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\begin{document}

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\section*{Quillen equivalence}

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

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

[[!include model category theory - 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
\noindent\hyperlink{TwoOutOfThree}{2-out-of-3}\dotfill \pageref*{TwoOutOfThree} \linebreak
\noindent\hyperlink{presentation_of_equivalence_of_categories}{Presentation of equivalence of $(\infty,1)$-categories}\dotfill \pageref*{presentation_of_equivalence_of_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

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

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

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

\begin{displaymath}
\mathbb{L} : Ho C \to Ho D
\end{displaymath}
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 \emph{\href{homotopy+category+of+a+model+category#DerivedFunctors}{homotopy category of a model category -- derived functors}} for more.

\begin{defn}
\label{}\hypertarget{}{}
A [[Quillen adjunction]] $(L \dashv R)$ is a \textbf{Quillen equivalence} if the following equivalent conditions are satisfied.

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


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


\item 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 $D$.


\item \begin{enumerate}%
\item 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 $C$,


\item 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 $D$.



\end{enumerate}


\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Not every equivalence between homotopy categories of model categories lifts to a Quillen equivalence. An interesting counterexample is given for instance in (\hyperlink{DuggerShipley09}{Dugger-Shipley 09}).

\end{remark}
Here are further characterizations:

\begin{prop}
\label{InCaseTheRightAdjointCreatesWeakEquivalences}\hypertarget{InCaseTheRightAdjointCreatesWeakEquivalences}{}
If in a [[Quillen adjunction]] $\itexarray{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}}$ the [[right adjoint]] $R$ ``creates weak equivalences'' (in that a morphism $f$ in $\mathcal{C}$ is a weak equivalence precisly 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]]

\begin{displaymath}
d \overset{\eta}{\longrightarrow} R (L (d))
\end{displaymath}
is a weak equivalence.

\end{prop}
\begin{proof}
Generally, $(L \dashv R)$ is a Quillen equivalence precisely if

\begin{enumerate}%
\item for every cofibrant object $d\in \mathcal{D}$, the ``derived adjunction unit'', hence the composite

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


\item for every fibrant object $c \in \mathcal{C}$, the ``derived adjunction counit'', hence the composite

\begin{displaymath}
L(Q(R(c))) 
    \overset{L(p_{R(c)})}{\longrightarrow} 
  L(R(c)) 
    \overset{\epsilon}{\longrightarrow}
  c
\end{displaymath}
(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 $D$.



\end{enumerate}
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

\begin{displaymath}
R(L(Q(R(c))))
    \overset{R(L(p_{R(c))})}{\longrightarrow} 
  R(L(R(c))) 
    \overset{R(\epsilon)}{\longrightarrow}
  R(c)
\end{displaymath}
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

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

\begin{displaymath}
\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)
  }
  \,.
\end{displaymath}
But $p_{R(c)}$ is a weak equivalence by definition of cofibrant replacement.

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

\hypertarget{TwoOutOfThree}{}\subsubsection*{{2-out-of-3}}\label{TwoOutOfThree}

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

\hypertarget{presentation_of_equivalence_of_categories}{}\subsubsection*{{Presentation of equivalence of $(\infty,1)$-categories}}\label{presentation_of_equivalence_of_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.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{TrivialQuillenEquivalence}\hypertarget{TrivialQuillenEquivalence}{}
\textbf{(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:

\begin{displaymath}
\mathcal{C}  
    \underoverset
      {\underset{id}{\longrightarrow}}
      {\overset{id}{\longleftarrow}}
      {\phantom{{}_{Qu}}\simeq_{Qu}}
  \mathcal{C}
\end{displaymath}
\end{example}
\begin{proof}
From \href{geometry+of+physics+--+categories+and+toposes#ComputationOfLeftRightDerivedFunctorsViaResolutions}{this prop.} 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]].

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Quillen adjunction]]


\item [[Quillen reflection]]


\item [[simplicial Quillen adjunction]]


\item \textbf{Quillen equivalence}


\item [[monoidal Quillen adjunction]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For standard references see at \emph{[[model category]]}.

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

\begin{itemize}%
\item [[Daniel Dugger]], [[Brooke Shipley]], \emph{A curious example of triangulated-equivalent model categories which are not Quillen equivalent}, Algebraic \& Geometric Topology 9 (2009) (\href{http://homepages.math.uic.edu/~bshipley/dugger.shipley.curious.example.pdf}{pdf})

\end{itemize}
[[!redirects Quillen equivalences]]



\end{document}