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\begin{document}

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\section*{model structure for quasi-categories}

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

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

[[!include model category theory - contents]]

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

[[!include quasi-category theory contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_a_cisinski_model_structure}{As a Cisinski model structure}\dotfill \pageref*{as_a_cisinski_model_structure} \linebreak
\noindent\hyperlink{GeneralProperties}{General properties}\dotfill \pageref*{GeneralProperties} \linebreak
\noindent\hyperlink{RelToInfGrpds}{Relation to the model structure for $\infty$-groupoids}\dotfill \pageref*{RelToInfGrpds} \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 [[quasi-category]] is a [[simplicial set]] satisfying [[weak Kan complex|weak Kan filler conditions]] that make it behave like the [[nerve]] of an [[(∞,1)-category]].

There is a [[model category]] structure on the category [[SSet]] -- the \textbf{[[Andre Joyal|Joyal]] model structure} or \textbf{model structure for quasi-categories} -- such that the fibrant objects are precisely the quasi-categories and the weak equivalences precisely the correct \emph{categorical equivalences} that generalize the notion of [[equivalence of categories]].

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

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{model structure for quasi-categories} or \textbf{Joyal model structure} $sSet_{Joyal}$ on [[sSet]] has

\begin{itemize}%
\item [[cofibrations]] are the [[monomorphisms]]


\item [[weak equivalences]] are those maps that are taken by the [[left adjoint]] of the [[homotopy coherent nerve]] to a weak equivalence in the [[model structure on simplicial categories]].



\end{itemize}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_a_cisinski_model_structure}{}\subsubsection*{{As a Cisinski model structure}}\label{as_a_cisinski_model_structure}

The model structure for quasi-categories is the [[Cisinski model structure]] on [[sSet]] whose class of weak equivalences is the localizer generated by the [[spine]] inclusions $\{Sp^n \hookrightarrow \Delta^n\}$. See (\hyperlink{Ara}{Ara}).

\hypertarget{GeneralProperties}{}\subsubsection*{{General properties}}\label{GeneralProperties}

\begin{prop}
\label{}\hypertarget{}{}
The model structure for quasi-categories is

\begin{itemize}%
\item a [[left proper model category]],


\item a [[combinatorial model category]].



\end{itemize}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
It is also a [[monoidal model category]] with respect to cartesian product and thus is naturally an [[enriched model category]] over itself, hence is $sSet_{Joyal}$-enriched (reflecting the fact that it tends to present an [[(infinity,2)-category]]). It is however \emph{not} $sSet_{Quillen}$-enriched and thus not a ``[[simplicial model category]]'' with respect to this enrichment.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $p \colon \mathcal{C} \to \mathcal{D}$ a morphism of [[simplicial sets]] such that $\mathcal{D}$ is a [[quasi-category]]. Then $p$ is a fibration in $sSet_{Joyal}$ precisely if

\begin{enumerate}%
\item it is an [[inner fibration]];


\item it is an ``[[isofibration]]'': for every [[equivalence in an (∞,1)-category|equivalence]] in $\mathcal{D}$ and a lift of its [[domain]] through $p$, there is also a lift of the whole equivalence through $p$ to an equivalence in $\mathcal{C}$.



\end{enumerate}
\end{prop}
This is due to [[Joyal]]. (\hyperlink{Lurie}{Lurie, cor. 2.4.6.5}).

So every [[fibration]] in $sSet_{Joyal}$ is an [[inner fibration]], but the converse is in general false. A notably exception are the fibrations to the point:

\begin{prop}
\label{}\hypertarget{}{}
The [[fibrant objects]] in $sSet_{Joyal}$ are precisely those that are [[inner fibration|inner fibrant]] over the point, hence those simplicial sets which are [[quasi-categories]].

\end{prop}
(\hyperlink{Lurie}{Lurie, theorem 2.4.6.1})

\hypertarget{RelToInfGrpds}{}\subsubsection*{{Relation to the model structure for $\infty$-groupoids}}\label{RelToInfGrpds}

The inclusion of [[(∞,1)-category|(∞,1)-catgeories]] [[∞Grpd]] $\stackrel{i}{\hookrightarrow}$ [[(∞,1)Cat]] has a left and a right [[adjoint (∞,1)-functor]]

\begin{displaymath}
(grpdfy \dashv i \dashv Core)
  \;\;
   :
  \;\;
  (\infty,1)Cat
  \stackrel{\overset{grpdfy}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\overset{Core}{\to}}}  
  \infty Grpd
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $Core$ is the operation of taking the [[core]], the maximal $\infty$-groupoid inside an $(\infty,1)$-category;


\item $grpdfy$ is the operation of \emph{groupoidification} that freely generates an $\infty$-groupoid on a given $(\infty,1)$-category



\end{itemize}
(see [[Higher Topos Theory|HTT, around remark 1.2.5.4]])

The adjunction $(grpdfy \dashv i)$ is modeled by the [[Bousfield localization of model categories|left Bousfield localization]]

\begin{displaymath}
(Id \dashv Id) \; :\;  sSet_{Joyal} \stackrel{\leftarrow}{\to}
  sSet_{Quillen}
  \,.
\end{displaymath}
Notice that the left [[derived functor]] $\mathbb{L} Id : (sSet_{Joyal})^\circ \to (sSet_{Quillen})^\circ$ takes a fibrant object on the left -- a [[quasi-category]] -- then does nothing to it but regarding it now as an object in $sSet_{Quillen}$ and then producing its fibrant replacement there, which is [[Kan fibrant replacement]]. This is indeed the operation of \emph{groupoidification} .

The other adjunction is given by the following

\begin{prop}
\label{}\hypertarget{}{}
There is a [[Quillen adjunction]]

\begin{displaymath}
(k_! \dashv k^!) \;\; : sSet_{Quillen}
  \stackrel{\overset{k^!}{\leftarrow}}{\overset{k_!}{\to}}
   sSet_{Joyal}
\end{displaymath}
which arises as [[nerve and realization]] for the [[cosimplicial object]]

\begin{displaymath}
k : \Delta \to sSet : [n] \mapsto \Delta'[n]
  \,,
\end{displaymath}
where $\Delta^'[n] = N(\{0 \stackrel{\simeq}{\to} 1 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n\})$ is the [[nerve]] of the [[groupoid]] freely generated from the linear [[quiver]] $[n]$.

This means that for $X \in SSet$ we have

\begin{itemize}%
\item $k^!(X)_n  = Hom_{sSet}(\Delta'[n],X)$.


\item and $k_!(X)_n = \int^{[k]} X_k \cdot \Delta'[k]$.



\end{itemize}
\end{prop}
This is (\hyperlink{JoyalTierney}{JoTi, prop 1.19})

The following proposition shows that $(k_! \dashv k^!)$ is indeed a model for $(i \dashv Core)$:

\begin{prop}
\label{}\hypertarget{}{}
\begin{itemize}%
\item For any $X \in sSet$ the canonical morphism $X \to k_!(X)$ is an acyclic cofibration in $sSet_{Quillen}$;


\item for $X \in sSet$ a [[quasi-category]], the canonical morphism $k^!(X) \to Core(X)$ is an acyclic fibration in $sSet_{Quillen}$.



\end{itemize}
\end{prop}
This is (\hyperlink{JoyalTierney}{JoTi, prop 1.20})

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

A similar model for [[(∞,n)-categories]] is discussed at

\begin{itemize}%
\item [[model structure on cellular sets]]

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

The original construction of the Joyal model structure is in

\begin{itemize}%
\item [[Andre Joyal]], \emph{Theory of quasi-categories I}

\end{itemize}
Unfortunately, this is still not publicly available, but see the lecture notes

\begin{itemize}%
\item [[Andre Joyal]], \emph{The theory of quasi-categories and its applications}, (\href{http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf}{pdf})

\end{itemize}
or the construction of the model structure in Cisinski's book

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Higher categories and homotopical algebra}, (\href{http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf}{pdf})

\end{itemize}
which closely follows Joyal's original construction.

A proof that proceeds via [[homotopy coherent nerve]] and [[simplicially enriched categories]] is given in detail following theorem 2.2.5.1 in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
The relation to the model structure for [[complete Segal space]]s is in

\begin{itemize}%
\item [[Andre Joyal]], [[Myles Tierney]], \emph{Quasi-categories vs. Segal spaces} (\href{http://arxiv.org/abs/math/0607820}{arXiv})

\end{itemize}
Discussion with an eye towards [[Cisinski model structures]] and the [[model structure on cellular sets]] is in

\begin{itemize}%
\item [[Dimitri Ara]], \emph{Higher quasi-categories vs higher Rezk spaces} (\href{http://arxiv.org/abs/1206.4354}{arXiv})

\end{itemize}
See also

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Alg\`e{}bre homotopique et cat\'e{}gories sup\'e{}rieures}, course notes, 2009, \href{http://www.math.univ-toulouse.fr/~dcisinsk/coursinftycat.pdf}{pdf}.

\end{itemize}
A model structure for [[(infinity,2)-sheaves]] with values in quasicategories is discussed in

\begin{itemize}%
\item [[Nicholas Meadows]], \emph{The Local Joyal Model Structure} (\href{http://arxiv.org/abs/1507.08723}{arXiv:1507.08723})

\end{itemize}
[[!redirects Joyal model structure]] [[!redirects Joyal model structure on simplicial sets]] [[!redirects model structure on quasicategories]] [[!redirects model structure on quasi-categories]] [[!redirects model structure for quasicategories]]



\end{document}