nLab
Quillen adjoint triple

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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 (infinity,1)-categories we still get an adjoint triple.

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 𝒞 1,𝒞 2,𝒟\mathcal{C}_1, \mathcal{C}_2, \mathcal{D} be model categories, where 𝒞 1\mathcal{C}_1 and 𝒞 2\mathcal{C}_2 share the same underlying category 𝒞\mathcal{C}, and such that the identity functor on 𝒞\mathcal{C} constitutes a Quillen equivalence

(1)𝒞 2 Qu QuAAidAAAAidAA𝒞 1 \mathcal{C}_2 \underoverset {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}} {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}} {{}_{\phantom{Qu}}\simeq_{Qu}} \mathcal{C}_1

Then a Quillen adjoint triple

𝒞 1 Qu QuL𝒟 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 2 Qu QuRC𝒟 \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

is a pair of Quillen adjunctions, as shown, together with a 2-morphism in the double category of model categories

(2)𝒟 ACA 𝒞 1 C id id 𝒞 2 AAidAA 𝒞 2 \array{ \mathcal{D} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 }

whose derived natural transformation Ho(id)Ho(id) (here) is invertible (a natural isomorphism).

If two Quillen adjoint triples overlap

𝒞 1 Qu QuL=A𝒟 1 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuC=L𝒟 1 \mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuA=RR=C𝒟 2 \mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2

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

𝒞AAAAL AAAAC AAAAR𝒟 \mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{R}{\longrightarrow} } \mathcal{D}

such that CC is both a left Quillen functor as well as a right Quillen functor for given model category-structures on the categories 𝒞\mathcal{C} and 𝒟\mathcal{D}. Then this is a Quillen adjoint triple (Def. ) of the form

𝒞 Qu QuL𝒟 \mathcal{C} \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 Qu QuRC𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
Proof

The condition of a Quillen equivalence (1) is trivially satisfied (this Prop.). Similarly the required 2-morphism (2)

𝒞 ACA 𝒟 C id id 𝒟 AAidAA 𝒟 \array{ \mathcal{C} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{D} &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{D} }

exists trivially. To check that its derived natural transformation (here) is a natural isomorphism we need to check (by this Prop.) that for every fibrant and cofibrant object d𝒟d \in \mathcal{D} the composite

QC(d)p C(d)C(d)j C(d)PC(C) Q C(d) \overset{ p_{C(d)} }{\longrightarrow} C(d) \overset{ j_{C(d)} }{\longrightarrow} P C(C)

is a weak equivalence. But this is trivially the case, by definition of fibrant resolution/cofibrant resolution (in fact, since CC is assumed to be both left and right Quillen, also C(d)C(d) is a fibrant and cofibrant objects and hence we may even take both p C(d)p_{C(d)} as well as j C(d)j_{C(d)} to be the identity morphism).

Homotopy (co-)limits

Example

(Quillen adjoint triple of homotopy limits/colimits of simplicial sets)

Let 𝒞\mathcal{C} be a small category, and write [𝒞 op,sSet Qu] proj/inj[\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} for the projective/injective model structure on simplicial presheaves over 𝒞\mathcal{C}, which participate in a Quillen equivalence of the form

[𝒞 op,sSet Qu] inj QuAAidAAAAidAA[𝒞 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\simeq_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}

(by this Prop.).

Moreover, the constant diagram-assigning functor

[𝒞 op,sSet]constsSet [\mathcal{C}^{op}, sSet] \overset{const}{\longleftarrow} sSet

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

sSet Qu const [𝒞 op,sSet Qu] proj const id id [𝒞 op,sSet Qu] inj id [𝒞 op,sSet Qu] inj \array{ sSet_{Qu} &\overset{const}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{const}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} }

Moreover, the derived natural transformation Ho(id)Ho(id) of this square is invertible, if for every Kan complex XX

QconstXconstXPconstX Q const X \overset{}{\longrightarrow} const X \longrightarrow P const X

is a weak homotopy equivalence (by this Prop.), which here is trivially the case.

Therefore we have a Quillen adjoint triple of the form

[𝒞 op,sSet Qu] proj Qu QulimsSet Qu [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{ \underset{\longrightarrow}{\lim} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu} } sSet_{Qu}
[𝒞 op,sSet Qu] inj Qu QulimconstsSet Qu [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{const}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu} } sSet_{Qu}

The induced adjoint triple of derived functors on the homotopy categories (via this Prop.) is the homotopy colimit/homotopy limit adjoint triple

Ho([𝒞 op,sSet])AA𝕃limAA AAconstAA AAlimAAHo(sSet) Ho([\mathcal{C}^{op}, sSet]) \; \array{ \overset{\phantom{AA}\mathbb{L}\underset{\longrightarrow}{\lim}\phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}const\phantom{AA}}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}\underset{\longleftarrow}{\lim}\phantom{AA}}{\longrightarrow} } \; Ho(sSet)

Homotopy Kan extension

More generally:

Example

(Quillen adjoint triple of homotopy Kan extension of simplicial presheaves)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be small categories, and let

𝒞AAFAA𝒟 \mathcal{C} \overset{\phantom{AA}F\phantom{AA}}{\longrightarrow} \mathcal{D}

be a functor between them. By Kan extension this induces an adjoint triple between categories of simplicial presheaves:

[𝒞 op,sSet]AAF !AA AAF *AA AAF *AA[𝒟 op,sSet] [\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot}{ \phantom{AA}F_!\phantom{AA} }{\longrightarrow} \\ \underoverset{\bot}{ \phantom{AA}F^\ast\phantom{AA} }{\longleftarrow} \\ \overset{ \phantom{AA}F_\ast\phantom{AA} }{\longrightarrow} } [\mathcal{D}^{op}, sSet]

where

F *XX(F()) F^\ast \mathbf{X} \;\coloneqq\; \mathbf{X}(F(-))

is the operation of precomposition with FF. This means that F *F^\ast preserves all objectwise cofibrations/fibrations/weak equivalences. Hence it is

  1. a right Quillen functor [𝒟 op,sSet] projF *[𝒞 op,sSet Qu] proj[\mathcal{D}^{op}, sSet]_{proj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{proj};

  2. a left Quillen functor [𝒟 op,sSet] injF *[𝒞 op,sSet Qu] inj[\mathcal{D}^{op}, sSet]_{inj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{inj};

and since

[𝒟 op,sSet] injAAAAidid[𝒟 op,sSet] proj [\mathcal{D}^{op}, sSet]_{inj} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} [\mathcal{D}^{op}, sSet]_{proj}

is also a Quillen adjunction, these imply that F *F^\ast is also

  • a right Quillen functor [𝒟 op,sSet] injF *[𝒞 op,sSet Qu] proj[\mathcal{D}^{op}, sSet]_{inj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}.

  • a left Quillen functor [𝒟 op,sSet] projF *[𝒞 op,sSet Qu] inj[\mathcal{D}^{op}, sSet]_{proj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{inj}.

In summary this means that we have 2-morphisms in the double category of model categories of the following form:

[𝒟 op,sSet Qu] proj AAF *AA [𝒞 op,sSet Qu] proj F * id id [𝒞 op,sSet Qu] inj id [𝒞 op,sSet Qu] injAAAandAAA[𝒟 op,sSet Qu] inj AAF *AA [𝒞 op,sSet Qu] proj F * id id [𝒞 op,sSet Qu] inj id [𝒞 op,sSet Qu] proj \array{ [\mathcal{D}^{op}, sSet_{Qu}]_{proj} &\overset{\phantom{AA}F^\ast\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{F^\ast}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} } \phantom{AAA} \text{and} \phantom{AAA} \array{ [\mathcal{D}^{op}, sSet_{Qu}]_{inj} &\overset{\phantom{AA}F^\ast\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{F^\ast}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} }

To check that the corresponding derived natural transformations Ho(id)Ho(id) are natural isomorphisms, we need to check (by this Prop.) that the composites

Q injF *Xp F *XF *Xj F *XP projF *X Q_{inj} F^\ast \mathbf{X} \overset{ p_{F^\ast \mathbf{X}} }{\longrightarrow} F^\ast \mathbf{X} \overset{ j_{F^\ast \mathbf{X}} }{\longrightarrow} P_{proj} F^\ast \mathbf{X}

are invertible in the homotopy category Ho([𝒞 op,sSet Qu] inj/proj)Ho([\mathcal{C}^{op}, sSet_{Qu}]_{inj/proj}), for all fibrant-cofibrant simplicial presheaves X\mathbf{X} in [𝒞 op,sSet Qu] proj/inj[\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}. 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)[𝒞 op,sSet Qu] proj/injAAF !AA AAF *AA AAF *AA[𝒟 op,sSet Qu] projAAAandAAA[𝒞 op,sSet Qu] proj/injAAF !AA AAF *AA AAF *AA[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow} \\ \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow} } [\mathcal{D}^{op}, sSet_{Qu}]_{proj} \phantom{AAA} \text{and} \phantom{AAA} [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow} \\ \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow} } [\mathcal{D}^{op}, sSet_{Qu}]_{inj}

The corresponding derived adjoint triple on homotopy categories (Prop. ) is that of homotopy Kan extension:

Ho([𝒞 op,sSet])A aA𝕃F !A aA A aAF *𝕃F *A AA aF *AHo([𝒟 op,sSet]) Ho([\mathcal{C}^{op}, sSet]) \array{ \underoverset{\bot \phantom{\simeq A_a}}{ \phantom{A}\mathbb{L}F_! \phantom{\simeq A_a}\phantom{A} }{\longrightarrow} \\ \underoverset{\phantom{\simeq A_a} \bot}{ \phantom{A}\mathbb{R}F^\ast \simeq \mathbb{L}F^\ast\phantom{A} }{\longleftarrow} \\ \overset{ \phantom{A} \phantom{A_a \simeq} \mathbb{R}F_\ast\phantom{A} }{\longrightarrow} } Ho([\mathcal{D}^{op}, sSet])
Example

(Quillen adjoint quadruple of homotopy Kan extension of simplicial presheaves along adjoint pair)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be small categories, and let

𝒞AARAAAALAA𝒟 \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longleftarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longrightarrow}} {\bot} \mathcal{D}

be a pair of adjoint functors. By Kan extension this induces an adjoint quadruple between categories of simplicial presheaves

[𝒞 op,sSet]A aL !A a L *R ! A aL *R * A aR *[𝒟 op,sSet] [\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot \phantom{\simeq A_a}}{ L_! \phantom{\simeq A_a} }{\longrightarrow} \\ \underoverset{\bot \phantom{\simeq} \bot }{ L^\ast \simeq R_! }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot}{ L_\ast \simeq R^\ast }{\longrightarrow} \\ \overset{ \phantom{A_a \simeq } R_\ast }{\longrightarrow} } [\mathcal{D}^{op}, sSet]

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

[𝒞 op,sSet Qu] proj Qu QuL !=A a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{= A_a}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] inj Qu QuL *=R ![𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast = R_!}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] inj Qu QuA=R *L *=R *[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{A=} R_\ast}{\longleftarrow}} {\overset{L_\ast = R^\ast}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}
Example

(Quillen adjoint quintuple of homotopy Kan extension of simplicial presheaves along adjoint triple)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be small categories and let

𝒞AAAAL AAAAC AAAAC𝒟 \mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} } \mathcal{D}

be a triple of adjoint functors. By Kan extension this induces an adjoint quintuple between categories of simplicial presheaves

(4)[𝒞 op,sSet]A aA aL !A aA a A aL *C !A a A aL *C *R ! A aA aA aC *R * A aA aA aC *R *[𝒟 op,sSet] [\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot \phantom{\simeq A_a \simeq A_a}}{ L_! \phantom{\simeq A_a \simeq A_a} }{\longrightarrow} \\ \underoverset{\bot \phantom{\simeq} \bot \phantom{\simeq A_a} }{ L^\ast \simeq C_! \phantom{\simeq A_a} }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot \phantom{\simeq} \bot}{ L_\ast \simeq C^\ast \simeq R_! }{\longrightarrow} \\ \underoverset{\phantom{A_a \simeq A_a } \bot}{ \phantom{A_a \simeq } C_\ast \simeq R^\ast }{\longrightarrow} \\ \underoverset{\phantom{A_a \simeq A_a } \phantom{\bot}}{ \phantom{A_a \simeq C_\ast \simeq} R_\ast }{\longrightarrow} } [\mathcal{D}^{op}, sSet]

By Example the top four functors in (4) form a Quillen adjoint quadruple ending in a right Quillen functor

[𝒞 op,sSet Qu] injC *R *[𝒞 op,sSet Qu] inj. [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \overset{C_\ast \simeq R^\ast}{\longrightarrow} [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \,.

But R *R^\ast 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

[𝒞 op,sSet Qu] projA Qu QuAL !A aA a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAL *C !A a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAL *C *R ![𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}
[𝒞 op,sSet Qu] injA Qu QuAA aA aR *A aC *R *[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}

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

[𝒞 op,sSet Qu] projL *[𝒟 op,sSet Qu] proj. [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \overset{ L^\ast }{\longleftarrow} [\mathcal{D}^{op}, sSet_{Qu}]_{proj} \,.

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

[𝒞 op,sSet Qu] projA Qu QuAL !A aA a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] projA Qu QuAL *C !A a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAL *C *R ![𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAA aA aR *A aC *R *[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}

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:

𝒞 1/2 Qu QuL Qu QuC AARAA 𝒟AAAAHo()AAAAHo(𝒞) Qu Qu𝕃L Qu Qu𝕃CC AARAA Ho(𝒟) \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})
Proof

This follows immediately from the fact that passing to homotopy categories of model categories is a double pseudofunctor from the double category of model categories to the double category of squares in Cat (this Prop.).

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 (,1)(\infty,1)-category theoretic arguments. But in what follows, we prove some basic facts about (,1)(\infty,1)-adjoint triples instead using model-categorical arguments.

Derived adjoint modality

Lemma

(derived adjunction units of Quillen adjoint triple)

Consider a Quillen adjoint triple (Def. )

𝒞 1 Qu QuL𝒟 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 2 Qu QuRC𝒟 \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

such that the two model structures 𝒞 1\mathcal{C}_1 and 𝒞 2\mathcal{C}_2 on the category 𝒞\mathcal{C} share the same class of weak equivalences.

Then:

  1. the derived adjunction unit of (LC)(L \dashv C) in 𝒞 1\mathcal{C}_1 differs only by a weak equivalence from the plain adjunction unit.

  2. the derived adjunction counit of (CR)(C \dashv R) differs only by a weak equivalence form the plain adjunction counit.

Proof

The derived adjunction unit is on cofibrant objects c𝒞 1c \in \mathcal{C}_1 given by

cη cCL(c)C(j L(c))CPL(c) c \overset{\eta_c}{\longrightarrow} C L (c) \overset{ C(j_{L(c)}) }{\longrightarrow} C P L (c)

Here the fibrant resolution-morphism j P(c)j_{P(c)} is an acyclic cofibration in 𝒟\mathcal{D}. Since CC is also a left Quillen functor 𝒟C𝒞 2\mathcal{D} \overset{C}{\to} \mathcal{C}_2, the comparison morphism C(j L(c))C(j_{L(c)}) is an acyclic cofibration in 𝒞 2\mathcal{C}_2, hence in particular a weak equivalence in 𝒞 2\mathcal{C}_2 and therefore, by assumption, also in 𝒞 1\mathcal{C}_1.

The derived adjunction counit of the second adjunction is

CQR(c)C(p R(c))CR(c)ϵ cc C Q R (c) \overset{ C(p_{R(c)}) }{\longrightarrow} C R (c) \overset{ \epsilon_c }{\longrightarrow} c

Here the cofibrant resolution-morphisms p R(c)p_{R(c)} is an acyclic fibration in 𝒟\mathcal{D}. Since CC is also a right Quillen functor 𝒟C𝒞 1\mathcal{D} \overset{C}{\to} \mathcal{C}_1, the comparison morphism C(p R(c))C(p_{R(c)}) is an acyclic fibration in 𝒞 1\mathcal{C}_1, hence in particular a weak equivalence there, hence, by assumption, also a weak equivalence in 𝒞 2\mathcal{C}_2.

Lemma

(fully faithful functors in Quillen adjoint triple)

Consider a Quillen adjoint triple (Def. )

𝒞 1 Qu QuL𝒟 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 2 Qu QuRC𝒟 \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

If LL and RR are fully faithful functors (necessarily jointly, by this Prop.), then so are their derived functors 𝕃L\mathbb{L}L and R\mathbb{R}R.

Proof

We discuss that RR being fully faithful implies that R\mathbb{R}R is fully faithful. Since also the derived functors form an adjoint triple (by Prop. ), this will imply the claim also for LL and 𝕃L\mathbb{L}L.

By Lemma the derived adjunction counit of CRC \dashv R is, up to weak equivalence, the ordinary adjunction counit. But the latter is an isomorphism, since RR is fully faithful (by this Prop.). In summary this means that the derived adjunction unit of (CR)(C \dashv R) is a weak equivalence, hence that its image in the homotopy category is an isomorphism. But the latter is the ordinary adjunction unit of 𝕃CR\mathbb{L}C \dashv \mathbb{R}R (by this Prop.), and hence the claim follows again by that Prop..

Lemma

(fully faithful functors in Quillen adjoint quadruple)

Given a Quillen adjoint quadruple (Def. )

𝒞 1 Qu QuL=A𝒟 1 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuC=L𝒟 1 \mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuA=RR=C𝒟 2 \mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2

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. )

𝒞 1 Qu QuL=A𝒟 1 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuC=L𝒟 1 \mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuA=RR=C𝒟 2 \mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2

then the corresponding derived functors form an adjoint quadruple

Ho(𝒞)AAAA𝕃L AAAA𝕃CC𝕃L AAAAR𝕃CC AAAARHo(𝒟) Ho(\mathcal{C}) \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{L}L }{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{L}C \simeq \mathbb{R}C \simeq \mathbb{L}L' }{\longleftarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{R}R \simeq \mathbb{L}C' \simeq \mathbb{R}C' }{\longrightarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{ \mathbb{R}R' }{\longleftarrow} } Ho(\mathcal{D})

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 𝒞\mathcal{C}

\bigcirc \dashv \Box \dashv \lozenge

or on 𝒟\mathcal{D}

\Box \dashv \bigcirc \dashv \triangle

then so do the corresponding derived functors on the homotopy categories, respectively.

Proof

The existence of the derived adjoint quadruple followy by Prop. and by uniqueness of adjoints (this Prop.).

The statement about fully faithful functors is Lemma . The reformulation in terms of adjoint modalities is by this Prop.

Last revised on July 21, 2018 at 13:51:20. See the history of this page for a list of all contributions to it.