\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*{canonical model structure}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\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{internalization}{Internalization}\dotfill \pageref*{internalization} \linebreak
\noindent\hyperlink{cofibrant_resolutions}{Cofibrant resolutions}\dotfill \pageref*{cofibrant_resolutions} \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}

In general, \textbf{canonical model structures} are [[model category]] structures on the categories of some flavor of [[n-category|n-categories]] for $1\le n\le \infty$ (note that $n=\infty$ or $\omega$ is allowed), which are intended to capture the correct ``categorical'' theory of these categories.

The archetypical examples are the \emph{[[canonical model structure on categories]]} and the \emph{[[canonical model structure on groupoids]]}.

Canonical model structures are sometimes called ``folk'' model structures, but the appropriateness of this term is very \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=649&page=1#Item_3}{questionable}, especially in the cases $n \gt 1$. Other alternatives are `endogenous', `standard', `natural', and `categorical'.

While ultimately the collection of all [[n-category|n-categories]] should form an $(n+1)$-category, restricting that to just invertible higher morphisms will yield an [[(n,r)-category|(n+1,1)-category]], and thus in particular an [[(∞,1)-category]]. It is this (∞,1)-category which is intended to be [[presentable (infinity,1)-category|presented]] by a canonical model structure. In particular, the [[weak equivalences]] in a canonical model structure should be the [[equivalence of categories|category-theoretic equivalences]].

This is to be contrasted with [[Thomason model structure|Thomason model structures]] in which the weak equivalences are the morphisms that induce a [[model structure on simplicial sets|weak homotopy equivalence]] of [[nerve|nerves]]. This amounts to regarding each category, or rather its [[nerve]], as a placeholder for its \emph{groupoidification} (Kan fibrant replacement) and then considering the standard notion of equivalence.

In a canonical model structure for some flavor of $n$-categories, usually

\begin{itemize}%
\item a \textbf{fibration} is a functor that lifts [[equivalence|equivalences]] in all dimensions,
\item an \textbf{acyclic fibration} is a functor which is [[k-surjective functor|k-surjective]] for all $0\le k\le n$,
\item a \textbf{weak equivalence} is a functor which is [[k-surjective functor|essentially k-surjective]] for all $0\le k\le n$, and
\item a \textbf{cofibration} is a functor which is injective on objects and ``relatively free'' on $k$-morphisms for $1\le k \lt n$. These can also be described as the morphisms [[cofibrantly generated model structure|generated]] by the inclusions $\partial G_k \hookrightarrow G_k$ of the boundary of the $k$-[[globe]] into the $k$-[[globe]] for $0\le k \lt \infty$.

\end{itemize}
\hypertarget{internalization}{}\subsection*{{Internalization}}\label{internalization}

A common problem is to transport the (a) model structure on plain $\omega$-categories, i.e. $\omega$-categories [[internal category|internal to]] $Sets$ to another [[internalization|internal context]], notably for the case that $Sets$ is replaced with some kind of category of $Spaces$. This is relevant for the discussion of the homotopy theory of topological and smooth $\omega$-categories.

Usually, such internalization of model structures has the consequence that some properties invoked in the description of the original model structure, notably some of the lifting properties, will only continue to hold ``locally''. One way to deal with this is to pass to a notion slightly weaker than that of a [[homotopy theory|model category]] called a [[category of fibrant objects]] as used in [[homotopical cohomology theory]].

But there are also full model structures for such situations. Notice that under a suitable [[nerve]] operation all [[n-category|n-categories]] usually embed into [[simplicial set]]s. The [[models for infinity-stack (infinity,1)-toposes]] given by the [[model structure on simplicial presheaves]] then serves to present the corresponding $(\infty,1)$-category of parameterized or internal $n$-categories. See for instance also [[smooth infinity-stack]].

\hypertarget{cofibrant_resolutions}{}\subsection*{{Cofibrant resolutions}}\label{cofibrant_resolutions}

In

\begin{itemize}%
\item [[Francois Métayer]], \emph{Cofibrant complexes are free} (\href{http://arxiv.org/abs/math.CT/0701746}{arXiv})

\end{itemize}
it is shown that \emph{cofibrant} $\omega$-categories with respect to the canonical model structure are precisely the ``free'' ones, where ``free'' here means ``generated from a [[polygraph]]'' as described in

\begin{itemize}%
\item [[Francois Métayer]], \emph{Resolutions by [[polygraphs]]} (\href{http://www.tac.mta.ca/tac/volumes/11/7/11-07.pdf}{tac})

\end{itemize}
We had some blog discussion about this at \href{http://golem.ph.utexas.edu/category/2008/10/freely_generated_categories.html}{Freely generated omega-categories}.

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

\begin{itemize}%
\item [[basic localizer]]


\item [[canonical model structure on Cat]]


\item [[canonical model structure on Grpd]]


\item [[canonical model structure on Operad]]


\item [[model structure on strict omega-categories]]


\item [[2-trivial model structure]]



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

The canonical model structure for 1-categories was known to experts for some time before being written down formally (this is the origin of the adjective ``folk'').

The canonical model structure for 1-groupoids is briefly described in section 5 of

\begin{itemize}%
\item D.W. Anderson, \emph{Fibrations and Geometric Relations}, Bulletin of the AMS, Vol. 84: 765-788, 1978.

\end{itemize}
It was apparently first published (for categories internal to a Grothendieck topos) by Joyal and Tierney, \emph{Strong Stacks and Classifying Spaces}, Category theory (Como, 1990) Springer LNM 1488, 213-236.

A more elementary writeup is in

\begin{itemize}%
\item [[Charles Rezk]] (\href{http://www.math.uiuc.edu/~rezk/cat-ho.dvi}{dvi})

\end{itemize}
A general internal version relative to a Grothendieck [[coverage]] can be found in

\begin{itemize}%
\item T. Everaert, R.W. Kieboom, T. Van der Linden, \emph{Model structures for homotopy of internal categories}, Theory and Applications of Categories, Vol. 15, CT2004, No. 3, pp 66-94. (\href{http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html}{tac}).

\end{itemize}
though it seems that the assumptions in this article apply to ambient categories that are [[semiabelian categories]], but do not apply to ambient categories like [[Top]].

A brief summary, together with a generalization when one assumes only weaker versions of the [[axiom of choice]], can be found at \emph{[[folk model structure on Cat]]}.

See also

\begin{itemize}%
\item [[Andre Joyal]], \emph{[[joyalscatlab:Model structures on Cat]]}

\end{itemize}
The canonical model structures for 2-categories and bicategories are due to

\begin{itemize}%
\item [[Steve Lack]],

\emph{A Quillen Model Structure for 2-Categories}, K-Theory 26: 171--205, 2002. (\href{http://web.science.mq.edu.au/~slack/papers/qmc2cat.html}{website})

\emph{A Quillen Model Structure for Bicategories}, K-Theory 33: 185-197, 2004. (\href{http://web.science.mq.edu.au/~slack/papers/qmcbicat.html}{website})



\end{itemize}
and for $n=3$ ([[Gray-categories]]) in

\begin{itemize}%
\item [[Steve Lack]], \emph{A Quillen model structure for Gray-categories} (\href{http://arxiv.org/abs/1001.2366}{arxiv:1001.2366})

\end{itemize}
For $n = \omega$ ([[strict omega-categories]])

\begin{itemize}%
\item Yves Lafont, [[Francois Métayer]], Krzysztof Worytkiewicz, \emph{A folk model structure on omega-cat} (\href{http://arxiv.org/abs/0712.0617}{arXiv})

\end{itemize}
For $n = \omega$ and all morphisms invertible, there is the [[model structure on strict omega-groupoids]]:

\begin{itemize}%
\item [[Ronnie Brown]] and M. Golasinski, \emph{A model structure for the homotopy theory of crossed complexes}, Cah. Top. G\'e{}om. Diff. Cat. 30 (1989) 61-82 (\href{http://www.bangor.ac.uk/~mas010/pdffiles/RB-golskyrev.pdf}{pdf})


\item [[Dimitri Ara]], [[Francois Metayer]], \emph{The Brown-Golasinski model structure on ∞-groupoids revisited} (pdf) Homology, Homotopy Appl. 13 (2011), no. 1, 121--142.



\end{itemize}
[[!redirects folk model structure]] [[!redirects canonical model structures]] [[!redirects folk model structures]]

[[!redirects natural model structure]]



\end{document}