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\begin{document}

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\section*{2-topos}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,2)$-Topos theory}}\label{topos_theory}

[[!include (infinity,2)-topos theory - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization_of_2sheaf_2toposes}{Characterization of 2-sheaf 2-toposes}\dotfill \pageref*{characterization_of_2sheaf_2toposes} \linebreak
\noindent\hyperlink{localic_2toposes}{$(n,r)$-Localic 2-toposes}\dotfill \pageref*{localic_2toposes} \linebreak
\noindent\hyperlink{InTermsOfInternalCategories}{In terms of internal categories}\dotfill \pageref*{InTermsOfInternalCategories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_archetypical_2topos}{The archetypical 2-topos}\dotfill \pageref*{the_archetypical_2topos} \linebreak
\noindent\hyperlink{internal_categories_in_a_topos}{Internal categories in a $(2,1)$-topos}\dotfill \pageref*{internal_categories_in_a_topos} \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 notion of \textbf{2-topos} is the generalization of the notion of [[topos]] from [[category theory]] to the [[higher category theory]] of [[2-categories]].

There are multiple conceivable such generalizations, depending in particular on whether one tries to generalize the notion of [[Grothendieck topos]] or of [[elementary topos]], and in the latter case what axioms one chooses to take as the basis for generalization.

In contrast, [[(n,1)-topos|(2,1)-toposes]] are much better understood.

A \emph{Grothendieck 2-topos} is a [[2-category]] of [[2-sheaves]] over a [[2-site]].

A \emph{Grothendieck (2,1)-topos} is a [[(2,1)-category]] of [[(2,1)-sheaves]] over a [[(2,1)-site]].

See also [[higher topos theory]].

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

\hypertarget{characterization_of_2sheaf_2toposes}{}\subsubsection*{{Characterization of 2-sheaf 2-toposes}}\label{characterization_of_2sheaf_2toposes}

The [[2-toposes]] of [[2-sheaves]] over a [[2-site]] are special among all 2-toposes, in direct generalization of how [[sheaf toposes]] (``[[Grothendieck toposes]]'') are special among all [[toposes]]. In that case, \emph{[[Giraud's theorem]]} famously characterizes sheaf toposes. This characterization has a 2-categorical analog: the \emph{[[2-Giraud theorem]]}.

\hypertarget{localic_2toposes}{}\subsubsection*{{$(n,r)$-Localic 2-toposes}}\label{localic_2toposes}

A [[2-sheaf]] [[2-topos]] is ``$(n,r)$-localic'' or ``$(n,r)$-truncated'' if it has an [[(n,r)-site]] of definition.

In particular a $(2,1)$-localic 2-topos is the same as a [[(2,1)-topos]].

\hypertarget{InTermsOfInternalCategories}{}\subsubsection*{{In terms of internal categories}}\label{InTermsOfInternalCategories}

Given a 2-topos $\mathcal{X}$, regard it is a [[2-site]] by equipping it with its [[canonical topology]].

\begin{defn}
\label{}\hypertarget{}{}
Write $Cat(\mathcal{X})$ for the 2-category of \emph{[[internal categories]]} in $\mathcal{X}$, precisely: the 2-category of [[2-congruences]] and internal [[anafunctors]] between them (see \href{http://ncatlab.org/nlab/show/2-congruence#2CategoryOf2Congruences}{here}).

\end{defn}
\begin{theorem}
\label{2SheavesAsInternalCategories}\hypertarget{2SheavesAsInternalCategories}{}
For $\mathcal{X}$ a 2-topos, there is an [[equivalence of 2-categories]]

\begin{displaymath}
\mathcal{X} \simeq Cat(\mathcal{X})
  \,.
\end{displaymath}
If $\mathcal{X}$ is $(2,1)$-localic, with a [[(2,1)-site]] of definition $C$, then there is already an equivalence

\begin{displaymath}
\mathcal{X} \simeq Cat(Sh_{(2,1)}(C))
\end{displaymath}
with the 2-category of categories internal to the underlying [[(2,1)-topos]].

If $\mathcal{X}$ is $1$-localic, with 1-site of definition, then there is even already an equivalence

\begin{displaymath}
\mathcal{X} \simeq Cat(Sh(C))
\end{displaymath}
with the internal categories in the underlying [[sheaf topos]].

\end{theorem}
\begin{proof}
By the [[2-Giraud theorem]], $\mathcal{X}$ is an [[exact 2-category]]. With this, the first statement is \href{http://ncatlab.org/nlab/show/2-congruence#nCongExidempotent}{this theorem} at \emph{[[2-congruence]]}.

By the discussion at [[n-localic 2-topos]], a 2-sheaf 2-topos has \emph{[[core in a 2-category|enough groupoids]]} precisely if it has a [[(2,1)-site]] of definition, and has \emph{[[core in a 2-category|enough discretes]]} precisely if it has a 1-site of definition. With this the second and third statement is \href{http://ncatlab.org/nlab/show/2-congruence#nCongOnGroupoidsAndDiscretes}{this theorem} at \emph{[[2-congruence]]}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The noteworthy point about theorem \ref{2SheavesAsInternalCategories} is that for an ambient context which is a $(2,1)$-localic [[(2,1)-topos]], the straightforward morphisms of [[internal categories]], hence the notion of [[internal functors]], needs no further [[localization]]. This is in stark contrast to the situation for an ambient [[1-category]]. The generalization of this phenomenon is discussed at \emph{[[category object in an (∞,1)-category]]}.

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

\hypertarget{the_archetypical_2topos}{}\subsubsection*{{The archetypical 2-topos}}\label{the_archetypical_2topos}

The archetypical 2-topos is [[Cat]]. This plays the role for 2-toposes as [[Set]] does for [[1-toposes]].

\hypertarget{internal_categories_in_a_topos}{}\subsubsection*{{Internal categories in a $(2,1)$-topos}}\label{internal_categories_in_a_topos}

Given any [[(2,1)-topos]] $\mathcal{X}$, the [[2-category]] $Cat(\mathcal{X})$ of [[internal (infinity,1)-category|internal categories]] in $\mathcal{X}$ ought to be a 2-topos. But it seems that at the moment there is no proof of this in the literature.

For literature on [[internal categories]] in [[1-toposes]] see at \emph{[[2-sheaf]]}.

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

\begin{itemize}%
\item [[(0,1)-topos]]


\item [[topos]], [[(∞,1)-topos]], [[(n,1)-topos]]


\item \textbf{2-topos}, [[(∞,2)-topos]]

\begin{itemize}%
\item [[elementary theory of the 2-category of categories]] ([[ETCC]])


\item [[n-localic 2-topos]]



\end{itemize}

\item [[n-topos]]


\item [[(∞,n)-topos]]



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

An introduction is in

\begin{itemize}%
\item [[Mike Shulman]], \emph{[[michaelshulman:What is a 2-topos]]?}

\end{itemize}
Early developments include

\begin{itemize}%
\item [[Ross Street]], \emph{Two dimensional sheaf theory}, J. Pure and Appl. Algebra 24 (1982) 2Opp.

\end{itemize}
A detailed discussion from the point of view of [[internal logic]] is at

\begin{itemize}%
\item [[Mike Shulman]], \emph{[[michaelshulman:2-categorical logic]]}

\end{itemize}
Discussion of the 2-categorical [[Giraud theorem]] for [[2-sheaf]] 2-toposes is in

\begin{itemize}%
\item [[Ross Street]], \emph{[[zoranskoda:Characterization of Bicategories of Stacks]]} Category theory (Gummersbach 1981) LNM 962, 1982, MR0682967 (84d:18006)


\item [[Mike Shulman]], \emph{[[michaelshulman:2-Giraud theorem]]}



\end{itemize}
Discussion of the [[elementary topos]]-analog of 2-toposes is in

\begin{itemize}%
\item [[Mark Weber]], \emph{Yoneda structures from 2-toposes} (\href{https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes}{pdf})

\end{itemize}
A notion of ``flat 2-functor'' (cf [[Diaconescu's theorem]]) perhaps relevant to the ``points'' of 2-toposes is in

\begin{itemize}%
\item M.E. Descotte, E.J. Dubuc, M. Szyld, \emph{On the notion of flat 2-functors}, arXiv:\href{https://arxiv.org/abs/1610.09429}{1610.09429}

\end{itemize}
[[!redirects 2-toposes]] [[!redirects 2-topoi]]

[[!redirects (2,1)-topos]] [[!redirects (2,1)-toposes]]

[[!redirects Grothendieck 2-topos]] [[!redirects Grothendieck 2-toposes]]



\end{document}