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\begin{document}

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\section*{Topos}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category Theory}}\label{category_theory}

[[!include category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_2categories}{Related 2-categories}\dotfill \pageref*{related_2categories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{from_topological_spaces_to_toposes}{From topological spaces to toposes}\dotfill \pageref*{from_topological_spaces_to_toposes} \linebreak
\noindent\hyperlink{from_toposes_to_higher_toposes}{From toposes to higher toposes}\dotfill \pageref*{from_toposes_to_higher_toposes} \linebreak
\noindent\hyperlink{AdjunctionToLocallyPresentable}{From locally presentable categories to toposes}\dotfill \pageref*{AdjunctionToLocallyPresentable} \linebreak
\noindent\hyperlink{Limits}{Limits and colimits}\dotfill \pageref*{Limits} \linebreak
\noindent\hyperlink{Colimits}{Colimits}\dotfill \pageref*{Colimits} \linebreak
\noindent\hyperlink{Pullbacks}{Pullbacks}\dotfill \pageref*{Pullbacks} \linebreak
\noindent\hyperlink{free_loop_spaces}{Free loop spaces}\dotfill \pageref*{free_loop_spaces} \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}

By \textbf{$Topos$} (or \textbf{$Toposes$}) is denoted the [[category]] of [[topos]]es. Usually this means:

\begin{itemize}%
\item [[object]]s are [[topos]]es;


\item [[morphism]]s are [[geometric morphism]]s of toposes.



\end{itemize}
This is naturally a [[2-category]], where

\begin{itemize}%
\item [[2-morphism]] are [[geometric transformation]]s

\end{itemize}
That is, a 2-morphism $f\to g$ is a [[natural transformation]] $f^* \to g^*$ (which is, by [[mate]] calculus, equivalent to a natural transformation $g_* \to f_*$ between [[direct image]]s). Thus, $Toposes$ is equivalent to both of

\begin{itemize}%
\item the (non-full) [[sub-2-category]] of $Cat^{op}$ on categories that are toposes and morphisms that are the inverse image parts of geometric morphisms, and
\item the (non-full) sub-2-category of $Cat^{co}$ on categories that are toposes and morphisms that are the direct image parts of geometric morphisms.

\end{itemize}
\hypertarget{related_2categories}{}\subsubsection*{{Related 2-categories}}\label{related_2categories}

\begin{itemize}%
\item There is also the sub-2-category $ShToposes = GrToposes$ of [[sheaf topos]]es (i.e. Grothendieck toposes).


\item Note that in some literature this 2-category is denoted merely $Top$, but that is also commonly used to denote [[Top|the category]] of [[topological spaces]].


\item We obtain a very different 2-category of toposes if we take the morphisms to be [[logical functors]]; this 2-category is sometimes denoted $Log$ or $LogTopos$.



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

\hypertarget{from_topological_spaces_to_toposes}{}\subsubsection*{{From topological spaces to toposes}}\label{from_topological_spaces_to_toposes}

The operation of forming [[categories of sheaves]]

\begin{displaymath}
Sh(-) : Top \to ShToposes
\end{displaymath}
embeds [[topological space]]s into toposes. For $f : X \to Y$ a [[continuous map]] we have that $Sh(f)$ is the [[geometric morphism]]

\begin{displaymath}
Sh(f) : Sh(X) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
  Sh(Y)
\end{displaymath}
with $f_*$ the [[direct image]] and $f^*$ the [[inverse image]].

Strictly speaking, this functor is not an [[embedding]] if we consider $Top$ as a 1-category and $Toposes$ as a 2-category, since it is then not [[fully faithful]] in the 2-categorical sense---there can be nontrivial 2-cells between geometric morphisms between toposes of sheaves on topological spaces.

However, if we regard $Top$ as a [[(1,2)-category]] where the 2-cells are inequalities in the [[specialization ordering]], then this functor does become a 2-categorically full embedding (i.e. an equivalence on hom-categories) if we restrict to the full subcategory $SobTop$ of [[sober spaces]]. This embedding can also be extended from $SobTop$ to the entire category of [[locales]] (which can be viewed as ``Grothendieck 0-toposes'').

\hypertarget{from_toposes_to_higher_toposes}{}\subsubsection*{{From toposes to higher toposes}}\label{from_toposes_to_higher_toposes}

There are similar full embeddings $ShTopos \hookrightarrow Sh 2 Topos$ and $ShTopos \hookrightarrow Sh(n,1)Topos$ of sheaf (1-)toposes into [[2-sheaf]] [[2-topos]]es and sheaf [[(n,1)-topos]]es for $2\le n\le \infty$. Note that these embeddings are \emph{not} the identity functor on underlying categories: a 1-topos is not itself an $n$-topos, instead we have to take $n$-sheaves on a suitable generating [[site]] for it.

\hypertarget{AdjunctionToLocallyPresentable}{}\subsubsection*{{From locally presentable categories to toposes}}\label{AdjunctionToLocallyPresentable}

There is a canonical [[forgetful functor]] $U : Topos \to$ [[Cat]] that lands, by definition, in the sub-2-category of [[locally presentable categories]] and [[functors]] which preserve all limits / are [[right adjoints]].

This [[2-functor]] has a [[2-adjunction|right 2-adjoint]] (\hyperlink{BungeCarboni}{Bunge-Carboni}).

\hypertarget{Limits}{}\subsubsection*{{Limits and colimits}}\label{Limits}

The 2-category $Topos$ is not all that well-endowed with [[limits]], but its [[slice categories]] are finitely complete [[2-limit|as 2-categories]], and $ShTopos$ is closed under finite limits in $Topos/Set$. In particular, the [[terminal object]] in $ShToposes$ is the topos [[Set]] $\simeq Sh(*)$.

\hypertarget{Colimits}{}\paragraph*{{Colimits}}\label{Colimits}

The supply with colimits is better:

\begin{prop}
\label{ColimitsByInverseImageLimits}\hypertarget{ColimitsByInverseImageLimits}{}
All small (indexed) [[2-colimit]]s in $ShTopos$ exists and are computed as (indexed) [[2-limit]]s in [[Cat]] of the underlying [[inverse image]] functors.

\end{prop}
This appears as (\hyperlink{Moerdijk}{Moerdijk, theorem 2.5})

\begin{prop}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
\itexarray{
    \mathcal{F} &\stackrel{p_2}{\to}& \mathcal{E}_2
    \\
    {}^{\mathllap{p_1}}\downarrow &\swArrow& \downarrow^{\mathrlap{f_2}}
    \\
    \mathcal{E}_1 &\underset{f_1}{\to}& \mathcal{E}
  }
\end{displaymath}
be a [[2-pullback]] in $Topos$ such that

\begin{itemize}%
\item $f_1, f_2$ are both [[pseudomonic morphism]]s


\item $\mathcal{E}_1 \coprod \mathcal{E}_2 \to \mathcal{E}$ is an [[effective epimorphism]];



\end{itemize}
then the diagram of [[inverse image]] functors

\begin{displaymath}
\itexarray{
    \mathcal{F} &\stackrel{p_2^*}{\leftarrow}& \mathcal{E}_2
    \\
    {}^{\mathllap{p_1^*}}\uparrow &\swArrow& \uparrow^{\mathrlap{f_2^*}}
    \\
    \mathcal{E}_1 &\underset{f_1^*}{\leftarrow}& \mathcal{E}
  }
\end{displaymath}
is a 2-pullback in [[Cat]] and so by the \hyperlink{ColimitsByInverseImageLimits}{above} the original square is also a 2-pushout.

\end{prop}
This appears as theorem 5.1 in (\hyperlink{BungeLack}{BungeLack})

\begin{prop}
\label{}\hypertarget{}{}
The 2-category $Topos$ is an [[extensive category]]. Same for toposes bounded over a base.

\end{prop}
This is in (\hyperlink{BungeLack}{BungeLack, proposition 4.3}).

\hypertarget{Pullbacks}{}\paragraph*{{Pullbacks}}\label{Pullbacks}

\begin{prop}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
\itexarray{
    && \mathcal{X}
    \\
    && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&  
    \mathcal{Z}
  }
\end{displaymath}
be a [[diagram]] of [[toposes]]. Then its [[pullback]] in the [[(2,1)-category]] version of $Topos$ is computed, roughly, by the [[pushout]] of their [[sites]] of definition.

More in detail: there exist [[sites]] $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with [[finite limit]]s and [[morphisms of sites]]

\begin{displaymath}
\itexarray{
    && \mathcal{D}
    \\
    && \uparrow^{\mathrlap{g}}
    \\
    \tilde \mathcal{D}
    &\stackrel{f}{\leftarrow}&  
    \mathcal{C}
  }
\end{displaymath}
such that

\begin{displaymath}
\left(
  \itexarray{
    && \mathcal{X}
    \\
    && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&  
    \mathcal{Z}
  }
  \right) 
  \,\,\,
   \simeq
  \,\,\,
  \left(
  \itexarray{
    && Sh(\mathcal{D})
    \\
    && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}}
    \\
    Sh(\tilde \mathcal{D})
    &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}&  
    Sh(\mathcal{C})
  }
  \right) 
  \,.
\end{displaymath}
Let then

\begin{displaymath}
\itexarray{
    \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D}
    \\
    {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}}
    \\
    \tilde \mathcal{D}
    &\stackrel{f}{\leftarrow}&  
    \mathcal{C}
  }
  \,\,\,\,\,
  \in
  Cat^{lex}
\end{displaymath}
be the [[pushout]] of the underlying [[categories]] in the [[full subcategory]] [[Cat]]${}^{lex} \subset Cat$ of categories with finite limits.

Let moreover

\begin{displaymath}
Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
  \hookrightarrow
  PSh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
\end{displaymath}
be the [[reflective subcategory]] obtained by [[localization]] at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the [[covering]]s of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.

Then

\begin{displaymath}
\itexarray{
    Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
    &\to& \mathcal{X}
    \\
    \downarrow && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&  
    \mathcal{Z}
  }
\end{displaymath}
is a [[pullback]] square.

\end{prop}
This appears for instance as (\hyperlink{Lurie}{Lurie, prop. 6.3.4.6}).

\begin{remark}
\label{}\hypertarget{}{}
For [[localic toposes]] this reduces to the statement of [[localic reflection]]: the pullback of toposes is given by the of the underlying [[locales]] which in turn is the [[pushout]] of the corresponding [[frames]].

\end{remark}
\hypertarget{free_loop_spaces}{}\subsubsection*{{Free loop spaces}}\label{free_loop_spaces}

The [[free loop space object]] of a topos in [[Topos]] is called the [[isotropy group of a topos]].

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

\begin{itemize}%
\item \textbf{Topos}


\item [[(∞,1)Topos]]



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

The characterization of colimits in $Topos$ is in

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{The classifying topos of a continuous groupoid. I} Transaction of the American mathematical society Volume 310, Number 2, (1988) (\href{http://www.ams.org/journals/tran/1988-310-02/S0002-9947-1988-0973173-9/S0002-9947-1988-0973173-9.pdf}{pdf})

\end{itemize}
The fact that $Topos$ is extensive is in

\begin{itemize}%
\item [[Marta Bunge]], [[Steve Lack]], \emph{van Kampen theorem for toposes} (\href{http://www.maths.usyd.edu.au/u/stevel/papers/vkt.ps.gz}{ps})

\end{itemize}
Limits and colimits of toposes are discussed in 6.3.2-6.3.4 of

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

\end{itemize}
There this is discussed for for [[(∞,1)-topos]]es, but the statements are verbatim true also for ordinary toposes (in the [[(2,1)-category]] version of $Topos$).

The adjunction between toposes and locally presentable categories is discussed in

\begin{itemize}%
\item [[Marta Bunge]], [[Aurelio Carboni]], \emph{The symmetric topos}, Journal of Pure and Applied Algebra 105:233-249, (1995)

\end{itemize}
category: category

[[!redirects Topos]] [[!redirects Topoi]] [[!redirects Toposes]]



\end{document}