Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
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With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
For and monoidal model categories, a lax monoidal Quillen adjunction
a Quillen adjunction between the underlying model categories;
equipped with the structure of a lax monoidal functor on with respect to the underlying monoidal categories
such that the induced structure of an oplax monoidal functor on satisfies:
for all cofibrant objects the oplax monoidal transformation
is a weak equivalence in
for some (hence any) cofibrant resolution of the monoidal unit object in , the composite
with the oplax monoidal counit is a weak equivalence in .
This is called a strong monoidal Quillen adjunction if is a strong monoidal functor. In this case the first condition above on is vacuous, and the second becomes vacuous if the unit object of 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 be a Quillen adjunction between monoidal model categories and let be equipped with the strcuture of a lax monoidal functor.
Then the following two conditions are sufficient for to be a lax monoidal Quillen adjunction:
for some (hence any) cofibrant resolution of the unit object in , the composite morphism
is a weak equivalence, (wher is the adjunct of );
the unit object detects weak equivalences in that for every weak equivalence between fibrant objects the morphism of hom-objects in the category of simplicial objects in is an equivalence of Kan complexes, for a cofibrant resolution in the Reedy model structure .
This is proposition 3.16 in (SchwedeShipley).
Lift to an adjunction on monoids
We discuss how a monoidal Quillen adjunction induces, under mild conditions, an adjunction on the corresponding categories of monoids. In the following secton we discuss how this is itself a Quillen adjunction
The lax monoidal functor induces (as described there) a functor on monoids (which by slight abuse of notation we denote by the same symbol). Write for the lax monoidal structure on . This induces canonically the structure of a oplax monoidal functor (as described there) on the left adjoint . Write for this oplax structure.
While will not extend to a functor on the category of monoids unless is a strong monoidal functor there is nevertheless an adjoint to .
As described at category of monoids, if has countable coproducts preserved by the tensor product, then we have a free functor/forgetful functor adjunction
where is the tensor algebra over the object in .
Let be a pair of adjoint functors between monoidal categories where is a lax monoidal functor and has all small colimits.
Then the functor has a left adjoint
given by forming the coequalizers
in of the following two morphisms
the first one is the image under of the adjunction counit ;
the second is the unique -monoid morphism that restricts to the -morphism
which is componentwise given by the oplax monoidal structure on induced by the lax monoidal structure on .
This is considered on p. 305 of (SchwedeShipley)
To see that first notice that a morphism of monoids
is by the definition of coequalizer a morphism of monoids satisfying a condition. By the free property of this in turn is a morphism in which by is a morphism in . So we need to show that the condition satisfied by is precisely the condition that makes a morphism of monoids in that
commutes. We insert the definition of the adjunct and the lax naturality square of 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 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 the morphism
is defined as a coequalizing morphism of monoids
This in turn is given by a morphism in
Take this to be given componentwise by the oplax counit .
This does coequalize then: for one route is
and the other
Lift to a Quillen equivalence on monoids
We now describe how the adjunction 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 is a monoidal Quillen eqivalence.
See model structure on monoids.
We assume for this section that the monoidal model category
Then by (SchwedeShipleyAlgebras) the transferred model structure on monoids in a monoidal model category exists.
Notice also that by cofibrant generation every cofibrant object in is a retract of a -cell object.
Let 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 and create model structures on monoids, then is a Quillen equivalence if is.
This is theorem 3.12 in (SchwedeShipley). Its proof uses the following technical lemmas.
Let 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).
induced by the oplax counit of the oplax monoidal functor is an isomorphism of monoids.
We have that and are the initial objects in and , respectively. Because 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 . 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 .
For every monoid which is an -cell object, the -adjunct
to the morphism underlying the unit is a weak equivalence.
This is proposition 5.1 in (SchwedeShipley).
We first show this for the tensor unit in , which in is the initial objects:
We claim hat the adjunction unit is the lax monoidal unit of .
To see this, use that by the previous lemma the -adjunct of is . Here the first morphism factors through the single power of , hence this is indeed .
Therefore by the axioms on monoidal Quillen adjunctions the -adjunct is a weak equivalence.
We now proceed from this by induction over the cells of the cell object .
So assume now that we have already shown that on some cell object the morphism is a weak equivalence. We want to deduce then that that after forming a new monoid by cell attachment, i.e. by a pushout
for a cofibration in , also is a weak equivalence.
Notice that since is left adjoint also
is a pushout in , and by the natural isomorphism from the above lemma so is
We claim that is cofibrant and that we can without restriction assume and to be cofibrant in .
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 is left adjoint and preserves pushouts in , we have that is also the pushout of the diagram
Since cofibrations are preserved by the Quillen left adjoint and under pushout, it follows that also is cofibrant if is a cofibration. So can be used in place of .
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
This is a colimit of a sequence of cofibrations
such that each morphism is a pushout in of a particular cofibration
By the coresponding disccussion of these pushouts under it follows that also is the colimit of a sequence of cofibrations betwen objects that are pushouts of these particular cofibrations.
And the morphism 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 is a weak equivalence.
This again works by proceeding as in category of monoids in the section free monoids.
If creates the model structure on and the unit in is cofibrant, then a cofibrant -monoid is also cofibrant as an object in .
We have now collected all prerequisites and turn to the proof of the theorem about lifted Quillen adjunctions.
Proof of the theorem
That 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 .
So the essential statement is that it is a Quillen equivalence of is.
First notice that since by assumption the model structure on monoids is created by it follows by definition that the cofibrant is a retract of a cell object in . Then the above lemma asserts that
is a weak equivalence.
To prove the theorem, we have to show for every cofibrant and fibrant that a morphism is a weak equivalence in (hence its underlying morphism in ) precisely if its adjunct is a weak equivalence in (hence its underlying morphism in ).
By definition of adjunct we have that
By the second lemma above we have that is cofibrant also in . Therefore, since is a Quillen equivalence between and , the right hand is a weak equivalence precisely if its -adjunct
is a weak equivalence in . But since is a weak equivalence, this is the case precisely if is a weak equivalence.
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