Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-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 $\infty$-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

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

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

$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$
$\mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$

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

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

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

If two Quillen adjoint triples overlap

$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$
$\mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$
$\mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2$

## Examples

### General

###### Example

(Quillen adjoint triple from left and right Quillen functor)

$\mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{R}{\longrightarrow} } \mathcal{D}$

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

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

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

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

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 $d \in \mathcal{D}$ the composite

$Q C(d) \overset{ p_{C(d)} }{\longrightarrow} C(d) \overset{ j_{C(d)} }{\longrightarrow} P C(C)$

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

### Homotopy (co-)limits

###### Example

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

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

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

(by this Prop.).

Moreover, the constant diagram-assigning functor

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

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

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

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

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

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

Therefore we have a Quillen adjoint triple of the form

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

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

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

### Homotopy Kan extension

More generally:

###### Example

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

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

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

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

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

where

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

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

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

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

and since

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

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

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

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

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

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

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

$Q_{inj} F^\ast \mathbf{X} \overset{ p_{F^\ast \mathbf{X}} }{\longrightarrow} F^\ast \mathbf{X} \overset{ j_{F^\ast \mathbf{X}} }{\longrightarrow} P_{proj} F^\ast \mathbf{X}$

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

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

$Ho([\mathcal{C}^{op}, sSet]) \array{ \underoverset{\bot \phantom{\simeq A_a}}{ \phantom{A}\mathbb{L}F_! \phantom{\simeq A_a}\phantom{A} }{\longrightarrow} \\ \underoverset{\phantom{\simeq A_a} \bot}{ \phantom{A}\mathbb{R}F^\ast \simeq \mathbb{L}F^\ast\phantom{A} }{\longleftarrow} \\ \overset{ \phantom{A} \phantom{A_a \simeq} \mathbb{R}F_\ast\phantom{A} }{\longrightarrow} } Ho([\mathcal{D}^{op}, sSet])$
###### Example

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

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

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

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

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

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

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

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

$\mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} } \mathcal{D}$

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

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

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

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

But $R^\ast$ 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

$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$
$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$
$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$
$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$

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

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

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

$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$
$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$
$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$
$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$

## Properties

###### 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:

$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})$
###### 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 $(\infty,1)$-category theoretic arguments. But in what follows, we prove some basic facts about $(\infty,1)$-adjoint triples instead using model-categorical arguments.

###### Lemma

Consider a Quillen adjoint triple (Def. )

$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$
$\mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$

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

Then:

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

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

###### Proof

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

$c \overset{\eta_c}{\longrightarrow} C L (c) \overset{ C(j_{L(c)}) }{\longrightarrow} C P L (c)$

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

$C Q R (c) \overset{ C(p_{R(c)}) }{\longrightarrow} C R (c) \overset{ \epsilon_c }{\longrightarrow} c$

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

###### Lemma

(fully faithful functors in Quillen adjoint triple)

Consider a Quillen adjoint triple (Def. )

$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$
$\mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$

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

###### Proof

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

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

###### Lemma

$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$
$\mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$
$\mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2$

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

$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$
$\mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$
$\mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2$

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

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 $\mathcal{C}$

$\bigcirc \dashv \Box \dashv \lozenge$

or on $\mathcal{D}$

$\Box \dashv \bigcirc \dashv \triangle$

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 October 13, 2021 at 14:45:28. See the history of this page for a list of all contributions to it.