\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{homotopy category of a model category}

\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{universal_property}{Universal property}\dotfill \pageref*{universal_property} \linebreak
\noindent\hyperlink{DerivedFunctors}{Derived functors}\dotfill \pageref*{DerivedFunctors} \linebreak
\noindent\hyperlink{further_properties}{Further properties}\dotfill \pageref*{further_properties} \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]] structure $(\mathcal{C}(W,Fib,Cof))$ on some [[category]] $\mathcal{C}$ is a means to guarantee the [[locally small category|local smallness]] and to improve the tractability of the [[homotopy category]] of the underlying [[category with weak equivalences]] $(\mathcal{C},W)$. In particular, if a [[category with weak equivalences]] admits a [[model category]] structure, then its [[homotopy category]] (in the sense of [[localization]] $\mathcal{C}[W^{-1}]$ at the class of [[weak equivalences]]) is [[equivalence of categories|equivalent]] to the category whose objects are those objects of $\mathcal{C}$ which are both fibrant cofibrant, and whose morphisms are the actual [[homotopy classes]] of morphisms between these objects ([[left homotopy]] or [[right homotopy]] [[equivalence classes]] in the sense of [[homotopy in a model category]]) .

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

\begin{defn}
\label{HomotopyCategoryOfAModelCategory}\hypertarget{HomotopyCategoryOfAModelCategory}{}
Let $\mathcal{C}$ be a [[model category]]. Write $Ho(\mathcal{C})$ for the [[category]] whose

\begin{itemize}%
\item [[objects]] are those objects of $\mathcal{C}$ which are both fibrant and cofibrant;


\item [[morphisms]] are the [[homotopy classes]] of morphisms of $\mathcal{C}$, hence the [[equivalence classes]] of morphism under [[left homotopy]].



\end{itemize}
\end{defn}
This is, up to [[equivalence of categories]], the \emph{homotopy category of the model category} $\mathcal{C}$.

\hypertarget{universal_property}{}\subsection*{{Universal property}}\label{universal_property}

We spell out that def. \ref{HomotopyCategoryOfAModelCategory} indeed satisfies the [[universal property]] that defines the [[homotopy category]] of a [[category with weak equivalences]].

\begin{lemma}
\label{WhiteheadTheoremInModelCategories}\hypertarget{WhiteheadTheoremInModelCategories}{}
\textbf{([[Whitehead theorem]] in model categories)}

Let $\mathcal{C}$ be a [[model category]]. A [[weak equivalence]] between two objects which are both fibrant and cofibrant is a [[homotopy equivalence]].

\end{lemma}
(e.g. \hyperlink{GoerssJardine99}{Goerss-Jardine 99, part I, theorem 1.10})

\begin{proof}
By the factorization axioms in $\mathcal{C}$, every weak equivalence $f\colon X \longrightarrow Y$ factors through an object $Z$ as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with $X$ and $Y$ both fibrant and cofibrant, so is $Z$, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.

So let $f \colon X \longrightarrow Y$ be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is [[formal dual|formally dual]]. Then in fact it has a genuine [[right inverse]] given by a lift $f^{-1}$ in the diagram

\begin{displaymath}
\itexarray{
    \emptyset &\rightarrow& X
    \\
    {}^{\mathllap{\in cof}}\downarrow 
     &{}^{{f^{-1}}}\nearrow& 
   \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib \cap W}}
    \\
    X &=& X
  }
  \,.
\end{displaymath}
To see that $f^{-1}$ is also a [[left inverse]] up to [[left homotopy]], let $Cyl(X)$ be any [[cylinder object]] on $X$, hence a factorization of the [[codiagonal]] on $X$ as a cofibration followed by a an acyclic fibration

\begin{displaymath}
X \sqcup X \stackrel{\iota_X}{\longrightarrow} Cyl(X) \stackrel{\sigma}{\longrightarrow} X
\end{displaymath}
and consider the square

\begin{displaymath}
\itexarray{
    X \sqcup X 
    &\stackrel{(f^{-1}\circ f, id)}{\longrightarrow}& X
    \\
    {}^{\mathllap{\iota_X}}{}_{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W \cap Fib}}
    \\
    Cyl(X) &\underset{f\circ \sigma}{\longrightarrow}& y
  }
  \,,
\end{displaymath}
which [[commuting square|commutes]] due to $f^{-1}$ being a genuine right inverse of $f$. By construction, this [[commuting square]] now admits a [[lift]] $\eta$, and that constitutes a [[left homotopy]] $\eta \colon f^{-1}\circ f \Rightarrow_L id$.

\end{proof}
\begin{defn}
\label{FibrantCofibrantReplacementFunctorToHomotopyCategory}\hypertarget{FibrantCofibrantReplacementFunctorToHomotopyCategory}{}
Given a [[model category]] $\mathcal{C}$, consider a \emph{choice} for each object $X \in \mathcal{C}$ of

\begin{enumerate}%
\item a factorization $\emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X$ of the initial morphism, such that when $X$ is already cofibrant then $p_X = id_X$;


\item a factorization $X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast$ of the terminal morphism, such that when $X$ is already fibrant then $j_X = id_X$.



\end{enumerate}
Write then

\begin{displaymath}
\gamma_{R,Q} 
  \;\colon\;
  \mathcal{C}
   \longrightarrow
  Ho(\mathcal{C})
\end{displaymath}
for the [[functor]] to the homotopy category, def. \ref{HomotopyCategoryOfAModelCategory}, which sends an object $X$ to the object $R Q X$ and sends a morphism $f \colon X \longrightarrow Y$ to the [[homotopy class]] of the result of first lifting in

\begin{displaymath}
\itexarray{
    \emptyset &\longrightarrow& Q Y
    \\
    {}^{\mathllap{i_X}}\downarrow &{}^{Q f}\nearrow& \downarrow^{\mathrlap{p_Y}}
    \\
    Q X &\underset{f\circ p_X}{\longrightarrow}& Y
  }
\end{displaymath}
and then lifting (here: [[extension|extending]]) in

\begin{displaymath}
\itexarray{ 
    Q X &\overset{j_{Q Y} \circ Q f}{\longrightarrow}& R Q Y
    \\
    {}^{\mathllap{j_{Q X}}}\downarrow &{}^{R Q f}\nearrow& \downarrow^{\mathrlap{q_{Q Y}}}
    \\
    R Q X
    &\longrightarrow&
    \ast
  }
  \,.
\end{displaymath}
\end{defn}
\begin{lemma}
\label{ConstructionOfLocalizationFunctorForModelCategoryIsWellDefined}\hypertarget{ConstructionOfLocalizationFunctorForModelCategoryIsWellDefined}{}
The construction in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} is indeed well defined.

\end{lemma}
\begin{proof}
First of all, the object $R Q X$ is indeed both fibrant and cofibrant (as well as related by a [[zig-zag]] of weak equivalences to $X$):

\begin{displaymath}
\itexarray{
     \emptyset 
     \\
     {}^{\mathllap{\in Cof}}\downarrow & \searrow^{\mathrlap{\in Cof}}
     \\
     Q X &\underset{\in W \cap Cof}{\longrightarrow}& R Q X &\underset{\in Fib}{\longrightarrow}& \ast
     \\
     {}^{\mathllap{\in W}}\downarrow
     \\
     X
  }
  \,.
\end{displaymath}
Now to see that the image on morphisms is well defined. First observe that any two choices $(Q f)_{i}$ of the first lift in the definition are left homotopic to each other, exhibited by lifting in

\begin{displaymath}
\itexarray{
    Q X \sqcup Q X &\stackrel{((Q f)_1, (Q f)_2 )}{\longrightarrow}& Q Y
    \\
    {}^{\mathllap{\in W \cap Cof}}\downarrow 
     && 
    \downarrow^{\mathrlap{p_{Y}}}_{\mathrlap{\in Fib}}
    \\
    Cyl(Q X)
    &\underset{f \circ p_{X} \circ \sigma_{Q X}}{\longrightarrow}& 
    Y
  }
  \,.
\end{displaymath}
Hence also the composites $j_{Q Y}\circ (Q_f)_i$ are [[left homotopy|left homotopic]] to each other, and since their domain is cofibrant, they are also [[right homotopy|right homotopic]] (via \href{homotopy+in+a+model+category#LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually}{this} lemma) by a right homotopy $\kappa$. This implies finally, by lifting in

\begin{displaymath}
\itexarray{
    Q X &\overset{\kappa}{\longrightarrow}& Path(R Q Y)
    \\
    {}^{\mathllap{\in W \cap Cof}}\downarrow 
     && 
    \downarrow^{\mathrlap{\in Fib}}
    \\
    R Q X &\underset{(R (Q f)_1, R (Q f)_2)}{\longrightarrow}&
    R Q Y \times R Q Y
  }
\end{displaymath}
that also $R (Q f)_1$ and $R (Q f)_2$ are right homotopic, hence that indeed $R Q f$ represents a well-defined [[homotopy class]].

Finally to see that the assignment is indeed [[functor|functorial]], observe that the commutativity of the lifting diagrams for $Q f$ and $R Q f$ imply that also the following diagram commutes

\begin{displaymath}
\itexarray{
    X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& R Q X
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}}
    \\
    Y &\underset{p_y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& R Q Y
  }
  \,.
\end{displaymath}
Now from the [[pasting]] composite

\begin{displaymath}
\itexarray{
    X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& R Q X
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}}
    \\
    Y &\underset{p_Y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& R Q Y
    \\
    {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{Q g}} && \downarrow^{\mathrlap{R Q g}}
    \\
    Z &\underset{p_Z}{\longleftarrow}& Q Z &\underset{j_{Q Z}}{\longrightarrow}& R Q Z
  }
\end{displaymath}
one sees that $(R Q g)\circ (R Q f)$ is a lift of $g \circ f$ and hence the same argument as above gives that it is homotopic to the chosen $R Q(g \circ f)$.

\end{proof}
\begin{defn}
\label{HomotopyCategoryOfACategoryWithWeakEquivalences}\hypertarget{HomotopyCategoryOfACategoryWithWeakEquivalences}{}
For $\mathcal{C}$ a [[category with weak equivalences]], its \emph{[[homotopy category]]} (or: \emph{[[localization]]} at the weak equivalences) is, if it exists, a [[category]] $Ho(\mathcal{C})$ equipped with a [[functor]]

\begin{displaymath}
\gamma \colon \mathcal{C} \longrightarrow Ho(C)
\end{displaymath}
which sends weak equivalences to [[isomorphisms]], and which is [[universal property|universal with this property]]:

for $F \colon \mathcal{C} \longrightarrow D$ any [[functor]] out of $\mathcal{C}$ into any [[category]], such that $F$ takes weak equivalences to [[isomorphisms]], it factors through $\gamma$ up to a [[natural isomorphism]]

\begin{displaymath}
\itexarray{
    \mathcal{C} && \overset{F}{\longrightarrow} && D
    \\
    & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}}
    \\
    && Ho(\mathcal{C})
  }
\end{displaymath}
and this factorization is unique up to unique isomorphism, in that for $(\tilde F_1, \rho_1)$ and $(\tilde F_2, \rho_2)$ two such factorizations, then there is a unique [[natural isomorphism]] $\kappa \colon \tilde F_1 \Rightarrow \tilde F_2$ making the evident diagram of natural isomorphisms commute.

\end{defn}
\begin{theorem}
\label{UniversalPropertyOfHomotopyCategoryOfAModelCategory}\hypertarget{UniversalPropertyOfHomotopyCategoryOfAModelCategory}{}
For $\mathcal{C}$ a [[model category]], the functor $\gamma_{R,Q}$ in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} (for any choice of $R$ and $Q$) exhibits $Ho(\mathcal{C})$ as indeed being the [[homotopy category]] of the underlying [[category with weak equivalences]], in the sense of def. \ref{HomotopyCategoryOfACategoryWithWeakEquivalences}.

\end{theorem}
\begin{proof}
First, to see that that $\gamma$ indeed takes weak equivalences to isomorphisms: By [[two-out-of-three]] applied to the [[commuting diagrams]] shown in the proof of lemma \ref{ConstructionOfLocalizationFunctorForModelCategoryIsWellDefined} the morphism $R Q f$ is a weak equivalence if $f$ is:

\begin{displaymath}
\itexarray{
    X &\underoverset{\simeq}{p_X}{\longleftarrow}& Q X &\underoverset{\simeq}{j_{Q X}}{\longrightarrow}& R Q X
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}}
    \\
    Y &\underoverset{p_y}{\simeq}{\longleftarrow}& Q Y &\underoverset{j_{Q Y}}{\simeq}{\longrightarrow}& R Q Y
  }
\end{displaymath}
With this the ``Whitehead theorem for model categories'', lemma \ref{WhiteheadTheoremInModelCategories}, implies that $R Q f$ represents an isomorphism in $Ho(\mathcal{C})$.

Now let $F \colon \mathcal{C}\longrightarrow D$ be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as

\begin{displaymath}
\itexarray{
    \mathcal{C} && \overset{F}{\longrightarrow} && D
    \\
    & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}}
    \\
    && Ho(\mathcal{C})
  }
\end{displaymath}
uniquely up to unique [[natural isomorphism]]. Now by construction of $R$ and $Q$ in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory}, $\gamma_{R,Q}$ is the identity on the [[full subcategory]] of fibrant-cofibrant objects. It follows that if $\tilde F$ exists at all, it must satisfy for all $X \stackrel{f}{\to} Y$ with $X$ and $Y$ both fibrant and cofibrant that

\begin{displaymath}
\tilde F([f]) \simeq F(f)
  \,.
\end{displaymath}
But by def. \ref{HomotopyCategoryOfAModelCategory} that already fixes $\tilde F$ on all of $Ho(\mathcal{C})$, up to unique [[natural isomorphism]]. Hence it only remains to check that with this definition of $\tilde F$ there exists any [[natural isomorphism]] $\rho$ filling the diagram above.

To that end, apply $F$ to the above [[commuting diagram]] to obtain

\begin{displaymath}
\itexarray{
    F(X) &\underoverset{iso}{F(p_X)}{\longleftarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(R Q X)
    \\
    {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{F(Q f)}} && \downarrow^{\mathrlap{F(R Q f)}}
    \\
    F(Y) &\underoverset{F(p_y)}{iso}{\longleftarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(R Q Y)
  }
  \,.
\end{displaymath}
Here now all horizontal morphisms are [[isomorphisms]], by assumption on $F$. It follows that defining $\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1}$ makes the required natural isomorphism:

\begin{displaymath}
\itexarray{
    \rho_X \colon & F(X) &\underoverset{iso}{F(p_X)^{-1}}{\longrightarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(R Q X)
   &=&
   \tilde F(\gamma_{R,Q}(X))
    \\
    & {}^{\mathllap{F(f)}}\downarrow 
     && 
     && \downarrow^{\mathrlap{F(R Q f)}}
     && \downarrow^{\tilde F(\gamma_{R,Q}(X))}
    \\
    \rho_Y\colon& F(Y) &\underoverset{F(p_y)^{-1}}{iso}{\longrightarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(R Q Y)
   &=&
   \tilde F(\gamma_{R,Q}(X))
  }
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{EssentialUniquenessOfLocalizationFunctorOfModelCategory}\hypertarget{EssentialUniquenessOfLocalizationFunctorOfModelCategory}{}
Due to theorem \ref{UniversalPropertyOfHomotopyCategoryOfAModelCategory} we may suppress the choices of cofibrant $Q$ and fibrant replacement $R$ in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} and just speak of [[generalized the|the]] [[localization functor]]

\begin{displaymath}
\gamma \;\colon\;
  \mathcal{C}
  \longrightarrow
  Ho(\mathcal{C})
\end{displaymath}
up to [[natural isomorphism]].

\end{remark}
While the construction of the homotopy category in def. \ref{HomotopyCategoryOfAModelCategory} combines the restriction to good (fibrant/cofibrant) objects with the passage to [[homotopy classes]] of morphisms, it is often useful to consider intermediate stages:

\begin{defn}
\label{FullSubcategoriesOfFibrantCofibrantObjects}\hypertarget{FullSubcategoriesOfFibrantCofibrantObjects}{}
Given a [[model category]] $\mathcal{C}$, write

\begin{displaymath}
\itexarray{
   && \mathcal{C}_{fc}
   \\
   & \swarrow && \searrow
   \\
   \mathcal{C}_c && && \mathcal{C}_f
   \\
   & \searrow && \swarrow
   \\
   && \mathcal{C}
  }
\end{displaymath}
for the system of [[full subcategory]] inclusions on the cofibrant objects ($\mathcal{C}_c$), the fibrant objects ($\mathcal{C}_f$) and the objects which are both fibrant and cofibrant ($\mathcal{C}_{fc}$), all regarded a [[categories with weak equivalences]], via the weak equivalences inherited from $\mathcal{C}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Of course the subcategories in def. \ref{FullSubcategoriesOfFibrantCofibrantObjects} inherit more structure than just that of [[categories with weak equivalences]] from $\mathcal{C}$. $\mathcal{C}_f$ and $\mathcal{C}_c$ each inherit ``half'' of the factorization axioms. One says that $\mathcal{C}_f$ has the structure of a ``[[fibration category]]'' called a ``[[category of fibrant objects]]'', while $\mathcal{C}_c$ has the structure of a ``[[cofibration category]]''.

\end{remark}
The proof of theorem \ref{UniversalPropertyOfHomotopyCategoryOfAModelCategory} immediately implies the following:

\begin{cor}
\label{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects}\hypertarget{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects}{}
For $\mathcal{C}$ a [[model category]], the restriction of the localization functor $\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})$ from def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} (using remark \ref{EssentialUniquenessOfLocalizationFunctorOfModelCategory}) to any of the sub-[[categories with weak equivalences]] of def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}

\begin{displaymath}
\itexarray{
   && \mathcal{C}_{fc}
   \\
   & \swarrow && \searrow
   \\
   \mathcal{C}_c && && \mathcal{C}_f
   \\
   & \searrow && \swarrow
   \\
   && \mathcal{C}
   \\
   && \downarrow^{\mathrlap{\gamma}}
   \\
   && Ho(\mathcal{C})
  }
\end{displaymath}
exhibits $Ho(\mathcal{C})$ equivalently as the [[homotopy category]] also of these subcategories. In particular there are [[equivalences of categories]]

\begin{displaymath}
Ho(\mathcal{C})
   \simeq
  Ho(\mathcal{C}_{f})
   \simeq
  Ho(\mathcal{C}_{c})
   \simeq
  Ho(\mathcal{C}_{fc})
  \,.
\end{displaymath}
\end{cor}
\hypertarget{DerivedFunctors}{}\subsection*{{Derived functors}}\label{DerivedFunctors}

\begin{defn}
\label{HomotopicalFunctor}\hypertarget{HomotopicalFunctor}{}
For $\mathcal{C}$ and $\mathcal{D}$ two [[categories with weak equivalences]], then a [[functor]] $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ is called \emph{[[homotopical functor]]} if it sends weak equivalences to weak equivalences.

\end{defn}
\begin{defn}
\label{DerivedFunctorOfAHomotopicalFunctor}\hypertarget{DerivedFunctorOfAHomotopicalFunctor}{}
Given a [[homotopical functor]] $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ (def. \ref{HomotopicalFunctor}) between [[categories with weak equivalences]] whose [[homotopy categories]] $Ho(\mathcal{C})$ and $Ho(\mathcal{D})$ exist (def. \ref{HomotopyCategoryOfACategoryWithWeakEquivalences}), then its \emph{[[derived functor]]} is the functor $Ho(F)$ between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. \ref{HomotopyCategoryOfACategoryWithWeakEquivalences}):

\begin{displaymath}
\itexarray{
    \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow 
    &\swArrow_{\simeq}& 
    \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
    \\
    Ho(\mathcal{C})
      &\underset{\exists \; Ho(F)}{\longrightarrow}&
    Ho(\mathcal{D})
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
While many functors of interest between [[model categories]] are not homotopical in the sense of def. \ref{HomotopicalFunctor}, many become homotopical after restriction to the [[full subcategories]] of fibrant object or of cofibrant objects, def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}. By corollary \ref{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects} this is just as good for the purpose of [[homotopy theory]].

\end{remark}
Therefore one considers the following generalization of def. \ref{DerivedFunctorOfAHomotopicalFunctor}:

\begin{defn}
\label{LeftAndRightDerivedFunctorsOnModelCategories}\hypertarget{LeftAndRightDerivedFunctorsOnModelCategories}{}
Consider a functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ out of a [[model category]] $\mathcal{C}$ into a [[category with weak equivalences]] $\mathcal{D}$.

\begin{enumerate}%
\item If the restriction of $F$ to the [[full subcategory]] $\mathcal{C}_f$ of fibrant object becomes a [[homotopical functor]] (def. \ref{HomotopicalFunctor}), then the [[derived functor]] of that restriction, according to def. \ref{DerivedFunctorOfAHomotopicalFunctor}, is called the \emph{[[right derived functor]]} of $F$ and denoted by $\mathbb{R}F$:

\begin{displaymath}
\itexarray{
    & \mathcal{C}_f &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D}
    \\
    & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow 
    &&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    &\swArrow_{\simeq}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
    \\
    \mathbb{R} F \colon & Ho(\mathcal{C}_f) &\simeq& Ho(\mathcal{C})
    &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D})
  }
  \,.
\end{displaymath}
Here the commuting square on the left is from corollary \ref{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects}, the square on the right is that of def. \ref{DerivedFunctorOfAHomotopicalFunctor}.


\item If the restriction of $F$ to the [[full subcategory]] $\mathcal{C}_c$ of cofibrant object becomes a homotopical functor (def. \ref{HomotopicalFunctor}), then the [[derived functor]] of that restriction, according to def. \ref{DerivedFunctorOfAHomotopicalFunctor}, is called the \emph{[[left derived functor]]} of $F$ and denoted by $\mathbb{L}F$:

\begin{displaymath}
\itexarray{
    & \mathcal{C}_c &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D}
    \\
    & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow 
    &&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    &\swArrow_{\simeq}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
    \\
    \mathbb{L} F \colon & Ho(\mathcal{C}_c) &\simeq& Ho(\mathcal{C})
    &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D})
  }
  \,.
\end{displaymath}
Here the commuting square on the left is from corollary \ref{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects}, the square on the right is that of def. \ref{DerivedFunctorOfAHomotopicalFunctor}.



\end{enumerate}
\end{defn}
The key fact that makes def. \ref{LeftAndRightDerivedFunctorsOnModelCategories} practically useful is the following

\begin{prop}
\label{KenBrownLemma}\hypertarget{KenBrownLemma}{}
\textbf{([[Ken Brown's lemma]])}

Let $\mathcal{C}$ be a [[model category]] with [[full subcategories]] $\mathcal{C}_f, \mathcal{C}_c$ [[category of fibrant objects|of fibrant objects]] and [[cofibration category|of cofibrant objects]] respectively (def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}). Let $\mathcal{D}$ be a [[category with weak equivalences]].

\begin{enumerate}%
\item A [[functor]]

\begin{displaymath}
F \;\colon\; \mathcal{C}_f \longrightarrow \mathcal{D}
\end{displaymath}


\end{enumerate}
is a [[homotopical functor]], def. \ref{HomotopicalFunctor}, already if it sends acylic fibrations to weak equivalences.

\begin{enumerate}%
\item A [[functor]]

\begin{displaymath}
F \;\colon\; \mathcal{C}_c \longrightarrow \mathcal{D}
\end{displaymath}


\end{enumerate}
is a [[homotopical functor]], def. \ref{HomotopicalFunctor}, already if it sends acylic cofibrations to weak equivalences.

\end{prop}
\begin{cor}
\label{LeftAndRightDerivedFunctors}\hypertarget{LeftAndRightDerivedFunctors}{}
Let $\mathcal{C}, \mathcal{D}$ be [[model categories]] and consider $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ a [[functor]]. Then:

\begin{enumerate}%
\item If $F$ preserves cofibrant objects and acyclic cofibrations between these, then its [[left derived functor]] (def. \ref{LeftAndRightDerivedFunctorsOnModelCategories}) $\mathbb{L}F$ exists, fitting into a [[diagram]]

\begin{displaymath}
\itexarray{
    \mathcal{C}_{c} &\overset{F}{\longrightarrow}& \mathcal{D}_{c}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow 
    &\swArrow_{\simeq}& 
    \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
    \\
    Ho(\mathcal{C}) &\overset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D})
  }
\end{displaymath}

\item If $F$ preserves fibrant objects and acyclic fibrants between these, then its [[right derived functor]] (def. \ref{LeftAndRightDerivedFunctorsOnModelCategories}) $\mathbb{R}F$ exists, fitting into a [[diagram]]

\begin{displaymath}
\itexarray{
    \mathcal{C}_{f} &\overset{F}{\longrightarrow}& \mathcal{D}_{f}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow 
    &\swArrow_{\simeq}& 
    \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
    \\
    Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D})
  }
  \,.
\end{displaymath}


\end{enumerate}
\end{cor}
In practice it turns out to be useful to arrange for the assumptions in corollary \ref{LeftAndRightDerivedFunctors} to be satisfied in the following neat way:

\begin{defn}
\label{QuillenAdjunction}\hypertarget{QuillenAdjunction}{}
Let $\mathcal{C}, \mathcal{D}$ be [[model categories]]. A pair of [[adjoint functors]] between them

\begin{displaymath}
(L \dashv R)
  \;\colon\;
  \mathcal{C}
    \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}}
  \mathcal{D}
\end{displaymath}
is called a \emph{[[Quillen adjunction]]} (and $L$,$R$ are called left/right \emph{Quillen functors}, respectively) if the following equivalent conditions are satisfied

\begin{enumerate}%
\item $L$ preserves cofibrations and $R$ preserves fibrations;


\item $L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations;


\item $L$ preserves cofibrations and acylic cofibrations;


\item $R$ preserves fibrations and acyclic fibrations.



\end{enumerate}
\end{defn}
\begin{prop}
\label{ConditionsOnQuillenAdjunctionAreIndeedEquivalent}\hypertarget{ConditionsOnQuillenAdjunctionAreIndeedEquivalent}{}
The conditions in def. \ref{QuillenAdjunction} are indeed all equivalent.

\end{prop}
\begin{proof}
Observe that

\begin{itemize}%
\item (i) \emph{A [[left adjoint]] $L$ between [[model categories]] preserves acyclic cofibrations precisely if its [[right adjoint]] $R$ preserves fibrations.}


\item (ii) \emph{A [[left adjoint]] $L$ between [[model categories]] preserves cofibrations precisely if its [[right adjoint]] $R$ preserves acyclic fibrations.}



\end{itemize}
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:

\begin{displaymath}
\itexarray{
    A &\longrightarrow& R(X)
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}}
    \\
    B &\longrightarrow& R(Y)
  }
  \;\;\;\;\;\;
  \,,
  \;\;\;\;\;\;
  \itexarray{
    L(A) &\longrightarrow& X
    \\
    {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}}
    \\
    L(B) &\longrightarrow& Y
  }
  \,.
\end{displaymath}
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.

\end{proof}
\begin{defn}
\label{QuillenEquivalence}\hypertarget{QuillenEquivalence}{}
For $\mathcal{C}, \mathcal{D}$ two [[model categories]], a [[Quillen adjunction]] (def.\ref{QuillenAdjunction})

\begin{displaymath}
(L \dashv R)
  \;\colon\;
  \mathcal{C}
    \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}}
  \mathcal{D}
\end{displaymath}
is called a \textbf{[[Quillen equivalence]]} if the following equivalent conditions hold.

\begin{enumerate}%
\item The [[right derived functor]] of $R$ (via prop. \ref{ConditionsOnQuillenAdjunctionAreIndeedEquivalent}, corollary \ref{LeftAndRightDerivedFunctors}) is an [[equivalence of categories]]

\begin{displaymath}
\mathbb{R}R \colon Ho(\mathcal{C}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{D})
  \,.
\end{displaymath}

\item The [[left derived functor]] of $L$ (via prop. \ref{ConditionsOnQuillenAdjunctionAreIndeedEquivalent}, corollary \ref{LeftAndRightDerivedFunctors}) is an [[equivalence of categories]]

\begin{displaymath}
\mathbb{L}L \colon Ho(\mathcal{D}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{C})
  \,.
\end{displaymath}

\item For every cofibrant object $d \in \mathcal{D}$ and every fibrant object $c \in \mathcal{C}$, a morphism $d \longrightarrow R(c)$ is a weak equivalence precisely if its [[adjunct]] morphism $L(c) \to d$ is

\begin{displaymath}
\frac{
     d \overset{\in W_{\mathcal{D}}}{\longrightarrow} R(c)
   }{
      L(d) \overset{\in W_{\mathcal{C}}}{\longrightarrow} c
   }
   \,.
\end{displaymath}

\item For every cofibrant object $d\in \mathcal{C}$, the [[derived adjunction unit]], hence the composite

\begin{displaymath}
d \longrightarrow R(L(d)) \longrightarrow R(L(d)^{fib})
\end{displaymath}
(of the [[adjunction unit]] with any [[fibrant replacement]]) is a [[weak equivalence]].


\item For every fibrant object $c \in \mathcal{C}$ the [[derived adjunction counit]], hence the composite

\begin{displaymath}
L(R(c)^{cof}) \longrightarrow L(R(c)) \longrightarrow c
\end{displaymath}
(of the [[adjunction counit]] with any [[cofibrant replacement]]) is a [[weak equivalence]].



\end{enumerate}
\end{defn}
\hypertarget{further_properties}{}\subsection*{{Further properties}}\label{further_properties}

This construction extends to a [[double pseudofunctor]]

\begin{displaymath}
Ho \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)
\end{displaymath}
on the [[double category of model categories]] (\href{double+category+of+model+categories#HomotopyDoublePseudofunctor}{this Prop.}).

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

\begin{itemize}%
\item [[monoidal homotopy category of a monoidal model category]]

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

The original account is due to

\begin{itemize}%
\item [[Daniel Quillen]], \emph{Axiomatic homotopy theory} in \emph{Homotopical algebra}, Lecture Notes in Mathematics, No. 43 43, Berlin (1967)

\end{itemize}
Review includes

\begin{itemize}%
\item [[William Dwyer]], J. Spalinski, \emph{[[Homotopy theories and model categories]]} (\href{http://folk.uio.no/paularne/SUPh05/DS.pdf}{pdf}) in [[Ioan Mackenzie James]] (ed.), \emph{[[Handbook of Algebraic Topology]]} 1995


\item [[Paul Goerss]], [[Rick Jardine]], section II.1 of \emph{[[Simplicial homotopy theory]]} Birkh\"a{}user 1999, 2009



\end{itemize}
[[!redirects homotopy categories of model categories]]



\end{document}