\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{(sub)object classifier in an (infinity,1)-topos}

\begin{quote}%
This page is about object classifier objects in [[(∞,1)-toposes]]. For the unrelated notion of the [[classifying topos]] of the [[theory of objects]] see at \emph{[[classifying topos for the theory of objects]]}.


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

\hypertarget{universes}{}\paragraph*{{Universes}}\label{universes}

[[!include universe - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - 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{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{DetailsSubObjClassf}{Subobject classifier}\dotfill \pageref*{DetailsSubObjClassf} \linebreak
\noindent\hyperlink{DetailsObjClassf}{Object classifier}\dotfill \pageref*{DetailsObjClassf} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ObjectClassifierInInfinityGroupoid}{Object classifier in $\infty Grpd$}\dotfill \pageref*{ObjectClassifierInInfinityGroupoid} \linebreak
\noindent\hyperlink{object_classifier_in_presheaf_toposes}{Object classifier in presheaf $(\infty,1)$-toposes}\dotfill \pageref*{object_classifier_in_presheaf_toposes} \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}

A crucial ingredient in a [[topos]] is a [[subobject classifier]]. From the point of view of [[homotopy theory]], that this has to do with [[subobjects]] turns out to be a coincidence of low dimensions: subobjects are [[(-1)-truncated]] morphisms.

As discussed also at [[stuff, structure, property]], the classifying objects in [[higher topos theory]] classify more general morphisms.

When one passes all the way to [[∞-toposes]], there should be objects that classify \emph{all} morphisms, subject to some bound on size. This is made precise in the context of [[(∞,1)-topos theory]].

One way to characterize an [[(∞,1)-topos]] is as

\begin{itemize}%
\item a [[presentable (∞,1)-category]]


\item with [[universal colimits]]


\item such that for all sufficiently large [[regular cardinal]]s $\kappa$ there is a \textbf{classifying object} for the class of all $\kappa$-compact morphisms in $X$.



\end{itemize}
This statement is originally due to [[Charles Rezk]]. It is reproduced as ([[Higher Topos Theory|Lurie HTT, theorem 6.1.6.8]]).

In terms of [[homotopy type theory]] these object classifiers are \emph{[[types of types]]}. See there for more details and see at \emph{[[relation between category theory and type theory]]}.

\begin{remark}
\label{ReflectonOnCharacterizationByObjectClassifier}\hypertarget{ReflectonOnCharacterizationByObjectClassifier}{}
An [[object classifier]] is a (small) \emph{self-reflection} of the $(\infty, 1)$-topos inside itself ([[type of types]], internal [[universe]]). It possesses an [[category object in an (infinity,1)-category|internal (∞,1)-topos ]] structure. See also ([[Science of Logic\#WesenAlsReflexionInIhmSelbst|WdL, book 2, section 1]]).

\end{remark}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

Let $\mathcal{C}$ be a [[(∞,1)-category]], and let $S$ be a class of [[1-morphisms]] of $\mathcal{C}$ which is stable under [[(∞,1)-pullback]]. Then an \textbf{$S$-classifier} is a [[terminal object in an (∞,1)-category|terminal object]] in the sub-category of arrows $\mathcal{C}^{\Delta^1}$ of $S$ whose morphisms are [[(∞,1)-pullback]] squares in $\mathcal{C}$.

Explicitly, an $S$-classifier consists of

\begin{itemize}%
\item a morphism $\widehat {S Type} \longrightarrow S Type$ in $S$


\item such that for each $X \stackrel{p}{\to} B$ in $S$, there exists an essentially unique [[(∞,1)-pullback]] square of the form

\begin{displaymath}
\itexarray{
     X &\longrightarrow& \widehat {S Type}
     \\
     \downarrow^{\mathrlap{p}} && \downarrow
     \\
     B &\stackrel{'X'}{\longrightarrow}& S Type
  }
  \,.
\end{displaymath}
in $\mathcal{C}$



\end{itemize}
Here $'X'$ may be called the \emph{name}, or the \emph{[[classifying morphism]]}, or the \emph{[[modulating morphism]]} or the \emph{internal reflection} of $X$ over $B$,

For example,

\begin{itemize}%
\item When $S$ is the class of \textbf{all monomorphisms} in $C$, an $S$-classifier is called a \textbf{subobject classifier}. For instance, every topos has a subobject classifier.


\item When $S$ is the class of \textbf{all morphisms} in $C$, an $S$-classifier is called an \textbf{object classifier}. However, due to size issues, interesting categories tend not to have such objects, which is one reason to be interested in the next example:


\item When $S$ is the class of \textbf{all relatively $\kappa$-compact morphisms} (for some regular cardinal $\kappa$--see below for the definition), an $S$-classifier is called a \textbf{$\kappa$-compact-object classifier}.



\end{itemize}
\textbf{Note on terminology:} In all cases, the ``things'' classified by an ``(adjectives) object classifier'' are \emph{arrows} -- this is no different from the most famous case of \emph{[[subobject classifiers]]}, which classify \emph{monos}. For each object $X$, a \emph{subobject classifier} classifies the \emph{subobjects of $X$}. For each object $X$, an \emph{object classifier} classifies the \emph{objects over $X$}.

So with that $\kappa$ fixed, we may write

\begin{displaymath}
\itexarray{
    \widehat Type
    \\
    \downarrow
    \\
    Type
  }
\end{displaymath}
for such a ``universal bundle of $\kappa$-small objects''. Intuitively this is easy to describe: a point in $Type$ corresponds to a $\kappa$-small object, hence is the ``name'' $'X'$ or ``code for'' a $\kappa$-small object, and the [[fiber]] in $\widehat Type$ over that point is the very object $X$ itself.

If one gives the projection of the universal object bundle $\widehat Type \to Type$ a name, such as $El$, and writes $El^{-1}(-)$ for its preimages then $X \simeq El^{-1}('X')$. This is, with the ${(-)}^{-1}$-suppressed, the notation used at \emph{\href{type+of+types#TarskiStyle}{Type universes a la Tarski}}.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{DetailsSubObjClassf}{}\subsubsection*{{Subobject classifier}}\label{DetailsSubObjClassf}

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be an [[(∞,1)-category]] and $S \in C_1$ a class of [[morphisms]] that is stable under [[(∞,1)-pullback]] in $C$.

Let $Cod_C$ be the [[codomain fibration]] of $X$, i.e. the [[(∞,1)-category of (∞,1)-functors]]

\begin{displaymath}
Cod_C := Func(\Delta[1], C)
\end{displaymath}
equipped with the [[Cartesian fibration]] $Cod_C \to C$ induced from the endpoint inclusion $\Delta[0] \to \Delta[1]$.

Write

\begin{itemize}%
\item $Cod_C^S$ for the full [[sub-(∞,1)-category]] of $Cod_C$ on the object in $S$;


\item $Cod_C^{(S)}$ the non-full subcategory whose objects are the elements of $S$, and whose morphisms are squares that are pullback diagrams.



\end{itemize}
Then evaluation at $\Delta[0] \to \Delta[1]$ yields

\begin{itemize}%
\item a [[Cartesian fibration]] $Cod_C^S \to C$;


\item a [[right fibration]] $Cod_C^{(S)} \to C$.



\end{itemize}
We say a morphism $f :x \to y$ in $C$ \emph{classifies} $S$ -- or simply that $y$ classifies $S$ -- if it is the [[terminal object]] in $Cod_C^{(S)}$.

\end{defn}
This is [[Higher Topos Theory|HTT, notation 6.1.3.4]] and [[Higher Topos Theory|HTT, def. 6.1.6.1 ]].

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{subobject classifier} for $C$ is an object that classifies the class $S$ of [[monomorphism in an (∞,1)-category|monomorphisms]]/[[(-1)-truncated]] morphisms in $C$.

\end{defn}
This is ([[Higher Topos Theory|HTT, def. 6.1.6.1 ]]).

\begin{example}
\label{}\hypertarget{}{}
The $(\infty,1)$-category [[∞Grpd]] has a a subobject classifier: the [[0-groupoid]]/[[set]] $\{\emptyset,*\}$ with two elements (the two [[(-1)-truncated]] $\infty$-groupoids).

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
Every [[(∞,1)-topos]] has a [[subobject classifier]].

\end{prop}
This appears as ([[Higher Topos Theory|HTT, prop. 6.1.6.3]]) and the remark below that.

\hypertarget{DetailsObjClassf}{}\subsubsection*{{Object classifier}}\label{DetailsObjClassf}

\textbf{Remark/Warning.} The point of having [[subobjects]] and hence [[monomorphisms]] classified by an object in an ordinary [[topos]] may be thought of as being solely due to the fact that in a 1-[[topos]], any object necessarily classifies a \emph{[[poset]]} i.e. a [[(0,1)-category]] of morphisms, and the point of subobjects/monomorphisms of a given object is that they do not have [[automorphisms]].

In an $(\infty,1)$-topos we thus expect an object that classifies \emph{all} morphisms, in that the assignment

\begin{displaymath}
c \mapsto Core(C_{/c})
\end{displaymath}
of an object $c\in C$ to the [[core]] of its [[over quasi-category|over (∞,1)-category]] yields a [[(∞,1)-functor]] $C^{op} \to \infty Grpd$ that is [[representable functor|representable]].

Indeed, this is \emph{essentially} the case -- up to size issues, that the following definitions take care of.

\begin{defn}
\label{RelativelyKappaCompact}\hypertarget{RelativelyKappaCompact}{}
For $\kappa$ some [[cardinal]], say a morphism $f : x \to y$ in $C$ is \textbf{[[relatively k-compact morphism in an (infinity,1)-category|relatively k-compact]]} if for all [[(∞,1)-pullbacks]] along $h : y' \to y$ to $\kappa$-[[compact object in an (∞,1)-category|compact object]]s, $y'$, the pulled back object $h^* x'$ is itself a $\kappa$-compact object.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
A [[presentable (∞,1)-category]] $C$ is an [[(∞,1)-topos]] precisely if

\begin{enumerate}%
\item it has [[universal colimits]];


\item for sufficiently large regular [[cardinal]]s $\kappa$, $C$ has a classifying object for relatively $\kappa$-compact morphisms.



\end{enumerate}
\end{theorem}
This is due to [[Charles Rezk]]. The statement appears as [[Higher Topos Theory|HTT, theorem 6.1.6.8]].

The proof essentially consists of showing that by the [[adjoint functor theorem]], the existence of object classifiers is equivalent to [[continuous functor|continuity]] of the [[core]] [[self-indexing]] $C^{op} \to \infty Gpd$ defined by $x\mapsto Core(C/x)$. In the presence of universal colimits, this latter condition is equivalent to all colimits being [[van Kampen colimits]], which in turn yields the connection to the Giraud-type exactness properties.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{ObjectClassifierInInfinityGroupoid}{}\subsubsection*{{Object classifier in $\infty Grpd$}}\label{ObjectClassifierInInfinityGroupoid}

We discuss that the $\kappa$-small object classifier in the $(\infty,1)$-topos [[∞Grpd]] of [[∞-groupoids]] is itself the [[core]] of the [[(∞,1)-category]] $\infty Grpd_\kappa$ of $\kappa$-small $\infty$-groupoids. Observing that the connected components of this are the [[delooping]] $B Aut(F)$ of the [[automorphism ∞-group]] of a given [[homotopy type]] $[F]$, and using that [[∞Grpd]] is [[presentable (∞,1)-category|presented]] by [[Top]] $\simeq$ [[sSet]] (see also at \emph{[[homotopy hypothesis]]}) this recovers classical theorems about the classification of [[fibrations]] in simplicial sets/topological spaces by a [[universal Kan fibration]], as listed in the \href{http://ncatlab.org/nlab/show/associated+infinity-bundle#References}{References} at \emph{[[associated ∞-bundle]]}.

\begin{prop}
\label{}\hypertarget{}{}
The $\kappa$-compact object classifier in [[∞Grpd]] is

\begin{displaymath}
Type_\kappa := Core(\infty Grpd_\kappa)
  \,,
\end{displaymath}
the [[core]] of the [[full sub-(∞,1)-category]] of [[∞Grpd]] on the $\kappa$-[[small ∞-groupoids]].

The corresponding [[universal bundle]] is presented by the map of [[simplicial sets]]

\begin{displaymath}
\widehat Type_\kappa \to Type_\kappa
\end{displaymath}
which is the [[pullback]] of simplicial sets

\begin{displaymath}
\itexarray{
    \widehat Type_\kappa &\to& Z_{\infty Grpd}
    \\
    \downarrow && \downarrow
    \\  
    Type_\kappa &\to& \infty Grpd 
  }
\end{displaymath}
of the [[universal right fibration]] along the defining inclusion of (the [[Kan complex]] presenting) $Type_\kappa$.

\end{prop}
\begin{lemma}
\label{RelativelyCompactInInfinityGroupods}\hypertarget{RelativelyCompactInInfinityGroupods}{}
In [[∞Grpd]] the relatively $\kappa$-compact morphisms, $X \to Y$, def. \ref{RelativelyKappaCompact}, are precisely those all whose [[homotopy fibers]]

\begin{displaymath}
X_{y} := X \times_{Y} \{y\}
\end{displaymath}
over all [[objects]] $y \in Y$ are $\kappa$-[[small infinity-groupoids]].

\end{lemma}
\begin{proof}
We may write $Y$ as an [[(∞,1)-colimit]] over itself (see there)

\begin{displaymath}
Y \simeq {\lim_{\to}}_{y \in Y} \{y\}
\end{displaymath}
and then use the fact that [[∞Grpd]] -- being an [[(∞,1)-topos]] -- has [[universal colimits]], to obtain the [[(∞,1)-pullback]] diagram

\begin{displaymath}
\itexarray{
    {\lim_{\to}}_{y \in Y} X_y 
      &\stackrel{\simeq}{\to}
     & X
    \\
    \downarrow && \downarrow
    \\
    {\lim_{\to}}_{y \in Y} \{y\}
    &\stackrel{\simeq}{\to}&
    Y
  }
\end{displaymath}
exhibiting $X$ as an $(\infty,1)$-colimit of $\kappa$-small objects over $Y$. By stability of $\kappa$-compact objects under $\kappa$-small colimits (see \href{http://ncatlab.org/nlab/show/compact+object+in+an+%28infinity%2C1%29-category#StabilityUnderColimits}{here}) it follows that $X$ is $\kappa$-compact if $Y$ is.

\end{proof}
\begin{proof}
Since [[right fibrations]] are stable under pullback (see \href{http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#PreservationByPullback}{here}), this is still a right fibration. Since, up to equivalence, every morphism into a [[Kan complex]] is a right fibration (see \href{http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#OverKanComplex}{here}), and since every morphism out of a Kan complex into $\infty Grpd_\kappa$ factors through the core $Type_\kappa$ it follows that $Type_\kappa$ classifies all morphisms $X \to Y$ in [[∞Grpd]] whose [[homotopy fibers]]

\begin{displaymath}
X_y \simeq X \times_Y \{y\}
\end{displaymath}
are $\kappa$-[[compact object in an (∞,1)-category|compact]].

The claim then follows with lemma \ref{RelativelyCompactInInfinityGroupods}.

\end{proof}
\hypertarget{object_classifier_in_presheaf_toposes}{}\subsubsection*{{Object classifier in presheaf $(\infty,1)$-toposes}}\label{object_classifier_in_presheaf_toposes}

Let $C$ be an [[(∞,1)-category]] and $\mathbf{H} = PSh_{\infty}(C)$ the [[(∞,1)-category of (∞,1)-presheaves]] over $C$.

By the [[(∞,1)-Yoneda lemma]], the $\kappa$-compact object classifier here should be the presheaf which assigns to $U \in C$ the $\infty$-groupoid of relatively $\kappa$-compact morphisms $X \to U$ in $PSh_\infty(C)$.

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

\begin{itemize}%
\item [[type of propositions]], [[subobject classifier]], [[partial map classifier]]


\item [[type of types]], [[univalence]]


\item [[classifying morphism]]


\item [[Awodey's conjecture]]



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

\begin{itemize}%
\item [[Jacob Lurie]], section 6.1.6 of \emph{[[Higher Topos Theory]]}

\end{itemize}
[[!redirects object classifier]] [[!redirects object classifiers]] [[!redirects (sub)object classifier in an (∞,1)-topos]] [[!redirects (sub)object classifiers in an (∞,1)-topos]] [[!redirects (sub)object classifier in an (infinity,1)-topos]] [[!redirects (sub)object classifiers in an (infinity,1)-topos]] [[!redirects object classifier in an (∞,1)-topos]] [[!redirects object classifiers in an (∞,1)-topos]] [[!redirects object classifier in an (infinity,1)-topos]] [[!redirects object classifiers in an (infinity,1)-topos]] [[!redirects (sub)object classifier in an (∞,1)-category]] [[!redirects (sub)object classifiers in an (∞,1)-category]] [[!redirects (sub)object classifier in an (infinity,1)-category]] [[!redirects (sub)object classifiers in an (infinity,1)-category]] [[!redirects object classifier in an (∞,1)-category]] [[!redirects object classifiers in an (∞,1)-category]] [[!redirects object classifier in an (infinity,1)-category]] [[!redirects object classifiers in an (infinity,1)-category]]



\end{document}