\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 an (infinity,1)-category}

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-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{in_terms_of_simplicially_enriched_categories}{In terms of simplicially enriched categories}\dotfill \pageref*{in_terms_of_simplicially_enriched_categories} \linebreak
\noindent\hyperlink{in_terms_of_complete_segal_spaces_and_segal_categories}{In terms of complete Segal spaces and Segal categories}\dotfill \pageref*{in_terms_of_complete_segal_spaces_and_segal_categories} \linebreak
\noindent\hyperlink{in_terms_of_quasicategories}{In terms of quasi-categories}\dotfill \pageref*{in_terms_of_quasicategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{BrownRepresentability}{Brown representability}\dotfill \pageref*{BrownRepresentability} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The [[homotopy category]] of an [[(∞,1)-category]] $\mathcal{C}$ is its [[decategorification]] to an ordinary [[category]] obtained by identifying 1-[[morphisms]] that are connected by a [[2-morphism]].

If the [[(∞,1)-category]] $\mathcal{C}$ is presented by a [[category with weak equivalences]] $C$ (for instance as the [[simplicial localization]] $\mathcal{C} = L C$) then the notion of [[homotopy category]] of $C$ (where the weak equivalences are universally turned into [[isomorphism]]s) coinicides with that of $\mathcal{C}$:

\begin{displaymath}
Ho(\mathcal{C}) \simeq Ho(C)
  \,.
\end{displaymath}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The component-wise definition depend on the chosen model for $(\infty,1)$-categories, as either

\begin{itemize}%
\item [[simplicially enriched category]]/ [[topologically enriched categories]]


\item [[complete Segal space]]


\item [[Segal category]]


\item [[quasi-category]]



\end{itemize}
\hypertarget{in_terms_of_simplicially_enriched_categories}{}\subsubsection*{{In terms of simplicially enriched categories}}\label{in_terms_of_simplicially_enriched_categories}

The \textbf{homotopy category} $h C$ of a [[sSet]]-[[enriched category]] $C$ (equivalently of a [[Top]]-[[enriched category]]) is hom-wise the image under the functor

\begin{displaymath}
\pi_0 : sSet \to Set
 \,,
\end{displaymath}
which sends each [[simplicial set]] to its 0th [[homotopy group|homotopy set]] of [[connected]] components, i.e. to the set of [[path component]]s:

\begin{displaymath}
Hom_{h C}(A,B) := \pi_0(Hom_C(A,B))
  \,.
\end{displaymath}
Sometimes it is useful to regard this after all as an enriched category, but now enriched over the [[homotopy category]] of the standard [[model structure on simplicial sets]] $sSet_{Quillen}$, which is the \emph{homotopy category of an $(\infty,1)$-category} of [[∞Grpd]].

Let $\mathbf{h}: sSet \to Ho(sSet)$ be the localization functor to the [[homotopy category]] (of a [[category with weak equivalences]]). This functor is a [[monoidal functor]] by the fact that the cartesian monoidal product is a left-[[Quillen bifunctor]] for $sSet$, which means that since every object in $sSet$ is cofibrant, it preserves weak equivalences in both arguments and hence descends to the homotopy category

\begin{displaymath}
\itexarray{
    sSet \times sSet &\stackrel{\times}{\to}& sSet
    \\
    \downarrow^{\mathrlap{\mathbf{h} \times \mathbf{h}}}
    &&
    \downarrow^{\mathrlap{\mathbf{h}}}
    \\
    Ho(sSet) \times Ho(sSet) &\stackrel{\mathbf{h}(\times)}{\to}&
    Ho(sSet)
  }
  \,.
\end{displaymath}
This inuces a canonical functor $h:sSet Cat\to Ho(sSet) Cat$ which is given by the identity on objects and: $Map_{h C}(A,B):=\mathbf{h} Map_{C}(A,B)$. Then since $Hom_C(A,B)=Hom_{sSet}(\Delta^0,Map_{C}(A,B))$, it is easy to see that $Hom_{hC}(A,B)=Hom_{Ho(sSet)}(\mathbf{h} \Delta^0, \mathbf{h} Map_C(A,B))=\pi_0 Map_C(A,B)$.

\hypertarget{in_terms_of_complete_segal_spaces_and_segal_categories}{}\subsubsection*{{In terms of complete Segal spaces and Segal categories}}\label{in_terms_of_complete_segal_spaces_and_segal_categories}

Similar, but more complicated, definitions work for [[complete Segal space]]s and [[Segal category|Segal categories]].

\hypertarget{in_terms_of_quasicategories}{}\subsubsection*{{In terms of quasi-categories}}\label{in_terms_of_quasicategories}

For [[quasi-category|quasi-categories]], one can write down a definition similar to those of $sSet$-enriched categories.

Viewing $C$ as a [[simplicial set]], the homotopy category $hC$ can also be described as its [[fundamental category]] $\tau_1(C)$, i.e. the image of $C$ by the [[left adjoint]] $\tau_1 : SSet \to Cat$ of the [[nerve]] functor $N$.

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

\hypertarget{BrownRepresentability}{}\subsubsection*{{Brown representability}}\label{BrownRepresentability}

The [[Brown representability theorem]] characterizes [[representable functors]] on homotopy categories of $(\infty,1)$-categories:

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category]], [[compactly generated (∞,1)-category|generated]] by a set

\begin{displaymath}
\{S_i \in \mathcal{C}\}_{i \in I}
\end{displaymath}
of [[compact object in an (infinity,1)-category|compact objects]] (i.e. every object of $\mathcal{C}$ is an [[(∞,1)-colimit]] of the objects $S_i$.)

If each $S_i$ admits the structure of a [[cogroup]] object in the [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(\mathcal{C})$, then a [[functor]]

\begin{displaymath}
F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
\end{displaymath}
(from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ to [[Set]])

is [[representable functor|representable]] precisely if it satisfies these two conditions:

\begin{enumerate}%
\item $F$ sends small [[coproducts]] to [[products]];


\item $F$ sends [[(∞,1)-pushouts]] $X \underset{Z}{\sqcup}Y$ to [[epimorphisms]], i.e. the canonical morphisms into the [[fiber product]]

\begin{displaymath}
F\left(X \underset{Z}{\sqcup}Y\right)
  \stackrel{epi}{\longrightarrow}
  F(X) \underset{F(Z)}{\times} F(Y)
\end{displaymath}
are [[surjections]].



\end{enumerate}
\end{prop}
(\hyperlink{LurieHigherAlgebra}{Lurie ``Higher Algebra'', theorem 1.4.1.2})

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

\begin{itemize}%
\item [[Jacob Lurie]], \href{http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=33}{Section 1.2.3, p. 33} of [[Higher Topos Theory]]


\item [[Jacob Lurie]], section 1.4.1 of \emph{[[Higher Algebra]]}



\end{itemize}
[[!redirects homotopy categories of (infinity,1)-categories]]

[[!redirects homotopy category of an (∞,1)-category]] [[!redirects homotopy categories of (∞,1)-categories]]



\end{document}