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\begin{document}

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\section*{Kan fibration}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{illustration}{Illustration}\dotfill \pageref*{illustration} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{minimal_kan_fibration}{Minimal Kan fibration}\dotfill \pageref*{minimal_kan_fibration} \linebreak
\noindent\hyperlink{quasifibration}{Quasi-fibration}\dotfill \pageref*{quasifibration} \linebreak
\noindent\hyperlink{left_and_right_kan_fibration}{Left and right Kan fibration}\dotfill \pageref*{left_and_right_kan_fibration} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{acyclic_kan_fibrations_and_weak_homotopy_equivalences}{Acyclic Kan fibrations and weak homotopy equivalences}\dotfill \pageref*{acyclic_kan_fibrations_and_weak_homotopy_equivalences} \linebreak
\noindent\hyperlink{pullback_and_homotopy_pullback}{Pullback and homotopy pullback}\dotfill \pageref*{pullback_and_homotopy_pullback} \linebreak
\noindent\hyperlink{on_nerves_of_groupoids}{On nerves of groupoids}\dotfill \pageref*{on_nerves_of_groupoids} \linebreak
\noindent\hyperlink{universal_kan_fibration}{Universal Kan fibration}\dotfill \pageref*{universal_kan_fibration} \linebreak
\noindent\hyperlink{relation_to_other_concepts}{Relation to other concepts}\dotfill \pageref*{relation_to_other_concepts} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A Kan fibration is one of the notions of [[fibrations of simplicial sets]].

A \emph{Kan fibration} is a [[morphism]] $\pi : Y \to X$ of [[simplicial set|simplicial sets]] with the [[weak factorization system|lifting property]] for all [[horn]] inclusions.

This means that for

\begin{displaymath}
\itexarray{
   \Lambda^k[n] &\to& Y
   \\
   \downarrow && \downarrow^\pi
   \\
   \Delta^n &\to& X
 }
\end{displaymath}
a commuting square with $n\ge 1$ and $0\le k\le n$, there always exists a lift

\begin{displaymath}
\itexarray{
   \Lambda^k[n] &\to& Y
   \\
   \downarrow &\nearrow& \downarrow^\pi
   \\
   \Delta^n &\to& X
 }
 \,.
\end{displaymath}
In terms of the canonical [[powering]] of simplicial sets over sets, this is equivalent to the morphisms

\begin{displaymath}
Y^{\Delta[n]}
  \to 
  Y^{\Lambda^k[n]} \times_{X^{\Lambda^k[n]}} X^{\Delta^k[n]}
\end{displaymath}
all being [[epimorphisms]]. (Here, for instance, $Y^{\Lambda^k[n]}$ is the set of tuples of $(n-1)$-cells in $Y$ that glue along their boundaries to an image of the $k$th $n$-[[horn]].)

\hypertarget{illustration}{}\subsection*{{Illustration}}\label{illustration}

Kan fibrations are combinatorial analogs of [[Serre fibration|Serre]] [[fibrations]] of [[topological spaces]]. In fact, under the [[Quillen equivalence]] of the standard [[model structure on topological spaces]] and the standard [[model structure on simplicial sets]], Kan fibrations map to Serre fibrations.

Recall the shape of the [[horns]] in low dimension.

\begin{itemize}%
\item -\textbf{$n=1$}- The horns $\Lambda^1_0$ and $\Lambda^1_1$ of the 1-[[simplex]] are just copies of the 0-[[simplex]] $\Delta^0$ regarded as the left and right endpoint of $\Delta^1$. For $n= 1$ the above condition says that for $\pi : Y \to X$ a Kan fibration we have

\begin{displaymath}
\itexarray{
     Y &\ni &
     y
     \\
    \downarrow^\pi
     \\
     X &\ni& \pi(y) &\stackrel{\forall f}{\to}& x
  }
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;  
  \itexarray{
     Y &\ni&
     y &\stackrel{\exists \hat f}{\to}& \exists \hat x
     \\
     \downarrow^\pi
     \\
     X &\ni& \pi(y) &\stackrel{f = \pi(\hat f)}{\to}& x = \pi(x)
  }
\end{displaymath}
corresponding to the lifting diagram

\begin{displaymath}
\itexarray{
   \Lambda_1^1 &\stackrel{y}{\to}& Y
   \\
   \downarrow &{}^{\hat f}\nearrow& \downarrow^\pi
   \\
   \Delta^1 &\stackrel{f}{\to}& X
 }
 \,.
\end{displaymath}

\item -\textbf{$n=2$}- the horn $\Lambda^2_1$ consists of the two top sides of a triangle. For this the Kan condition says that for any two composable 1-cells in $Y$ that have a ``composite up to a 2-cell'' in $X$, there exists a corresponding ``composite up to a 2-cell'' in $Y$ that projects down to the one in $X$:

\begin{displaymath}
\itexarray{
  &&&&& y_2
  \\
  &&&& \nearrow && \searrow
  \\
  Y &\ni& & y_1 &&&& y_3
  \\
  \downarrow^\pi 
  \\
  X &\ni& &&& \pi(y_2)
  \\
  &&& & \nearrow &\Downarrow^{\forall h}& \searrow
  \\
  &&& \pi(y_1) &&\to&& \pi(y_2)
}
\;\;\;\;\;\;
\Rightarrow
\;\;\;\;\;\;
\itexarray{
  &&&&& y_2
  \\
  &&&& \nearrow &\Downarrow^{\exists \hat h}& \searrow
  \\
  Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3
  \\
  \downarrow^\pi 
  \\
  X &\ni& &&& \pi(y_2)
  \\
  &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow
  \\
  &&& \pi(y_1) &&\to&& \pi(y_2)
}
\end{displaymath}
This corresponds to the lifting diagram

\begin{displaymath}
\itexarray{
   \Lambda_2^1 &\stackrel{y}{\to}& Y
   \\
   \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi
   \\
   \Delta^2 &\stackrel{h}{\to}& X
 }
 \,.
\end{displaymath}
\begin{itemize}%
\item Crucial is this condition for the \emph{outer horns} $\Lambda^n_0$ and $\Lambda^n_n$, where it says that the above works not only when edges are composable, but also when they meet just at their sources or their targets. For instance for the horn $\Lambda^2_2$ the picture is\begin{displaymath}
\itexarray{
  &&&&& y_2
  \\
  &&&& && \searrow
  \\
  Y &\ni& & y_1 &&\to&& y_3
  \\
  \downarrow^\pi 
  \\
  X &\ni& &&& \pi(y_2)
  \\
  &&& & \nearrow &\Downarrow^{\forall h}& \searrow
  \\
  &&& \pi(y_1) &&\to&& \pi(y_2)
}
\;\;\;\;\;\;
\Rightarrow  
\;\;\;\;\;\;
\itexarray{
  &&&&& y_2
  \\
  &&&& {}^\exists\nearrow &\Downarrow^{\exists \hat h}
  &    \searrow
  \\
  Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3
  \\
  \downarrow^\pi 
  \\
  X &\ni& &&& \pi(y_2)
  \\
  &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow
  \\
  &&& \pi(y_1) &&\to&& \pi(y_2)
}
\end{displaymath}
This corresponds to the lifting diagram

\begin{displaymath}
\itexarray{
   \Lambda_2^2 &\stackrel{y}{\to}& Y
   \\
   \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi
   \\
   \Delta^2 &\stackrel{h}{\to}& X
 }
 \,.
\end{displaymath}


\end{itemize}


\end{itemize}
\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

\hypertarget{minimal_kan_fibration}{}\subsubsection*{{Minimal Kan fibration}}\label{minimal_kan_fibration}

A Kan fibration $p : E \to B$ is called a \textbf{[[minimal Kan fibration]]} if for all cells $x,y : \Delta[n] \to E$ the condition $p(x) = p(y)$ and $\partial_i x = \partial_i y$ implies for all $k$ that $\partial_k x = \partial_k y$.

\hypertarget{quasifibration}{}\subsubsection*{{Quasi-fibration}}\label{quasifibration}

A \textbf{quasi-fibration} or \textbf{weak Kan fibration} or \textbf{inner Kan fibration} of simplicial sets is defined as above, but with the lifting property only imposed in \emph{inner horns}: $\Lambda^n_k$ with $0 \lt k \lt (n-1)$, not the \emph{outer horns} $\Lambda^n_0$ and $\Lambda^n_n$.

This weakened condition then says that \emph{composition} of cells may be lifted through the quasi-fibration, but not necessarily [[inverse|inversion]] of 1-cells. See [[fibrations of quasi-categories]] for more details.

\hypertarget{left_and_right_kan_fibration}{}\subsubsection*{{Left and right Kan fibration}}\label{left_and_right_kan_fibration}

Similarly, a \textbf{left Kan fibration} is one that has the lifting property for all horns except possibly the last one. and a \textbf{right Kan fibration} is one that has the lifting property for all horns except possibly the first one. See [[fibrations of quasi-categories]] for more details.

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

\hypertarget{acyclic_kan_fibrations_and_weak_homotopy_equivalences}{}\subsubsection*{{Acyclic Kan fibrations and weak homotopy equivalences}}\label{acyclic_kan_fibrations_and_weak_homotopy_equivalences}

\begin{theorem}
\label{}\hypertarget{}{}
The [[acyclic Kan fibrations]] morphisms $f : X \to Y$ of Kan complexes that are both Kan fibrations as well as [[model structure on simplicial sets|weak equivalences]] in that they induce isomorphisms on all [[simplicial homotopy groups]] (i.e. the \textbf{[[acyclic fibrations]]} of Kan complexes) are precisely the morphisms that have the [[weak factorization system|right lifting property]] with respect to all [[boundary of a simplex|boundary inclusions]] $\partial \Delta^n \hookrightarrow \Delta^n$:

\begin{displaymath}
\itexarray{
      \partial \Delta[n] &\to& X
      \\
      \downarrow &{}^\exists\nearrow& \downarrow^f 
      \\
      \Delta[n] &\to& Y
    }
    \,.
\end{displaymath}
\end{theorem}
e.g. (\hyperlink{GoerssJardine96}{Goerss-Jardine, chapter I})

\begin{cor}
\label{}\hypertarget{}{}
Kan fibrations and acyclic Kan fibrations are both stable under [[pullback]].

\end{cor}
\begin{proof}
Because every class of morphisms defined by a [[weak factorization system|right lifting property]] is stable under pullback.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
From this it follows readily that [[Kan complexes]] form a Brownian [[category of fibrant objects]].

\end{remark}
\begin{prop}
\label{AcyclicFibrationsDetectedFiberwise}\hypertarget{AcyclicFibrationsDetectedFiberwise}{}
A Kan fibration $f\colon X\to Y$ is acyclic precisely if the [[fiber]] $f^{-1}(y)$ over each vertex $y$ is [[contractible]].

\end{prop}
Purely combinatorial proofs of this statement include (\hyperlink{Joyal}{Joyal, prop. 8.23}, \hyperlink{RiehlVerity13}{Riehl-Verity 13, lemma 5.4.16})

\hypertarget{pullback_and_homotopy_pullback}{}\subsubsection*{{Pullback and homotopy pullback}}\label{pullback_and_homotopy_pullback}

\begin{lemma}
\label{PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence}\hypertarget{PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence}{}
Let $p \colon X \longrightarrow Y$ be a [[Kan fibration]], def. \ref{KanFibration}, and let $f_1,f_2 \colon A \longrightarrow X$ be two morphisms. If there is a [[left homotopy]] $f_1 \rightarrow f_2$ between these maps, then there is a fiberwise [[homotopy equivalence]], between the [[pullback]] fibrations $f_1^\ast X \simeq f_2^\ast X$.

\end{lemma}
(e.g. \hyperlink{GoerssJardine96}{Goerss-Jardine 96, chapter I, lemma 10.6})

See also at \emph{[[homotopy pullback]]}.

\hypertarget{on_nerves_of_groupoids}{}\subsubsection*{{On nerves of groupoids}}\label{on_nerves_of_groupoids}

\begin{theorem}
\label{}\hypertarget{}{}
A [[functor]] $F \colon C \to D$ between [[groupoids]] is [[k-surjective functor|k-surjective]] for all $k$ and hence a surjective [[equivalence of categories]] precisely if under the [[nerve]] $N(F) : N(C) \to N(D)$ it induces an acyclic fibration of Kan complexes;

\end{theorem}
\begin{proof}
We know that both $N(C)$ and $N(D)$ are Kan complexes. By the above theorem it suffices to show that $N(f)$ being a surjective equivalence is the same as having all lifts

\begin{displaymath}
\itexarray{
    \delta \Delta[n] &\to& N(C)
    \\
    \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    \Delta[n] &\to& N(D)
  }
  \,.
\end{displaymath}
We check successively what this means for increasing $n$:

\begin{itemize}%
\item $n= 0$. In degree 0 the boundary inclusion is that of the empty set into the [[nLab:point|point]] $\emptyset \hookrightarrow {*}$. The lifting property in this case amounts to saying that every point in $N(D)$ lifts through $N(F)$.

\begin{displaymath}
\itexarray{
    \emptyset &\to& N(C)
    \\
    \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    {*} &\to& N(D)
  }
  \Leftrightarrow
  \itexarray{
    && N(C)
    \\
    &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    {*} &\to& N(D)
  }
  \,.
\end{displaymath}
This precisely says that $N(F)$ is surjective on 0-cells and hence that $F$ is surjective on objects.


\item $n=1$. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval $\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}$. The lifting property here evidently is equivalent to saying that for all objects $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ are hit. Hence that $F$ is a [[nLab:full functor|full functor]].


\item $n=2$. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map $\partial \Delta[2] \to N(C)$ sends the top left edge of the triangle to an identity. Then the lifting property here evidently is equivalent to saying that for all objects $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a [[nLab:faithful functor|faithful functor]].

\begin{displaymath}
\left(
  \itexarray{
    && a 
    \\
    & {}^{Id_a}\nearrow && \searrow^{f}
    \\
    a &&\stackrel{g}{\to}&& b
  }
  \right)
  \stackrel{N(F)}{\mapsto}
  \left(
  \itexarray{
    && a 
    \\
    & {}^{Id_a}\nearrow &\Downarrow^=& \searrow^{F(f)}
    \\
    a &&\stackrel{F(g)}{\to}&& b
  }
  \right)
\end{displaymath}


\end{itemize}
\end{proof}
\hypertarget{universal_kan_fibration}{}\subsubsection*{{Universal Kan fibration}}\label{universal_kan_fibration}

See at \emph{[[universal Kan fibration]]}.

\hypertarget{relation_to_other_concepts}{}\subsection*{{Relation to other concepts}}\label{relation_to_other_concepts}

\begin{itemize}%
\item Kan fibrations and quasi-fibrations are fibrations in two common [[model structure on simplicial sets|model structures on simplicial sets]].


\item Recall that the [[horn]] $\Lambda^k[n]$ is the boundary of the $n$-[[simplex]] $\Delta^n$ with one face removed. If in the above definition one replaces horns with the full boundaries of simplices, one obtaines the definition of a [[hypercover]], the acyclic fibrations in the classical [[model structure on simplicial sets]].


\item A simplicial set $X$ for which the unique morphism $X \to pt$ to the [[terminal object|terminal simplicial set]] is a Kan fibration is called a [[Kan complex]].


\item A simplicial set $X$ for which the unique morphism $X \to pt$ to the [[terminal object|terminal simplicial set]] is a quasi-fibration/weak Kan fibration is called a [[quasi-category]].


\item Just as the underlying simplicial set of a [[simplicial group]] is a [[Kan complex]] (see algorithm at [[simplicial group]]), so also given any simplicial morphism $f : G\to H$ of simplicial groups for which in each dimension, $n$, the homomorphism $f_n : G_n \to H_n$ is an [[epimorphism]], then the underlying simplicial map of simplicial sets is a Kan fibration. (Apart from a careful choice of section in each dimension, the proof can be constructed from the algorithm given in [[simplicial group]].)


\item A morphism of simplicial sets that has the left [[lifting property]] with respect to all Kan fibrations is called an [[anodyne morphism]].



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

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

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

\end{itemize}

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


\item [[inner fibration]]


\item [[Cartesian fibration]]


\item [[simplicial homotopy theory]]



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

A standard textbook account is

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], \emph{[[Simplicial homotopy theory]]}, 1996

\end{itemize}
That [[geometric realization]] takes Kan fibrations to [[Serre fibrations]] is due to

\begin{itemize}%
\item [[Dan Quillen]], \emph{The geometric realization of a Kan fibration is a Serre fibration, Proc. AMS 19 (1968), 1499--1500}

\end{itemize}
See also

\begin{itemize}%
\item [[Andre Joyal]] \emph{Theory of Quasi-Categories and its applications} \href{http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf}{pdf}


\item [[Emily Riehl]], [[Dominic Verity]], \emph{Homotopy coherent adjunctions and the formal theory of monads}, (\href{http://arxiv.org/abs/1310.8279}{arXiv:1310.8279})



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



\end{document}