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\begin{document}

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\section*{subtopos}

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

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

[[!include topos 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{sheaves_localization_closure_and_reflection}{Sheaves, localization, closure and reflection}\dotfill \pageref*{sheaves_localization_closure_and_reflection} \linebreak
\noindent\hyperlink{for_classifying_toposes}{For classifying toposes}\dotfill \pageref*{for_classifying_toposes} \linebreak
\noindent\hyperlink{LatticeOfSubtoposes}{The lattice of subtoposes}\dotfill \pageref*{LatticeOfSubtoposes} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{subtopos} of a topos is a generalization of the concept of a subspace of a [[topological space]].

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

For $\mathcal{E}$ a [[topos]], a \textbf{subtopos} is another topos $\mathcal{F}$ equipped with a [[geometric embedding]] $\mathcal{F} \hookrightarrow \mathcal{E}$.

If this is an [[open geometric morphism]] (or an [[essential geometric morphism]]) one speaks of an \textbf{[[open subtopos]]} (an \textbf{[[essential subtopos]]}, respectively, also called a \emph{[[level of a topos]]}).

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

\hypertarget{sheaves_localization_closure_and_reflection}{}\subsubsection*{{Sheaves, localization, closure and reflection}}\label{sheaves_localization_closure_and_reflection}

If $\mathcal{E}$ is an [[elementary topos]] then subtoposes correspond to [[Lawvere-Tierney topologies]] $j$ on $\mathcal{E}$, to [[localization|localizations]] of $\mathcal{E}$ as well as to [[closure operator|universal closure operators]] on $\mathcal{E}$.

\hypertarget{for_classifying_toposes}{}\subsubsection*{{For classifying toposes}}\label{for_classifying_toposes}

Every Grothendieck topos $\mathcal{E}$ over $Set$ is (equivalent to) the [[classifying topos]] of some [[geometric theory]] $T$ and it can be shown that subtoposes of $\mathcal{E}$ correspond precisely to \textbf{deductively closed quotient theories} of $T$ (\hyperlink{Caramello09}{Caramello (2009)}; thm. 3.6) i.e. passage to a subtopos corresponds to adding further geometric [[axioms]] to $T$ - localizing geometrically amounts to theory refinement logically.

\hypertarget{LatticeOfSubtoposes}{}\subsubsection*{{The lattice of subtoposes}}\label{LatticeOfSubtoposes}

The inclusions induce an ordering on the subtoposes of $\mathcal{E}$ that makes them a [[lattice]] with a [[co-Heyting algebra]] structure i.e. the [[join]] operator has a [[left adjoint]] \emph{[[subtraction]]} operator. This corresponds to a dual [[Heyting algebra]] structure on the [[Lawvere-Tierney topologies]].

The lattice structure is further analyzed in (\hyperlink{Johnstone02}{Johnstone (2002)}, \hyperlink{Caramello09}{Caramello (2009)}). We contend us to report

\begin{prop}
\label{subtopos_meet}\hypertarget{subtopos_meet}{}
Let $Sh_j(\mathcal{E})$, $Sh_k(\mathcal{E})$ be two subtoposes of a topos $\mathcal{E}$ and $j,k$ the corresponding topologies with $j\vee k$ their join in the lattice of topologies. Then the following holds: $Sh_j(\mathcal{E})\wedge Sh_k(\mathcal{E})=Sh_{j\vee k}(\mathcal{E})=Sh_j(\mathcal{E})\cap Sh_k(\mathcal{E})$.

\end{prop}
(cf. \hyperlink{Johnstone02}{Johnstone (2002)}, p.217). In other words, the meet of two subtoposes is just their intersection (and is equivalently given by the subtopos corresponding to the join of their topologies).

\begin{prop}
\label{BooleantoposesAreAtoms}\hypertarget{BooleantoposesAreAtoms}{}
The \emph{[[atoms]] in the lattice of subtoposes of $\mathcal{E}$ are precisely the two-valued [[Boolean topos|Boolean]] subtoposes} of $\mathcal{E}$.

\end{prop}
(\hyperlink{Caramello09}{Caramello (2009)}; prop. 10.1). This follows from the fact that two-valued and Boolean toposes are opposite extremes when it comes to dense subtoposes: in a two-valued topos every non-trivial subtopos is dense, whereas a Boolean topos has no non-trivial dense subtopos (cf. at [[dense subtopos]] for further details).

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[open subtopos]]


\item [[closed subtopos]]


\item [[locally closed subtopos]]


\item [[dense subtopos]]


\item [[level]]


\item [[co-Heyting boundary]]


\item [[Artin gluing]]



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

\begin{itemize}%
\item [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], \emph{Th\'e{}orie des Topos et Cohomologie Etale des Sch\'e{}mas ([[SGA4]])}, LNM \textbf{269} Springer Heidelberg 1972. (expos\'e{} IV, section 9, pp.431ff)


\item [[Francis Borceux|F. Borceux]], M. Korostenski, \emph{Open Localizations} , JPAA \textbf{74} (1991) pp.229-238.


\item [[Olivia Caramello|O. Caramello]], pp.15,58 of \emph{Lattices of theories}, (2009). (\href{http://arxiv.org/abs/0905.0299}{arXiv:0905.0299})


\item H. Forssell, \emph{Subgroupoids and quotient theories} , TAC \textbf{28} no.18 (2013) pp.541-551. (\href{http://www.emis.de/journals/TAC/volumes/28/18/28-18.pdf}{pdf})


\item [[Peter Johnstone]], \emph{Sketches of an [[Elephant]] I}, Oxford UP 2002. (pp.195-223)


\item [[Peter Johnstone]], \emph{Topos Theory} , Academic Press New York 1977. (Dover reprint 2014, section 3.5)


\item [[G. M. Kelly]], [[William Lawvere|F. W. Lawvere]], \emph{On the Complete Lattice of Essential Localizations} , Bull. Soc. Math. de Belgique \textbf{XLI} (1989) pp.261-299.



\end{itemize}
[[!redirects Subtopos]] [[!redirects subtoposes]] [[!redirects Subtoposes]]

[[!redirects lattice of subtoposes]] [[!redirects lattices of subtoposes]]

[[!redirects subtopos lattice]] [[!redirects subtopos lattices]]



\end{document}