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\begin{document}

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\section*{fibrations of quasi-categories}

\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{trivial_fibration}{Trivial fibration}\dotfill \pageref*{trivial_fibration} \linebreak
\noindent\hyperlink{kan_fibration}{Kan fibration}\dotfill \pageref*{kan_fibration} \linebreak
\noindent\hyperlink{leftright_fibration}{(Left/)Right fibration}\dotfill \pageref*{leftright_fibration} \linebreak
\noindent\hyperlink{cocartesian_fibration}{(co)Cartesian fibration}\dotfill \pageref*{cocartesian_fibration} \linebreak
\noindent\hyperlink{categorical_fibration}{Categorical fibration}\dotfill \pageref*{categorical_fibration} \linebreak
\noindent\hyperlink{inner_fibration}{Inner fibration}\dotfill \pageref*{inner_fibration} \linebreak
\noindent\hyperlink{minimal_fibration}{Minimal fibration}\dotfill \pageref*{minimal_fibration} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{trivial_fibration_2}{Trivial fibration}\dotfill \pageref*{trivial_fibration_2} \linebreak
\noindent\hyperlink{kan_fibration_2}{Kan fibration}\dotfill \pageref*{kan_fibration_2} \linebreak
\noindent\hyperlink{leftright_fibration_2}{(Left/)Right fibration}\dotfill \pageref*{leftright_fibration_2} \linebreak
\noindent\hyperlink{homotopy_lifting_property}{Homotopy lifting property}\dotfill \pageref*{homotopy_lifting_property} \linebreak
\noindent\hyperlink{as_fibrations_in_groupoids}{As fibrations in $\infty$-groupoids}\dotfill \pageref*{as_fibrations_in_groupoids} \linebreak
\noindent\hyperlink{PropRightAnodyne}{(Left/)Right anodyne moprphisms}\dotfill \pageref*{PropRightAnodyne} \linebreak
\noindent\hyperlink{cocartesian_fibration_2}{(co)Cartesian fibration}\dotfill \pageref*{cocartesian_fibration_2} \linebreak
\noindent\hyperlink{categorical_fibration_2}{Categorical fibration}\dotfill \pageref*{categorical_fibration_2} \linebreak
\noindent\hyperlink{inner_fibrations}{Inner fibrations}\dotfill \pageref*{inner_fibrations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For ordinary [[categories]] there is the notion of

\begin{itemize}%
\item [[Grothendieck fibration]] between two categories.


\item and the special case of a [[category fibered in groupoids]].



\end{itemize}
The analog of this for [[quasi-categories]] are

\begin{itemize}%
\item [[Cartesian fibrations]]


\item [[left fibrations]]/[[right fibrations]] of quasi-categories.



\end{itemize}
There are more types of fibrations between the [[simplicial sets]] underlying the [[quasi-category]]

\begin{itemize}%
\item [[inner fibrations]] -- these correspond to \textbf{bundles of [[quasi-categories]]} : an inner fibration $E \to \Delta[1]$ over the interval is the quasi-categorical analog of a [[cograph of a profunctor]]: it characterizes the fibers $C, D$ over the endpoints $0,1 \in \Delta[1]$ as [[quasi-categories]]. Notably having an inner fibration $C \to \Delta[0]$ over the point says precisely that $C$ is a [[quasi-category]].


\item categorical fibrations -- these appear as the fibrations in the sense of [[model category]] theory in the Joyal [[model structure for quasi-categories]] $sSet_{Joyal}$ . But they have no particular intrinsic meaning in [[higher category theory]]. In fact, there is also the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]] which is [[Quillen equivalence|Quillen equivalent]] to $sSet_{Joyal}$ and in which the model-theoretic fibrations coincide precisely with the [[Cartesian fibration]]s that do have an intrinsic category theoretic meaning.


\item [[minimal Joyal fibrations]]



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

We list the different definitions in the order of their generality. The examples of each definition are also examples of the following definitions.

All morphisms in the following are morphisms of [[simplicial set]]s.

\hypertarget{trivial_fibration}{}\subsubsection*{{Trivial fibration}}\label{trivial_fibration}

A \textbf{trivial fibration} (trivial [[Kan fibration]]) is a morphism that has the [[right lifting property]] with respect to the boundary inclusions $\partial \Delta[n] \hookrightarrow \Delta[n], n \geq 1$.

\hypertarget{kan_fibration}{}\subsubsection*{{Kan fibration}}\label{kan_fibration}

\begin{itemize}%
\item for the moment see [[Kan fibration]]

\end{itemize}
A morphism with [[left lifting property]] against all Kan fibrations is called \textbf{anodyne}.

\hypertarget{leftright_fibration}{}\subsubsection*{{(Left/)Right fibration}}\label{leftright_fibration}

A [[morphism]] of [[simplicial set]]s $f : X \to S$ is a \textbf{left fibration} or \textbf{[[left Kan fibration]]} if it has the [[right lifting property]] with respect to all [[horn]] inclusions except the right outer ones. It is a \textbf{right fibration} or \textbf{right Kan fibration} if its extends against all horns except the left outer ones.

\begin{displaymath}
\itexarray{
    \Lambda[n]_{k \gt 0} &\to& X
    \\
    \downarrow &{}^{\exists}\nearrow& \downarrow
    \\
    \Delta[n] &\to& S
  } 
  \,.
\end{displaymath}
Morphisms with the [[left lifting property]] against all left/right fibrations are called \textbf{left/right [[anodyne morphism]]} maps.

Write

\begin{displaymath}
RFib(S) \subset sSet/S
\end{displaymath}
for the full [[SSet]]-[[subcategory]] of the [[overcategory]] of [[SSet]] over $S$ on those morphisms that are right fibrations.

This is a [[Kan complex]]-enriched category and as such an incarnation of the \textbf{[[(∞,1)-category]] of right fibrations}. It is modeled by the [[model structure for left fibrations|model structure for right fibrations]]. For details on this see the discussion at [[(∞,1)-Grothendieck construction]].

\hypertarget{cocartesian_fibration}{}\subsubsection*{{(co)Cartesian fibration}}\label{cocartesian_fibration}

A [[Cartesian fibration]] is an [[inner fibration]] $p : X \to S$ such that

\begin{itemize}%
\item for every edge $f : X \to Y$ of $S$


\item and every lift $\tilde{y}$ of $y$ (that is, $p(\tilde{y})=y$),



\end{itemize}
there is a [[cartesian morphism|Cartesian edge]] $\tilde{f} : \tilde{x} \to \tilde{y}$ in $X$ lifting $f$.

(HTT, def 2.4.2.1)

see also

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


\item [[(∞,1)-Grothendieck construction]]



\end{itemize}
\hypertarget{categorical_fibration}{}\subsubsection*{{Categorical fibration}}\label{categorical_fibration}

A \textbf{categorical fibration} is a fibration in the [[model structure for quasi-categories]]: morphism $f : X \to S$ with the [[right lifting property]] against all monomorphic \emph{categorical equivalences} .

(HTT, p. 81).

\hypertarget{inner_fibration}{}\subsubsection*{{Inner fibration}}\label{inner_fibration}

A [[morphism]] of [[simplicial set]]s $f : X \to S$ is an \textbf{inner fibration} or \textbf{[[inner Kan fibration]]} if its has the [[right lifting property]] with respect to all inner [[horn]] inclusions.

The morphisms with the [[left lifting property]] against all inner fibrations are called \textbf{inner anodyne}.

\hypertarget{minimal_fibration}{}\paragraph*{{Minimal fibration}}\label{minimal_fibration}

\begin{itemize}%
\item [[minimal Kan fibration]]


\item [[minimal Joyal fibration]]



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

\begin{remark}
\label{}\hypertarget{}{}
By the [[small object argument]] we have that every morphism $f : X \to Y$ of simplicial sets may be factored as

\begin{displaymath}
f : X \to Z \to Y
\end{displaymath}
with $X \to Z$ a left/right/inner anodyne cofibration and $Z \to Y$ accordingly a left/right/inner Kan fibration.

\end{remark}
\hypertarget{trivial_fibration_2}{}\subsubsection*{{Trivial fibration}}\label{trivial_fibration_2}

\ldots{}

\hypertarget{kan_fibration_2}{}\subsubsection*{{Kan fibration}}\label{kan_fibration_2}

\ldots{}

\hypertarget{leftright_fibration_2}{}\subsubsection*{{(Left/)Right fibration}}\label{leftright_fibration_2}

\begin{remark}
\label{}\hypertarget{}{}
Under the operation of forming the [[opposite quasi-category]], left fibrations turn into right fibrations, and vice versa: if $p : C \to D$ is a left fibration then $p^{op} : C^{op} \to D^{op}$ is a right fibration.

Therefore it is sufficient to list properties of only one type of these fibrations, that for the other follows.

\end{remark}
\hypertarget{homotopy_lifting_property}{}\paragraph*{{Homotopy lifting property}}\label{homotopy_lifting_property}

In classical [[homotopy theory]], a continuous map $p : E \to B$ of [[topological spaces]] is said to have the [[homotopy lifting property]] if it has the [[right lifting property]] with respect to all morphisms $Y \stackrel{(Id, 0)}{\to} Y \times I$ for $I = [0,1]$ the standard [[interval]] and every commuting diagram

\begin{displaymath}
\itexarray{
    Y &\to& E
    \\
    \downarrow && \downarrow
    \\
    Y \times I &\to& B
  }
\end{displaymath}
there exists a lift $\sigma : Y \times I \to E$ making the two triangles

\begin{displaymath}
\itexarray{
    Y &\to& E
    \\
    \downarrow &{}^\sigma\nearrow& \downarrow
    \\
    Y \times I &\to& B
  }
\end{displaymath}
commute. For $Y = *$ the [[point]] this can be rephrased as saying that the universal morphism $E^I \to B^I \times_B E$ induced by the commuting square commuting square

\begin{displaymath}
\itexarray{
    E^I &\to& E
    \\
    \downarrow && \downarrow
    \\
    B^I &\to& B
  }
\end{displaymath}
is an [[epimorphism]]. If it is even an [[isomorphism]] then the lift $\sigma$ exists \emph{uniquely} . This is the situation that the following proposition generalizes:

\begin{prop}
\label{}\hypertarget{}{}
A morphism $p : X \to S$ of simplicial sets is a left fibration precisely if the canonical morphism

\begin{displaymath}
X^{\Delta[1]} \to X^{\{0\}} \times_{S^{\{0\}}} S^{\Delta^1}
\end{displaymath}
is a trivial Kan fibration.

\end{prop}
\begin{proof}
This is a corollary of the characterization of left anodyne morphisms in \hyperlink{PropRightAnodyne}{Properties of left anodyne maps} by [[Andre Joyal]], recalled in [[Higher Topos Theory|HTT, corollary 2.1.2.10]].

\end{proof}
\hypertarget{as_fibrations_in_groupoids}{}\paragraph*{{As fibrations in $\infty$-groupoids}}\label{as_fibrations_in_groupoids}

The notion of right fibration of quasi-categories generalizes the notion of [[category fibered in groupoids]]. This follows from the following properties.

\begin{prop}
\label{}\hypertarget{}{}
For $C \to *$ a right (left) fibration over the [[point]], $C$ is a [[Kan complex]], i.e. an [[∞-groupoid]].

\end{prop}
\begin{proof}
Due to [[Andre Joyal]]. Recalled at [[Higher Topos Theory|HTT, prop. 1.2.5.1]].

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Right (left) fibrations are preserved by [[pullback]] in [[sSet]].

\end{prop}
\begin{corollary}
\label{}\hypertarget{}{}
It follows that the fiber $X_c$ of every right fibration $X \to C$ over every point $c \in C$, i.e. the [[pullback]]

\begin{displaymath}
\itexarray{
    X_c &\to& X
    \\
    \downarrow && \downarrow
    \\
    \{c\} &\to& C
  }
\end{displaymath}
is a [[Kan complex]].

\end{corollary}
\begin{prop}
\label{}\hypertarget{}{}
For $C$ and $D$ quasi-categories that are ordinary [[categories]] (i.e. simplicial sets that are [[nerve]]s of ordinary categories), a morphism $C \to D$ is a right fibration precisely if the correspunding ordinary [[functor]] exhibits $C$ as a [[category fibered in groupoids]] over $D$.

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 2.1.1.3]].

\end{proof}
A canonical class of examples of a [[fibered category]] is the [[codomain fibration]]. This is actually a [[bifibration]]. For an ordinary category, a bifiber of this is just a set. For an $(\infty,1)$-category it is an $\infty$-groupoid. Hence fixing only one fiber of the bifibration should yield a fibration in $\infty$-groupoids. This is asserted by the following statement.

\begin{prop}
\label{}\hypertarget{}{}
Let $p : K \to C$ be an arbitrary morphism to a [[quasi-category]] $C$ and let $C_{p/}$ be the corresponding [[over quasi-category|under quasi-category]]. Then the canonical projection $C_{p/} \to C$ is a left fibration.

\end{prop}
\begin{proof}
Due to [[Andre Joyal]]. Recalled as [[Higher Topos Theory|HTT, prop 2.1.2.2]].

\end{proof}
\hypertarget{PropRightAnodyne}{}\paragraph*{{(Left/)Right anodyne moprphisms}}\label{PropRightAnodyne}

\begin{prop}
\label{}\hypertarget{}{}
The collection of left anodyne morphisms (those with [[left lifting property]] against left fibrations) is equivalently $LAn = LLP(RLP(LAn_0))$ for the following choices of $LAn_0$:

$LAn_0 =$

\begin{itemize}%
\item the collection of all left [[horn]] inclusions

$\{ \Lambda[n]_{i} \to \Delta[n] | 0 \leq i \lt n \}$;


\item blah-blah


\item blah-blah



\end{itemize}
\end{prop}
\begin{proof}
This is due to [[Andre Joyal]], recalled as [[Higher Topos Theory|HTT, prop 2.1.2.6]].

\end{proof}
\hypertarget{cocartesian_fibration_2}{}\subsubsection*{{(co)Cartesian fibration}}\label{cocartesian_fibration_2}

\hypertarget{categorical_fibration_2}{}\subsubsection*{{Categorical fibration}}\label{categorical_fibration_2}

\hypertarget{inner_fibrations}{}\subsubsection*{{Inner fibrations}}\label{inner_fibrations}

\begin{prop}
\label{}\hypertarget{}{}
A [[simplicial set]] $K$ is the [[nerve]] of an ordinary [[category]] $C$, $K \simeq_{iso} N(C)$ precisely if the terminal morphism $K \to \Delta[0]$ is an inner fibration with \emph{unique} inner [[horn]] fillers, i.e. precisely if for all morphisms

\begin{displaymath}
\Lambda[n]_i \to K
\end{displaymath}
with $n \in \mathbb{N}$ and $0 \lt i \lt n$ there is a \emph{unique} morphism $\Delta[n] \to K$ making the diagram

\begin{displaymath}
\itexarray{
    \Lambda[n]_i &\to& K
    \\
    \downarrow & \nearrow
    \\
    \Delta[n]
  }
\end{displaymath}
commute.

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 1.1.2.2]]

\end{proof}
\begin{corollary}
\label{}\hypertarget{}{}
It follows that under the [[nerve]] \emph{every} functor $f : C \to D$ between ordinary [[categories]] is an inner fibration.

\end{corollary}
\begin{proof}
This is immediate, but let's spell it out:

In any commutative diagram

\begin{displaymath}
\itexarray{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow && \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] && \to& N(D)
  }
\end{displaymath}
by the above the bottom morphism is already uniquely specified by the remaining diagram.

\begin{displaymath}
\itexarray{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow && \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] && N(D)
  }
  \,.
\end{displaymath}
By the above there exists a unique lift into $N(C)$

\begin{displaymath}
\itexarray{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] && N(D)
  }
\end{displaymath}
and by uniqueness of lifts into $N(D)$ this must also make the lower square commute

\begin{displaymath}
\itexarray{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] &\to& N(D)
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Jacob Lurie]], chapter 2 of \emph{[[Higher Topos Theory]]}
\item [[David Ayala]] and [[John Francis]], \emph{Fibrations of ∞-categories}, (\href{https://arxiv.org/abs/1702.02681}{arxiv:1702.02681})
\item [[Clark Barwick]], [[Jay Shah]], \emph{Fibrations in ∞-category theory}, (\href{https://arxiv.org/abs/1607.04343}{arXiv:1607.04343})

\end{itemize}
[[!redirects fibration of quasi-categories]] [[!redirects fibrations of quasi-categories]] [[!redirects fibration of simplicial sets]] [[!redirects fibrations of simplicial sets]]

[[!redirects weak fibration]] [[!redirects weak fibrations]]

[[!redirects fibration of quasi-categories]]



\end{document}