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\begin{document}

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\section*{Cartesian fibration}

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

\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{general_properties}{General properties}\dotfill \pageref*{general_properties} \linebreak
\noindent\hyperlink{BehaviourUnderPullback}{Behaviour under pullback}\dotfill \pageref*{BehaviourUnderPullback} \linebreak
\noindent\hyperlink{relation_to_other_kinds_of_fibrations}{Relation to other kinds of fibrations}\dotfill \pageref*{relation_to_other_kinds_of_fibrations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{cartesian_fibrations_over_one_morphism_and_adjunctions}{Cartesian fibrations over one morphism and Adjunctions}\dotfill \pageref*{cartesian_fibrations_over_one_morphism_and_adjunctions} \linebreak
\noindent\hyperlink{CartOverSimplex}{Cartesian fibrations over simplices}\dotfill \pageref*{CartOverSimplex} \linebreak
\noindent\hyperlink{the_universal_cartesian_fibration}{The universal Cartesian fibration}\dotfill \pageref*{the_universal_cartesian_fibration} \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}

Cartesian fibrations are one of the types of [[fibrations of quasi-categories]].

A \emph{Cartesian fibration} of quasi-categories -- or more generally of [[simplicial set]]s -- is a morphism that generalizes the notion of [[Grothendieck fibration]] from [[category theory]] to [[(∞,1)-category theory]], specifically with [[(∞,1)-categories]] incarnated as [[quasi-categories]]:

It is an [[inner fibration]] that lifts also all right \emph{outer} [[horn]] inclusions whose last edge is a [[cartesian morphism]], and such that there is a sufficient supply of [[cartesian morphisms]].

An [[inner fibration]] $p : C \to D$ may be thought of as a family of [[(infinity,1)-categories]] $(C_d)_{d \in D}$ which is [[functorial]] in $d$ only in the sense of [[correspondences]]. Then the condition of $p$ being a cartesian fibration ensures that the family is actually [[functorial]]. More precisely, if an [[(∞,1)-functor]] $p : C \to D$ is a Cartesian fibration, then it is possible to interpret its value over any [[morphism]] $f : d_1 \to d_2$ in $D$ as an [[(∞,1)-functor]] $p^{-1}(f) : p^{-1}(d_2) \to p^{-1}(d_1)$ between the [[fibers]] $p^{-1}(d_2)$ and $p^{-1}(d_1)$ over its source and target [[objects]].

This way a Cartesian fibration $p : C \to D$ determines and is determined by an [[(∞,1)-functor]] $D^{op} \to (\infty,1)Cat$ into the [[(∞,1)-category of (∞,1)-categories]]. This is the content of the [[(∞,1)-Grothendieck construction]].

Cartesian fibrations over $S$ are the fibrant objects in the [[model structure on marked simplicial over-sets]] over $S$.

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

\begin{udefn}
A morphism $p : X \to Y$ in [[sSet]] is a \textbf{Cartesian fibration} (or \textbf{Grothendieck fibration}) if

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


\item for every edge $f : x \to y$ of $Y$ and for every lift $\hat {y}$ of $y$ (that is: $p(\hat{y})=y$) there is a [[cartesian morphism|Cartesian edge]] $\hat{f} : \hat{x} \to \hat{y}$ in $X$ lifting $f$ (that is: such that $p(\hat f) = f$)



\end{itemize}
The morphism is a \textbf{cocartesian fibration} (or \textbf{Cartesian opfibration}, \textbf{Grothendieck opfibration}) if the [[opposite quasi-category|opposite]] $p^{op} : X^{op} \to Y^{op}$ is a Cartesian fibration.

\end{udefn}
This is [[Higher Topos Theory|HTT, def. 2.4.2.1]].

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

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

\begin{uprop}
We have:

\begin{itemize}%
\item Every [[isomorphism]] of [[simplicial sets]] is a Cartesian fibration.


\item The composite of two Cartesian fibrations is again a Cartesian fibration.



\end{itemize}
\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 2.4.2.3 ]].

\end{proof}
\begin{uprop}
Every Cartesian fibration is a fibration in the [[Andre Joyal|Joyal]]-[[model structure for quasi-categories]].

\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 3.3.1.7]].

\end{proof}
\hypertarget{BehaviourUnderPullback}{}\subsubsection*{{Behaviour under pullback}}\label{BehaviourUnderPullback}

Since a Cartesian fibration is in particular an [[inner fibration]] and since inner fibrations are stable under [[pullback]] in [[sSet]], it follows that for $p : E \to C$ a Cartesian fibration, the fiber $E_x$ over every point $x \in C_0$ is a [[quasi-category]]

\begin{displaymath}
\itexarray{
    E_x &\to& E
    \\
    \downarrow && \downarrow
    \\
    \{x\} &\to& C
  }
  \,.
\end{displaymath}
The difference between inner fibrations and Cartesian fibrations is that only for Cartesian fibrations is it generally guaranteed that these fibers over the points are \emph{functorially} related over the morphisms in $C$. This is the content of the [[(∞,1)-Grothendieck construction]].

But moreover:

\begin{uprop}
The pullback of a Cartesian fibration in [[sSet]] is again a Cartesian fibration.

\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 2.4.2.3]].

We know from the discussion at [[inner fibration]] that the pullback is an inner fibration. It remains to check if it has enough [[Cartesian morphism]]s. By [[Higher Topos Theory|HTT, prop 2.4.1.3]] we have that in a [[pullback]] diagram

\begin{displaymath}
\itexarray{
    E' &\stackrel{q}{\to}& E
    \\
    {}^{\mathllap{p'}}\downarrow && \downarrow^{\mathrlap{p}}
    \\ 
    C' &\stackrel{k}{\to}& C
  }
\end{displaymath}
a morphism $f \in E'$ is $p'$-Cartesian if $q(f)$ is $p$-Cartesian. Since the morphisms of $E'$ are pairs of morphisms $(\gamma, \hat f) \in C'_1 \times E_1$ and since by assumption $p$ is a Cartesian fibration, there is for $\gamma \in C'_1$ and $y \in E'_0$ such that $p'(y)$ is the target of the morphism $\gamma$ a Cartesian lift $\hat f \in E$ of $k(\gamma)$ such that $q(y)$ is the target of $\hat f$. Hence a Cartesian lift $(\gamma, \hat f)$ of $\gamma$ in $E'$ having $y$ as target.

\end{proof}
We can test locally if a morphism is a Cartesian fibration:

\begin{uprop}
An [[inner fibration]] $p : X \to Y$ is Cartesian precisely if for each $n \leq 2$ and for every $n$-[[simplex]] $\sigma : \Delta[n] \to Y$ the [[sSet]]-[[pullback]] $\sigma^* p : X \times_Y \Delta[n] \to \Delta[n]$ in

\begin{displaymath}
\itexarray{
    X \times_Y \Delta[n] &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \Delta[n] &\stackrel{\sigma}{\to}& Y
  }
\end{displaymath}
is a Cartesian fibration.

\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, cor. 2.4.2.10]].

\end{proof}
\begin{uprop}
The [[pullback]] in [[sSet]] of a weak equivalence in the [[Andre Joyal|Joyal]]-[[model structure for quasi-categories]] along a Cartesian fibration is again a Joyal-weak equivalence

\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop 3.3.1.3]]

\end{proof}
\begin{ulemma}
Equivalences in $sSet_{Joyal}$ are stable under pullback along Cartesian fibrations:

if

\begin{displaymath}
\itexarray{
    X \times_S T &\to & X
    \\
    \downarrow && \downarrow
    \\ 
    T &\stackrel{\simeq}{\to}& S 
  }
\end{displaymath}
is a [[pullback]] square in [[sSet]] with $T \to S$ a weak equivalence in $sSet_{Joyal}$ and $X \to S$ a Cartesian fibration, then $X \times_S T \to X$ is also a weak equivalence.

\end{ulemma}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 3.3.1.3]].

\end{proof}
The following proposition asserts that the ordinary pullback (in [[sSet]]) of Cartesian fibrations already models the correct [[homotopy pullback]].

\begin{uprop}
Let

\begin{displaymath}
\itexarray{
    X &\to& X'
    \\
    \downarrow && \downarrow
    \\
    S &\to & S'
  }
\end{displaymath}
be an [[pullback]] [[diagram]] in [[sSet]] of quasi-categories, where $X' \to S'$ is a Cartesian fibration. Then this is already a [[homotopy pullback]] diagram with respect to the [[model structure for quasi-categories]].

\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop 3.3.1.4]]. We factor the bottom morphism as $S \stackrel{\simeq}{\to} T \to S'$ into a weak equivalence and a fibration in $sSet_{Joyal}$. Then the right square in

\begin{displaymath}
\itexarray{
    X &\to& T \times_{S'} X'& \to& X'
    \\
    \downarrow && \downarrow && \downarrow
    \\
    S &\to & T &\to& S'
  }
\end{displaymath}
is the ordinary pullback over a fibrant replacement of the original diagram hence is a homotopy pullback. The claim follows thus if $X \to T \times_{S'} X'$ is a weak equivalence, which it is by one of the above lemmas.

\end{proof}
\hypertarget{relation_to_other_kinds_of_fibrations}{}\subsubsection*{{Relation to other kinds of fibrations}}\label{relation_to_other_kinds_of_fibrations}

The notion of [[right fibration]] is a special case of that of Cartesian fibration:

\begin{uprop}
The following are equivalent:

\begin{itemize}%
\item $p : X \to Y$ is a Cartesian fibration and every edge in $X$ is $p$-Cartesian


\item $p$ is a [[right fibration]];


\item $p$ is a Cartesian fibration and every fiber is a [[Kan complex]].



\end{itemize}
\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 2.4.2.4]].

\end{proof}
There are also the ``categorical fibrations'', the fibrations in the Joyal [[model structure for quasi-categories]] on $sSet$. These turn out not to have much of an intrinsic category theoretic meaning. By the following proposition one can understand the notion of Cartesian fibration as a suitable refinement of the notion of categorical fibration

\begin{uprop}
Every Cartesian fibration $p : X \to Y$ in [[sSet]] is a fibration with respect to the [[Andre Joyal|Joyal]] [[model structure for quasi-categories]] on [[sSet]].

\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 3.3.1.7]].

\end{proof}
However, when the base is an $\infty$-groupoid, then the difference between Cartesian fibrations and categorical fibrations disappears:

\begin{uprop}
Let $p : X \to Y$ be a morphism of [[simplicial set]]s where $Y$ is a [[Kan complex]]. Then the following are equivalent:

\begin{enumerate}%
\item $p$ is a Cartesian fibration


\item $p$ is a coCartesian fibration


\item $p$ is a [[Joyal model structure on simplicial sets|categorical fibration]]



\end{enumerate}
\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 3.3.1.8]].

\end{proof}
There is however another [[model category]] structure, which does model Cartesian fibrations.

\begin{uprop}
Let $sSet^+/S$ be the [[overcategory]] of the category of [[marked simplicial set]]s over $S$, equipped with the [[model structure on marked simplicial over-sets]].

An object $X \to S$ is fibrant in that model category precisely if

\begin{itemize}%
\item the underlying morphism of simplicial sets $X \to S$ is a Cartesian fibration;


\item the marked edges in $X$ are precisely the [[Cartesian morphism]]s.



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

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{cartesian_fibrations_over_one_morphism_and_adjunctions}{}\subsubsection*{{Cartesian fibrations over one morphism and Adjunctions}}\label{cartesian_fibrations_over_one_morphism_and_adjunctions}

By the [[(infinity,1)-Grothendieck construction]] a Cartesian fibration $K \to \Delta[1]$ corresponds to a morphism $\Delta[1]^{op} \to (\infty,1)Cat$, hence to an [[(infinity,1)-functor]] $f : C \to D$.

Obtaining this through the straightening functor above is tedious, but there is a more immediate way to characterize $f$:

\begin{udef}
For $p : K \to \Delta[1]$ a Cartesian fibration with specified equivalences

\begin{displaymath}
h_0  : C \stackrel{\simeq}{\to} p^{-1}(0)
\end{displaymath}
and

\begin{displaymath}
h_1 : D \stackrel{\simeq}{\to} p^{-1}(1)
\end{displaymath}
an [[(infinity,1)-functor]] $f : D \to C$ is \textbf{associated} to $p$ if there exists a commuting diagram

\begin{displaymath}
\itexarray{
    D \times \Delta[1] &&\stackrel{s}{\to}&& K
    \\
    & \searrow && \swarrow
    \\
    && \Delta[1]
  }
\end{displaymath}
such that

\begin{itemize}%
\item $s|_{D \times \{1\}} = h_1$;


\item $s|_{D \times \{0\}} = h_0 \circ f$


\item for every object $x$ of $D$ we have that $s|_{\{x\} \times \Delta[1]}$ is a p-[[Cartesian morphism]] of $K$.



\end{itemize}
\end{udef}
This is [[Higher Topos Theory|HTT, def. 5.2.1.1]].

If $p : K \to \Delta[1]$ is both a Cartesian fibration as well as a coCartesian fibration, then it determines $(\infty,1)$-functors in both directions

\begin{displaymath}
C
    \stackrel{\leftarrow}{\to} 
  D
  \,.
\end{displaymath}
Such a pair is a pair of [[adjoint (infinity,1)-functor]]s.

\hypertarget{CartOverSimplex}{}\subsubsection*{{Cartesian fibrations over simplices}}\label{CartOverSimplex}

\ldots{} for the moment see [[Higher Topos Theory|HTT, section 3.2.2]] \ldots{}

\hypertarget{the_universal_cartesian_fibration}{}\subsubsection*{{The universal Cartesian fibration}}\label{the_universal_cartesian_fibration}

for the moment see

\begin{itemize}%
\item [[universal fibration of (∞,1)-categories]].

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

\begin{itemize}%
\item [[Kan fibration]], [[anodyne morphism]]


\item [[right/left Kan fibration]], [[right/left anodyne map]]


\item [[inner fibration]]


\item \textbf{Cartesian fibration}, [[Cartesian fibration of dendroidal sets]]


\item [[coCartesian fibration of (∞,1)-operads]]



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

\begin{itemize}%
\item [[Jacob Lurie]], ection 2.4.2 in \emph{[[Higher Topos Theory]]}


\item [[Aaron Mazel-Gee]], \emph{A user's guide to co/cartesian fibrations} (\href{http://arxiv.org/abs/1510.02402}{arXiv:1510.02402})



\end{itemize}
[[!redirects cartesian fibration]] [[!redirects coCartesian fibration]] [[!redirects co-cartesian fibration]] [[!redirects cocartesian fibration]] [[!redirects Cartesian fibrations]] [[!redirects cartesian fibrations]] [[!redirects coCartesian fibrations]] [[!redirects co-cartesian fibrations]] [[!redirects cocartesian fibrations]]



\end{document}