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\begin{document}

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\section*{essential sublocale}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_the_internal_language}{In the internal language}\dotfill \pageref*{in_the_internal_language} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

Essential [[sublocales]] are a generalization of [[locally connected geometric morphism|locally connected sublocales]].

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

\begin{defn}
\label{}\hypertarget{}{}
A [[sublocale]] $X_j$ of a [[locale]] $X$, given by a [[nucleus]] $j : \mathcal{O}(X) \to \mathcal{O}(X)$, is called \emph{essential} (sometimes also \emph{principal}) if and only if the following equivalent conditions are satisfied:

\begin{enumerate}%
\item There is a monotone map $b : \mathcal{O}(X) \to \mathcal{O}(X)$ which is [[left adjoint]] to $j$.
\item The nucleus $j$ preserves arbitrary (not only finite) meets.
\item For any $u \in \mathcal{O}(X)$, there is a smallest $v \in \mathcal{O}(X)$ such that $u \preceq j(v)$.
\item The pullback functor $\mathrm{Sh}(X) \to \mathrm{Sh}(X_j)$ possesses a left adjoint (it always has a right adjoint).
\item The [[geometric embedding]] $\mathrm{Sh}(X_j) \hookrightarrow \mathrm{Sh}(X)$ is an [[essential geometric morphism]].

\end{enumerate}
\end{defn}
\begin{proof}
\begin{itemize}%
\item The equivalence of (1) and (2) is by the [[adjoint functor theorem]] for complete lattices, which furthermore gives an explicit formula for the left adjoint $b$, namely\begin{displaymath}
b(u) \coloneqq inf \{ v \in \mathcal{O}(X) \,|\, u \preceq j(v) \}.
\end{displaymath}
This also shows the equivalence of (1) and (3).


\item The equivalence of (4) and (5) is by definition.
\item Recall that the pullback of the representable sheaf $Hom_{\mathcal{O}(X)}(\cdot,u)$ is $Hom_{\mathcal{O}(X)}(\cdot,j(u))$. Therefore [[continuous functor|continuity]] of the pullback functor translates to continuity of $j$. This shows that (4) implies (2).
\item Conversely, assume (3). Then the explicit description of the pullback functor given below (which is valid under this assumption) shows that the pullback functor preserves arbitrary limits. By the adjoint functor theorem for Grothendieck toposes, statement (4) follows.

\end{itemize}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The left adjoint $b$ automatically satisfies $b(u) \preceq u$, $b(b(u)) = u$, $j(b(u)) = j(u)$, and $b(j(u)) = b(u)$ for all $u \in \mathcal{O}(X)$. These relations follow from playing around with the adjunction.

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

\begin{itemize}%
\item Any [[open sublocale]] is essential: The nucleus for an open sublocale is of the form $j = (u \rightarrow -)$. Its left adjoint is $b = (- \wedge u)$.


\item More generally, any [[locally connected geometric morphism|locally connected sublocale]] is essential.



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

\begin{itemize}%
\item For a general sublocale $i : X_j \hookrightarrow X$ and a sheaf $\mathcal{F}$ on $X$, the pullback $i^{-1} \mathcal{F}$ is the [[sheafification]] of the presheaf $u \mapsto \colim_{u \preceq j(v)} \mathcal{F}(v)$. In the case that $X_j$ is an essential sublocale, this presheaf is simply given by the formula $u \mapsto \mathcal{F}(b(u))$ and is already a sheaf, so sheafification is not necessary.


\item A sublocale of the one-point locale is essential if and only if it is [[open morphism|open]]. This is because the extra [[Frobenius reciprocity]] condition demanded by openness is automatically satisfied (in classical mathematics and in impredicative [[constructive mathematics]]).


\item The lattice of essential sublocales of a given locale is [[complete lattice|complete]]. Suprema are calculated as in the lattice of all sublocales; infima, however, are not. There are counterexamples in (\hyperlink{KL89}{Kelly and Lawvere (1989)}), however in the slightly different context of essential [[localizations of categories]].



\end{itemize}
\hypertarget{in_the_internal_language}{}\subsection*{{In the internal language}}\label{in_the_internal_language}

Let $X_j \hookrightarrow X$ be a sublocale. From the [[internal language|internal point of view]] of the [[topos]] $Sh(X)$, this sublocale looks like a sublocale of the one-point locale and it's interesting to compare the properties of $X_j$ with this [[internal locale]].

\begin{prop}
\label{}\hypertarget{}{}
The following statements are equivalent:

\begin{enumerate}%
\item $Sh(X) \models X_j \hookrightarrow 1 \,\text{is an essential sublocale}$.
\item $Sh(X) \models X_j \hookrightarrow 1 \,\text{is an open sublocale}$.
\item $X_j \hookrightarrow X$ is an open sublocale.

\end{enumerate}
Note that none of these statements are equivalent to $X_j \hookrightarrow X$ being an essential sublocale.

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

\begin{itemize}%
\item [[G. M. Kelly]], [[F. W. Lawvere]], \emph{On the complete lattice of essential localizations}, Bull. Soc. Math. de Belgique \textbf{XLI} (1989) pp. 289--319.


\item Guilherme Frederico Lima, \emph{\href{https://www.youtube.com/watch?v=YsoGN91Rh_s}{From essential inclusions to local geometric morphisms}}, talk at Topos \`a{} l'IH\'E{}S in September 2015.



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

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


\item [[essential geometric morphism]]



\end{itemize}


\end{document}