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\begin{document}

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\section*{right/left Kan 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{GrothGrpdFibsAreRightKanFibs}{Motivation: ordinary fibrations in groupoids are right Kan fibrations}\dotfill \pageref*{GrothGrpdFibsAreRightKanFibs} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_lifting_property}{Homotopy lifting property}\dotfill \pageref*{homotopy_lifting_property} \linebreak
\noindent\hyperlink{AsFibrationsInInfinityGroupoids}{As fibrations in $\infty$-groupoids}\dotfill \pageref*{AsFibrationsInInfinityGroupoids} \linebreak
\noindent\hyperlink{PropRightAnodyne}{(Left/)Right anodyne morphisms}\dotfill \pageref*{PropRightAnodyne} \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}

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

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


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



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

\begin{enumerate}%
\item [[Cartesian fibration]]s


\item the special case of left/right (Kan-) [[fibrations of quasi-categories]]



\end{enumerate}
respectively.

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

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 $\Lambda[n]_k \to \Delta[n]$ except possibly the right outer ones: $0 \leq k \lt n$.

It is a \textbf{right fibration} or \textbf{right Kan fibration} if its extends against all horns except possibly the left outer ones: $0 \lt k \leq n$.

So $X \to S$ is a left fibration precisely if for all commuting squares

\begin{displaymath}
\itexarray{
    \Lambda[n]_{k} &\to& X
    \\
    \downarrow &{}^{\exists}\nearrow& \downarrow
    \\
    \Delta[n] &\to& S
  }
\end{displaymath}
for $n \in \mathbb{N}$ and $0 \leq k \lt n$, a diagonal lift exists as indicated.

Morphisms with the [[left lifting property]] against all left/right fibrations are called \textbf{left/right anodyne} 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{GrothGrpdFibsAreRightKanFibs}{}\subsection*{{Motivation: ordinary fibrations in groupoids are right Kan fibrations}}\label{GrothGrpdFibsAreRightKanFibs}

Ordinary [[fibration fibered in groupoids|categories fibered in groupoids]] have a simple characterization in terms of their [[nerve]]s. Let $N : Cat \to sSet$ be the [[nerve]] functor and for $p : E \to B$ a morphism in [[Cat]] (a [[functor]]), let $N(p) : N(E) \to N(B)$ be the corresponding morphism in [[sSet]].

Then

\begin{prop}
\label{}\hypertarget{}{}
The functor $p : E \to B$ is an [[fibration fibered in groupoids|fibration in groupoids]] precisely if the morphism $N(p) : N(E) \to N(B)$ is a right Kan fibration of simplicial sets

\end{prop}
To see this, first notice the following facts:

\begin{lemma}
\label{}\hypertarget{}{}
For $C$ a [[category]], the [[nerve]] $N(C)$ is [[coskeletal|2-coskeletal]]. In particular all $n$-spheres for $n \geq 3$ have unique fillers

\begin{displaymath}
\itexarray{
    \partial \Delta[n] &\stackrel{\forall}{\to}& N(C)
    \\
    \downarrow & \nearrow_{\mathrlap{\exists !}}
    \\
    \Delta[n]
  }
    \;\;\;\;\;
    (n \geq 3)
\end{displaymath}
and (implied by that) all $n$-horns for $n \gt 3$ have fillers

\begin{displaymath}
\itexarray{
    \partial \Lambda[n] &\stackrel{\forall}{\to}& N(C)
    \\
    \downarrow & \nearrow_{\mathrlap{\exists }}
    \\
    \Delta[n]
  }
  \;\;\;\;\;
  (n \gt 3)
  \,.
\end{displaymath}
\end{lemma}
This is discussed at [[nerve]].

\begin{lemma}
\label{}\hypertarget{}{}
If $p : E \to B$ is an ordinary [[functor]], then $N(f) : N(E) \to N(B)$ is an [[inner fibration]], meaning that its has the [[right lifting property]] with respect to all \emph{inner} horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$ for $0 \lt i \lt n$.

\end{lemma}
This is discussed at [[inner fibration]].

\begin{proof}
From the above lemmas it follows that $N(p) : N(E) \to N(B)$ is a right fibration already precisely if it has the right lifting property with respect only to the three horn inclusions

\begin{displaymath}
\{  \Lambda[n]_n \hookrightarrow \Delta[n] | n = 1,2,3\}
  \,.
\end{displaymath}
So we check explicitly what these three conditions amount to

\begin{itemize}%
\item $n=1$ -- The existence of all fillers

\begin{displaymath}
\itexarray{
    \Lambda[1]_1 = \Delta^{\{1\}} &\stackrel{e}{\to}& N(E)
    \\
    \downarrow & {}^{{\hat f}}\nearrow  & \downarrow^{\mathrlap{N(p)}}
    \\
    \Delta^{\{0 \to 1\}} &\stackrel{f}{\to}& N(B)
  }
\end{displaymath}
means that for all objects $e \in E$ and morphism $f : b \to p(e)$ in $B$, there exists a morphism $\hat f : \hat b \to e$ in $E$ such that $p(\hat f) = f$.


\item $n=2$ -- The existence of fillers

\begin{displaymath}
\itexarray{
    \Lambda[2]_2 &\stackrel{e}{\to}& N(E)
    \\
    \downarrow & {}^{{\hat f}}\nearrow  & \downarrow^{\mathrlap{N(p)}}
    \\
    \Delta[2] &\stackrel{f}{\to}& N(B)
  }
\end{displaymath}
means that for all diagrams

\begin{displaymath}
\itexarray{
     && e_1 
     \\
     &&& \searrow^{\mathrlap{\epsilon_{12}}}
     \\
     e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2
  }
\end{displaymath}
in $E$ and commuting triangles

\begin{displaymath}
\itexarray{
     && p(e_1) 
     \\
     & {}^{\mathllap{f}}\nearrow&& \searrow^{\mathrlap{p(\epsilon_{12}})}
     \\
     p(e_0) &&\stackrel{p(\epsilon_{02})}{\to}&& p(e_2)
  }
\end{displaymath}
in $B$, there is a commuting triangle

\begin{displaymath}
\itexarray{
     && e_1 
     \\
     &{}^{\mathllap{\hat f}}\nearrow&& \searrow^{\mathrlap{\epsilon_{12}}}
     \\
     e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2
  }
\end{displaymath}
in $E$, such that $p(\hat f ) = f$.


\item $n=3$ -- \ldots{}

Consider first the case of degenrate 3-simplices on $N(B)$, on 2-simplices as above.

Suppose in the above situation two lifts $(\hat f)_1$ and $(\hat f)_2$ are found. Together these yield a $\Lambda[3]_3$-horn in $N(E)$. The filler condition says this can be filled, which implies that $(\hat f)_1 = (\hat f)_2$.

So the $n=3$-condition implies that the lift whose existence is guaranteed by the $n=2$-condition is unique.

By similar reasoning one sees that this is all the $n=3$-condition yields.



\end{itemize}
In total, these three lifting conditions are precisely those for a Grothendieck fibration in groupoids.

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

\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 fib rations, that for the other follows.

\end{remark}
\hypertarget{homotopy_lifting_property}{}\subsubsection*{{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

\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{AsFibrationsInInfinityGroupoids}{}\subsubsection*{{As fibrations in $\infty$-groupoids}}\label{AsFibrationsInInfinityGroupoids}

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{OverKanComplex}\hypertarget{OverKanComplex}{}
Over a [[Kan complex]] $T$, left fibrations $S \to T$ are automatically [[Kan fibration]]s.

\end{prop}
\begin{proof}
This appears as [[Higher Topos Theory|HTT, prop. 2.1.3.3]].

\end{proof}
As an important special case of this we have

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

\end{corollary}
\begin{proof}
This is originally due to [[Andre Joyal]]. Recalled at [[Higher Topos Theory|HTT, prop. 1.2.5.1]].

\end{proof}
\begin{prop}
\label{PreservationByPullback}\hypertarget{PreservationByPullback}{}
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 propjection $C_{p/} \to C$ is a left fibration.

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

\hypertarget{PropRightAnodyne}{}\subsubsection*{{(Left/)Right anodyne morphisms}}\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{enumerate}%
\item the collection of all left [[horn]] inclusions

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

\begin{enumerate}%
\item the collection of all inclusions of the form

\begin{displaymath}
(\Delta[m] \times \{0\})
  \coprod_{\partial \Delta[m] \times \{0\}}
  (\partial \Delta[m] \times \Delta[1])
  \hookrightarrow
  \Delta[m] \times \Delta[1]
\end{displaymath}

\item the collection of all inclusions of the form

\begin{displaymath}
(S' \times \{0\})
  \coprod_{S \times \{0\}}
  (S \times \Delta[1])
  \hookrightarrow
  S' \times \Delta[1]
\end{displaymath}
for all inclusions of simplicial sets $S \hookrightarrow S'$.



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

\begin{proof}
\ldots{}

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
For $i : A \to A'$ left-anodyne and $j : B \to B'$ a cofibration in the [[model structure for quasi-categories]], the canonical morphism

\begin{displaymath}
(A \times B') \coprod_{A \times B} (A' \times B)
  \to 
  A' \times B'
\end{displaymath}
is left-anodyne.

\end{cor}
This appears as [[Higher Topos Theory|HTT, cor. 2.1.2.7]].

\begin{cor}
\label{}\hypertarget{}{}
For $p : X \to S$ a left fibration and $i : A \to B$ a cofibration of simplicial sets, the canonical morphism

\begin{displaymath}
q : X^B \to X^A \times_{S^A} S^B
\end{displaymath}
is a left fibration. If $i$ is furthermore left anodyne, then it is an acyclic [[Kan fibration]].

\end{cor}
This appears as [[Higher Topos Theory|HTT, cor. 2.1.2.9]].

\begin{prop}
\label{}\hypertarget{}{}
For $f : A_0 \to A$ and $g : B_0 \to B$ two inclusions of [[simplicial set]]s with $f$ left anodyne, we have that the canonical morphism

\begin{displaymath}
(A_0 \star B ) \coprod_{A_0 \star B_0} (A \star B_0)
  \to
  A \star B
\end{displaymath}
into the [[join of simplicial sets]] is left anodyne.

\end{prop}
This is due to [[Andre Joyal]]. It appears as [[Higher Topos Theory|HTT, lemma 2.1.4.2]].

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(restriction of over-quasi-categories along left anodyne inclusions)}

Let $p : B \to S$ be a morphism of [[simplicial set]]s and $i : A \to B$ a left anodyne morphism, then the restriction morphism of [[over quasi-categories|under quasi-categories]]

\begin{displaymath}
S_{/p} \to S_{/p|_A}
\end{displaymath}
is an acyclic [[Kan fibration]].

\end{prop}
This is a special case of what appears as [[Higher Topos Theory|HTT, prop. 2.1.2.5]], which is originally due to [[Andre Joyal]].

\begin{prop}
\label{}\hypertarget{}{}
Let $p : X \to S$ be a morphism of simplicial sets with [[section]] $s : S \to X$. If there is a fiberwise simplicial [[homotopy]] $X \times \Delta[1] \to S$ from $s \circ p$ to $Id_X$ then $s$ is left anodyne.

\end{prop}
This appears as [[Higher Topos Theory|HTT, prop. 2.1.2.11]].

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

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


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

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


\item [[universal right fibration]]



\end{itemize}

\item [[inner fibration]]


\item [[Cartesian fibration]]


\item [[right/left fibration of simplicial spaces]]



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

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


\item [[David Ayala]], [[John Francis]], \emph{Fibrations of $\infty$-Categories} (\href{https://arxiv.org/abs/1702.02681}{arXiv:1702.02681})



\end{itemize}
[[!redirects left fibration]] [[!redirects left fibrations]] [[!redirects left Kan fibration]] [[!redirects left Kan fibrations]]

[[!redirects right fibration]] [[!redirects right fibrations]] [[!redirects right Kan fibration]] [[!redirects right Kan fibrations]]

[[!redirects left anodyne morphism]] [[!redirects right anodyne morphism]]

[[!redirects left anodyne morphisms]] [[!redirects right anodyne morphisms]]

[[!redirects left anodyne map]] [[!redirects right anodyne map]]

[[!redirects left anodyne maps]] [[!redirects right anodyne maps]]

[[!redirects right/left anodyne map]]



\end{document}