Let $S$ be a Boolean topos.


Let ${f : E \rightarrow S}$ be a hyperconnected geometric morphism.


For $s$ in $S$ and $x$ in $E$ let ${\pi : f^\* s \times x \rightarrow x}$ be the obvious projection in $E$.


Let ${u \rightarrow f^\* s \times x}$ be a complemented subobject of ${f^\* s \times x}$.


Is the image of $u$ along $\pi$ a complemented subobject of $x$?


(Note: this is a continuation of [Images of complemented subobjects in toposes](https://mathoverflow.net/questions/395966/images-of-complemented-subobjects-in-toposes))



Images of complemented subobjects in hyperconnected toposes over Boolean bases
No.


Take $S$ to be Sets, then for any set $s$, $f^\* s \times x$ is the coproduct of $s$-copies of $X$, and a complemented subobject of $f^\* s \times x$ is the same as an $s$ indexed collection of complemented subobject of $x$. The the image by the projection $f^\*s \times x \to x$ is their union.


So, for $S= Set$, your question translate into: Are complemented subobject closed under union in hyperconnected topos ? This is not the case.


For example, the [Jonsson-Tarski](https://ncatlab.org/nlab/show/J%C3%B3nsson-Tarski+topos) topos is an hyperconnected topos, which has a slice equivalent to Sh(K) for K the Cantor space. In particular it has an object $X$ such that the subobject of $X$ identifies with the open subset of the Cantor space and every open subset of the Cantor is a union of clopen (i.e. complemented open)


Edit: It was asked in the comment to give a description of the left adjoint $f^\*$ from Sets to JT of the global section functor $f\_\*$ topos. It happens that there is a fairly explicit description of it. The trick is that the object $X$ such that $JT/X = Sh(K)$ is actually the free JT algebra on one generator. Meaning that given an object $Z$ of JT one recover the corresponding JT-algebra by looking at $Hom(X,Z)$.


So what you are asking is what is $Hom(X,f^\*s)$, Let me call $h$ the geometric morphisms $JT/X \to JT$ and $g:JT/X = Sh(K) \to Sets$ the unique geometric morphisms. Now, as $h$ is étale, it has an exceptional further left adjoint $h\_!$ and $X= h\_! 1$, so, $Hom(X,Z)$ can be more generally written as $Hom(h\_! 1 ,Z ) = Hom(1,h^\*Z)$ and hence $Hom(X,f^\*s) = Hom(1,h^\* f^\* s) = Hom(1,g^\* s) $


And $g^\*s$ is the sheaf of locally constant functions with values in $s$ on $K$, so $Hom(1,g^\*s)$ is the set of locally constant functions from $K$ to $s$.


So at the end of the day one has:


$$ f^\* (s) = \{ \text{locally constant functions } K \to s\} $$


where $K=2^\mathbb{N}$ is the Cantor space. With some additional calculation (related to how $JT/X$ identifies with $Sh(K)$), one can show that the JT algebra structure on $f^\*(s)$ is obtained from the homeomorphisms $K = K \coprod K$ obtained by separating along the first component of $K = 2^\mathbb{N}$


