Redirected from "Quillen adjoint quadruples".
Contents
Context
Model category theory
model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
for ∞-groupoids
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
Contents
Idea
Just as a Quillen adjunction is a model-categorical version of an adjunction, a Quillen adjoint triple should be a model-categorical version of an adjoint triple.
However, there are various levels of generality at which this could be defined. The simplest would be an adjoint triple between model categories in which both adjunctions are Quillen adjunctions. However, this would require the middle functor to be both Quillen left adjoint and Quillen right adjoint for the same model structures, which (though not impossible) is a strong restriction. A more general notion is obtained by allowing the model structures on one or both of the categories to vary between the two Quillen adjunctions, but keeping the same Quillen equivalence type so that when we pass to homotopy categories or derived -categories we still get an adjoint triple (such as in 2Ho(CombModCat)).
Currently, this page is primarily about a very restrictive case where only one of the model structures is allowed to vary, and the Quillen equivalence between the two model structures on that side must be an identity functor. However, more general versions could certainly also be defined.
Definition
Definition
(Quillen adjoint triple)
Let be model categories, where and share the same underlying category , and such that the identity functor on constitutes a Quillen equivalence
(1)
Then a Quillen adjoint triple
is a pair of Quillen adjunctions, as shown, together with a 2-morphism in the double category of model categories
(2)
whose derived natural transformation (here) is invertible (a natural isomorphism).
If two Quillen adjoint triples overlap
we speak of a Quillen adjoint quadruple, and so forth.
Examples
General
Example
(Quillen adjoint triple from left and right Quillen functor)
Given an adjoint triple
such that is both a left Quillen functor as well as a right Quillen functor for given model category-structures on the categories and . Then this is a Quillen adjoint triple (Def. ) of the form
Homotopy (co-)limits
Example
(Quillen adjoint triple of homotopy limits/colimits of simplicial sets)
Let be a small category, and write for the projective/injective model structure on simplicial presheaves over , which participate in a Quillen equivalence of the form
(by this Prop.).
Moreover, the constant diagram-assigning functor
is clearly a left Quillen functor for the injective model structure, and a right Quillen functor for the projective model structure.
Together this means that in the double category of model categories we have a 2-morphism of the form
Moreover, the derived natural transformation of this square is invertible, if for every Kan complex
is a weak homotopy equivalence (by this Prop.), which here is trivially the case.
Therefore we have a Quillen adjoint triple of the form
The induced adjoint triple of derived functors on the homotopy categories (via this Prop.) is the homotopy colimit/homotopy limit adjoint triple
Homotopy Kan extension
More generally:
Example
(Quillen adjoint triple of homotopy Kan extension of simplicial presheaves)
Let and be small categories, and let
be a functor between them. By Kan extension this induces an adjoint triple between categories of simplicial presheaves:
where
is the operation of precomposition with . This means that preserves all objectwise cofibrations/fibrations/weak equivalences. Hence it is
-
a right Quillen functor ;
-
a left Quillen functor ;
and since
is also a Quillen adjunction, these imply that is also
-
a right Quillen functor .
-
a left Quillen functor .
In summary this means that we have 2-morphisms in the double category of model categories of the following form:
To check that the corresponding derived natural transformations are natural isomorphisms, we need to check (by this Prop.) that the composites
are invertible in the homotopy category , for all fibrant-cofibrant simplicial presheaves in . But this is immediate, since the two factors are weak equivalences, by definition of fibrant/cofibrant resolution.
Hence we have a Quillen adjoint triple (Def. ) of the form
(3)
The corresponding derived adjoint triple on homotopy categories (Prop. ) is that of homotopy Kan extension:
Example
(Quillen adjoint quadruple of homotopy Kan extension of simplicial presheaves along adjoint pair)
Let and be small categories, and let
be a pair of adjoint functors. By Kan extension this induces an adjoint quadruple between categories of simplicial presheaves
By Example the top three as well as the bottom three of these form Quillen adjoint triples (Def. ) in two ways (3). If for the top three we choose the first version, and for the bottom three the second version from (3), then these combine to a Quillen adjoint quadruple of the form
Example
(Quillen adjoint quintuple of homotopy Kan extension of simplicial presheaves along adjoint triple)
Let and be small categories and let
be a triple of adjoint functors. By Kan extension this induces an adjoint quintuple between categories of simplicial presheaves
(4)
By Example the top four functors in (4) form a Quillen adjoint quadruple ending in a right Quillen functor
But here is also a left Quillen functor (as in Example ), and hence this continues by one more Quillen adjoint triple via Example to a Quillen adjoint quintuple of the form
Alternatively, we may regard the bottom four functors in (4) as a Quillen adjoint quadruple via example , whose top functor is then the left Quillen functor
But this is also a right Quillen functor (as in Example ) and hence we may continue by one more Quillen adjoint triple upwards (via Example ) to obtain a Quillen adjoint quintuple, now of the form
Properties
Derived adjoint triple
Proposition
(Quillen adjoint triple induces adjoint triple of derived functors on homotopy categories)
Given a Quillen adjoint triple (Def. ), the induced derived functors on the homotopy categories form an ordinary adjoint triple:
A similar argument should show that we get an adjoint triple between the (infinity,1)-categories presented by the model categories. We can therefore apply all -category theoretic arguments. But in what follows, we prove some basic facts about -adjoint triples instead using model-categorical arguments.
Derived adjoint modality
Proof
The derived adjunction unit is on cofibrant objects given by
Here the fibrant resolution-morphism is an acyclic cofibration in . Since is also a left Quillen functor , the comparison morphism is an acyclic cofibration in , hence in particular a weak equivalence in and therefore, by assumption, also in .
The derived adjunction counit of the second adjunction is
Here the cofibrant resolution-morphisms is an acyclic fibration in . Since is also a right Quillen functor , the comparison morphism is an acyclic fibration in , hence in particular a weak equivalence there, hence, by assumption, also a weak equivalence in .
Lemma
(fully faithful functors in Quillen adjoint quadruple)
Given a Quillen adjoint quadruple (Def. )
if any of the four functors is fully faithful functor, then so is its derived functor.
Proof
Observing that each of the four functors is either the leftmost or the rightmost adjoint in the top or the bottom adjoint triple within the adjoint quadruple, the claim follows by Lemma .
In summary
Proposition
(derived adjoint modalities from fully faithful Quillen adjoint quadruples)
Given a Quillen adjoint quadruple (Def. )
then the corresponding derived functors form an adjoint quadruple
Moreover, if one of the functors in the Quillen adjoint quadruple is a fully faithful functor, then so is the corresponding derived functor.
Hence if the original adjoint quadruple induces an adjoint modality on
or on
then so do the corresponding derived functors on the homotopy categories, respectively.