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\begin{document}

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\section*{exponentiable topos}

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

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

[[!include topos theory - contents]]

\hypertarget{topos_theory_2}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory_2}

[[!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{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{remarks_on_duality}{Remarks on duality}\dotfill \pageref*{remarks_on_duality} \linebreak
\noindent\hyperlink{ramifications}{Ramifications}\dotfill \pageref*{ramifications} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An \textbf{exponentiable topos} is a generalization of the notion of an exponentiable [[locale]] and can be viewed as a (topological) ``space'' $X$ that behaves well with respect to the construction of mapping spaces $Y^X$.

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

\begin{defn}
\label{}\hypertarget{}{}
A [[Grothendieck topos]] $\mathcal{E}$ is called \emph{exponentiable} (in the [[2-category]] of Grothendieck toposes $GrTop$) if the 2-functor ${}_{-}\times\mathcal{E}$ has a right 2-adjoint $(_{-})^\mathcal{E}$.

\end{defn}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item More concretely, $\mathcal{E}$ is exponentiable if there exists a functor $(_{-})^\mathcal{E}$ such that for all toposes $\mathcal{F},\mathcal{G}$ $Hom(\mathcal{F}\times\mathcal{E},\mathcal{G})$ is (naturally) equivalent as a category to $Hom(\mathcal{F},\mathcal{G}^\mathcal{E})$.


\item The concept generalizes to [[higher topos theory]] (cf. \hyperlink{AJ18}{Anel-Lejay 2018}, \hyperlink{Lurie18}{Lurie 2018}).



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

Interestingly, in the category of locales exponentiability of a locale $X$ hinges on the existence of the single exponential $S^X$ where $S$ is the [[Sierpinski space]]: $Y^X$ exists for all $Y$ iff $S^X$ exists.

In $GrTop$ the [[classifying topos for the theory of objects|object classifier]] $\mathcal{S}[\mathbb{O}]$ takes over the role of the Sierpinski space and we have the following

\begin{prop}
\label{}\hypertarget{}{}
A [[Grothendieck topos]] $\mathcal{E}$ is exponentiable iff the [[exponential object|exponential]] $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ exists.

\end{prop}
This result is due to Johnstone-Joyal (\hyperlink{JJ82}{1982}, p.282) and occurs as theorem 4.3.1 of Johnstone (\hyperlink{J02}{2002, vol.1 p.433}).

The following theorem pursues this analogy and generalizes a result of [[Martin Hyland]] on locales (\hyperlink{Hyland81}{1981}).

\begin{prop}
\label{}\hypertarget{}{}
A [[Grothendieck topos]] $\mathcal{E}$ is an exponentiable object in the 2-category of Grothendieck toposes and [[geometric morphisms]] iff $\mathcal{E}$ is a [[continuous category]].

\end{prop}
This result is due to Johnstone-Joyal (\hyperlink{JJ82}{1982}) and occurs as theorem 4.4.5 of Johnstone (\hyperlink{J02}{2002, p.748}).

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

\begin{itemize}%
\item Since [[locally finitely presentable categories]] are [[continuous category|continuous]] and [[coherent topos|coherent toposes]] are locally finitely presentable (cf. Johnstone (\hyperlink{J02}{2002, p.915})) it follows that \emph{coherent toposes are exponentiable}. This can be viewed as an avatar of the fact that (locally) compact topological spaces behave well with respect to mapping spaces.


\item By the same reasoning all functor categories $Set^{\mathcal{C}}$ for $\mathcal{C}$ a [[small category]] are exponentiable since they are [[locally finitely presentable category|locally finitely presentable]]. This includes in particular all [[presheaf toposes]] on small categories.



\end{itemize}
\hypertarget{remarks_on_duality}{}\subsection*{{Remarks on duality}}\label{remarks_on_duality}

It is worthwhile to muse a bit about $\mathcal{S}[\mathbb{O}]^\mathcal{E}$:

First of at all, it is [[injective topos|injective]] since $\mathcal{S}[\mathbb{O}]$ is injective and $(_-)^\mathcal{E}$ preserves inclusions.

This and the various universal properties of the toposes involved imply that an exponentiable topos

\begin{displaymath}
\mathcal{E}\cong Hom(\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S},\mathcal{S}[\mathbb{O}]^\mathcal{E})
\end{displaymath}
is (up to equivalence) the \emph{category of points of an injective topos}.

Now consider an arbitrary topos $\mathcal{E}$ classifying a geometric theory $\mathbb{T}$.

Then from the [[geometric theory\#FunctorialDefinition|functorial perspective on logic]] the 2-functor $\mathbb{T}^*:GrTop^{op}\to CAT$ assigning to $\mathcal{F}$ the category of models $\mathbb{T}^*(\mathcal{F}):=\mathcal{F}\times\mathcal{E}=\mathcal{F}\times\mathcal{S}[\mathbb{T}]$ is called the \textbf{dual theory} of $\mathbb{T}$.

It need not be geometric i.e. have a [[classifying topos]] but when it is, $\mathbb{T}$ being called \emph{dualizable} in that case, the following

\begin{displaymath}
\mathbb{T}^*(\mathcal{F})=\mathcal{F}\times\mathcal{E}\cong Hom(\mathcal{F}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{F},\mathcal{S}[\mathbb{O}]^\mathcal{E})
\end{displaymath}
shows that $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ has precisely the properties required for $\mathcal{S}[\mathbb{T}^*]$.

In other words, $\mathbb{T}^*$ is geometric (and classified by $\mathcal{S}[\mathbb{O}]^\mathcal{E}$) precisely iff $\mathcal{E}=\mathcal{S}[\mathbb{T}]$ is exponentiable!

\hypertarget{ramifications}{}\subsection*{{Ramifications}}\label{ramifications}

The notion of a [[tiny object]] in a cartesian closed category suggests the following

\begin{defn}
\label{}\hypertarget{}{}
An exponentiable Grothendieck topos $\mathcal{E}$ is called \emph{tiny} (or \emph{infinitesimal}) if the 2-functor $(_{-})^\mathcal{E}$ has a right 2-adjoint $(_{-})_\mathcal{E}$.

\end{defn}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[exponential object]]


\item [[continuous category]]


\item [[reflexive object]]


\item [[injective topos]]


\item [[ind-object]]


\item [[classifying topos for the theory of objects]]


\item [[convenient category of spaces]]



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

\begin{itemize}%
\item [[Mathieu Anel]], Damien Lejay, \emph{Exponentiable Higher Toposes} , arXiv:1802.10425 (2018). (\href{https://arxiv.org/abs/1802.10425}{abstract})


\item [[Andreas Blass]], \emph{The interaction of category theory and set theory} , Cont. Math. \textbf{30} (1984) pp.5-29. (\href{http://www.math.lsa.umich.edu/~ablass/interact.pdf}{draft})


\item [[Martin Hyland]], \emph{Function spaces in the category of locales} , Springer LNM \textbf{871} (1981) pp.264-281.


\item [[Peter Johnstone]], [[André Joyal]], \emph{Continuous categories and exponentiable toposes} , JPAA \textbf{25} (1982) pp.255-296.


\item [[Peter Johnstone]], \emph{Sketches of an Elephant vols. 1\&2} , CUP 2002. (sections B4.3 pp.432-438, C4.4 pp.745-754)


\item [[Jacob Lurie]], \emph{Spectral Algebraic Geometry} , ms. Harvard University 2018. (section 21.1.6)


\item [[Susan Niefield]], \emph{Exponentiable Morphisms: posets, spaces, locales and Grothendieck toposes} , TAC \textbf{8} (2001) pp.16-32. (\href{http://www.tac.mta.ca/tac/volumes/8/n2/8-02abs.html}{abstract})



\end{itemize}
[[!redirects Exponentiable topos]] [[!redirects exponential topos]]



\end{document}