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\begin{document}

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\section*{minimal inner 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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The condition that aa [[quasicategory]] has ``no non-trivial cells'' above degree $n$ (which makes it a particularly strict model of an [[(n,1)-category]]) is not invariant under [[Joyal model structure|categorical equivalence]]. Hence there is no intrinsic characterization of the class of the simplicial sets which are ``(n+1)-[[simplicial skeleton|coskeletal]]'' in this sense.

(Warning: in \hyperlink{Lurie}{Lurie, Def. 2.3.4.1} such an ``(n+1)-coskeletal'' quasi-category is called an ``$n$-category'', but this is not the intrinsic notion of [[(n,1)-category]].)

However there is such a description of the class of quasi-categories which are \emph{equivalent} to such $(n+1)$-coskeletal quasicategories. To make this more concrete the notion of a \emph{minimal inner fibration} can be used (a quasi-categorical analog of [[minimal Kan fibrations]], see also at \emph{[[minimal fibration]]}). This is an [[inner fibration]] of simplicial sets satisfying a relative homotopy condition and that of a \emph{minimal quasi-category} .

Every quasi-category is equivalent to a minimal quasi-category.

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

\begin{defn}
\label{homotopic}\hypertarget{homotopic}{}
Let

\begin{displaymath}
\itexarray{
A&\stackrel{u}{\to}&X
\\
\downarrow^i&&\downarrow^p
\\
B&\stackrel{v}{\to}&S}
\end{displaymath}
denote a lifting problem. Then putative solutions $f,g$ of this lifting problem are called \emph{homotopic relative $A$ over $S$} if they are equivalent as objects in the fiber of the map

\begin{displaymath}
X^B\to X^A\times_{S^A}S^B
\end{displaymath}
Equivalently $f,g$ are homotopic relative $A$ over $B$ if there is a map

\begin{displaymath}
F:B\times \Delta[1]\to X
\end{displaymath}
such that

$F|B\times\{0\}=f$

$F|B\times\{1\}=g$

$p\circ F=v\circ \pi_B$

$F\circ(i\times id_{\Delta[1]})=u\circ\pi_A$

$F|\{b\}\times \Delta[1]$

and $F|\{b\}\times\Delta[1]$ is an equivalence in the $(\infty,1)$-category $X_{v(b)}$ for every vertex $b$ of $B$.

\end{defn}
\begin{udefn}
Let $p : X \to S$ be an inner fibration of simplicial sets. $p$ is called \emph{minimal inner fibration} if $f = f^\prime$ for every pair of maps $f , f ^\prime : \Delta[n] \to X$ which are homotopic relative to $\partial \Delta[n]$ over $S$ .

An $(\infty,1)$-category $C$ is called \emph{minimal $(\infty,1)$-category} if $C\to *$ is minimal.

\end{udefn}
\begin{uprop}
Let $C$ be an $(\infty,1)$-category and let $n\ge -1$. The the following statements are equivalent:

\begin{enumerate}%
\item There exists a minimal model $C^\prime\subseteq C$ such that $C^\prime$ is an $(n+1)$-coskeletal quasi-category.


\item There exists a categorical equivalence $D\to C$, where $D$ is an $(n+1)$-coskeletal quasi-category.


\item For every pair of objects $X,Y\in C$, the mapping space $Map_C(X,Y)\in H$ is $(n-1)$-truncated.



\end{enumerate}
\end{uprop}
\begin{ucor}
Let $X$ be a [[Kan complex]]. Then is is equivalent to an $(n+1)$-coskeletal quasi-category iff it is $n$[[truncation|-truncated]].

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

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


\item [[inner fibration]], [[left fibration]], [[right fibration]]



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

Section 2.3.3 and section 2.3.4 of

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

\end{itemize}
[[!redirects minimal Joyal fibration]] [[!redirects minimal Joyal fibrations]]

[[!redirects minimal inner fibrations]]



\end{document}