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\begin{document}

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\section*{universal fibration of (infinity,1)-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{for_categories}{For $(\infty,1)$-categories}\dotfill \pageref*{for_categories} \linebreak
\noindent\hyperlink{RestInfGrpd}{For $\infty$-Groupoids}\dotfill \pageref*{RestInfGrpd} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{universal fibration of [[(infinity,1)-category|(∞,1)-categories]] is the [[generalized universal bundle]] of $(\infty,1)$-categories in that it is [[Cartesian fibration]]}

\begin{displaymath}
p : Z \to (\infty,1)Cat^{op}
\end{displaymath}
over the [[opposite category]] of the [[(∞,1)-category of (∞,1)-categories]] such that

\begin{itemize}%
\item its fiber $p^{-1}(C)$ over $C \in (\infty,1)Cat$ is just the $(\infty,1)$-category $C$ itself;


\item every [[Cartesian fibration]] $p : C \to D$ arises as the [[pullback]] of the universal fibration along an [[(∞,1)-functor]] $S_p : D \to (\infty,1)Cat^{op}$.



\end{itemize}
Recall from the discussion at [[generalized universal bundle]] and at [[stuff, structure, property]] that for [[n-category|n-categories]] at least for low $n$ the corresponding universal object was the $n$-category $n Cat_*$ of [[pointed object|pointed]] $n$-categories. $Z$ should at least morally be $(\infty,1)Cat_*$.

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

\hypertarget{for_categories}{}\subsubsection*{{For $(\infty,1)$-categories}}\label{for_categories}

\ldots{}see section 3.3.2 of [[Higher Topos Theory|HTT]]\ldots{}

\hypertarget{RestInfGrpd}{}\subsubsection*{{For $\infty$-Groupoids}}\label{RestInfGrpd}

\begin{defn}
\label{}\hypertarget{}{}
The universal fibration of $(\infty,1)$-categories restricts to a [[Cartesian fibration]] $Z|_{\infty Grpd} \to \infty Grpd^{op}$ over [[∞Grpd]] by [[pullback]] along the inclusion morphism $\infty Grpd \hookrightarrow (\infty,1)Cat$

\begin{displaymath}
\itexarray{
    Z|_{\infty Grpd} &\to& Z
    \\
    \downarrow && \downarrow
    \\
    \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op}
  }
  \,.
\end{displaymath}
\end{defn}
This is the \emph{[[universal Kan fibration]]}.

\begin{remark}
\label{}\hypertarget{}{}
The [[∞-functor]] $Z|_{\infty Grpd} \to \infty Grpd^{op}$ is even a [[right fibration]] and it is the \emph{universal right fibration}. In fact it is (when restricted to small objects) the [[object classifier]] in the [[(∞,1)-topos]] [[∞Grpd]], see at \href{%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos#ObjectClassifierInInfinityGroupoid}{object classifier -- In ∞Grpd}.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The following are equivalent:

\begin{itemize}%
\item An [[∞-functor]] $p : C \to D$ is a [[right Kan fibration]].


\item Every functor $S_p : D \to (\infty,1)Cat$ that classifies $p$ as a [[Cartesian fibration]] factors through [[∞-Grpd]].


\item There is a functor $G_p : D \to \infty Grpd$ that classifies $p$ as a [[right Kan fibration]].



\end{itemize}
\end{prop}
\begin{proof}
This is proposition 3.3.2.5 in [[Higher Topos Theory|HTT]].

\end{proof}
\hypertarget{models}{}\subsection*{{Models}}\label{models}

For concretely constructing the relation between [[Cartesian fibration]]s $p : E \to C$ of [[(infinity,1)-categories|(∞,1)-categories]] and [[(∞,1)-functor]]s $F_p : C \to (\infty,1)Cat$ one may use a [[Quillen equivalence]] between suitable [[model category|model categories]] of [[marked simplicial set]]s.

For $C$ an [[(∞,1)-category]] regarded as a [[quasi-category]] (i.e. as a [[simplicial set]] with certain properties), the two model categories in question are

\begin{itemize}%
\item the projective [[global model structure on simplicial presheaves]] on $[C,SSet]$ -- this models the [[(∞,1)-category of (∞,1)-functors]] $(\infty,1)Func(C,(\infty,1)Cat)$.


\item the \emph{covariant model structure} on the [[over category]] $SSet/C$ -- this models the $(\infty,1)$-category of [[Cartesian fibration]]s over $C$.



\end{itemize}
The [[Quillen equivalence]] between these is established by the [[relative nerve]] construction

\begin{displaymath}
N_{-}(C) : [C,SSet] \to SSet/C
  \,.
\end{displaymath}
By the [[adjoint functor theorem]] this functor has a [[left adjoint]]

\begin{displaymath}
F_{-}(C) : SSet/C \to [C,SSet]
  \,.
\end{displaymath}
For $p : E \to C$ a [[left Kan fibration]] the functor $F_p(C) : C \to SSet$ sends $c \in Obj(C)$ to the [[fiber]] $p^{-1}(c) := E \times_C \{c\}$

\begin{displaymath}
F_p(C) : c \mapsto p^{-1}(c)
  \,.
\end{displaymath}
(See remark 3.2.5.5 of [[Higher Topos Theory|HTT]]).

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

The universal fibration as such is discussed in section 3.3.2 of

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

\end{itemize}
The concrete description in terms of model theory on marked simplicial sets is in section 3.2. A simpler version of this is in section 2.2.1

[[!redirects universal fibration of (∞,1)-categories]]

[[!redirects universal right fibration]]

[[!redirects universal Cartesian fibration]]

[[!redirects universal fibration of ∞-groupoids]]



\end{document}