Could not include topos theory - contents
A geometric morphism is locally connected if it behaves as though its fibers are locally connected spaces. In particular, a Grothendieck topos $E$ is locally connected iff the unique geometric morphism to Set (the terminal Grothendieck topos, i.e. the point in the category Toposes of toposes) is locally connected.
A geometric morphism $(f^* \dashv f_*) : F \underoverset{f_*}{f^*}{\leftrightarrows} E$ is locally connected if it satisfies the following equivalent conditions:
It is essential, i.e. $f^*$ has a left adjoint $f_!$, and moreover $f_!$ can be made into an $E$-indexed functor.
For every $A\in E$, the functor $f^* \colon E/A \to F/f^*A$ is cartesian closed.
$f^*$ 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.
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 if and only if $f_!$ preserves the terminal object. However, not every connected geometric morphism is locally connected.
Over the base topos $E =$ Set every connected topos which is essential is automatically locally connected.
This is because the required Frobenius reciprocity condition
is automatically satisfied, using that cartesian product with a set is equivalently a coproduct
that the left adjoint $f_!$ preserves coproducts, and that for $f^*$ full and faithful we have $f_! f^* \simeq Id$.
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 an natural isomorphism
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
is a bijection. By adjunction this is the same as
Again by Yoneda, this is a bijection precisely if
is an isomorphism. But this is the Frobenius reciprocity condition on $f^*$.
Locally connected toposes are coreflective in Topos. See (Funk).
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 locally connected $S$-topos.
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.
The canonical textbook reference is section C3.3 of
Further references include
J. Funk, The locally connected coclosure of a Grothendieck topos, Journal of Pure and Applied Algebra Volume 137, Issue 1, 7 (1999), Pages 17-27
Ieke Moerdijk, Gavin Wraith, Connected and locally connected toposes are path connected (JSTOR)