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\begin{document}

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\section*{model structure for left fibrations}

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

\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{Motivation}{Motivation}\dotfill \pageref*{Motivation} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{WeakEquivalences}{Weak equivalences}\dotfill \pageref*{WeakEquivalences} \linebreak
\noindent\hyperlink{ChangeOfBase}{Change of base}\dotfill \pageref*{ChangeOfBase} \linebreak
\noindent\hyperlink{GrothendieckConstruction}{Grothendieck construction}\dotfill \pageref*{GrothendieckConstruction} \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}

The \textbf{model structure for right fibrations} $(SSet/S)_{rfib}$ is a [[model category]] structure on the [[overcategory]] $SSet/S$ of [[simplicial set]]s over a given [[quasi-category]] $S$, that [[presentable (∞,1)-category|presents]] the [[(∞,1)-category]] of [[right Kan fibration]]s [[fibrations of quasi-categories|of quasi-categories]] over $S$

\begin{displaymath}
((SSet/S)_{rfib})^\circ \simeq RFib(S)
  \,.
\end{displaymath}
This is the $(\infty,1)$-analog of the $(2,1)$-category $Fib_{grpd}(S)$ of [[category fibered in groupoids|categories fibered in groupoids]] over a category $S$.

Similarly there is an analogous \textbf{model structure for left fibrations} that models [[left Kan fibration]]s, i.e. op-fibrations in [[∞-groupoid]]s

\begin{displaymath}
((SSet/S)_{lfib})^\circ \simeq LFib(S)
  \,.
\end{displaymath}
The extension of this from right fibrations to [[Cartesian fibration]]s and from left fibrations to [[coCartesian fibration]]s is the [[model structure for coCartesian fibrations]].

The [[(∞,1)-Grothendieck construction]] relates this to the [[global model structure on functors]] that presents the [[(∞,1)-category of (∞,1)-functors]] $Fun(S,\infty Grpd)$ (for left fibrations) and $Fun(S^{op},\infty Grpd)$ (for right fibrations).

\hypertarget{Motivation}{}\subsection*{{Motivation}}\label{Motivation}

The following [[model category]] structure is best understood with the [[(∞,1)-Grothendieck construction]] in mind, which it serves to model.

Recall from the discussion there that given a morphism $p : X \to S$ of [[quasi-categories]], the [[(∞,1)-functor]] $S^{op} \to \infty Grpd$ that the left adjoint to the Grothendieck construction extracts from it is all encoded in the [[pushout]] $X^{\triangleleft} \coprod_X S$ in

\begin{displaymath}
\itexarray{
    X &\stackrel{p}{\to}& S
    \\
    \downarrow && \downarrow
    \\
    X^{\triangleleft} &\to& X^{\triangleleft} \coprod_X S
  }
  \,,
\end{displaymath}
where $X^\triangleleft = (*) \star X$ is the [[join of quasi-categories|join]] of $X$ with the point, i.e. $X$ with an [[initial object]] freely adjoined to it.

More discussion of why this is the case is at \href{http://ncatlab.org/nlab/show/Grothendieck+construction#adjunction}{Adjoints to the Grothendieck construction}.

The model category structure described below declares that a morphism in the [[overcategory]] $sSet/S$ is a weak equivalence if it induces a weak equivalence of the quasi-categories given by these pushouts. So this is effectively saying that we regard a morphism of right fibration of quasi-categories as a weak equivalences, if under the left adjoint to the $(\infty,1)$-Grothendieck construction it induces a weak equivalences of the $(\infty,1)$-functors that classify these fibrations.

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

For $f : X \to S$ a morphism of [[simplicial set]]s, write $C^{\triangleleft}(f)$ for the [[pushout]]

\begin{displaymath}
\itexarray{
    X &\hookrightarrow& X^{\triangleleft}
    \\
    \downarrow && \downarrow
    \\
    S &\to& S \coprod_{X} X^{\triangleleft} & =: C^{\triangleleft}(f) 
  }
\end{displaymath}
in the category [[sSet]] of simplicial sets. Call this the \textbf{left cone} over $f$.

The [[colimit]]s in [[sSet]] are computed componentwise, so that the set of vertices $C^{\triangleleft}(f)_0$ is the disjoint union of the vertices of $S$ and one extra vertex $v$, the \textbf{cone point}.

\begin{defn}
\label{}\hypertarget{}{}
([[Higher Topos Theory|HTT, def. 2.1.4.5]])

The \textbf{model structure for left fibrations} or \textbf{covariant model structure} $(sSet/S)_{lfib}$ on $SSet/S$ is given by

A morphism $f : X \to Y$ is

\begin{itemize}%
\item a cofibration if the underlying morphism of simplicial sets is a cofibration in the standard [[model structure on simplicial sets]], i.e. a [[monomorphism]];


\item a weak equivalence if the induced morphism of cones

\begin{displaymath}
X^{\triangleleft} \coprod_X S
  \to 
  Y^{\triangleleft} \coprod_Y S
\end{displaymath}
is a weak equivalence in the Joyal [[model structure for quasi-categories]], where $X^{\triangleleft}$ is the [[join of simplicial sets|join]] $X^{\triangleleft} := {*} \star X$.



\end{itemize}
\end{defn}
\begin{prop}
\label{BasicProperties}\hypertarget{BasicProperties}{}
This is a

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


\item [[simplicial model category|simplicial]]


\item [[combinatorial model category]].



\end{itemize}
([[Higher Topos Theory|HTT, prop 2.1.4.7, 2.1.4.8]])

We have

\begin{itemize}%
\item The fibrant objects are precisely the [[left fibration]]s ([[Higher Topos Theory|HTT, corollary 2.2.3.12]])


\item Every left anodyne morphism is an acyclic cofibration. ([[Higher Topos Theory|HTT, prop 2.1.4.9]])



\end{itemize}
\end{prop}
An alternative characterization of this model structure is:

\begin{theorem}
\label{AsLocalization}\hypertarget{AsLocalization}{}
The model structure for left fibrations is the left [[Bousfield localization]] of the [[model structure on an overcategory|overcategory]] model structure on $SSet/X$ induced by the [[model structure for quasicategories]] on $SSet$ at the set of maps $\{ \Delta^n_0 \hookrightarrow \Delta^n | n\ge 0, \Delta^n \to X \}$ indexed by all the simplices of $X$.

\end{theorem}
This is mentioned in \hyperlink{HeutsMoerdijk13}{Heuts-Moerdijk}, p.5; see also \href{http://nforum.ncatlab.org/discussion/915/model-structure-for-left-fibrations/?Focus=53814#Comment_53814}{this discussion}.

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

\hypertarget{WeakEquivalences}{}\subsubsection*{{Weak equivalences}}\label{WeakEquivalences}

\begin{prop}
\label{}\hypertarget{}{}
Let $S$ be any [[simplicial set]]. Every morphism

\begin{displaymath}
\itexarray{
    X &&\to&& Y
    \\
    & \searrow && \swarrow
    \\
    && S
  }
\end{displaymath}
in $sSet_S$ for which $X\to Y$ is [[left fibration|left anodyne]] is a weak equivalence in the model structure for left fibrations.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 2.1.4.6]].

\begin{proof}
Recall from  that left anodyne morphisms are the weakly saturated class generated by the [[horn]] inclusions (i.e. under [[transfinite composition]] of [[retract]]s of [[pushout]]s). Therefore it is sufficient to check the statement for these generating morphisms.

By the definition of weak equivalences above, this means that we need to check that

\begin{displaymath}
(\Lambda[n]_i)^{\triangleleft} \coprod_{\Lambda[n]_i} S
  \to
  (\Delta[n])^{\triangleleft} \coprod_{\Delta[n]_i} S
\end{displaymath}
is a weak equivalence in $sSet_{Quillen}$.

Observe that this is a [[pushout]]

\begin{displaymath}
\itexarray{
       \Lambda[n+1]_{i+1} &\to& (\Lambda[n]_i)^{\triangleleft} \coprod_{\Lambda^n_i} S
      \\
       \downarrow && \downarrow 
       \\
       \Delta[n+1] &\to& (\Delta[n])^{\triangleleft} \coprod_{\Delta^n} S
   }
\end{displaymath}
of the [[inner fibration|inner anodyne morphism]] $\Lambda[n+1]_{i+1} \to \Delta[n+1]$ and therefore a weak equivalence.

To illustrate the above pushout property set $n = 2$ for example. Start with a 2-simplex $\sigma$ in $S$. Then $(\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S$ is the original simplicial set $S$ together with a tetrahedron $\Delta^3$ built over $\sigma$. One face of the tetrahedron is the original 2-simplex $\sigma$ in $S$, the three others ``stick out'' of $S$:

The simplicial set $(\Lambda^2_1)^{\triangleleft} \coprod_{\Lambda^2_1} S$ is accordingly the simplicial set $S$ with only two of the three faces of this tetrahedron over $\sigma$ erected.

The map $(\Lambda^3_2) \to (\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S$ identifies the horn of this tetrahedron given by these two new faces and the original face $\sigma$.

The pushout therefore glues in the remaining face of the tetrahedron and fills it with a 3-cell.

\end{proof}
\hypertarget{ChangeOfBase}{}\subsubsection*{{Change of base}}\label{ChangeOfBase}

For every morphism $j : S \to S'$ we have the corresponding [[adjunction]] on [[overcategories]]

\begin{displaymath}
(j_! \dashv j^*)
  :
  sSet/S \stackrel{\overset{f_!}{\to}}{\underset{j^*}{\leftarrow}}
  sSet/{S'}
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $j_!$ is given by postcomposition of $j$;


\item $j^*$ is given by [[pullback]] along $j$.



\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(change of base)}

This is a [[Quillen adjunction]] with respect to the model structures for left fibrations over $S$ and $S'$, respectively. ([[Higher Topos Theory|HTT, prop. 2.1.4.10]])

If $j$ is a [[model structure for quasi-categories|weak equivalence in]] $sSet_{Joyal}$, then this is a [[Quillen equivalence]]. ([[Higher Topos Theory|HTT, remark. 2.1.4.11]])

\end{prop}
\hypertarget{GrothendieckConstruction}{}\subsubsection*{{Grothendieck construction}}\label{GrothendieckConstruction}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{($(\infty,1)$-Grothendieck construction)}

For $C$ a [[simplicially enriched category]] and $S = N(C)$ its [[homotopy coherent nerve]], there is a [[Quillen equivalence]]

\begin{displaymath}
(sSet/S)_{lfib} \stackrel{\leftarrow}{\to}
  [C, sSet_{Quillen}]
\end{displaymath}
between the model structure for left fibrations over $S$ and the [[global model structure on functors|global model structure on sSet-functors]] on $C$ with values in [[sSet]] equipped with the standard [[model structure on simplicial sets]].

\end{prop}
For more on this see [[(∞,1)-Grothendieck construction]].

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

The [[operad|operadic]] generalization is the

\begin{itemize}%
\item [[model structure for dendroidal left fibrations]].

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

This is the content of section 2.1.4 of

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

\end{itemize}
There the model structure $(sSet/S)_{lfib}$ is called the \textbf{covariant model structure} and the model structure $(sSet/S)_{rfib}$ the \textbf{contravariant model structure}.

The alternative construction as a localization is mentioned in

\begin{itemize}%
\item [[Gijs Heuts]], [[Ieke Moerdijk]], \emph{Left fibrations and homotopy colimits}. \href{http://dx.doi.org/10.1007/s00209-014-1390-7}{DOI}, (\href{http://arxiv.org/abs/1308.0704}{arXiv:1308.0704})

\end{itemize}
Further discussion is in

\begin{itemize}%
\item [[Danny Stevenson]], \emph{Covariant Model Structures and Simplicial Localization} (\href{http://arxiv.org/abs/1512.04815}{arXiv:1512.04815})

\end{itemize}
[[!redirects model structure for right fibrations]] [[!redirects covariant model structure]] [[!redirects contravariant model structure]]



\end{document}