model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 $C$ and $D$ monoidal model categories, a lax monoidal Quillen adjunction
is
a Quillen adjunction $(L \dashv R)$ between the underlying model categories;
equipped with the structure of a lax monoidal functor on $R$ with respect to the underlying monoidal categories
such that the induced structure of an oplax monoidal functor on $L$ satisfies:
for all cofibrant objects $x,y \in D$ the oplax monoidal transformation
is a weak equivalence in $C$
for some (hence any) cofibrant resolution $q : \hat I_D \stackrel{\simeq}{\to} I_D$ of the monoidal unit object in $D$, the composite
with the oplax monoidal counit is a weak equivalence in $C$.
This is called a strong monoidal Quillen adjunction if $L$ is a strong monoidal functor. In this case the first condition above on $L$ is vacuous, and the second becomes vacuous if the unit object of $D$ is cofibrant.
If a monoidal Quillen adjunction is also a Quillen equivalence it is called a monoidal Quillen equivalence.
Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a Quillen adjunction between monoidal model categories and let $R$ be equipped with the strcuture of a lax monoidal functor.
Then the following two conditions are sufficient for $(L \dashv R)$ to be a lax monoidal Quillen adjunction:
for some (hence any) cofibrant resolution $q : \hat I_D \stackrel{\simeq}{\to} I_D$ of the unit object in $D$, the composite morphism
is a weak equivalence, (wher $\tilde i$ is the adjunct of $i : I_D \to R(I_C)$);
the unit object $I_D$ detects weak equivalences in that for every weak equivalence $f : X \to Y$ between fibrant objects the morphism $D^{\Delta^{op}}(Q I_D, f)$ of hom-objects in the category of simplicial objects in $D$ is an equivalence of Kan complexes, for $Q I_D$ a cofibrant resolution in the Reedy model structure $D^{\Delta^{op}}_{Reedy}$.
This is proposition 3.16 in (SchwedeShipley).
We discuss how a monoidal Quillen adjunction $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ induces, under mild conditions, an adjunction $(L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} Mon(D)$ on the corresponding categories of monoids. In the following section we discuss how this is itself a Quillen adjunction
The lax monoidal functor $R : C \to D$ induces (as described there) a functor $R : Mon(C) \to Mon(D)$ on monoids (which by slight abuse of notation we denote by the same symbol). Write $\nabla_{X,Y} : R X \otimes R Y \to R(X \otimes Y)$ for the lax monoidal structure on $R$. This induces canonically the structure of a oplax monoidal functor (as described there) on the left adjoint $L : D \to C$. Write $\tilde\nabla : L(X \otimes Y) \to L X \otimes L Y$ for this oplax structure.
While $L$ will not extend to a functor on the category of monoids unless $R$ is a strong monoidal functor there is nevertheless an adjoint $L^{mon}$ to $R : Mon(C) \to Mon(D)$.
As described at category of monoids, if $C$ has countable coproducts preserved by the tensor product, then we have a free functor/forgetful functor adjunction
where $F(X)$ is the tensor algebra over the object $X$ in $(C, \otimes)$.
Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a pair of adjoint functors between monoidal categories where $R$ is a lax monoidal functor and $D$ has all small colimits.
Then the functor $R : Mon(C) \to Mon(D)$ has a left adjoint
given by forming the coequalizers
in $Mon(C)$ of the following two morphisms
the first one is the image under $F_C \circ L$ of the adjunction counit $F_D U_D B \to B$;
the second is the unique $C$-monoid morphism that restricts to the $C$-morphism
which is componentwise given by the oplax monoidal structure on $L$ induced by the lax monoidal structure on $R$.
This is considered on p. 305 of (SchwedeShipley)
To see that $(L^{mon} \dashv R)$ first notice that a morphism of monoids
is by the definition of coequalizer a morphism of monoids $f : F_C L X \to Y$ satisfying a condition. By the free property of $F_C L X$ this in turn is a morphism $f_1 : L X \to Y$ in $C$ which by $(L \dashv R)$ is a morphism $\tilde f_1 : X \to R Y$ in $C$. So we need to show that the condition satisfied by $f$ is precisely the condition that makes $\tilde f_1$ a morphism of monoids in that
commutes. We insert the definition of the adjunct $\tilde f_1$ and the lax naturality square of $R$ to get
The adjunct of the left/bottom composite is
while the adjunct of the top/right composite is that of the diagonal, which is
This in turn is by the definition of $f$ in terms of its components equal to
The coequalizer property says indeed precisely that these two adjuncts are equal.
This is considered on p. 305 of (SchwedeShipley).
On a monoid $K$ the morphism
is defined as a coequalizing morphism of monoids
This in turn is given by a morphism in $C$
Take this to be given componentwise by the oplax counit $\tilde e$.
This does coequalize then: for one route is
and the other
We now describe how the adjunction $(L^{mon} \dashv R)$ established above becomes a Quillen adjunction for the transferred model structures on the categories of monoids, transferred along the forgetful/free functor adjunction
and how it becomes a Quillen equivalence if $(L \dashv R)$ is a monoidal Quillen eqivalence.
See model structure on monoids.
We assume for this section that the monoidal model category $C$
satisfies the monoid axiom in a monoidal model category.
Then by (SchwedeShipleyAlgebras) the transferred model structure on monoids in a monoidal model category $Mon(C)$ exists.
Notice also that by cofibrant generation every cofibrant object in $Mon(C)$ is a retract of a $(F \dashv U)$-cell object.
=–
Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a lax monoidal Quillen adjunction between monoidal model categories with cofibrant unit obects.
Then also the adjunction
from above is a Quillen adjunction between the transferred model structures on monoids.
If the forgetful functors $U_C$ and $U_D$ create model structures on monoids, then $(L^{mon} \dashv R)$ is a Quillen equivalence if $(L \dashv R)$ is.
This is theorem 3.12 in (SchwedeShipley). Its proof uses the following technical lemmas.
Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a monoidal Quillen adjunction between monoidal model categories with cofibrant unit objects.
Suppose the adjunction
described above exists (just as an adjunction, not yet assumed to be a Quillen adjunction).
The morphism
induced by the oplax counit $\tilde e : L I_D \to I_C$ of the oplax monoidal functor is an isomorphism of monoids.
We have that $I_D$ and $I_C$ are the initial objects in $Mon(D)$ and $Mon(C)$, respectively. Because $L^{mon}$ 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
is a morphism of monoids, because
commutes. So we have to show that this morphism coequalizes the two morphisms in the definition of $L^{mon} I_D$. By the same argument as in the above proof this is equivalent to showing that
commutes. This follows from the unitality of the lax monoidal functor $R$.
For every monoid $B \in Mon(D)$ which is an $(F \dashv U)$-cell object, the $(L \dashv R)$-adjunct
to the morphism underlying the unit $B \to R L^{mon} B$ is a weak equivalence.
This is proposition 5.1 in (SchwedeShipley).
We first show this for $B = I_D$ the tensor unit in $D$, which in $Mon(D)$ is the initial objects:
We claim hat the adjunction unit $I_C \to R L^{mon} I_D \stackrel{\simeq}{\to} R(I_C)$ is the lax monoidal unit $e$ of $R$.
To see this, use that by the previous lemma the $(L \dashv R)$-adjunct of $I \to R L^{mon} I \to R I$ is $L I \to L^{mon} I \stackrel{\coprod_n \mu^n {\tilde e}^{\otimes n}}{\to} I$. Here the first morphism factors through the single power of $L I$, hence this is indeed $\tilde e : L I_D \to I_C$.
Therefore by the axioms on monoidal Quillen adjunctions the $(L \dashv R)$-adjunct $\chi_I$ is a weak equivalence.
We now proceed from this by induction over the cells of the cell object $B$.
So assume now that we have already shown that on some cell object $B$ the morphism $\chi_B$ is a weak equivalence. We want to deduce then that that after forming a new monoid $P$ by cell attachment, i.e. by a pushout
for $K \to K'$ a cofibration in $D$, also $\chi_P : L P \to L^{mon} P$ is a weak equivalence.
Notice that since $L^{mon}$ is left adjoint also
is a pushout in $Mon(C)$, and by the natural isomorphism from the above lemma so is
We claim that $B$ is cofibrant and that we can without restriction assume $K$ and $K'$ to be cofibrant in $D$.
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 $F$ is left adjoint and preserves pushouts in $D$, we have that $P$ is also the pushout of the diagram
Since cofibrations are preserved by the Quillen left adjoint $F$ and under pushout, it follows that also $B \coprod_K K'$ is cofibrant if $K \to K'$ is a cofibration. So $B \to B \coprod_K K'$ can be used in place of $K \to K'$.
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 $P$
This $P$ is a colimit of a sequence of cofibrations
such that each morphism $P_{n-1} \hookrightarow P_n$ is a pushout in $D$ of a particular cofibration $Q_n(K,K', B) \hookrightarrow (B \otimes K')^{\otimes n} \otimes B$
By the coresponding disccussion of these pushouts under $L^{mon}$ it follows that also $L^{mon} P$ is the colimit of a sequence of cofibrations betwen objects $R_n$ that are pushouts of these particular cofibrations.
And the morphism $\chi_P$ respects all that and sends
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 $\chi_P$ is a weak equivalence.
This again works by proceeding as in category of monoids in the section free monoids.
If $U_C$ creates the model structure on $Mon(C)$ and the unit in $C$ is cofibrant, then a cofibrant $C$-monoid is also cofibrant as an object in $C$.
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.
That $(L^{mon} \dashv R)$ 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 $R$.
So the essential statement is that it is a Quillen equivalence of $(L \dashv R)$ is.
First notice that since by assumption the model structure on monoids $Mon(D)$ is created by $U_D$ it follows by definition that the cofibrant $B$ is a retract of a cell object in $Mon(D)$. Then the above lemma asserts that
is a weak equivalence.
To prove the theorem, we have to show for every cofibrant $B \in Mon(D)$ and fibrant $Y \in Mon(C)$ that a morphism $B \to R Y$ is a weak equivalence in $Mon(D)$ (hence its underlying morphism in $D$) precisely if its adjunct $L^{mon} B \to Y$ is a weak equivalence in $Mon(C)$ (hence its underlying morphism in $C$).
By definition of adjunct we have that
By the second lemma above we have that $B$ is cofibrant also in $C$. Therefore, since $(L \dashv R)$ is a Quillen equivalence between $C$ and $D$, the right hand is a weak equivalence precisely if its $(L \dashv R)$-adjunct
is a weak equivalence in $D$. But since $\chi_B$ is a weak equivalence, this is the case precisely if $L^{mon}B \to Y$ is a weak equivalence.
(stabilization in stable homotopy theory)
The stabilization adjunction
between the classical homotopy category $Ho(Spaces)$ and the stable homotopy category $Ho(Spectra)$ is a monoidal adjunction, since the left adjoint $\Sigma^\infty(-)_+$ (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
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.).
$\,$
Examples arise in the monoidal Dold-Kan correspondence. See there for details.
The quivalence between module spectra and chain complexes is exhibited by monoidal Quillen equivalences. See module spectrum for details.
monoidal Quillen adjunction
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