Let $\mathcal{E}$ be a topos and let $X$ be an object of $\mathcal{E}$. Let $S \to X$ be a subobject of $X$. We only know that the category of the subobject of $X$ is a Heyting Algebra, so we do not know if $S \vee \neg S \to X$ is $X \to X$. Let us assume that there is an arrow $M \to X$ that does not factor through $S$.
My generic question is: what do we need to have, in order to conclude that $M \to X$ factors through $\neg S \to X$?
If $\mathcal{E}=\textbf{Set}$, then we know that the answer is: we need to have that $M$ is the terminal object (that is, $M\to X$ is an element of $X$), that is (being $\mathcal{E}=\textbf{Set}$), we need to have that $M$ is an object of a family of generators of $\mathcal{E}$.
Seen this, I think my question turns into the following: can we conclude that $M \to X$ factors through $\neg S \to X$ when $M$ is an object of a family of generators of $\mathcal{E}$?
I wonder if this is true because, in some ways, arrows from objects of a family of generators are, in a topos, what is most similar to the notion of "elements of a set".
Best Answer
The object $M$ must be connected and the subobject $S \to X$ must be complemented. Then $X$ would be the coproduct of $S$ and $\neg S$, and the functor $\hom{(M, -)}$ would then preserve coproducts by the definition of connected object in an extensive category. It thus follows that every arrow from $M$ to $X$ must either factor through $S$ or through $\neg S$, but not both.