Quillen adjoint triple

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#Contents#
* table of contents
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## Idea

Just as a [[Quillen adjunction]] is a [[model category|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|$\infty$-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

+-- {: .num_defn #QuillenAdjointTriple}
###### Definition
**(Quillen adjoint triple)**

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]]

\[
  \label{EquivForQuillenAdjointTriple}
  \mathcal{C}_2
    \underoverset
      {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}}
      {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}}
      {{}_{\phantom{Qu}}\simeq_{Qu}}
  \mathcal{C}_1
\]

Then a [[Quillen adjoint triple]]

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

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

\[
  \label{Comparison2MorphismForQuillenAdjointTriple}
  \array{
    \mathcal{D}
     &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}&
    \mathcal{C}_1
    \\
    {}^{\mathllap{C}}
    \Big\downarrow
      &\swArrow_{\mathrlap{id}}&
    \Big\downarrow{}^{ \mathrlap{ id } }
    \\
    \mathcal{C}_2
    &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}&
    \mathcal{C}_2
  }
\]

whose [[derived natural transformation]] $Ho(id)$ ([here](double+category+of+model+categories#DerivedNaturalTransformation)) is invertible (a [[natural isomorphism]]).

If two Quillen adjoint triples overlap

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

we speak of a _Quillen [[adjoint quadruple]]_, and so forth.

=--

## Examples

### General

+-- {: .num_example #QuillenAdjointTripleFromLeftAndRightQuillenFunctor}
###### Example
**([[Quillen adjoint triple]] from left and right [[Quillen functor]])**

Given an [[adjoint triple]]

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

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. \ref{QuillenAdjointTriple}) 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}
###### Proof

The condition of a [[Quillen equivalence]] (eq:EquivForQuillenAdjointTriple) is trivially satisfied ([this Prop.](Quillen+equivalence#TrivialQuillenEquivalence)). Similarly the required [[2-morphism]] (eq:Comparison2MorphismForQuillenAdjointTriple)

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

exists trivially. To check that its [[derived natural transformation]] ([here](double+category+of+model+categories#DerivedNaturalTransformation)) is a [[natural isomorphism]] we need to check (by [this Prop.](double+category+of+model+categories#DerivedNaturalTransformationUpToIsos)) 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

+-- {: .num_example #HomotopyLimitQuillenAdjointTriple}
###### Example
**([[Quillen adjoint triple]] of [[homotopy limits]]/[[homotopy colimits|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.](geometry+of+physics+--+categories+and+toposes#ModelCategoriesOfSimplicialPresheaves)).

Moreover, the [[constant functor|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.](double+category+of+model+categories#DerivedNaturalTransformationUpToIsos)), 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 category of a model category|homotopy categories]] (via [this Prop.](geometry+of+physics+--+categories+and+toposes#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories)) 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:

+-- {: .num_example #QuillenAdjointTripleHomotopyKanExtension}
###### 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]] $[\mathcal{D}^{op}, sSet]_{proj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$;

1. 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.](double+category+of+model+categories#DerivedNaturalTransformationUpToIsos)) 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 of a model category|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 resolution|fibrant/cofibrant resolution]].

Hence we have a [[Quillen adjoint triple]] (Def. \ref{QuillenAdjointTriple}) of the form

\[
  \label{QuillenAdjointTripleFromKanExtensionOfSimplicialPresheaves}
  [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}
    \array{
       \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow}
       \\
       \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow}
       \\
       \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow}
    }
  [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
  \phantom{AAA}
  \text{and}
  \phantom{AAA}
  [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}
    \array{
       \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow}
       \\
       \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow}
       \\
       \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow}
    }
  [\mathcal{D}^{op}, sSet_{Qu}]_{inj}
\]

The corresponding derived [[adjoint triple]] on [[homotopy category of a model category|homotopy categories]] (Prop. \ref{QuillenAdjointTripleInducesAdjointTripleOfDerivedFunctors}) is that of _[[homotopy Kan extension]]_:

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

=--

+-- {: .num_example #QuillenAdjointQuadrupleOfHomotopyKanExtensionAlongAdjointPair}
###### Example
**([[Quillen adjoint quadruple]] of [[homotopy Kan extension]] of [[simplicial presheaves]] along [[adjoint pair]])**

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 [[adjoint pair|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 \ref{QuillenAdjointTripleHomotopyKanExtension} the top three as well as the bottom three of these form [[Quillen adjoint triples]] (Def. \ref{QuillenAdjointTriple}) in two ways (eq:QuillenAdjointTripleFromKanExtensionOfSimplicialPresheaves). If for the top three we choose the first version, and for the bottom three the second version from (eq:QuillenAdjointTripleFromKanExtensionOfSimplicialPresheaves), then these combine to a Quillen [[adjoint quadruple]] of the form

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

=--

+-- {: .num_example #QuillenAdjointQuintupleOfHomotopyKanExtensionAlongAdjointTriple}
###### 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 [[adjoint triple|triple]] of [[adjoint functors]]. By [[Kan extension]] this induces an [[adjoint quintuple]] between [[categories of simplicial presheaves]]

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

By Example \ref{QuillenAdjointQuadrupleOfHomotopyKanExtensionAlongAdjointPair} the top four functors in (eq:AdjointQuintupleFromKanExtensionInDiscussionOfQuillenAdjointQuintuple) 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 \ref{QuillenAdjointTripleHomotopyKanExtension}), and hence this continues by one more Quillen adjoint triple via Example \ref{QuillenAdjointTripleFromLeftAndRightQuillenFunctor} 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 (eq:AdjointQuintupleFromKanExtensionInDiscussionOfQuillenAdjointQuintuple) as a [[Quillen adjoint quadruple]] via example \ref{QuillenAdjointQuadrupleOfHomotopyKanExtensionAlongAdjointPair}, 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 \ref{QuillenAdjointTripleHomotopyKanExtension}) and hence we may continue by one more [[Quillen adjoint triple]] upwards (via Example \ref{QuillenAdjointTripleFromLeftAndRightQuillenFunctor}) 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

### Derived adjoint triple
 {#DerivedAdjointTriple}

+-- {: .num_prop #QuillenAdjointTripleInducesAdjointTripleOfDerivedFunctors}
###### Proposition
**([[Quillen adjoint triple]] induces [[adjoint triple]] of [[derived functors]] on [[homotopy category of a model category|homotopy categories]])**

Given a [[Quillen adjoint triple]] (Def. \ref{QuillenAdjointTriple}), the induced [[derived functors]] on the [[homotopy category of a model category|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}
###### 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.](double+category+of+model+categories#HomotopyDoublePseudofunctor)).

=--

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.

### Derived adjoint modality
 {#DerivedAdjointModality}

+-- {: .num_lemma #DerivedAdjunctionUnitOfQuillenAdjointTriple}
###### Lemma
**([[derived adjunction units]] of [[Quillen adjoint triple]])**

Consider a [[Quillen adjoint triple]] (Def. \ref{QuillenAdjointTriple})

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

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]].

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

=--

+-- {: .proof}
###### 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 $\mathcal{C}_1$.

The [[derived adjunction counit]] of the second adjunction is

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

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 $\mathcal{C}_2$.

=--

+-- {: .num_lemma #DerivedModalityFromQuillenAdjointTriple}
###### Lemma
**([[fully faithful functors]] in [[Quillen adjoint triple]])**

Consider a [[Quillen adjoint triple]] (Def. \ref{QuillenAdjointTriple})

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

If $L$ and $R$ are [[fully faithful functors]] (necessarily jointly, by [this Prop.](adjoint+triple#FullyFaithful)), then so are their [[derived functors]] $\mathbb{L}L$ and $\mathbb{R}R$.

=--

+-- {: .proof}
###### 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. \ref{QuillenAdjointTripleInducesAdjointTripleOfDerivedFunctors}), this will imply the claim also for $L$ and $\mathbb{L}L$.

By Lemma \ref{DerivedAdjunctionUnitOfQuillenAdjointTriple} 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.](adjoint+functor#FullyFaithfulAndInvertibleAdjoints)). 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.](geometry+of+physics+--+categories+and+toposes#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories)), and hence the claim follows again by [that Prop.](adjoint+functor#FullyFaithfulAndInvertibleAdjoints).

=--

+-- {: .num_lemma #DerivedAdjointModalityFromQuillenAdjointQuadruple}
###### Lemma
**([[fully faithful functors]] in [[Quillen adjoint quadruple]])**

Given a [[Quillen adjoint quadruple]] (Def. \ref{QuillenAdjointTriple})

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

if any of the four functors is [[fully faithful functor]], then so is its [[derived functor]].

=--

+-- {: .proof}
###### 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 \ref{DerivedAdjointModalityFromQuillenAdjointQuadruple}.

=--

In summary

+-- {: .num_prop #DerivedAdjointModalitiesFromFullyFaithfulQuillenAdjointQuadruples}
###### Proposition
**([[derived functor|derived]] [[adjoint modalities]] from [[fully faithful functor|fully faithful]] [[Quillen adjoint quadruples]])**

Given a [[Quillen adjoint quadruple]] (Def. \ref{QuillenAdjointTriple})

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

then the corresponding [[derived functors]] form an [[adjoint quadruple]]

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

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 category of a model category|homotopy categories]], respectively.

=--

+-- {: .proof}
###### Proof

The existence of the derived [[adjoint quadruple]] followy by Prop. \ref{QuillenAdjointTripleInducesAdjointTripleOfDerivedFunctors} and by uniqueness of adjoints ([this Prop.](adjoint+functor#UniquenessOfAdjoints)).

The statement about fully faithful functors is Lemma \ref{DerivedAdjointModalityFromQuillenAdjointQuadruple}. The reformulation in terms of [[adjoint modalities]] is by [this Prop.](geometry+of+physics+--+categories+and+toposes#ModalOperatorsEquivalentToReflectiveSubcategories)

=--

## Related concepts

* [[adjoint triple]]

* [[adjoint quadruple]]

[[!redirects Quillen adjoint triples]]

[[!redirects Quillen adjoint quadruple]]
[[!redirects Quillen adjoint quadruples]]

[[!redirects Quillen adjoint quintuple]]
[[!redirects Quillen adjoint quintuples]]


Quillen adjunction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Quillen adjunctions are one convenient notion of [[morphisms]] between [[model categories]].  They present [[adjoint (∞,1)-functors]] between the [[(∞,1)-category|(∞,1)-categories]] [[presentable (infinity,1)-category|presented]] by the model categories.

## Definition
 {#Definition}

+-- {: .num_defn #QuillenAdjunction}
###### Definition

For $C$ and $D$ two [[model category|model categories]], a pair $(L,R)$

$$
  (L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}}
  D
$$

of [[adjoint functors]] (with $L$ [[left adjoint]] and $R$ [[right adjoint]]) is a **Quillen adjunction** if the following equivalent conditions are satisfied:

1. $L$ preserves [[cofibrations]] and [[acyclic cofibrations]];

1. $R$ preserves [[fibrations]] and [[acyclic fibrations]];

1. $L$ preserves [[cofibrations]] and $R$ preserves [[fibrations]];

1. $L$ preserves [[acyclic cofibrations]] and $R$ preserves [[acyclic fibrations]].

=--

+-- {: .num_prop}
###### Proposition

The conditions in def. \ref{QuillenAdjunction} are indeed all equivalent.

=--

+-- {: .proof}
###### Proof

Observe that

* (i) _A [[left adjoint]] $L$ between [[model categories]] preserves acyclic cofibrations precisely if its [[right adjoint]] $R$ preserves fibrations.

* (ii) _A [[left adjoint]] $L$ between [[model categories]] preserves cofibrations precisely if its [[right adjoint]] $R$ preserves acyclic fibrations.

We discuss statement (i), statement (ii) is [[formal dual|formally dual]].  So let $f\colon A \to B$ be an acyclic cofibration in $\mathcal{D}$ and $g \colon X \to Y$ a fibration in $\mathcal{C}$. Then for every [[commuting diagram]] as on the left of the following, its $(L\dashv R)$-[[adjunct]] is a commuting diagram as on the right here:

$$
  \array{
    A &\longrightarrow& R(X)
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}}
    \\
    B &\longrightarrow& R(Y)
  }
  \;\;\;\;\;\;
  \,,
  \;\;\;\;\;\;
  \array{
    L(A) &\longrightarrow& X
    \\
    {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}}
    \\
    L(B) &\longrightarrow& Y
  }
  \,.
$$

If $L$ preserves acyclic cofibrations, then the diagram on the right has a [[lift]], and so the $(L\dashv R)$-[[adjunct]] of that lift is a lift of the left diagram. This shows that $R(g)$ has the [[right lifting property]] against all acylic cofibrations and hence is a fibration.
Conversely, if $R$ preserves fibrations, the same argument run from right to left gives that $L$ preserves acyclic fibrations.

Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.

=--

+-- {: .num_remark}
###### Remark

Quillen adjunctions that are analogous to an [[equivalence of categories]] are called _[[Quillen equivalences]]_.

In an [[enriched model category]] one speaks of _[[enriched Quillen adjunction]]_.

=--

## Properties

### Derived adjunction

+-- {: .num_prop}
###### Proposition
**([[Ken Brown's lemma]])**

Given a [[Quillen adjunction]] $(L \dashv R)$ (def. \ref{QuillenAdjunction}), then

* the [[left adjoint]] $L$ preserves weak equivalences between cofibrant objects;

* the [[right adjoint]] $R$ preserves weak equivalences between fibrant objects.

=--

+-- {: .proof}
###### Proof

To show this for instance for $R$, we may argue as in a
[[category of fibrant objects]] and apply the _[[factorization lemma]]_
which shows that every weak equivalence between fibrant objects may be
factored, up to [[homotopy]], as a [[span]] of acyclic fibrations.

These weak equivalences are preserved by the right Quillen property of $R$ and hence the claim follows by [[category with weak equivalences|2-out-of-3]].

For $L$ we apply the [[formal dual|formally dual]] argument.

=--

+-- {: .num_prop #QuillenAdjunctionInducesAdjunctionOnHomotopyCategories}
###### Proposition
**([[derived adjunction]])**

For $\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}$ a [[Quillen adjunction]] between [[model categories]], also the corresponding left and right [[derived functors]] form a pair of [[adjoint functors]]

$$
  Ho(\mathcal{C})
    \underoverset
      {\underset{\mathbb{R}}{\longrightarrow}}
      {\overset{\mathbb{L}}{\longleftarrow}}
      {\bot}
  Ho(\mathcal{D})
$$

between the corresponding [[homotopy category of a model category|homotopy categories]].

Moreover, the [[adjunction unit]] and [[adjunction counit]] of this derived adjunction are the images of the [[derived adjunction unit]] and [[derived adjunction counit]] of the original [[Quillen adjunction]].

=--

([Quillen 67, I.4 theorem 3](model+category#Quillen67))

### Behaviour under Bousfield localization
 {#BehaviourUnderLocalization}

+-- {: .num_prop #DescendToLeftBousfieldLocalization}
###### Proposition

Given a [[Quillen adjunction]]

$$
  (L \dashv R)
  \;\;\;\colon\;\;\;
  \mathcal{C}
  \underoverset
     {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}}
     {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}}
     {\bot_{\mathrlap{{}_{Qu}}}}
  \mathcal{D}
$$

and $S \subset Mor(\mathcal{D})$ is a [[set]] of [[morphisms]] such that

1. the [[Bousfield localization of model categories|left Bousfield localization]] of $\mathcal{D}$ at $S$ exists,

1. the [[derived functor|derived]] image $\mathbb{L}L(S)$ of $S$ lands in the [[weak equivalences]] of $\mathcal{C}$,

then the Quillen adjunction descends to the [[Bousfield localization of model categories|Bousfield localization]] $\mathcal{D}_S$

$$
  (L \dashv R)
  \;\;\;\colon\;\;\;
  \mathcal{C}
  \underoverset
     {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}}
     {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}}
     {\bot_{\mathrlap{{}_{Qu}}}}
  \mathcal{D}_S
  \,.
$$

=--

This appears as ([Hirschhorn, prop. 3.3.18](#Hirschhorn))

### Of $sSet$-enriched adjunctions {#sSet}

Of particular interest are [[SSet]]-[[enriched category theory|enriched]] adjunctions between [[simplicial model categories]]: [[simplicial Quillen adjunctions]].

These present [[adjoint (∞,1)-functors]], as the first proposition below asserts.

+-- {: .num_prop}
###### Proposition

Let $C$ and $D$ be [[simplicial model categories]] and let

$$
  (L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D
$$

be an [[sSet]]-[[enriched category theory|enriched]] [[adjunction]] whose underlying ordinary adjunction is a Quillen adjunction. Let $C^\circ$ and $D^\circ$ be the [[(∞,1)-categories]] presented by $C$ and $D$ (the [[Kan complex]]-enriched full [[sSet]]-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of [[adjoint (∞,1)-functors]]

$$
  (\mathbb{L} \dashv \mathbb{R}) : C^\circ \stackrel{\leftarrow}{\to} D^{\circ}
  \,.
$$

On the [[decategorification|decategorified]] level of the [[homotopy category|homotopy categories]] these are the total left and right [[derived functors]], respectively, of $L$ and $R$.

=--

+-- {: .proof}
###### Proof

This is proposition 5.2.4.6 in [[Higher Topos Theory|HTT]].

=--

The following proposition states conditions under which a Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).

+-- {: .num_prop #RecognitionOfSimplicialQuillenAdjunctions}
###### Proposition
**(recognition of [[simplicial Quillen adjunctions]])**

If $C$ and $D$ are [[simplicial model categories]] and $D$ is a left [[proper model category]], then an [[sSet]]-enriched adjunction

$$
  (L \dashv R) : C \stackrel{\leftarrow}{\to} D
$$

is a Quillen adjunction already if $L$ preserves cofibrations and $R$ just fibrant objects.

=--

This appears as [[Higher Topos Theory|HTT, cor. A.3.7.2]].

See [[simplicial Quillen adjunction]] for more details.

### Associated (infinity,1)-adjunction
 {#AssociatedInfinityAdjunction}

+-- {: .un_theorem}
###### Theorem
Let $F : C \rightleftarrows D : G$ be a Quillen adjunction between [[model categories]] (which are not assumed to admit functorial factorizations or infinite (co)limits).  Then there is an induced [[adjunction of (infinity,1)-categories]]
  $$ F : C[W_C^{-1}] \rightleftarrows D[W_D^{-1}] : G $$
where $C[W_C^{-1}]$ and $D[W_D^{-1}]$ denote the respective [[simplicial localizations]] at the respective classes of [[weak equivalences]].
=--

See ([Mazel-Gee 16](#MazelGee16), Theorem 2.1).  (This is also asserted as ([Hinich 14](#Hinich14), Proposition 1.5.1), but it is not completely proved there -- see ([Mazel-Gee 16](#MazelGee16), Remark 2.3).)

For [[simplicial model categories]] with [[sSet]]-[[enriched functor|enriched]] [[Quillen adjunctions]] between them, this is also in ([Lurie, prop. 5.2.4.6](#Lurie)).

See also at _[derived functor -- As functors on infinity-categories](derived%20functor#OnInftyCats)_

## Related concepts

* [[Quillen reflection]]

* [[Quillen adjoint triple]]

* [[simplicial Quillen adjunction]]

* [[Quillen equivalence]]

* [[monoidal Quillen adjunction]]

## References

See the references at _[[model category]]_. For instance

* [[Philip Hirschhorn]], _Model categories and their localization_
 {#Hirschhorn}

* [[Peter May]], [[Kate Ponto]], Section 16.2 of: _[[More concise algebraic topology]]_,   University of Chicago Press (2012) ([ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf))

The proof that a Quillen adjunction of [[model categories]] induces an [[adjunction of (infinity,1)-categories]] is recorded in

* {#MazelGee16} [[Aaron Mazel-Gee]], _Quillen adjunctions induce adjunctions of quasicategories_, New York Journal of Mathematics Volume 22 (2016) 57-93 ([arXiv:1501.03146](https://arxiv.org/abs/1501.03146), [publisher](http://nyjm.albany.edu/j/2016/22-4.html))

and this question is also partially addressed in

* {#Hinich14} [[Vladimir Hinich]], *Dwyer-Kan Localization Revisited*, Homology, Homotopy and Applications Volume 18 (2016) Number 1 ([arXiv:1311.4128](https://arxiv.org/abs/1311.4128), [doi:10.4310/HHA.2016.v18.n1.a3](https://dx.doi.org/10.4310/HHA.2016.v18.n1.a3))

The case for [[simplicial model categories]] is also in

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects Quillen adjunctions]]

[[!redirects right Quillen functor]]
[[!redirects left Quillen functor]]
[[!redirects right Quillen functors]]
[[!redirects left Quillen functors]]

[[!redirects Quillen functor]]
[[!redirects Quillen functors]]
[[!redirects Quillen pair]]
[[!redirects Quillen pairs]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop}
###### Proposition
**([[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]])**

Let $G$ be a [[finite group]].

The $G$-[[equivariant PL de Rham complex]]-construction is the [[left adjoint]] in a [[Quillen adjunction]] between

* the [[opposite model structure|opposite]] of the [[projective model structure on equivariant connective dgc-algebras]]

* the [[model structure on equivariant simplicial sets]]

  (i.e.: the global projective [[model structure on functors]] from the [[opposite category|opposite]] of the [[orbit category]] to the [[classical model structure on simplicial sets]])

$$
  \big(
    G dgcAlgebras^{\geq 0}_{k}
  \big)^{op}_{proj}
  \underoverset
    {
      \underset
        {\;\;\; exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\;\Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot_{\mathrlap{Qu}}}
  G SimplicialSets_{Qu}
$$

=--

([Scull 08, Prop. 5.1](#Scull08))

## Related statements

* [[fundamental theorem of equivariant dg-algebraic rational homotopy theory]]

* [[Quillen adjunction between simplicial sets and connective dgc-algebras]]

## References

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop}
###### Proposition
**([[Quillen adjunction between simplicial sets and connective dgc-algebras]])**

The [[PL de Rham complex]]-construction is the [[left adjoint]] in a [[Quillen adjunction]] between

* the [[opposite model structure|opposite]] of the [[projective model structure on connective dgc-algebras]]

* the [[classical model structure on simplicial sets]]

$$
  \big(
    DiffGradedCommAlgebras^{\geq 0}_{k}
  \big)^{op}_{proj}
  \underoverset
    {
      \underset
        {\;\;\; exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\;\Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot_{\mathrlap{Qu}}}
  SimplicialSets_{Qu}
$$

=--

+-- {: .proof}
###### Proof

That the [[PL de Rham complex]] [[functor]] preserves [[cofibrations]], hence sends injections of simplicial sets to surjections of dgc-algebras, is immediate from its construction.

That its [[right adjoint]] preserves fibrations, hence sends cofibrations of dgc-algebras to [[Kan fibrations]], is the statement of [Bousfield-Gugenheim 76, Lemma 8.2](#BousfieldGugenheim76).

=--

## Related statements

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

* [[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]]

* [[rational homotopy theory]], [[real homotopy theory]]

## References

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))


Quillen bifunctor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

A _(left) Quillen [[bifunctor]]_ is a [[functor]] of two variables between [[model category|model categories]] that respects combined [[cofibrations]] in its two arguments in a suitable sense.

The notion of Quillen bifunctor enters the definition of *[[monoidal model category]]* and of *[[enriched model category]]*.

## Definition

\begin{definition}
\label{QuillenBifunctor}
**(Quillen bifunctor)**
\linebreak
Let $C, D, E$ be [[model categories]]. A
[[functor]] $F \,\colon\, C \times D \to E$ (out of the [[product category]] of $C$ with $D$) is a **Quillen bifunctor**  if it satisfies the following two conditions:

1. it [[preserved colimit|preserves]] [[colimits]] separately in each variable,

1. (**[[pushout-product axiom]]**):

   for any

   * [[cofibration]] $\;i \,\colon\, c \to c'$ in $C$

   * [[cofibration]] $\;j \,\colon\, d \to d'$ in $D$,

   the induced [[pushout product]]-morphism

   $$
     F(c', d)
     \overset
       {F(c,d)}
       {\amalg}
     F(c,d')
       \longrightarrow
     F(c', d')
   $$

   is a [[cofibration]] in $E$, which is a [[weak equivalence]] if either $i$ or $j$ is a weak equivalence.

\end{definition}

\begin{remark}
\label{UnwindingTheDefinition}
In more detail, the [[pushout]] appearing in the first condition in Def. \ref{QuillenBifunctor} is the one sitting in the following pushout [[square]]:

$$
  \array{
    F(c,d) &\stackrel{F(Id,j)}{\to}& F(c,d')
    \\
    \;\;\downarrow^{F(i,Id)} && \downarrow
    \\
    F(c',d) &\stackrel{}{\to}&
    F(c', d) \coprod_{F(c,d)} F(c,d')
  }
  \,.
$$

In particular, if $i = (\varnothing \hookrightarrow c)$ (for $\varnothing$ denoting the [[initial object]]) we have $F(\varnothing, d) = F(\varnothing, d') = \varnothing$ (since the [[initial object]] is the [[colimit]] over the [[empty diagram]] and $F$ is assumed to preserve colimits) and the above pushout diagram reduces to

$$
  \array{
    \emptyset &{\to}& \emptyset
    \\
    \;\;\downarrow && \downarrow
    \\
    F(c,d) &\stackrel{}{\to}&
    F(c,d)
  }
  \,.
$$

Therefore:

* for $c$ a [[cofibrant object]] the condition is that $F(c,-) \,\colon\, D \to E$ preserves [[cofibrations]] and [[acyclic cofibrations]];

* for $d$ a [[cofibrant object]], the condition is that $F(-,d) \,\colon\, C \to E$ preserves [[cofibrations]] and [[acyclic cofibrations]].

\end{remark}

## Properties {#Properties}

+-- {: .un_prop}
###### Proposition

Let $\otimes \colon C \times D \to E$ be an [[adjunction of two variables]] between [[model categories]] and assume that $C$ and $D$ are [[cofibrantly generated model categories]]. Then $\otimes$ is a Quillen bifunctor precisely if it satisfies its axioms on generating (acyclic) cofibrations, i.e. if for $f \colon c_1 \to c_2$ and $g \colon d_1 \to d_2$ we have for the morphism

$$
 (c_1 \otimes d_2) \coprod_{c_1 \otimes d_1} (c_2 \otimes d_1)
  \to
  c_2 \otimes d_2
$$

is

* a cofibration if both $f$ and $g$ are generating cofibrations;

* an acyclic cofibration if one is a generating cofibration and the other a generating acyclic cofibration.

=--

This appears for instance as Corollary 4.2.5 in

* [[Mark Hovey]], _Model Categories_

## Applications ##

### Monoidal and enriched model categories ###

* In a [[monoidal model category]] $C$ the [[tensor product]] $\otimes : C \times C \to C$
  is required to be a Quillen bifunctor.

* An [[enriched model category]] $D$ over the [[monoidal model category]] $C$ is one that is [[power]]ed and [[copower]]ed over $D$ such that the [[copower]] $\otimes : D \times C \to D$ is a Quillen bifunctor.

### Lift to coends over tensors ###
 {#CoendsOverTensors}

The following proposition asserts that under mild conditions
a Quillen bifunctor on $C \times D$ lifts to a Quillen bifunctor
on [[functor category|functor categories]] of functors to $C$ and $D$.

+-- {: .un_prop}
###### Proposition

Let $\otimes : C \times D \to E$ be a Quillen functor. Let

* $S$ be a [[Reedy category]]
  and take the [[functor category|functor categories]]
  $[S,C]$ and $[S^{op},C]$ be equipped with the corresponding[[Reedy model structure]].

* _or_ assume that $C$ and $D$ are [[combinatorial model category|combinatorial model categories]]
  and let $[S,C]$ and $[S^{op},A]$ be equipped, respectively with the projective and
  the injective globel [[global model structure on functors|model structure on functor categories]].

Then the [[coend]] [[functor]]

$$
  \int^{S} (- \otimes -) : [S,C]\times [S^{op},D] \to E
$$

is again a ´Quillen bifunctor.

=--

This [Lurie, prop. A.2.9.26 with remark A.2.9.27](#Lurie).

It follows that the corresponding left [[derived functor]] computes the corresponding [[homotopy coend]].

### Bousfield-Kan type homotopy colimits ###

This is an application of the above application.

Let $C$ be a [[category]] and $A$ be a [[simplicial model category]]. Let $F : C \to A$ be a functor and let ${*} : C^{op} \to A$ be the functor constant on the terminal object.

Consider the [[global model structure on functors]] $[C^{op},SSet]_{proj}$ and $[C^{op},A]_{inj}$ and let
$Q({*})_{proj}$ be a cofibrant replacement for ${*}$ in $[C^{op},Set]_{proj}$ and $Q_{inj}(F)$ a cofibrant replacement for $F$ in $[C,A]_{inj}$.

One show that the [[homotopy colimit]] over $F$ is computed as the [[coend]] or [[weighted limit]]

$$
  hocolim F = \int Q_{proj}({*}) \cdot Q_{inj}(F)
  \,.
$$

One possible choice is

$$
  Q_{proj}({*})
  =
  N(-/C)^{op}
  \,.
$$

That this is indeed a projectively cofibrant resulution of the constant on the [[terminal object]] is for instance  shown in [Hirschhorn (2002), Prop 14.8.9](model+category#Hirschhorn02).

For the case that $C = \Delta^{op}$ (the [[opposite category|opposite]] of the [[simplex category]]) this is the classical choice in the discussion of the *[[Bousfield-Kan map]]*.

Assume that $A$ takes values in [[cofibrant objects]] of $A$, then it is already cofibrant in the [[injective model structure on functors]] $[C,A]_{inj}$  and we can take $Q_{inj}(F) = F$. Then the above says that

$$
  hocolim F
  \,=\,
  \int N(-/C)^\op \cdot F
  \,.
$$

For $C = $ [[simplex category|$\Delta$]] this is the classical prescription by Bousfield-Kan for [[homotopy colimits]], see also the discussion at *[[weighted limit]]*.

Using the above proposition, it follows in particular explicitly that the homotopy colimit preserves degreewise cofibrations of functors over which it is taken.

A nice discussion of this is in [Gambino (2010)](#Gambino10).

## Related concepts

* [[two-variable adjunction]]

* [[Quillen functor]]

* [[monoidal model category]], [[enriched model category]]

* [[pushout-product axiom]], [[pullback-power axiom]]

* [[Joyal-Tierney calculus]]

## References

* {#Hovey99} [[Mark Hovey]], Def. 4.2.1 in: *[[Model Categories]]*, Mathematical Surveys and Monographs, **63** AMS (1999) [[ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover)]

* {#Lurie} [[Jacob Lurie]], Appendix A.2 of: _[[Higher Topos Theory]]_ (2009)

* {#Gambino10} [[Nicola Gambino]], Def. 2.3 in: *Weighted limits in simplicial homotopy theory*, Journal of Pure and Applied Algebra **214** 7 (2010) 1193–1199 [[doi:10.1016/j.jpaa.2009.10.006](https://doi.org/10.1016/j.jpaa.2009.10.006)]

[[!redirects Quillen bifunctors]]
[[!redirects left Quillen bifunctor]]
[[!redirects left Quillen bifunctors]]
[[!redirects right Quillen bifunctor]]
[[!redirects right Quillen bifunctors]]


Quillen equivalence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] is a context in which we can do  [[homotopy theory]] or some generalization thereof; two model categories are 'the same' for this purpose if they are Quillen equivalent.  For example, the classic version of homotopy theory can be done using either [[topological space|topological spaces]] or [[simplicial sets]].  There is a model category of topological spaces, and a model category of simplicial sets, and they are Quillen equivalent.

In short, Quillen equivalence is the right notion of [[equivalence]] for [[model category|model categories]] --- and most importantly, this notion is weaker than [[equivalence of categories]].  The work of Dwyer--Kan, Bergner and others has shown that Quillen equivalent model categories [[presentable (infinity,1)-category|present]] equivalent [[(infinity,1)-category|(infinity,1)-categories]].

## Definition

Let $C$ and $D$ be [[model category|model categories]] and let

$$
  (L \dashv R)
  \;\colon\;
  C
    \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}}
  D
$$

be a [[Quillen adjunction]] with $L$ [[left adjoint]] to $R$.

Write $Ho C$ and $Ho D$ for the corresponding [[homotopy category of a model category|homotopy categories]].

By the discussion there, $Ho C$ may be regarded as obtained by first passing to the full [[subcategory]] on [[cofibrant objects]] and then [[localization|inverting]] [[weak equivalences]], and $L$ (being a left Quillen adjoint) preserves weak equivalences between cofibrant objects.  Thus, $L$ induces a functor

$$
  \mathbb{L} : Ho C \to Ho D
$$

between the [[homotopy category|homotopy categories]], called its (total) left [[derived functor]].  Similarly (but dually), $R$ induces a (total) right derived functor $\mathbb{R} : Ho D \to Ho C$. See at _[homotopy category of a model category -- derived functors](homotopy+category+of+a+model+category#DerivedFunctors)_ for more.

+-- {: .num_defn}
###### Definition

A [[Quillen adjunction]] $(L \dashv R)$ is a **Quillen equivalence** if the following equivalent conditions are satisfied:

* The total left [[derived functor]] $\mathbb{L} \colon Ho(C) \to Ho(D)$ is an [[equivalence of categories|equivalence]] of the [[homotopy categories]];

* The total right [[derived functor]] $\mathbb{R} \colon Ho(D) \to Ho(C)$ is an [[equivalence of categories|equivalence]] of the [[homotopy categories]];

* {#AdjunctOfWeakEquivalence} For every [[cofibrant object]] $c \in C$ and every [[fibrant object]] $d \in D$, a [[morphism]] $c \to R(d)$ is a [[weak equivalence]] in $C$ precisely when the [[adjunct]] morphism $L(c) \to d$ is a weak equivalence in $D$.

* The following two conditions hold:

  1. The [[derived adjunction unit]] is a weak equivalence, in that for every cofibrant object $c\in C$, the composite $c \overset{\eta_c}{\to} R(L(c)) \to R(L(c)^{fib})$ (of the [[adjunction unit]] with a [[fibrant replacement]] $R(L(c) \stackrel{\simeq}{\to} L(c)^{fib})$) is a weak equivalence in $C$,

  1. The [[derived adjunction counit]] is a weak equivalence, in that for every fibrant object $d\in D$, the composite $L(R(d)^{cof}) \to L(R(d)) \overset{\epsilon_d}{\to} d$ (of the [[adjunction counit]] with [[cofibrant replacement]] $L(R(d)^{cof} \stackrel{\simeq}{\to} R(d))$) is a weak equivalence in $D$.

=--

+-- {: .num_remark}
###### Remark

Not every equivalence between homotopy categories of model categories lifts to a Quillen equivalence. An interesting [[counterexample]] is given for instance by [Dugger & Shipley (2009)](#DuggerShipley09).

=--

Here are further characterizations:

+-- {: .num_prop #InCaseTheRightAdjointCreatesWeakEquivalences}
###### Proposition

If in a [[Quillen adjunction]]  $ \array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}}$ the [[right adjoint]] $R$ "[[created limit|creates]] weak equivalences" (in that a morphism $f$ in $\mathcal{C}$ is a weak equivalence precisely if $R(f)$ is) then $(L \dashv R)$ is a Quillen equivalence precisely already if for all cofibrant objects $d \in \mathcal{D}$ the plain [[adjunction unit]]

$$
  d \overset{\eta}{\longrightarrow} R (L (d))
$$

is a weak equivalence.

=--

(e.g. [Erdal-Ilhan 19, Lemma 3.3.](#ErdalIlhan19))

+-- {: .proof}
###### Proof

Generally, $(L \dashv R)$ is a Quillen equivalence precisely if

1. for every cofibrant object $d\in \mathcal{D}$, the "derived adjunction unit", hence the composite

   $$
     d
       \overset{\eta}{\longrightarrow}
     R(L(d))
       \overset{R(j_{L(d)})}{\longrightarrow}
     R(P(L(d)))
   $$

   (of the [[adjunction unit]] with image under $R$ of any fibrant replacement $L(d) \underoverset{\in W}{j_{L(d)}}{\longrightarrow} P(L(d))$) is a weak equivalence;

1. for every fibrant object $c \in \mathcal{C}$, the "derived adjunction counit", hence the composite

   $$
     L(Q(R(c)))
       \overset{L(p_{R(c)})}{\longrightarrow}
     L(R(c))
       \overset{\epsilon}{\longrightarrow}
     c
   $$

   (of the [[adjunction counit]] with the image under $L$ of any cofibrant replacement $Q R(c)\underoverset{\in W}{p_{R(c)}}{\longrightarrow} R(c)$  is a weak equivalence in $D$.

Consider the first condition: Since $R$ preserves the weak equivalence $j_{L(d)}$, by [[two-out-of-three]] the composite in the first item is a weak equivalence precisely if $\eta$ is.

Hence it is now sufficient to show that in this case the second condition above is automatic.

Since $R$ also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image

$$
  R(L(Q(R(c))))
    \overset{R(L(p_{R(c))})}{\longrightarrow}
  R(L(R(c)))
    \overset{R(\epsilon)}{\longrightarrow}
  R(c)
$$

under $R$ is.

Moreover, assuming, by the above, that $\eta_{Q(R(c))}$ on the cofibrant object $Q(R(c))$ is a weak equivalence, then by [[two-out-of-three]] this composite is a weak equivalence precisely if the further composite with $\eta$ is

$$
  Q(R(c))
    \overset{\eta_{Q(R(c))}}{\longrightarrow}
  R(L(Q(R(c))))
    \overset{R(L(p_{R(c))})}{\longrightarrow}
  R(L(R(c)))
    \overset{R(\epsilon)}{\longrightarrow}
  R(c)
  \,.
$$

But by the formula for [[adjuncts]], this composite is the $(L\dashv R)$-adjunct of the original composite, which is just $p_{R(c)}$

$$
  \frac{
     L(Q(R(c)))
       \overset{L(p_{R(c)})}{\longrightarrow}
     L(R(c))
       \overset{\epsilon}{\longrightarrow}
     c
  }{
     Q(R(C)) \overset{p_{R(c)}}{\longrightarrow} R(c)
  }
  \,.
$$

But $p_{R(c)}$ is a weak equivalence by definition of cofibrant replacement.

=--

## Properties

### 2-out-of-3
  {#TwoOutOfThree}

Since [[equivalence of categories|equivalences of categories]] enjoy the [[category with weak equivalences|2-out-of-3-property]], so do Quillen equivalences.

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

[[sSet]]-[[enriched functor|enriched]] Quillen equivalences between [[combinatorial model categories]] present equivalences between the corresponding [[locally presentable (infinity,1)-categories]]. And every equivalence between these is presented by a Zig-Zag of Quillen equivalences. See there for more details.

## Examples
 {#Examples}

+-- {: .num_example #TrivialQuillenEquivalence}
###### Example
**(trivial [[Quillen equivalence]])**

Let $\mathcal{C}$ be a [[model category]]. Then the [[identity functor]] on $\mathcal{C}$ constitutes a [[Quillen equivalence]] from $\mathcal{C}$ to itself:

$$
  \mathcal{C}
    \underoverset
      {\underset{id}{\longrightarrow}}
      {\overset{id}{\longleftarrow}}
      {\phantom{{}_{Qu}}\simeq_{Qu}}
  \mathcal{C}
$$

=--

+-- {: .proof}
###### Proof

From [this prop.](geometry+of+physics+--+categories+and+toposes#ComputationOfLeftRightDerivedFunctorsViaResolutions) it is clear that in this case the [[derived functors]] $\mathbb{L}id$ and $\mathbb{R}id$ both are themselves the [[identity functor]] on the [[homotopy category of a model category]], hence in particular are an [[equivalence of categories]].

=--

\begin{prop}\label{LeftBaseChangeQuillenEquivalence}
**([[left base change Quillen equivalence]])** \linebreak

Let $\mathcal{C}$ be a [[model category]],
and $\phi \colon S \overset{ \in \mathrm{W} }{\longrightarrow} T$ be a [[weak equivalence]] in $\mathcal{C}$.

Then the [left base change Quillen adjunction](slice+model+structure#BaseChangeQuillenAdjunction) along $\phi$ is a Quillen equivalence

$$
  \mathcal{C}_{/T}
    \underoverset
      {\underset{\phi^*}{\longrightarrow}}
      {\overset{\phi_!}{\longleftarrow}}
      {\phantom{{}_{Qu}}\simeq_{Qu}}
  \mathcal{C}_{/S}
$$

if and only if $\phi$ has this property:

$(\ast)$ *The [[pullback]] ([[base change]]) of $\phi$ along any [[fibration]] is still a [[weak equivalence]].*
\end{prop}
Notice that the property $(\ast)$ of $\phi$ is implied as soon as either:

* $\mathcal{C}$ is [[right proper model category|right proper]], or

* $\phi$ is an [[acyclic fibration]], or

* both $S$ and $T$ are [[fibrant objects]]

(for the first this follows by definition; for the second by the fact that $\phi^\ast$ is a [[right Quillen functor]] by [this Prop.](slice+model+structure#LeftBaseChangeQuillenAdjunction); for the third by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback) on recognizing [[homotopy pullbacks]]).
\begin{proof}
Using the characterization of Quillen equivalences by derived adjuncts ([here](#AdjunctOfWeakEquivalence)), the base change adjunction is a Quillen equivalence iff for

* any [[cofibrant object]] $X \to S$ in the slice over $S$ (i.e. $X$ is cofibrant in $\mathcal{C}$)

* and a [[fibrant object]] $p \colon Y \to T$ in the slice over $T$ (i.e. $p$ is a [[fibration]] in $\mathcal{C}$),

we have that

(1) $X \to \phi^*(Y) = S \times_T Y$ is a weak equivalence

iff

(2) $\phi_!(X) \to Y$ is a weak equivalence.

But the latter morphism is the top composite in the following [[commuting diagram]]:

$$
  \array{
    X
      &\longrightarrow&
    S \times_T Y
      &\overset{p^\ast \phi}{\longrightarrow}&
    Y
    \\
    &\searrow&
    \big\downarrow
    &{}^{_{(pb)}}&
    \big\downarrow {}^{\mathrlap{p \in Fib}}
    \\
    && S &\underset{\phi \in \mathrm{W} }{\longrightarrow}& T
  }
$$

Hence the [[two-out-of-three]]-property says that (1) is equivalent to (2) if $p^\ast \phi$ is a weak equivalence.

Conversely, taking $X \to \phi^\ast(X)$ to be a weak equivalence (hence a [[cofibrant resolution]] of $\phi^\ast(X)$), [[two-out-of-three]] implies that if $(\phi_! \dashv \phi^\ast)$ is a Quillen equivalence, then $p^\ast \phi$ is a weak equivalence.

\end{proof}

## Related concepts

* [[Quillen adjunction]]

* [[Quillen reflection]]

* [[simplicial Quillen adjunction]]

* **Quillen equivalence**

* [[monoidal Quillen adjunction]]

## References

For standard references see at _[[model category]]_.

An example of an equivalence of [[homotopy categories]] of model categories which does not lift to a Quillen equivalence is in

* {#DuggerShipley09} [[Daniel Dugger]], [[Brooke Shipley]], _A curious example of triangulated-equivalent model categories which are not Quillen equivalent_, Algebraic & Geometric Topology 9 (2009) ([pdf](http://homepages.math.uic.edu/~bshipley/dugger.shipley.curious.example.pdf))

The characterization of Quillen equivalences in the case that one of the adjoints creates equivalences appears for instance in

* {#ErdalIlhan19} Mehmet Akif Erdal, Aslı Güçlükan İlhan, _A model structure via orbit spaces for equivariant homotopy_, Journal of Homotopy and Related Structures volume 14, pages 1131–1141 (2019) ([arXiv:1903.03152](https://arxiv.org/abs/1903.03152), [doi:10.1007/s40062-019-00241-4](https://doi.org/10.1007/s40062-019-00241-4))

[[!redirects Quillen equivalences]]
[[!redirects Quillen equivalent]]
[[!redirects Quillen equivalent model categories]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

Write

* $Groups(sSet)$ for the [[category]] of [[simplicial groups]]

* $Groups(sSet)_{proj}$ for the [[model structure on simplicial groups]] whose [[weak equivalences]] and [[fibrations]] are those of the underlying morphisms in the [[classical model structure on simplicial sets]] (hence the underlying [[simplicial weak equivalences]] and [[Kan fibrations]], respectively);

* $sSet_{\geq 1}$ for the [[category]] of [[reduced simplicial sets]];

* $(sSet_{\geq 1})$ for the [[model structure on reduced simplicial sets]] whose [[weak equivalences]] and [[cofibrations]] are those of the underlying morphisms in the [[classical model structure on simplicial sets]].

\begin{proposition}\label{QuillenEquivalenceBetweenSimplicialGroupsAndReducedSimplicialSets}
**(Quillen equivalence between simplicial groups and reduced simplicial sets)** \linebreak

The

* [[simplicial classifying space]]-construction $\overline{W}(-)$ is the [[right adjoint]]

* [[simplicial loop space]]-construction $\Omega(-)$ is the [[left adjoint]]

in a [[Quillen equivalence]] between the projective [[model structure on simplicial groups]] and the injective [[model structure on reduced simplicial sets]].

$$
  Groups(sSet)_{proj}
  \underoverset
    {\underset{\overline{W}}{\longrightarrow}}
    {\overset{ \Omega }{\longleftarrow}}
    {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;}
  (sSet_{\geq 0})_{inj}
  \,.
$$

\end{proposition}

(e.g. [Goerss & Jardine 09, V Prop. 6.3](#GoerssJardine09))

## Related concepts

* [[looping and delooping]]

* [[May recognition theorem]]

## References

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.4 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

[[!redirects Quillen equivalence between reduced simplicial sets and simplicial groups]]


Quillen exact category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

## Via embedding

A full [[additive category|additive]] [[subcategory]] $A$ of an [[abelian category]] $B$ is called **Quillen exact category** if it is closed under [[extensions]] (if in extension $0\to X\stackrel{j}\to Y\stackrel{p}\to Z\to 0$, $X$ and $Z$ are in $A$ then $Y$ is in $A$). It is viewed as a pair $(A,E)$ where $E$ is a class of all short exact sequences in $A$ which are [[exact sequence|exact]] in $B$.

All $j$ which appear as $j$ in an exact sequence as above are called __inflation__s or __admissible monomorphism__s. All $p$ which appear in an exact sequence as above are called __deflation__s or __admissible epimorphism__s.

## Via exact structure

A **Quillen exact category** is a pair $(A,E)$ of an [[additive category]] $A$ and a class of sequences $E$ called 'exact'.
The following axioms are required for $(A,E)$:

(QE1) The class of 'exact' sequences is closed under [[isomorphism]]s and it contains all split extensions.
For any 'exact' sequence the deflation is the [[cokernel]] of inflation and the inflation is the [[kernel]] of the deflation.

(QE2) The class of deflations is closed under [[composition]] and [[base change]] by arbitrary maps. The class of inflations is closed under compositions and [[cobase change]] by arbitrary maps.

(QE3) If a morphism $M\to M'$ having a [[kernel]] can factor a deflation $N\to M'$ as $N\to M\to M'$ then it is a deflation. If a morphism $I\to I'$ having a cokernel can factor an inflation $I\to J$ as $I\to I'\to J$ then it is also an inflation.

## Properties

### Quillen-Gabriel embedding theorem

For every small exact category in the sense of a pair $(A,E)$, there is an embedding $A\hookrightarrow B$ into an [[abelian category]] such that $E$ is a class of all sequences which are (short) exact in $B$.

### Relation to Waldhausen categories and algebraic K-theory

Every Quillen exact category can be made into a [[Waldhausen category]]. However some information is lost in the process. Moreover, not every Waldhausen category comes from a Quillen exact category. Both Quillen exact categories and Waldhausen categories are devised in order to do [[algebraic K-theory]]. The K-theory spectrum based on [[Quillen's Q-construction]] and an exact category agrees with the K-theory spectrum based on the [[Waldhausen S-construction]] of the K-theory spectrum from its associated [[Waldhausen category]].

## References

Quillen introduced exact categories in above sense in the article

* [[Daniel Quillen]], "Higher algebraic K-theory", in Higher K-theories, pp. 85--147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer  1973.

A nonadditive generalization of exact categories has been introduced by Dyckerhoff and Kapranov and named a [[proto-exact category]].

[[Alexander Rosenberg]]  introduced one sided generalizations of Quillen exact categories: right 'exact' categories involving deflations, and left 'exact' categories involving inflations. One of the motivations an alternative definition of higher K-theory of (right exact) categories not involving spectra. In this setup the K-theory is an example of a derived functor in nonabelian homological algebra utilizing roughly the left 'exact' structure on the category of essentially small right 'exact' categories. It is not known if this K-theory when restricted to the category of essentially small Quillen exact categories agrees with Quillen K-theory. But it has the standard properties of Quillen K-theory (devissage, exactness and so on).
The one-sided generalization inspired by ideas introduced by Keller and Vossieck in the build up of the theory of [[suspended category|suspended categories]].

A right 'exact' category is a category with an initial object and a Grothendieck pretopology consisting of single maps which are [[strict epimorphism]]s. The distinguished class of strict epimorphisms is called a right 'exact' structure, or the class of _deflations_. The construction of derived functors in this generality involves a version of [[satellite|satellites]].

* Dmitry Kaledin, Wendy Lowen, _Cohomology of exact categories and (non-)additive sheaves_, Adv. Math. __272__ (2015) 652--698 [arXiv:1102.5756](https://arxiv.org/abs/1102.5756)

[[!redirects Quillen exact categories]]
[[!redirects Quillen-exact category]]
[[!redirects Quillen exactness]]
[[!redirects Quillen-Gabriel embedding theorem]]


Quillen negation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Definition

\begin{definition}\label{QuillenNegation}
**(weak orthogonal class/Quillen negation)**
\linebreak
Given a [[class]] $M \;\subset\; Mor(\mathcal{C})$ of [[morphisms]] in a [[category]] $\mathcal{C}$, its

* *left weak orthogonal class* or *left [[Quillen negation]]*

  $\multiscripts{^⧄}{M}{} \;\subset\; Mor(\mathcal{C})$

  (also denoted $M Proj$ or similar, for *$M$-[[projective morphisms]]*)

or

* *right weak orthogonal class* or *right [[Quillen negation]]*

  $\multiscripts{}{M}{^⧄} \;\subset\; Mor(\mathcal{C})$

  (also denoted $M Inj$ or similar, for *$M$-[[injective morphisms]]*)

is the [[class]] of [[morphisms]] in $\mathcal{C}$ which have the left, respectively right, [[lifting property]] with respect to each morphism in the class $M$:

$$
  M^{⧄}
  \;\coloneqq\;
  \Big\{
    p \,\in\, Mor(\mathcal{C})
    \;\big\vert\;
   \underset{i \in M}{\forall}
   \; i \,⧄\, p
  \Big\},
  \;\;\;\;\;
  {}^{⧄}M
  \;\coloneqq\;
  \Big\{
    i \,\in\, Mor(\mathcal{C})
    \;\big\vert\;
   \underset{p \in M}{\forall}
   \; i \,⧄\, p
  \Big\}
  \,.
$$

\end{definition}

## Properties

\begin{proposition}
**(closure properties)**
\linebreak
  A right weak-orthogonal class $C^{⧄}$ is always closed under  [[retracts]], [[pullbacks]], (small) [[product|products]] (whenever they exist in the category) and [[composition]] of morphisms, and contains all [[isomorphisms]] of C.

Meanwhile, a left weak-orthogonal class ${}^{⧄}M$ is closed under [[retracts]], [[pushout]], (small) [[coproduct]] and [[transfinite composition]] ([[filtered colimits]]) of morphisms (whenever they exist in the category), and also contains all [[isomorphisms]].
\end{proposition}
This **proof** is straightforward, see for instance [here](ClosurePropertiesOfInjectiveAndProjectiveMorphisms) at *[[injective or projective morphism]]*.

\begin{remark}
It is clear that
$$
  C \;\subset\;
  \big({}^{⧄}M\big)^{⧄}
  \,,
  \;\;\;
  C \;\subset\;
  \multiscripts{^⧄}{\big(M^{⧄}\big)}{}
  \,.
$$
\end{remark}

## Related entries

* [[lifting property]]

* [[separation axioms in terms of lifting properties]]

## Examples

\begin{example}
In the category [[Set]] of [[sets]], the right [[Quillen negation]]
of the "simplest non-[[surjection]]" $\varnothing \to \ast$ (the unique [[map]] from the [[empty set]] to the [[singleton set]]) is the class of [[surjections]]:

$$
  \big\{
    \varnothing \to \ast
  \big\}^{⧄}
  \;=\;
  \big\{
    f \;\big\vert\; f \; \text{is surjective}
  \big\}
  \,.
$$

The left and right Quillen negations of $\{x_1,x_2\} \to \ast$ (the "simplest non-[[injection]]") are both precisely the class of [[injections]],
$$
  {}^{⧄}
  \big\{
    \{x_1,x_2\}
    \longrightarrow
    \ast
  \big\}
  \;=\;
  \big\{
    \{x_1,x_2\}  \longrightarrow \ast
  \big\}^{⧄}
  \;=\;
  \big\{ f \;\big\vert\; f \; \text{ is injective } \big\}
  \,.
$$

\end{example}

## References

Left/right weak orthogonal classes are discussed, for instance, in any text on [[model category]]-theory, see the references [there](model+category#References).

Discussion with an eye towards [[separation axioms in terms of lifting properties]] and amplifying the logical aspect of conceptual negation:

* [[Misha Gavrilovich]], *Remarks on Shelah's classification theory and Quillen's negation* ([arXiv:2010.08791](https://arxiv.org/abs/2010.08791))

[[!redirects Quillen negations]]

[[!redirects orthogonal class]]
[[!redirects orthogonal classes]]

[[!redirects left orthogonal class]]
[[!redirects left orthogonal classes]]
[[!redirects right orthogonal class]]
[[!redirects right orthogonal classes]]

[[!redirects weak orthogonal class]]
[[!redirects weak orthogonal classes]]

[[!redirects left weak orthogonal class]]
[[!redirects left weak orthogonal classes]]
[[!redirects right weak orthogonal class]]
[[!redirects right weak orthogonal classes]]


Quillen plus construction

#Contents#
* table of contents
{:toc}

## Idea

The _Quillen plus construction_ is a method for simplifying the [[fundamental group]] of the [[homotopy type]] of a [[topological space]] without changing its [[ordinary homology]] and [[ordinary cohomology]] groups.

It was introduced by [[Kervaire]] (1969), but was crucially used by [[Daniel Quillen]] to define the [[algebraic K-theory]] groups by applying it to the [[classifying space]] of the [[stable general linear group]], $GL(A)$, of a ring $A$.

The plus construction doesn't change the [[homology]] of the space but 'kills' a [[perfect group|perfect]] normal subgroup of $G$ within the classifying space $BG$. For algebraic K-theory, that subgroup is the [[Steinberg group|stable elementary linear group]], $E(A)$ within $GL(A)$.

## Related concepts

* [[group completion theorem]]

* [[perfect group]]

* [[algebraic K-theory]]

## References

* [[Michel Kervaire]], *Smooth homology spheres and their fundamental groups*, Trans. Amer. Math. Soc., 144 (1969) pp. 67–72

* {#Weibel} [[Charles Weibel]], Section IV.1 of _The K-Book: An introduction to algebraic K-theory_ ([web](http://www.math.rutgers.edu/~weibel/Kbook.html))

Relation to the [[group completion theorem]]:

* [[Thomas Nikolaus]], _The group completion theorem via localizations of ring spectra_, 2017 ([pdf](https://people.mpim-bonn.mpg.de/thoni/Papers/Group_completion.pdf))

See also:

* [[Dustin Clausen]], _Plus construction considerations_, [MO:a/121351](http://mathoverflow.net/a/121351/2503)

Discussion in [[(∞,1)-category theory]] in relation to [[epimorphism in an (infinity,1)-category|$(\infty,1)$-epimorphisms]]:

* {#Raptis17} [[George Raptis]], *Some characterizations of acyclic maps*, Journal of Homotopy and Related Structures volume 14, pages 773–785 (2019) 2017 ([arxiv:1711.08898](https://arxiv.org/abs/1711.08898), [doi:10.1007/s40062-019-00231-6](https://doi.org/10.1007/s40062-019-00231-6))

* {#Hoyois19} [[Marc Hoyois]], *On Quillen's plus construction*, 2019 ([pdf](http://www.mathematik.ur.de/hoyois/papers/acyclic.pdf), [[Hoyois_PlusConstruction.pdf:file]])


Quillen Q-construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Quillen Q-construction_ ([Quillen 72](#Quillen72)) is a tool for producing the [[algebraic K-theory]] of a [[Quillen exact category]] $\mathcal{C}$.  The Quillen Q-construction is generalized ([Waldhausen 83, section 1.9](#Waldhausen83)) by the [[Waldhausen S-construction]] which applies more generally to [[Waldhausen categories]].
(However, the [[Quillen Dévissage theorem]] does not generalize to [[Waldhausen categories]].)

Both these constructions appear in [[stable homotopy theory]] as special cases of the concept of [[algebraic K-theory of a stable (∞,1)-category]] ([Haugseng 10](#Haugseng10), [Barwick 13](#Barwick13)).

## Related concepts

* [[Waldhausen S-construction]]

## References

The construction is due to

* {#Quillen72} [[Daniel Quillen]], _Higher algebraic K-theory I_, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lecture
Notes in Math., Vol. 341

The generalization to the [[Waldhausen S-construction]] is due to

* {#Waldhausen83} [[Friedhelm Waldhausen]], _Algebraic K-theory of spaces_, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419 ([pdf](http://www.math.uni-bielefeld.de/~fw/LNM1126_318-419.pdf))

Refinement of the construction to [[stable (∞,1)-categories]] and [[exact (infinity,1)-categories]] is discussed in

* {#Haugseng10} [[Rune Haugseng]], _the Q-construction for stable $\infty$-Categories_, 2010 (can be found [here](http://folk.ntnu.no/runegha/notes.html))

* {#Barwick13} [[Clark Barwick]], _On the Q construction for exact quasicategories_ ([arXiv:1301.4725](http://arxiv.org/abs/1301.4725))

* [[Clark Barwick]], _On exact infinity-categories and the Theorem of the Heart_, [arXiv:1212.5232](http://arxiv.org/abs/1212.5232).

See also

* _Motivation/interpretation for Quillen's Q-construction?_, [MO/1006](http://mathoverflow.net/questions/1006/motivation-interpretation-for-quillens-q-construction).

[[!redirects Q-construction]]

[[!redirects Q-constructions]]

[[!redirects Quillen's Q-construction]]
[[!redirects Quillen's Q-constructions]]


Quillen reflection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analog of a [[reflective subcategory]]-inclusion as [[adjoint functor|adjunctions of functors]] are replaced by [[Quillen adjunctions]].

## Definition

+-- {: .num_defn #QuillenReflection}
###### Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[model categories]], and let

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

be a [[Quillen adjunction]] between them. Then this may be called

1. a _Quillen reflection_ if the [[derived adjunction counit]] is componentwise a [[weak equivalence]];

1. a _Quillen co-reflection_ if the [[derived adjunction unit]] is componentwise a [[weak equivalence]].

=--

+-- {: .num_example #BousfieldLocalizationIsQuillenReflection}
###### Example
**([[left Bousfield localization]] is [[Quillen reflection]])**

1. A [[left Bousfield localization]] is a Quillen reflection.

1. A [[right Bousfield localization]] is a Quillen coreflection.

=--

+-- {: .proof}
###### Proof

We consider the case of [[left Bousfield localizations]], the other case is [[formal duality|formally dual]].

A left Bousfield localization is a [[Quillen adjunction]] by [[identity functors]] ([this Remark](Bousfield+localization+of+model+categories#ImmediateImpiciationsOfLeftBousfieldLocalization))

$$
  \mathcal{D}_{loc}
    \underoverset
      {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}}
      {\overset{id}{\longleftarrow}}
      {{}_{\phantom{Qu}} \bot_{Qu}}
  \mathcal{D}
$$

This means that the ordinary [[adjunction counit]] is the [[identity morphism]] and hence that the [[derived adjunction counit]] on a [[fibrant object]] $c$ is just a [[cofibrant resolution]]-morphism

$$
  Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c
$$

but regarded in the model structure $\mathcal{D}_{loc}$. Hence it is sufficient to see that [[acyclic fibrations]] in $\mathcal{D}$ remain weak equivalences in the left Bousfield localized model structure. In fact they even remain acyclic fibrations, by [this Remark](Bousfield+localization+of+model+categories#ImmediateImpiciationsOfLeftBousfieldLocalization).

=--

## Properties

+-- {: .num_prop }
###### Proposition

Let

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

be a [[Quillen adjunction]] and write

$$
  Ho(\mathcal{C})
    \underoverset
      {\underset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow}}
      {\overset{\mathbb{L}L}{\longleftarrow}}
      {\bot_{Qu}}
  Ho(\mathcal{D})
$$

for the induced [[adjoint pair]] of [[derived functors]] on the [[homotopy category of a model category|homotopy categories]] ([this Prop.](geometry+of+physics+--+categories+and+toposes#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories)).

Then

1. $(L \underset{Qu}{\dashv} R)$ is a Quillen reflection precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a [[reflective subcategory]]-inclusion;

1. $(L \underset{Qu}{\dashv} R)$ is a Quillen co-reflection precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a [[co-reflective subcategory]]-inclusion;

1. $(L \underset{Qu}{\dashv} R)$ is a [[Quillen equivalence]] precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is an [[equivalence of categories]].

=--

+-- {: .proof}
###### Proof

By [this Prop.](geometry+of+physics+--+categories+and+toposes#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories) the components of the [[adjunction unit]]/[[adjunction counit|counit]] of $(\mathbb{L}L \dashv \mathbb{R}R)$ are precisely the images under [[localization]] of the [[derived adjunction unit]]/[[derived adjunction counit|counit]] of $(L \underset{Qu}{\dashv} R)$. Moreover, by [this Prop.](geometry+of+physics+--+categories+and+toposes#MorphismIsWeakEquivalenceIfIsoInHomotopyCategoryForQuillen) the localization functor of a [[model category]] inverts precisely the [[weak equivalences]]. Hence the adjunction (co-)unit of $(\mathbb{L}L \dashv \mathbb{R}R)$ is an isomorphism if and only if the derived (co-)unit of $(L \underset{Qu}{\dashv} R)$ is a weak equivalence, respectively.

With this the statement reduces to the characterization of (co-)reflections via invertible units/counits, respectively ([this Prop.](adjoint+functor#FullyFaithfulAndInvertibleAdjoints)).

=--

[[!redirects Quillen reflections]]

[[!redirects Quillen coreflection]]
[[!redirects Quillen coreflections]]

[[!redirects Quillen co-reflection]]
[[!redirects Quillen co-reflections]]


Quillen's theorem

[[Daniel Quillen]] has proven many things, and various theorems carry his name:

* [[Quillen's theorem A]]

* [[Quillen's theorem B]]

* [[Quillen's theorem on MU]]

* [Quillen's theorem on the image of J](J-homomorphism#ImageOfJTradtionalFormulation)

[[!redirects Quillen's theorems]]

[[!redirects Quillen theorem]]
[[!redirects Quillen theorems]]


Quillen-Suslin theorem

#Contents#
* table of contents
{:toc}

## Statement

The most memorable formulation of the Quillen-Suslin theorem states that for a [[field]] $k$ (or more generally a [[principal ideal domain]]), [[finitely generated object|finitely generated]] [[projective modules]] over a finitary [[polynomial algebra]] $A = k[x_1, \ldots, x_r]$ are [[free object|free]].

In Serre's [FAC](#Serre1955) appears the sentence "It is not known if there exist projective A-modules of finite type which are not free." This question became known as Serre's problem or Serre's conjecture (over repeated objections from Serre). Serre had made partial progress by proving that f.g. projective $A$-modules are [[stably free module|stably free]], but the question remained unresolved until 1976 when an affirmative solution was produced by [[Daniel Quillen]] and independently by [[Andrei Suslin]].

A later simplified proof was given by Leonid Vaserstein; this is recounted in Lang's *Algebra*.

## References

* [[Jean-Pierre Serre]], *Faisceaux algébriques cohérents*, Annals of Mathematics, Second Series, 61 (2), 197–278 (1955). ([doi](http://www.jstor.org/stable/1969915?origin=crossref&seq=1#page_scan_tab_contents))
 {#Serre1955}

* [[Daniel Quillen]], *Projective modules over polynomial rings*, Inventiones Mathematicae 36 (1) (1976), 167–171. ([doi](http://link.springer.com/article/10.1007%2FBF01390008))

* [[Andrei Suslin]], *Проективные модули над кольцами многочленов свободны*, Doklady Akademii Nauk SSSR 229 (5) (1976), 1063-1066. Translated as *Projective modules over polynomial rings are free*, Soviet Mathematics 17 (4) (1976),  1160–1164.

* [[Serge Lang]], *Algebra*, Graduate Texts in Mathematics 211 (Revised third ed.), Springer-Verlag New York, 2002.


quintessence

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[dark energy]]

* [[phantom dark energy]]

* [[inhomogeneous cosmology]]

* [[Witten's Dark Fantasy]]

## References

### General

See also

* Wikipedia, _Quintessence_

### As higher curvature corrections

Discussion of [[higher curvature corrections]] to [[D=4 supergravity]] appearing as [[quintessence]]:

* [[Fotis Farakos]], _Quintessence from higher curvature supergravity_ ([arXiv:2003.09366](https://arxiv.org/abs/2003.09366))


quintet construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

# The Quintet Construction
* table of contents
{:toc}

## Idea

A *quintet construction* is an operation that takes a "globular" categorical structure and produces a "cubical" one (see [[geometric shape for higher structures]]) in which the squares

$$ \array{ A & \overset{f}{\to} & B \\
^g\downarrow  & \swArrow & \downarrow^h\\
C & \underset{k}{\to} & D }$$

are morphisms between composites $\alpha:h\circ f \to k\circ g$.  The name arises because a square in the resulting cubical structure is formally a quintet $(f,g,h,k,\alpha)$ (since just knowing the globular cell $\alpha$ does not determine the decomposition of its domain and codomain as composites).

Hence given a [[2-category]] $\mathcal{C}$, it induces a [[double category]] $Sq(\mathcal{C})$ whose

* [[objects]] are the objects of $\mathcal{C}$;

* [[horizontal morphisms]] are the [[1-morphisms]] of $\mathcal{C}$;

* [[vertical morphisms]] are also the [[1-morphisms]] of $\mathcal{C}$;

* [[2-morphisms]] are the [[2-morphisms]] of $\mathcal{C}$ between the [[compositions]] of the [[1-morphisms]].

## Examples

* Quintets in a [[strict 2-category]] form a [[strict double category]].

* Quintets in a [[bicategory]] form a [[double bicategory]].

* Quintets in a [[Gray-category]] form an [[intercategory]].

* By a certain stretch of terminology, the singular [[cubical set]] of a [[topological space]] might be called a quintet construction.

## Applications

A quintet construction is a [[left adjoint]] to a functor that picks out the [[companion pairs]] in a cubical structure (see Theorem 1.7 in [Grandis and Paré](#GrandisPare04)).  Thus, functors *out* of quintet constructions are nothing new; but some interesting applications involve functors *into* quintet constructions from other cubical structures.  For instance:

* The assignment of [[homotopy categories]] and [[derived functors]] to [[model categories]] can be made into a [[double pseudofunctor]] from the [[double category of model categories]] to the double category of quintets in [[Cat]] ([this Prop.](double+category+of+model+categories#HomotopyDoublePseudofunctor)).  This implies that [[mates]] are preserved even when they relate composites of left and right [[Quillen functors]].

## Properties

* The quintet construction admits a [[left adjoint]] (see Proposition 8 in [Multiple Functors IV](#BastianiEhresmann79)).

## References

The concept is due to

* [[Charles Ehresmann]], _Catégorie double des quintettes; applications covariantes, C. R. A. S. Paris 256 (1963), 1891-1894._ and in [volume III](http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_III_1.pdf) in his collected works.

It appears spelled out also in

* {#BastianiEhresmann74} A. Bastiani, [[Charles Ehresmann]], pages 272-273 of  _Multiple functors. I. Limits relative to double categories_, Cah. Top. Géom. Différ. Catég. 15 (1974) 215–292

* {#BastianiEhresmann79} [[Andrée Ehresmann]] and [[Charles Ehresmann]]. _Multiple functors IV. Monoidal closed structures on $Cat_n$_, Cahiers de topologie et géométrie différentielle, Volume 20 (1979) no. 1, pp. 59-104. ([link](http://www.numdam.org/item/CTGDC_1979__20_1_59_0/))

* {#GrandisPare04} Grandis, Marco, and Robert Paré. "Adjoint for double categories." Cahiers de topologie et géométrie différentielle catégoriques 45.3 (2004): 193-240. [link](http://www.numdam.org/item/CTGDC_2004__45_3_193_0/)

[[!redirects quintet]]
[[!redirects quintets]]
[[!redirects quintet constructions]]
[[!redirects quintets in a 2-category]]
[[!redirects double category of quintets]]

[[!redirects double category of squares]]
[[!redirects double categories of squares]]


quintuple

A **quintuple** is an [[n-tuple]] for $n = 5$.

[[!redirects quintuples]]