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\begin{document}

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\section*{locally connected geometric morphism}

\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{relation_to_connectedness}{Relation to connectedness}\dotfill \pageref*{relation_to_connectedness} \linebreak
\noindent\hyperlink{over_}{Over $Set$}\dotfill \pageref*{over_} \linebreak
\noindent\hyperlink{StrongAdjunctions}{Strong adjunctions}\dotfill \pageref*{StrongAdjunctions} \linebreak
\noindent\hyperlink{coreflectivity}{Coreflectivity}\dotfill \pageref*{coreflectivity} \linebreak
\noindent\hyperlink{other_characterizations}{Other characterizations}\dotfill \pageref*{other_characterizations} \linebreak
\noindent\hyperlink{variations_in_the_context_of_the_nullstellensatz}{Variations in the context of the Nullstellensatz}\dotfill \pageref*{variations_in_the_context_of_the_nullstellensatz} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

A [[geometric morphism]] is \emph{locally connected} if it behaves as though its [[fiber]]s are [[locally connected space]]s. In particular, a [[Grothendieck topos]] $E$ is [[locally connected topos|locally connected]] iff the unique [[geometric morphism]] to [[Set]] (the terminal Grothendieck topos, i.e. the [[point]] in the [[category]] [[Topos]] of toposes) is locally connected.

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

A [[geometric morphism]] $(f^* \dashv f_*) : F \underoverset{f_*}{f^*}{\leftrightarrows} E$ is \textbf{locally connected} if it satisfies the following equivalent conditions:

\begin{enumerate}%
\item It is [[essential geometric morphism|essential]], i.e. $f^*$ has a [[left adjoint]] $f_!$, and moreover $f_!$ can be made into an $E$-[[indexed functor]].


\item For every $A\in E$, the functor $f^* \colon E/A \to F/f^*A$ is [[cartesian closed functor|cartesian closed]].


\item $f^*$ commutes with [[dependent products]] -- For any morphism $h\colon A\to B$ in $E$, the canonically defined [[natural transformation]] $f^* \circ \Pi_h \to \Pi_{f^*h} \circ f^*$ is an [[isomorphism]].



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

\hypertarget{relation_to_connectedness}{}\subsubsection*{{Relation to connectedness}}\label{relation_to_connectedness}

If $f$ is locally connected, then it makes sense to think of the left adjoint $f_!$ as assigning to an object of $F$ its ``set of connected components'' in $E$. In particular, if $f$ is locally connected, then it is moreover [[connected geometric morphism|connected]] if and only if $f_!$ preserves the [[terminal object]]. However, not every connected geometric morphism is locally connected.

\hypertarget{over_}{}\subsubsection*{{Over $Set$}}\label{over_}

Over the [[base topos]] $E =$ [[Set]] every [[connected topos]] which is [[essential geometric morphism|essential]] is automatically locally connected.

This is because the required [[Frobenius reciprocity]] condition

\begin{displaymath}
f_!(A \times f^* (B)) \simeq f_!(A) \times B
\end{displaymath}
is automatically satisfied, using that [[cartesian product]] with a [[set]] is equivalently a [[coproduct]]

\begin{displaymath}
A \times B = \coprod_{a \in A} B
  \,,
\end{displaymath}
that the [[left adjoint]] $f_!$ preserves coproducts, and that for $f^*$ [[full and faithful]] we have $f_! f^* \simeq Id$.

\hypertarget{StrongAdjunctions}{}\subsubsection*{{Strong adjunctions}}\label{StrongAdjunctions}

The pair of [[adjoint functors]] $(f_! \dashv f^*)$ in a locally connected geometric morphisms forms a ``strong adjunction'' in that it holds also for the [[internal homs]] in the sense that there is a [[natural isomorphism]]

\begin{displaymath}
[f_!(X), A] \simeq f_* [X, f^* A]
\end{displaymath}
for all $X, A$. This follows by duality from the [[Frobenius reciprocity]] that characterizes $f_*$ as being a [[cartesian closed functor]]:

by the [[Yoneda lemma]], the morphism in question is an [[isomorphism]] if for all objects $A,B, X$ the morphism

\begin{displaymath}
Hom(X, [f_!(A), B]) 
   \stackrel{}{\to}
  Hom(X,f_*[A,f^*(B)])
\end{displaymath}
is a bijection. By [[adjunction]] this is the same as

\begin{displaymath}
Hom(X \times f_!(A), B) \stackrel{\simeq}{\to}
  Hom(f_!(f^*(X) \times A), B)
  \,.
\end{displaymath}
Again by Yoneda, this is a bijection precisely if

\begin{displaymath}
f_!(f^*(X) \times A) \to X \times f_!(A)
\end{displaymath}
is an [[isomorphism]]. But this is the [[Frobenius reciprocity]] condition on $f^*$.

\hypertarget{coreflectivity}{}\subsubsection*{{Coreflectivity}}\label{coreflectivity}

Locally connected toposes are [[coreflective subcategory|coreflective]] in [[Topos]]. See (\hyperlink{Funk}{Funk (1999)}).

\hypertarget{other_characterizations}{}\subsubsection*{{Other characterizations}}\label{other_characterizations}

\begin{itemize}%
\item Let $(\mathcal{C}, J)$ be a [[site]] and $S$ be a [[sieve]] on the object $U$. $S$ is called \emph{connected} when $S$ viewed as a full subcategory of $\mathcal{C}/U$ is connected. The \textbf{site} is called \emph{locally connected} if every sieve is connected. For a [[bounded geometric morphism]] $p:\mathcal{E}\to\mathcal{S}$ the following holds: \emph{$p$ is locally connected iff there exists a locally connected internal site in $\mathcal{S}$ such that $\mathcal{E}\simeq Sh(\mathcal{C},J)$.} (cf. \hyperlink{elephant}{Johnstone (2002)}, pp.656-658)


\item \hyperlink{Cara12}{Caramello (2012)} gives syntactic characterizations of [[geometric theories]] whose [[classifying topos]] is locally connected.



\end{itemize}
The same paper also contains the following characterization:

\begin{itemize}%
\item A Grothendieck topos is locally connected iff it has a [[separator|separating set]] of (coproduct) indecomposable objects.

\end{itemize}
\hypertarget{variations_in_the_context_of_the_nullstellensatz}{}\subsubsection*{{Variations in the context of the Nullstellensatz}}\label{variations_in_the_context_of_the_nullstellensatz}

\hyperlink{JS11}{Johnstone (2011)} studies several subclasses of locally connected geometric morphisms in the context of [[William Lawvere|Lawvere]]`s theory of [[cohesion]] and the [[Nullstellensatz]]. He calls a locally connected morphism $p$ \emph{stably locally connected} if $p_!$ preserves finite products. According to the \hyperlink{connect}{above remark} this implies that $p$ is connected. Slightly stronger is the preservation of all finite limits by $p_!$: these $p$ are called [[totally connected geometric morphisms]].

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

\begin{itemize}%
\item If the terminal [[global section]] geometric morphism $E \to Set$ is locally connected, one calls $E$ a [[locally connected topos]]. More generally, if $E\to S$ is locally connected, we may call $E$ a \emph{locally connected $S$-topos}.


\item Let $X$ be a [[topological space]] (or a [[locale]]) and $U\subseteq X$ an [[open subset]], with corresponding [[geometric embedding]] $j\colon Sh(U)\to Sh(X)$. Then any $A\in Sh(X)$ can be identified with a space (or locale) $A$ equipped with a [[local homeomorphism]] $A\to X$, in such a way that $Sh(X)/A \simeq Sh(A)$. Moreover, $j^*A \in Sh(U)$ can be identified with the pullback of $A\to X$ along $U$, and so $Sh(U)/j^*A \simeq Sh(j^*A)$ similarly. Noting that $j^*A \to A$ is again the inclusion of an open subset, and using the fact that the inverse image part of any open [[geometric embedding]] is cartesian closed, we see that $(j/A)^*\colon Sh(X)/A \to Sh(U)/j^*A$ is cartesian closed for any $A$. Hence $j$ is locally connected.



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

\begin{itemize}%
\item [[connected geometric morphism]]


\item [[local geometric morphism]]


\item [[totally connected geometric morphism]]



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

The case of $Sh(X)$ for a topological space $X$ was an exercise (p.417) in [[SGA4]]:

\begin{itemize}%
\item [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], \emph{Th\'e{}orie des Topos et Cohomologie Etale des Sch\'e{}mas ([[SGA4]])}, Springer LNM vol.269 (1972).

\end{itemize}
The concept relative to other bases was introduced in the following paper:

\begin{itemize}%
\item [[Michael Barr]], [[Robert Paré]], \emph{Molecular Toposes} , JPAA \textbf{17} (1980) pp.127-152.

\end{itemize}
The standard reference is section C3.3 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]} , Oxford UP 2002.

\end{itemize}
Further references include

\begin{itemize}%
\item [[Olivia Caramello]], \emph{Syntactic Characterizations of Properties of Classifying Toposes} , TAC \textbf{26} no.6 (2012) pp.176-193. (\href{http://www.tac.mta.ca/tac/volumes/26/6/26-06.pdf}{pdf})


\item [[Jonathon Funk]], \emph{The locally connected coclosure of a Grothendieck topos}, JPAA \textbf{137} (1999) pp.17-27.


\item [[Peter Johnstone]], \emph{Remarks on Punctual Local Connectedness} , TAC \textbf{25} no.3 (2011) pp.51-63. (\href{http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html}{pdf})


\item [[Ieke Moerdijk]], \emph{Continuous fibrations and inverse limits of toposes} , Comp. Math. \textbf{58} (1986) pp.45-72. (\href{http://archive.numdam.org/article/CM_1986__58_1_45_0.pdf}{pdf})


\item [[Ieke Moerdijk]], [[Gavin Wraith]], \emph{Connected and locally connected toposes are path connected} , Trans. AMS \textbf{295} (1986) pp.849-859. (\href{http://www.ams.org/journals/tran/1986-295-02/S0002-9947-1986-0833712-3/S0002-9947-1986-0833712-3.pdf}{pdf})



\end{itemize}
[[!redirects locally connected geometric morphisms]]



\end{document}