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\begin{document}

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\section*{model structure for Cartesian 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{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{marked_simplicial_sets}{Marked simplicial sets}\dotfill \pageref*{marked_simplicial_sets} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{as_a_quasitopos}{As a quasi-topos}\dotfill \pageref*{as_a_quasitopos} \linebreak
\noindent\hyperlink{CartesianClosure}{Cartesian closure}\dotfill \pageref*{CartesianClosure} \linebreak
\noindent\hyperlink{model_structure_on_marked_simplicial_sets}{Model structure on marked simplicial sets}\dotfill \pageref*{model_structure_on_marked_simplicial_sets} \linebreak
\noindent\hyperlink{cartesian_weak_equivalences}{Cartesian weak equivalences}\dotfill \pageref*{cartesian_weak_equivalences} \linebreak
\noindent\hyperlink{the_model_structure}{The model structure}\dotfill \pageref*{the_model_structure} \linebreak
\noindent\hyperlink{MarkedAnodyne}{Marked anodyne morphisms}\dotfill \pageref*{MarkedAnodyne} \linebreak
\noindent\hyperlink{as_a_model_for_the_category_of_categories}{As a model for the $(\infty,1)$-category of $(\infty,1)$-categories}\dotfill \pageref*{as_a_model_for_the_category_of_categories} \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 [[model category]] structure on the [[category]] $SSet^+/S$ of [[marked simplicial set]]s over a given [[simplicial set]] $S$ is a [[presentable (infinity,1)-category|presentation]] for the [[(∞,1)-category]] of [[Cartesian fibration]]s over $S$. Every object is cofibrant and the fibrant objects of $SSet^+/S$ are precisely the [[Cartesian fibration]]s over $S$.

Notably for $S = {*}$ this is a presentation of the [[(∞,1)-category of (∞,1)-categories]]: as a plain [[model category]] this is [[Quillen equivalence|Quillen equivalent]] to the [[model structure for quasi-categories]], but it is indeed an $sSet_{Quillen}$-[[enriched model category]] (i.e. enriched over the ordinary [[model structure on simplicial sets]] that models [[∞-groupoid]]s).

The $(\infty,1)$-categorical [[Grothendieck construction]] that exhibits the correspondence between [[Cartesian fibration]]s and [[(∞,1)-presheaf|(∞,1)-presheaves]] is in turn modeled by a [[Quillen equivalence]] between the model structure on marked simplicial over-sets and the projective [[global model structure on simplicial presheaves]].

\hypertarget{marked_simplicial_sets}{}\subsection*{{Marked simplicial sets}}\label{marked_simplicial_sets}

Marked simplicial sets are [[simplicial set]]s with a little bit of extra [[stuff, structure, property|structure]]: a marking that remembers which edges are supposed to be [[Cartesian morphism]]s.

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

\hypertarget{in_components}{}\paragraph*{{In components}}\label{in_components}

\begin{udef}
A \textbf{marked simplicial set} is

\begin{itemize}%
\item a pair $(S,E)$ consisting of

\begin{itemize}%
\item a [[simplicial set]] $S$


\item and a subset $E \subset S_1$ of edges of $S$, called the \emph{marked edges},



\end{itemize}

\item such that

\begin{itemize}%
\item all degenerate edges are marked edges.

\end{itemize}


\end{itemize}
A morphism $(S,E) \to (S',E')$ of marked simplicial sets is a morphism $f : S \to S'$ of [[simplicial set]]s that carries marked edges to marked edges in that $f(E) \subset E'$.

\end{udef}
\begin{udef}
\begin{itemize}%
\item The category of marked simplicial sets is denoted $sSet^+$.


\item for $S$ a [[simplicial set]] let

\begin{itemize}%
\item $S^\flat$ or $S^{min}$ be the minimally marked simplicial set: only the degenerate edges are marked;


\item $S^\sharp$ or $S^{max}$ be the maximally marked simplicial set: every edge is marked.



\end{itemize}

\item for $p : X \to S$ a [[Cartesian fibration]] of [[simplicial set]]s let

\begin{itemize}%
\item $X^\natural$ or $X^{cart}$ be the cartesian marked simplicial set: precisely the $p$-[[cartesian morphism]]s are marked

\end{itemize}


\end{itemize}
\end{udef}
\hypertarget{as_a_quasitopos}{}\paragraph*{{As a quasi-topos}}\label{as_a_quasitopos}

Simple as the above definition is, for seeing some of its properties it is useful to think of $sSet^+$ in a more abstract way.

\begin{udef}
Let $\Delta^+$ be the [[category]] defined as the [[simplex category]] $\Delta$, but with one more object $[1^+]$ that factors the unique morphism $[1] \to [0]$ in $\Delta$

\begin{displaymath}
\itexarray{
    [0] &\stackrel{\to}{\to}& [1]
    \\
    {}^{\mathllap{=}}\downarrow &\swarrow& \downarrow^{\mathrlap{p}}
    \\
    [0] &\leftarrow& [1^+]  
}
  \,.
\end{displaymath}
Equip this category with a [[coverage]] whose only non-trivial covering family is $\{p : [1] \to [1]^+\}$.

\end{udef}
\begin{ulemma}
The category $sSet^+$ is the [[quasi-topos]] of [[separated presheaves]] on $\Delta^+$:

\begin{displaymath}
sSet^+ \simeq SepPSh(\Delta^+)
  \,.
\end{displaymath}
\end{ulemma}
\begin{proof}
A [[presheaf]] $X : (\Delta^+)^{op} \to Set$ is separated precisely if the morphism

\begin{displaymath}
X(p) : X_{1^+} \to X_1
\end{displaymath}
is a [[monomorphism]], hence if $X_{1^+}$ is a subset of $X_1$. By functoriality this subset contains all the degenerate 1-cells

\begin{displaymath}
\itexarray{
      X_0
      \\
      \downarrow & \searrow^{\mathrlap{\sigma}}
      \\
      X_{1^+} &\hookrightarrow& X_1
  }
  \,.
\end{displaymath}
Therefore we may naturally identify $X$ as a simplicial set equipped with a subset of $X_1$ that contains all degenerate 1-cells.

Moreover, a morphism of separated preseheaves on $\Delta^+$ is by definition just a [[natural transformation]] between them, which means it is under this interpretation precisely a morphism of simplicial sets that respects the marked 1-simplices.

\end{proof}
Notice that $sSet^+$ is a genuine quasi-topos:

\begin{ulemma}
$sSet^+$ is not a [[topos]].

\end{ulemma}
\begin{proof}
The canonical morphisms $X^\flat \to X^\sharp$ are [[monomorphism]]s and [[epimorphism]]s, but not [[isomorphism]]s. Therefore $sSet^+$ is not a [[balanced category]], hence cannot be a topos.

\end{proof}
\hypertarget{CartesianClosure}{}\subsubsection*{{Cartesian closure}}\label{CartesianClosure}

\begin{ulemma}
The category $sSet^+$ is a [[cartesian closed category]].

\end{ulemma}
\begin{proof}
This is an immediate consequence of the above \hyperlink{Quasitopos}{observation} that $sSet^+$ is a [[quasitopos]]. But it is useful to spell out the Cartesian closure in detail.

By the general logic of the [[closed monoidal structure on presheaves]] we have that $PSh(\Delta^+)$ is cartesian closed. It remains to check that if $X,Y \in PSh(\Delta^+)$ are marked simplicial sets in that $X_{1^+} \to X_1$ is a monomorphism and similarly for $Y$, that then also $Y^X$ has this property.

We find that the marked edges of $Y^X$ are

\begin{displaymath}
(Y^X)_{1^+} \simeq
  Hom_{PSh(\Delta^+)}([1^+], Y^X) \simeq Hom_{PSh(\Delta^+)}([1^+] \times X, Y)
\end{displaymath}
and the morphism $(Y^X)_{1^+} \to (Y^X)_1$ sends $X \times [1^+] \stackrel{\eta}{\to} Y$ to

\begin{displaymath}
X \times [1] \stackrel{(Id,p)}{\to} X \times [1^+]
  \stackrel{\eta}{\to} Y
  \,.
\end{displaymath}
Now, by construction, every non-identity morphism $U \to [1^+]$ in $\Delta^+$ factors through $U \to [1]$, which implies that if the components of $p^* \eta_1$ and $p^* \eta_2$ coincide on $U \neq [1^+]$, then already the components of $\eta_1$ and $\eta_2$ on $U$ coincided. By assumption on $X$ the values of $\eta_1$ and $\eta_2$ on $U = [1^+]$ are already fixed, due to the inclusion $X_{1^+} \times [1^+]_{1^+} \hookrightarrow X_{1} \times [1^+]_{1}$. Hence $p^*$ is injective, and so $Y^X$ formed in $PSh(\Delta^+)$ is itself a marked simplicial set.

\end{proof}
\begin{udef}
\begin{itemize}%
\item For $X$ and $Y$ marked simplicial sets let

\begin{itemize}%
\item $Map^\flat(X,Y)$ be the [[simplicial set]] underlying the [[cartesian closed category|cartesian]] [[internal hom]] $Y^X \in sSet^+$


\item $Map^\sharp(X,Y)$ the simplicial set consisting of all simplices $\sigma \in Map^\flat(X,Y)$ such that every edge of $\sigma$ is a marked edge of $Y^X$.



\end{itemize}


\end{itemize}
\end{udef}
\begin{ucorolary}
These mapping complexes are characterized by the fact that we have natural bijections

\begin{displaymath}
Hom_{sSet}(K, Map^\flat(X,Y)) 
  \simeq
  Hom_{sSet^+}(K^\flat, Y^X)
  \simeq
  Hom_{sSet^+}(K^\flat \times X, Y)
\end{displaymath}
and

\begin{displaymath}
Hom_{sSet}(K, Map^\sharp(X,Y)) 
   \simeq 
  Hom_{sSet^+}(K^\sharp, Y^X)
   \simeq
  Hom_{sSet^+}(K^\sharp \times X, Y)
\end{displaymath}
for $K \in sSet$ and $X,Y \in sSet^+$. In particular

\begin{displaymath}
Map^\flat(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\flat, Y)
\end{displaymath}
and

\begin{displaymath}
Map^\sharp(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\sharp, Y)
  \,.
\end{displaymath}
\end{ucorolary}
In words we have

\begin{itemize}%
\item The $n$-simplices of the internal hom $Y^X$ are simplicial maps $X \times \Delta[n] \rightarrow Y$ such that when you restrict $X_1 \times \Delta[n]_1 \rightarrow Y_1$ to $E \times \Delta[n]_0$ (where $E$ is the set of marked edges of $X$), this morphism factors through the marked edges of $Y$.


\item The marked edges of $Y^X$ are those simplicial maps $X \times \Delta[1] \rightarrow Y$ such that the restriction of $X_1 \times \Delta[1]_1 \rightarrow Y_1$ to $E \times \Delta[1]_1$ factors though the marked edges of $Y$. In the presence of the previous condition, this says that when you apply the homotopy $X \times \Delta[1] \rightarrow Y$ to a marked edge of $X$ paired with the identity at $[1]$, the result should be marked.



\end{itemize}
\begin{udef}
We generalize all this notation from $sSet^+$ to the [[overcategory]] $sSet^+/S := sSet^+/(S^\sharp)$ for any given (plain) simplicial set $S$, by declaring

\begin{displaymath}
Map_S^\flat(X,Y) \subset Map^\flat(X,Y)
\end{displaymath}
and

\begin{displaymath}
Map_S^\sharp(X,Y) \subset Map^\sharp(X,Y)
\end{displaymath}
to be the subcomplexes spanned by the cells that respect that map to the base $S$.

\end{udef}
\begin{ulemma}
Let $Y \to S$ be a [[Cartesian fibration]] of simplicial sets, and $X^\natural$ as above the marked simplicial set with precisely the [[Cartesian morphism]]s marked.

Then

\begin{itemize}%
\item $Map_S^\flat(X,Y^\natural)$ is an [[quasi-category]];


\item $Map_S^\sharp(X, Y^\natural)$ is its [[core]], the maximal [[Kan complex]] inside it.



\end{itemize}
\end{ulemma}
This is [[Higher Topos Theory|HTT, remark 3.1.3.1]].

\begin{proof}
The $n$-cells of $Map_S^\flat(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\flat \to Y^\natural$ over $S$. This means that for fixed $x \in X_0$, $\Delta[n]$ maps into a [[fiber]] of $Y\to S$. But fibers of Cartesian fibrations are fibers of [[inner fibration]]s, hence are quasi-categories.

Similarly, the $n$-cells of $Map_S^\sharp(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\sharp \to Y^\natural$ over $S$. Again for fixed $x \in X_0$, $\Delta[n]$ maps into a [[fiber]] of $Y\to S$, but now only hitting Cartesian edges there. But (as discussed at [[Cartesian morphism]]), an edge over a point is Cartesian precisely if it is an equivalence.

\end{proof}
\begin{uprop}
We have a sequence of [[adjoint functor]]s

\begin{displaymath}
(-)^{\flat} \dashv (-)_{\flat} \dashv (-)^{\sharp} \dashv (-)_{\sharp}
  :
  \itexarray{ 
    & \stackrel{(-)^{\flat}}{\to} & 
    \\ 
    & \stackrel{(-)_{\flat}}{\leftarrow} & 
    \\ 
    sSet & \stackrel{(-)^{\sharp}}{\to} & sSet^+ 
    \\ 
    & \stackrel{(-)_{\sharp}}{\leftarrow} & 
  }
\end{displaymath}
\end{uprop}
\hypertarget{model_structure_on_marked_simplicial_sets}{}\subsection*{{Model structure on marked simplicial sets}}\label{model_structure_on_marked_simplicial_sets}

\hypertarget{cartesian_weak_equivalences}{}\subsubsection*{{Cartesian weak equivalences}}\label{cartesian_weak_equivalences}

Observe that weak equivalences in the [[model structure for quasi-categories]] may be characterized as follows.

\begin{ulemma}
A morphism $f : C \to D$ between simplicial sets that are [[quasi-categories]] is a weak equivalence in the [[model structure for quasi-categories]] precisely if the following equivalent coditions hold:

\begin{itemize}%
\item For every simplicial set $K$, the morphism $sSet(K,f) : sSet(K,C) \to sSet(K,D)$ is a weak equivalence in the model structure for quasi-categories.


\item For every simplicial set $K$, the morphism $Core(sSet(K,f)) : Core(sSet(K,C)) \to Core(sSet(K,D))$ on the [[core]]s, the maximal [[Kan complex]]es inside, is a weak equivalence in the standard [[model structure on simplicial sets]], hence a [[homotopy equivalence]].



\end{itemize}
\end{ulemma}
\begin{proof}
This is [[Higher Topos Theory|HTT, lemma 3.1.3.2]].

\end{proof}
This may be taken as motivation for the following definition.

\begin{udef}
For every [[Cartesian fibration]] $Z \to S$, we have that

\begin{displaymath}
Map_S^\flat(X, Z^{\natural})
\end{displaymath}
is a [[quasi-category]] and

\begin{displaymath}
Map_S^\sharp(X,Z^\natural) = Core(Map_S^\flat(X,Z^\natural))
\end{displaymath}
is the maximal Kan complex inside it.

A morphism $p : X \to Y$ in $sSet^+/S$ is a \textbf{Cartesian equivalence} if for every Cartesian fibration $Z$ we have

\begin{itemize}%
\item The induced morphism $Map_S^\flat(Y,Z^{\natural}) \to Map_S^\flat(X,Z^{\natural})$ is an [[equivalence of quasi-categories]];

\end{itemize}
Or equivalently:

\begin{itemize}%
\item The induced morphism $Map_S^\sharp(Y,Z^{\natural}) \to Map_S^\sharp(X,Z^{\natural})$ is a [[model structure on simplicial sets|weak equivalence of Kan complexes]].

\end{itemize}
\end{udef}
This is [[Higher Topos Theory|HTT, prop. 3.1.3.3]] with [[Higher Topos Theory|HTT, remark 3.1.3.1]].

\begin{uprop}
Let

\begin{displaymath}
\itexarray{
    X &&\stackrel{p}{\to}&& Y
    \\
    & \searrow && \swarrow
    \\
    && S
  }
\end{displaymath}
be a morphism in $sSet/S$ such that both vertical maps to $S$ are Cartesian fibrations. Then the following are equivalent:

\begin{itemize}%
\item $p$ is a [[homotopy equivalence]].


\item The induced morphism $X^\natural \to Y^\natural$ in $sSet^+/S$ is a Cartesian equivalence.


\item The induced morphism on each fiber $X_s \to Y_{p(s)}$ is a weak equivalence in the [[model structure for quasi-categories]].



\end{itemize}
\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, lemma 3.1.3.5]].

\end{proof}
\hypertarget{the_model_structure}{}\subsubsection*{{The model structure}}\label{the_model_structure}

The \textbf{model structure on marked simplicial over-sets} $Set^+/S$ over $S \in SSet$ -- also called the \textbf{Cartesian model structure} since it models [[Cartesian fibration]]s -- is defined as follows.

\begin{udef}
\textbf{(Cartesian model structure on $sSet^+/S$)}

The category $SSet^+/S$ of [[marked simplicial set]]s over a marked simplicial set $S$ carries a structure of a [[proper model category|proper]] [[combinatorial simplicial model category]] defined as follows.

The [[SSet]]-[[enriched category|enrichment]] is given by

\begin{displaymath}
sSet^+/S(X,Y) := Map_S^\sharp(X,Y)
  \,.
\end{displaymath}
A morphism $f : X \to X'$ in $SSet^+/S$ of [[marked simplicial set]]s is

\begin{itemize}%
\item a \emph{cofibration} precisely if underlying morphism of [[simplicial set]]s is a cofibration in the standard [[model structure on simplicial sets]] (i.e. a [[monomorphism]]).

(def. 3.1.2.2 of [[Higher Topos Theory|HTT]])


\item a weak equivalences precisely if it is a Cartesian equivalence, as defined above.



\end{itemize}
\end{udef}
\begin{proof}
The model structure is proposition 3.1.3.7 in [[Higher Topos Theory|HTT]]. The simplicial enrichment is corollary 3.1.4.4.

\end{proof}
\begin{uremark}
Using $Map_S^\flat(X,Y)$ for the mapping objects makes $sSet^+/S$ a $sSet_{Joyal}$-[[enriched model category]] (i.e. enriched in the [[model structure for quasi-categories]]). This is [[Higher Topos Theory|HTT, remark 3.1.4.5]].

\end{uremark}
Notice that trivially every object in this model structure is cofibrant. The following proposition shows that the above model structure indeed presents the $(\infty,1)$-category $CartFib(S)$ of [[Cartesian fibration]]s.

\begin{prop}
\label{FibrantObjects}\hypertarget{FibrantObjects}{}
An object $p : X \to S$ in $sSet^+/S$ is \emph{fibrant} with respect to the above model structure precisely if it is isomorphic to an object of the form $Y^\natural$, for $Y \to S$ a [[Cartesian fibration]] in [[sSet]].

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 3.1.4.1]].

\end{proof}
In particular, the fibrant objects of $sSet^+ \cong sSet^+/*$ are precisely the quasicategories in which the marked edges are precisely the [[equivalence in a quasi-category|equivalences]]. Note that the Cartesian model structure on $sSet^+/S$ is \emph{not} the [[model structure on an over category]] induced on $sSet^+/S$ from the Cartesian model structure on $sSet^+$!

\begin{udef}
\textbf{(coCartesian model structure on $sSet^+/S$)}

There is another such model structure, with [[Cartesian fibration]]s replaced everywhere by \textbf{coCartesian fibrations.}

\end{udef}
\hypertarget{MarkedAnodyne}{}\subsubsection*{{Marked anodyne morphisms}}\label{MarkedAnodyne}

A class of morphisms with [[left lifting property]] again some class of fibrations is usually called \emph{anodyne} . For instance a left/right/inner anodyne morphism of simplicial sets is one that has the left lifting property against all [[left fibration|left/right fibration]]s or [[inner fibration]]s, respectively.

The class of \emph{marked anodyne} morphisms in $sSet^+$ as defined in the following is something that comes close to having the left lifting property against all Cartesian fibrations. It does not quite, but is still useful for various purposes.

\begin{udefn}
The collection of \textbf{marked anodyne morphisms} in $SSet^+/S$ is the class of morphisms $An^+ = LLP(RLP(An^+_0))$ where the generating set $An^+_0$ consists of

\begin{itemize}%
\item for $0 \lt i \lt n$ the minimally marked [[horn]] inclusions

\begin{displaymath}
(\Lambda[n]_i)^\flat \to \Delta[n]^\flat
\end{displaymath}

\item for $i = n$ the horn inclusion with the last edge marked:

\begin{displaymath}
(\Lambda[n]_n, \mathcal{E} \cap (\Lambda[n]_n)_1)
  \to 
  (\Delta[n], \mathcal{E} )    
  \,,
\end{displaymath}
where $\mathcal{E}$ is the union of all degenerate edges in $\Delta[n]$ together with the edge $\Delta^{\{n-1,n\}} \to \Delta[n]$.


\item the inclusion

\begin{displaymath}
(\Lambda[2]_1)^\sharp \coprod_{(\Lambda[2]_1)^\flat}
  (\Delta[2])^\flat
  \to
  (\Delta[2])^\sharp
  \,.
\end{displaymath}

\item for every [[Kan complex]] $K$ the morphism $K^\flat \to K^\sharp$.



\end{itemize}
\end{udefn}
The crucial property of marked anodyne morphisms is the following characterization of morphisms that have the right lifting property with respect to them.

\begin{uprop}
A morphism $p : X \to S$ in $SSet^+$ has the [[right lifting property]] with respect to the class $An^+$ of marked anodyne maps precisely if

\begin{enumerate}%
\item $p$ is an [[inner fibration]]


\item an edge $e$ of $X$ is marked precisely if it is a $p$-[[Cartesian morphism]] and $p(e)$ is marked in $S$


\item for every object $y$ of $X$ and every marked edge $\bar e : \bar x \to p(y)$ in $S$ there exists a marked edge $e : x \to y$ of $X$ with $p(e) = \bar e$.



\end{enumerate}
\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 3.1.1.6]]

\end{proof}
\begin{uremark}
Thus, if $(X, E_X) \to (S,E_S)$ is a morphism in $sSet^+$ with RLP against marked anodyne morphisms, then its underlying morphism $X\to S$ in $sSet$ is almost a [[Cartesian fibration]]: it may fail to be such only due to missing markings in $E_S$.

However, if \emph{all} morphisms in $S$ are marked, then $(X,E_X) \to S^\sharp$ has the RLP against marked anodyne morphisms precisely when the underlying morphism $X\to S$ is a [[Cartesian fibration]] and exactly the [[Cartesian morphism]]s are marked in $X$, $(X,E_X) = X^\natural$ --- in other words, precisely if it is a fibrant object in the model structure on $sSet^+/S$.

\end{uremark}
See also [[Higher Topos Theory|HTT, remark. 3.1.1.11]].

The following stability property of marked anodyne morphisms is important in applications. Recall that a cofibration in $sSet^+$ is a morphism whose underlying morphism in [[sSet]] is a [[monomorphism]].

\begin{uprop}
\textbf{(stability under smash product with cofibrations)}

Marked anodyne morphisms are stable under ``[[smash product]]'' with cofibrations:

for $f : X \to X'$ marked anodyne, and $g : Y \to Y'$ a cofibration, the induced morphism

\begin{displaymath}
(X \times Y') \coprod_{X \times Y} (X' \times Y)
  \to
  X' \times Y'
\end{displaymath}
out of the [[pushout]] in $sSet^+$ is marked anodyne.

\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 3.1.2.3]].

\end{proof}
\hypertarget{as_a_model_for_the_category_of_categories}{}\subsubsection*{{As a model for the $(\infty,1)$-category of $(\infty,1)$-categories}}\label{as_a_model_for_the_category_of_categories}

The Joyal [[model structure for quasi-categories]] $sSet_{Joyal}$ is an [[enriched category]] enriched over itself. So it is \emph{not} a [[simplicial model category]] in the standard sense, which means $sSet_{Quillen}$-enriched.

Indeed, the full sSet-enriched subcategory $(sSet_{Joyal})^\circ$ on fibrant-cofibrant objects is a model for the [[(∞,2)-category]] [[(∞,1)Cat]] of [[(∞,1)-categories]]. For many applications it is more convenient to work just with the [[(∞,1)-category of (∞,1)-categories]] inside that, obtained by taking in each [[hom-object]] [[quasi-category]] the maximal [[Kan complex]].

The resulting [[(∞,1)-category]] should have a presentation by a [[simplicial model category]]. And the model structure on marked simplicial sets does accomplish this.

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

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

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

Marked simplicial sets are introduced in section 3.1 of

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

\end{itemize}
The model structure on marked simplicial oversets is described in section 3.1.3

[[!redirects marked simplicial set]] [[!redirects marked simplicial sets]] [[!redirects model structure for coCartesian fibrations]] [[!redirects model structure on marked simplicial over-sets]] [[!redirects model structure on marked simplicial oversets]]

[[!redirects model structure on marked simplicial sets]]



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