& \downarrow \\ && R(L X \otimes L Y) &\stackrel{R f_1}{\to}& R ( Y \otimes Y) \\ \downarrow && && \downarrow \\ X & &\stackrel{\tilde f_1}{\to}&& R Y } \,.

The adjunct of the left/bottom composite is

L(XX)LXf 1Y L(X\otimes X) \to L X \stackrel{f_1}{\to} Y

while the adjunct of the top/right composite is that of the diagonal, which is

L(XX)˜LXLXf 1f 1YYY. L(X \otimes X) \stackrel{\tilde \nabla}{\to} L X \otimes L X \stackrel{f_1 \otimes f_1}{\to} Y \otimes Y \to Y \,.

This in turn is by the definition of ff in terms of its components equal to

L(XX)˜LXLXf 2Y. L(X \otimes X) \stackrel{\tilde \nabla}{\to} L X \otimes L X \stackrel{f_2}{\to}Y \,.

The coequalizer property says indeed precisely that these two adjuncts are equal.


There is a natural isomorphism

L monF DF CL. L^{mon} \circ F_D \simeq F_C \circ L \,.

This is considered on p. 305 of (SchwedeShipley).


On a monoid KK the morphism

L monFKFLK L^{mon} F K \to F L K

is defined as a coequalizing morphism of monoids

FLFKFLK. F L F K \to F L K \,.

This in turn is given by a morphism in CC

LFKFLK. L F K \to F L K \,.

Take this to be given componentwise by the oplax counit e˜\tilde e.

This does coequalize then: for one route is

L((K)(K))L(KK)˜L(K)L(K) L( (K) \otimes (K) ) \to L(K \otimes K) \stackrel{\tilde \nabla}{\to} L(K) \otimes L(K)

and the other

K((K)(K))˜LKLKIdLKLK. K( (K) \otimes (K) ) \stackrel{\tilde \nabla}{\to} L K \otimes L K \stackrel{Id}{\to} L K \otimes L K \,.

Lift to a Quillen equivalence on monoids

We now describe how the adjunction (L monR)(L^{mon} \dashv R) established above becomes a Quillen adjunction for the transferred model structures on the categories of monoids, transferred along the forgetful/free functor adjunction

(F CU C):Mon(C)UFC (F_C \dashv U_C) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C

and how it becomes a Quillen equivalence if (LR)(L \dashv R) is a monoidal Quillen eqivalence.

See model structure on monoids.


We assume for this section that the monoidal model category CC

Then by (SchwedeShipleyAlgebras) the transferred model structure on monoids in a monoidal model category Mon(C)Mon(C) exists.

Notice also that by cofibrant generation every cofibrant object in Mon(C)Mon(C) is a retract of a (FU)(F \dashv U)-cell object.



Let (LR):CRLD(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D be a lax monoidal Quillen adjunction between monoidal model categories with cofibrant unit obects.

Then also the adjunction

(L monR):Mon(C)RL monMon(D), (L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L^{mon}}{\leftarrow}}{\underset{R}{\to}} Mon(D) \,,

from above is a Quillen adjunction between the transferred model structures on monoids.

If the forgetful functors U CU_C and U DU_D create model structures on monoids, then (L monR)(L^{mon} \dashv R) is a Quillen equivalence if (LR)(L \dashv R) is.

This is theorem 3.12 in (SchwedeShipley). Its proof uses the following technical lemmas.

Let (LR):CRLD(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D be a monoidal Quillen adjunction between monoidal model categories with cofibrant unit objects.

Suppose the adjunction

(L monR):Mon(C)RLMon(D) (L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} Mon(D)

described above exists (just as an adjunction, not yet assumed to be a Quillen adjunction).


The morphism

L monI DI C L^{mon} I_D \to I_C

induced by the oplax counit e˜:LI DI C\tilde e : L I_D \to I_C of the oplax monoidal functor is an isomorphism of monoids.


We have that I DI_D and I CI_C are the initial objects in Mon(D)Mon(D) and Mon(C)Mon(C), respectively. Because L monL^{mon} is left adjoint, it preserves these initial objects, so that there is some isomorphism as claimed. It is hence sufficient to show that the oplax counit induces a morphism of monoids at all, by the universal property of the initial object it will be an isomorphism.

It is clear that

nμ ne˜ n:FLI DI C \coprod_n \mu^n {\tilde e}^{\otimes n} : F L I_D \to I_C

is a morphism of monoids, because

(LI) k(LI) (nk) μ I ke˜ kμ I nke˜ (nk) II (LI) n μ I ne˜ n I \array{ (L I)^{\otimes k} \otimes (L I)^{\otimes (n-k)} &\stackrel{\mu_I^k {\tilde e}^{\otimes k} \otimes \mu_I^{n-k}{\tilde e}^{\otimes (n-k)}}{\to}& I \otimes I \\ \downarrow && \downarrow \\ (L I)^{\otimes n} &\stackrel{\mu^n_I {\tilde e}^{n}}{\to}& I }

commutes. So we have to show that this morphism coequalizes the two morphisms in the definition of L monI DL^{mon} I_D. By the same argument as in the above proof this is equivalent to showing that

I CI C ee RI CRI C R(I CI C) I C e RI C \array{ I_C \otimes I_C &\stackrel{e \otimes e}{\to}& R I_C \otimes R I_C \\ \downarrow && \downarrow \\ && R (I_C \otimes I_C) \\ \downarrow && \downarrow \\ I_C &\stackrel{e}{\to}& R I_C }

commutes. This follows from the unitality of the lax monoidal functor RR.


For every monoid BMon(D)B \in Mon(D) which is an (FU)(F \dashv U)-cell object, the (LR)(L \dashv R)-adjunct

χ B:LBL monB \chi_B : L B \to L^{mon} B

to the morphism underlying the unit BRL monBB \to R L^{mon} B is a weak equivalence.

This is proposition 5.1 in (SchwedeShipley).


We first show this for B=I DB = I_D the tensor unit in DD, which in Mon(D)Mon(D) is the initial objects:

  • We claim hat the adjunction unit I CRL monI DR(I C)I_C \to R L^{mon} I_D \stackrel{\simeq}{\to} R(I_C) is the lax monoidal unit ee of RR.

    To see this, use that by the previous lemma the (LR)(L \dashv R)-adjunct of IRL monIRII \to R L^{mon} I \to R I is LIL monI nμ ne˜ nIL I \to L^{mon} I \stackrel{\coprod_n \mu^n {\tilde e}^{\otimes n}}{\to} I. Here the first morphism factors through the single power of LIL I, hence this is indeed e˜:LI DI C\tilde e : L I_D \to I_C.

    Therefore by the axioms on monoidal Quillen adjunctions the (LR)(L \dashv R)-adjunct χ I\chi_I is a weak equivalence.

We now proceed from this by induction over the cells of the cell object BB.

So assume now that we have already shown that on some cell object BB the morphism χ B\chi_B is a weak equivalence. We want to deduce then that that after forming a new monoid PP by cell attachment, i.e. by a pushout

FK FK B P \array{ F K &\to& F K' \\ \downarrow && \downarrow \\ B &\to& P }

for KKK \to K' a cofibration in DD, also χ P:LPL monP\chi_P : L P \to L^{mon} P is a weak equivalence.

Notice that since L monL^{mon} is left adjoint also

L monFK L monFK L monB L monP \array{ L^{mon} F K &\to& L^{mon} F K' \\ \downarrow && \downarrow \\ L^{mon} B &\to& L^{mon} P }

is a pushout in Mon(C)Mon(C), and by the natural isomorphism from the above lemma so is

FLK FLK L monB L monP. \array{ F L K &\to& F L K' \\ \downarrow && \downarrow \\ L^{mon} B &\to& L^{mon} P } \,.
  • We claim that BB is cofibrant and that we can without restriction assume KK and KK' to be cofibrant in DD.

    The first statement follows from an inductive application of the construction of pushouts as discussed at category of monoids in the section free monoids. For the second statement notice that since FF is left adjoint and preserves pushouts in DD, we have that PP is also the pushout of the diagram

    (FB F(B KK) B P)=( FK FK FB FB B P). \left( \array{ F B &\to& F( B \coprod_K K' ) \\ \downarrow && \downarrow \\ B &\to& P } \right) = \left( \array{ && F K &\to& F K' \\ && \downarrow && \downarrow \\ F B &\to& F B \\ \downarrow &&&& \downarrow \\ B && \to && P } \right) \,.

    Since cofibrations are preserved by the Quillen left adjoint FF and under pushout, it follows that also B KKB \coprod_K K' is cofibrant if KKK \to K' is a cofibration. So BB KKB \to B \coprod_K K' can be used in place of KKK \to K'.

Notice that this means that our pushout square is in fact a homotopy pushout square (as discussed there). In particular a weak equivalence of these pushout diagrams will induce a weak equivalence of the pushouts, so that is what we will establish.

We now proceed as in category of monoids in the section free monoids for getting the following statement about the object underlying PP

This PP is a colimit of a sequence of cofibrations

Plim (B:=P 0P 1P 2) P \simeq \lim_{\to} ( B := P_0 \hookrightarrow P_1 \hookrightarrow P_2 \hookrightarrow \cdots )

such that each morphism P n1hookrightarowP nP_{n-1} \hookrightarow P_n is a pushout in DD of a particular cofibration Q n(K,K,B)(BK) nBQ_n(K,K', B) \hookrightarrow (B \otimes K')^{\otimes n} \otimes B

By the coresponding disccussion of these pushouts under L monL^{mon} it follows that also L monPL^{mon} P is the colimit of a sequence of cofibrations betwen objects R nR_n that are pushouts of these particular cofibrations.

And the morphism χ P\chi_P respects all that and sends

χ P n:LP nL monR n \chi_{P_n} : L P_n \to L^{mon} R_n

at each stage of the cell attachments. So it is sufficient to show that the three components of these maps on the pushout squares are weak equivalences. Since we showed above that our pushout squares are actually homotopy pushout squares, this will imply that also χ P\chi_P is a weak equivalence.

This again works by proceeding as in category of monoids in the section free monoids.


If U CU_C creates the model structure on Mon(C)Mon(C) and the unit in CC is cofibrant, then a cofibrant CC-monoid is also cofibrant as an object in CC.


This is once more a consequence of the lemma on pushouts at at category of monoids in the section free monoids.

We have now collected all prerequisites and turn to the proof of the theorem about lifted Quillen adjunctions.

Proof of the theorem

That (L monR)(L^{mon} \dashv R) is a Quillen adjunction is clear, as the model structure on monoids has fibrations and acyclic fibrations those in the underlying category, and these are preserved by RR.

So the essential statement is that it is a Quillen equivalence of (LR)(L \dashv R) is.

First notice that since by assumption the model structure on monoids Mon(D)Mon(D) is created by U DU_D it follows by definition that the cofibrant BB is a retract of a cell object in Mon(D)Mon(D). Then the above lemma asserts that

χ B:LBL monB \chi_B : L B \to L^{mon} B

is a weak equivalence.

To prove the theorem, we have to show for every cofibrant BMon(D)B \in Mon(D) and fibrant YMon(C)Y \in Mon(C) that a morphism BRYB \to R Y is a weak equivalence in Mon(D)Mon(D) (hence its underlying morphism in DD) precisely if its adjunct L monBYL^{mon} B \to Y is a weak equivalence in Mon(C)Mon(C) (hence its underlying morphism in CC).

By definition of adjunct we have that

(BRY)=(BRL monBRY). (B \to R Y) = ( B \to R L^{mon} B \to R Y) \,.

By the second lemma above we have that BB is cofibrant also in CC. Therefore, since (LR)(L \dashv R) is a Quillen equivalence between CC and DD, the right hand is a weak equivalence precisely if its (LR)(L \dashv R)-adjunct

LBχ BL monBY L B \stackrel{\chi_B}{\to} L^{mon} B \to Y

is a weak equivalence in DD. But since χ B\chi_B is a weak equivalence, this is the case precisely if L monBYL^{mon}B \to Y is a weak equivalence.



(stabilization in stable homotopy theory)

The stabilization adjunction

(Σ () +Ω ):Ho(Spectra)Ω Σ ()Ho(Spaces) \left( \Sigma^\infty(-)_+ \dashv \Omega^\infty \right) \;\colon\; Ho(Spectra) \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty(-)}{\longleftarrow}} {\bot} Ho(Spaces)

between the classical homotopy category Ho(Spaces)Ho(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. Hence this is a monoidal adjunction.

In fact this is the derived functors of what is even a monoidal Quillen adjunction

(Σ orth () +Ω orth ):OrthSpec stableΩ orth Σ orth () +Top Quillen (\Sigma^\infty_{orth}(-)_+ \dashv \Omega^\infty_{orth}) \;\colon\; OrthSpec_{stable} \underoverset {\underset{\Omega_{orth}^\infty}{\longrightarrow}} {\overset{\Sigma_{orth}^\infty(-)_+}{\longleftarrow}} {} Top_{Quillen}

between the classical model structure on topological spaces and the stable model structure on orthogonal spectra (this cor.) which implies (strong) modality of the derived functors on homotopy categories (this prop.).



The notion of strong monoidal Quillen adjunction is def. 4.2.16 in

The lax monoidal version is considered as definition 3.6 of

The statements involving pushouts along free monoid morphisms are discussed in lemma 6.2 of

Revised on June 22, 2017 02:30:03 by Urs Schreiber (