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\begin{document}

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\section*{Quillen reflection}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{QuillenReflection}\hypertarget{QuillenReflection}{}
Let $\mathcal{C}$ and $\mathcal{D}$ be [[model categories]], and let

\begin{displaymath}
\mathcal{C}  
    \underoverset
      {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\bot_{Qu}}
  \mathcal{D}
\end{displaymath}
be a [[Quillen adjunction]] between them. Then this may be called

\begin{enumerate}%
\item a \emph{Quillen reflection} if the [[derived adjunction counit]] is componentwise a [[weak equivalence]];


\item a \emph{Quillen co-reflection} if the [[derived adjunction unit]] is componentwise a [[weak equivalence]].



\end{enumerate}
\end{defn}
\begin{example}
\label{BousfieldLocalizationIsQuillenReflection}\hypertarget{BousfieldLocalizationIsQuillenReflection}{}
\textbf{([[left Bousfield localization]] is [[Quillen reflection]])}

\begin{enumerate}%
\item A [[left Bousfield localization]] is a Quillen reflection.


\item A [[right Bousfield localization]] is a Quillen coreflection.



\end{enumerate}
\end{example}
\begin{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]] (\href{Bousfield+localization+of+model+categories#ImmediateImpiciationsOfLeftBousfieldLocalization}{this Remark})

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

\begin{displaymath}
Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c
\end{displaymath}
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 \href{Bousfield+localization+of+model+categories#ImmediateImpiciationsOfLeftBousfieldLocalization}{this Remark}.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
\mathcal{C}  
    \underoverset
      {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\bot_{Qu}}
  \mathcal{D}
\end{displaymath}
be a [[Quillen adjunction]] and write

\begin{displaymath}
Ho(\mathcal{C})
    \underoverset
      {\underset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow}}
      {\overset{\mathbb{L}L}{\longleftarrow}}
      {\bot_{Qu}}
  Ho(\mathcal{D})
\end{displaymath}
for the induced [[adjoint pair]] of [[derived functors]] on the [[homotopy category of a model category|homotopy categories]] (\href{geometry+of+physics+--+categories+and+toposes#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories}{this Prop.}).

Then

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


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


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



\end{enumerate}
\end{prop}
\begin{proof}
By \href{geometry+of+physics+--+categories+and+toposes#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories}{this Prop.} 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 \href{geometry+of+physics+--+categories+and+toposes#MorphismIsWeakEquivalenceIfIsoInHomotopyCategoryForQuillen}{this Prop.} 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 (\href{adjoint+functor#FullyFaithfulAndInvertibleAdjoints}{this Prop.}).

\end{proof}
[[!redirects Quillen reflections]]

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

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



\end{document}