\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 in 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_lemmas}{Basic lemmas}\dotfill \pageref*{basic_lemmas} \linebreak
\noindent\hyperlink{equivalence_relation}{Equivalence relation}\dotfill \pageref*{equivalence_relation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The extra structure of a [[model category]] over a [[category with weak equivalences]] induces concrete constructions for expressing [[homotopy]] between [[morphisms]]. These lead in particular to an explicit construction of the [[homotopy category of a model category]].

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

\begin{defn}
\label{PathAndCylinderObjectsInAModelCategory}\hypertarget{PathAndCylinderObjectsInAModelCategory}{}
Let $\mathcal{C}$ be a [[model category]] and $X \in \mathcal{C}$ an [[object]].

\begin{itemize}%
\item A \textbf{[[path object]]} $Path(X)$ for $X$ is a factorization of the [[diagonal]] $\nabla_X \colon X \to X \times X$ as

\end{itemize}
\begin{displaymath}
\nabla_X 
  \;\colon\;
   X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X
  \,.
\end{displaymath}
where $X\to Path(X)$ is a weak equivalence. This is called a \textbf{good path object} if in addition $Path(X) \to X \times X$ is a fibration.

\begin{itemize}%
\item A \textbf{[[cylinder object]]} $Cyl(X)$ for $X$ is a factorization of the [[codiagonal]] (or ``fold map'') $\Delta_X X \sqcup X \to X$ as

\end{itemize}
\begin{displaymath}
\Delta_X
  \;\colon\;
  X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X
  \,.
\end{displaymath}
where $Cyl(X) \to X$ is a weak equivalence. This is called a \textbf{good cylinder object} if in addition $X \sqcup X \to Cyl(X)$ is a cofibration.

\end{defn}
\begin{remark}
\label{RemarkOnChoicesOfNonGoodPathAndCylinderObjects}\hypertarget{RemarkOnChoicesOfNonGoodPathAndCylinderObjects}{}
By the factorization axioms every object in a model category has both a good path object and as well as a good cylinder object according to def. \ref{PathAndCylinderObjectsInAModelCategory}. But in some situations one is genuinely interested in using non-good such objects.

For instance in the [[classical model structure on topological spaces]], the obvious object $X\times [0,1]$ is a cylinder object, but not a good cylinder unless $X$ itself is cofibrant (a [[cell complex]] in this case).

More generally, the path object $Path(X)$ of def. \ref{PathAndCylinderObjectsInAModelCategory} is analogous to the [[powering]] $\pitchfork(I,X)$ with an [[interval object]] and the cyclinder object $Cyl(X)$ is analogous to the [[tensoring]] with a cylinder object $I\odot X$. In fact, if $\mathcal{C}$ is a $V$-[[enriched model category]] and $X$ is fibrant/cofibrant, then these powers and copowers are in fact examples of (good) path and cylinder objects if the [[interval object]] is sufficiently good.

\end{remark}
\begin{defn}
\label{LeftAndRightHomotopyInAModelCategory}\hypertarget{LeftAndRightHomotopyInAModelCategory}{}
Let $f,g \colon X \longrightarrow Y$ be two [[parallel morphisms]] in a [[model category]].

\begin{itemize}%
\item A \textbf{left homotopy} $\eta \colon f \Rightarrow_L g$ is a morphism $\eta \colon Cyl(X) \longrightarrow Y$ from a [[cylinder object]] of $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, such that it makes this [[commuting diagram|diagram commute]]:

\end{itemize}
\begin{displaymath}
\itexarray{
    X &\longrightarrow& Cyl(X) &\longleftarrow& X
    \\
    & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}}
    \\
    && 
    Y
  }
  \,.
\end{displaymath}
\begin{itemize}%
\item A \textbf{right homotopy} $\eta \colon f \Rightarrow_R g$ is a morphism $\eta \colon X \to Path(Y)$ to some [[path object]] of $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, such that this [[commuting diagram|diagram commutes]]:

\end{itemize}
\begin{displaymath}
\itexarray{
    && X
    \\
    & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} 
    \\
    Y &\longleftarrow& Path(Y) &\longrightarrow& Y
  }
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{basic_lemmas}{}\subsubsection*{{Basic lemmas}}\label{basic_lemmas}

\begin{lemma}
\label{ComponentsOfGoodCylinderOfCofibrantAreAcyclicCofibrationsAndDually}\hypertarget{ComponentsOfGoodCylinderOfCofibrantAreAcyclicCofibrationsAndDually}{}
If $X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X$ is a good [[cylinder object]] for a cofibrant object $X$ def. \ref{PathAndCylinderObjectsInAModelCategory}, then both components $i_0, i_1 \colon X \to Cyl(X)$ are acyclic cofibrations.

Dually, if $X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X$

is a good path object for a fibrant object $X$, then both component $p_0,p_1 \colon Path(X)\to X$ are acyclic fibrations.

\end{lemma}
\begin{proof}
We discuss the first case, the second is [[formal dual|formally dual]]. First observe that the two inclusions $X \to X \sqcup X$ are cofibrations, since they are the [[pushout]] of the cofibration $\emptyset \to X$. This implies that $i_0$ and $i_1$ are composites of two cofibrations

\begin{displaymath}
i_0, i_1 
   \;\colon\;
  X \overset{\in Cof}{\longrightarrow} X\sqcup X  
  \overset{\in Cof}{\longrightarrow}
\end{displaymath}
and hence are themselves cofibrations. That they are in addition weak equivalences follows from [[two-out-of-three]] applied to the identity

\begin{displaymath}
id_X \;\colon\;
  X \overset{\in W}{\longrightarrow}
  Cyl(X)
  \overset{i_0}{\longrightarrow}
  X
  \,.
\end{displaymath}
implied by the fact that the cylinder by definition factors the [[codiagonal]].

\end{proof}
The following says that the choice of cylinder/path objects in def. \ref{LeftAndRightHomotopyInAModelCategory} is irrelevant as long it is ``good''.

\begin{lemma}
\label{GoodCylinderObjectsSupportEveryLeftHomotopyAndDually}\hypertarget{GoodCylinderObjectsSupportEveryLeftHomotopyAndDually}{}
For $\eta \colon f \Rightarrow_L g \colon X \to Y$ a [[left homotopy]] in some [[model category]], def. \ref{LeftAndRightHomotopyInAModelCategory}, such that $Y$ is a fibrant object, then for $Cyl(X)$ any choice of \emph{good} [[cylinder object]] for $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, there is a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    X &\longrightarrow& Cyl(X) &\longleftarrow& X
    \\
    & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\tilde \eta}}& \swarrow_{\mathrlap{g}}
    \\
    && 
    Y
  }
  \,.
\end{displaymath}
Dually, for $\eta \colon f \Rightarrow_R g \colon X \to Y$ a [[right homotopy]], def. \ref{LeftAndRightHomotopyInAModelCategory}, such that $X$ is cofibrant, then for $Path(X)$ any choice of \emph{good} [[path object]] for $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, there is a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    && X
    \\
    & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\tilde \eta}} & \searrow^{\mathrlap{g}} 
    \\
    Y &\longleftarrow& Path(Y) &\longrightarrow& Y
  }
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
We discuss the first statement, the second is [[formal dual|formally dual]]. Let $\eta \colon \hat X \longrightarrow Y$ be the given left homotopy with respect to a given cylinder object $\hat X$ of $X$. Factor it as

\begin{displaymath}
\eta 
    \;\colon\; 
  \hat X \overset{\in Cof}{\longrightarrow} Z \overset{\in W \cap Fib}{\longrightarrow} 
  Y
  \,.
\end{displaymath}
Then find liftings $\ell$ and $k$ in the following two [[commuting diagrams]]

\begin{displaymath}
\itexarray{
    X \sqcup X &\overset{}{\longrightarrow}& \hat X &\longrightarrow& Z
    \\
    \downarrow && & {}^{\mathllap{\ell}}\nearrow & \downarrow
    \\
    Cyl(X) &\longrightarrow& &\longrightarrow& Y 
  }
  \;\;\;\;\;
  \,,
  \;\;\;\;\;
  \itexarray{
    \hat X &\overset{\eta}{\longrightarrow}& Y
    \\
    \downarrow &{}^{\mathllap{k}}\nearrow& \downarrow
    \\
    Z &\longrightarrow& \ast 
  }
  \,.
\end{displaymath}
Now the composite $\eta \coloneqq k \circ \ell$ is of the required kind,

\begin{displaymath}
\itexarray{
    X \sqcup X &\overset{}{\longrightarrow}& \hat X &\longrightarrow& Z &\overset{k}{\longrightarrow}& Y
    \\
    \downarrow &&&  {}^{\mathllap{\ell}}\nearrow &
    \\
    Cyl(X) &\longrightarrow& 
  }
  \,.
\end{displaymath}
\end{proof}
\begin{lemma}
\label{LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually}\hypertarget{LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually}{}
Let $f,g \colon X \to Y$ be two [[parallel morphisms]] in a [[model category]].

\begin{itemize}%
\item Let $X$ be cofibrant. If there is a [[left homotopy]] $f \Rightarrow_L g$ then there is also a [[right homotopy]] $f \Rightarrow_R g$ (def. \ref{LeftAndRightHomotopyInAModelCategory}) with respect to any chosen path object.


\item Let $Y$ be fibrant. If there is a [[right homotopy]] $f \Rightarrow_R g$ then there is also a [[left homotopy]] $f \Rightarrow_L g$ with respect to any chosen cylinder object.



\end{itemize}
\end{lemma}
\begin{proof}
We discuss the first case, the second is [[formal dual|formally dual]]. Let $\eta \colon Cyl(X) \longrightarrow Y$ be the given left homotopy. By lemma \ref{GoodCylinderObjectsSupportEveryLeftHomotopyAndDually} we may assume without restriction that $Cyl(X)$ is \emph{good} in the sense of def. \ref{PathAndCylinderObjectsInAModelCategory}, for otherwise replace it by one that is. With this, lemma \ref{ComponentsOfGoodCylinderOfCofibrantAreAcyclicCofibrationsAndDually} implies that we have a lift $h$ in the following [[commuting diagram]]

\begin{displaymath}
\itexarray{
    X &\overset{i \circ f}{\longrightarrow}& Path(Y)
    \\
    {}^{\mathllap{i_0}}_{\mathllap{\in W \cap Cof}}\downarrow 
    &{}^{\mathllap{h}}\nearrow& 
    \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}}
    \\
    Cyl(X)
    &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y
  }
  \,,
\end{displaymath}
where on the right we have the chosen path space object. Now the composite $\tilde \eta \coloneqq h \circ i_1$ is a right homotopy as required.

\begin{displaymath}
\itexarray{
    && && Path(Y)
    \\
    && &{}^{\mathllap{h}}\nearrow& 
    \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}}
    \\
    X &\overset{i_1}{\longrightarrow}& Cyl(X)
    &\underset{(f \circ p,\eta)}{\longrightarrow}&
    Y \times Y
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{equivalence_relation}{}\subsubsection*{{Equivalence relation}}\label{equivalence_relation}

\begin{prop}
\label{}\hypertarget{}{}
For $X$ a cofibrant object in a [[model category]] and $Y$ a [[fibrant object]], then the [[relations]] of [[left homotopy]] $f \Rightarrow_L g$ and of [[right homotopy]] $f \Rightarrow_R g$ (def. \ref{LeftAndRightHomotopyInAModelCategory}) on the [[hom set]] $Hom(X,Y)$ coincide and are both [[equivalence relations]].

\end{prop}
\begin{proof}
That both relations coincide under the (co-)fibrancy assumption follows directly from lemma \ref{LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually}.

To see that left homotopy with domain $X$ is a [[transitive relation]] first use lemma \ref{GoodCylinderObjectsSupportEveryLeftHomotopyAndDually} to obtain that every left homotopy is exhibited by a \emph{good} cylinder object $Cyl(X)$ and then lemma \ref{ComponentsOfGoodCylinderOfCofibrantAreAcyclicCofibrationsAndDually} to see that the cofiber coproduct $Cyl(X)\underset{X}{\sqcup} Cyl(X)$ in

\begin{displaymath}
\itexarray{
     && && X
     \\
     && && \downarrow^{\mathrlap{i_0}}
     \\
     && X &\underoverset{\in W}{i_1}{\longrightarrow}& Cyl(X)
     \\
     && {}^{\mathllap{i_0}}\downarrow &(po)& \downarrow
     \\
     X &\underset{i_1}{\longrightarrow}& 
    Cyl(X) &\underset{\in W}{\longrightarrow}& Cyl(X) \underset{X}{\sqcup} Cyl(X)
     \\
     && &{}_{\mathllap{}}\searrow& & \searrow
     \\
     && && \underset{\in W}{\longrightarrow} &\longrightarrow& X
  }
\end{displaymath}
is again a [[cylinder object]], def. \ref{PathAndCylinderObjectsInAModelCategory}. The [[symmetric relation|symmetry]] and [[reflexive relation|reflexivity]] of the relation is obvious.

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

See the references at \emph{[[model category]]}.



\end{document}