About the complement of a subobject in a topos

Question

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".

Answer 1

[C]an we conclude that $M\to X$ factors through $\neg S\to X$ when $M$ is an object of a family of generators of $\mathcal{E}$?

No. Consider the category of directed multigraphs (i.e., $\mathbf{Set}^{\bullet\rightrightarrows\bullet}$) and the graph $G$ with two vertices $a,b$, and two edges, a loop on $a$ and one between $a$ and $b$ (the direction doesn't matter). In this category the generators are the graph with a single vertex and no edges (call this $D$), and the graph with two vertices and one edge between them (call this $A$).

So let $S$ be the subgraph of $G$ consisting only of the vertex $a$. $\neg S$ is then the graph consisting only of the vertex $b$. So neither $S$ nor its Heyting pseudocomplement is "empty," but the morphism $A\to G$ that maps $\bullet\to\bullet$ onto the loop on $a$ factors through neither subobject.

There's a similar example to show that this can even fail in the (Boolean!) $\neg\neg$-sheaves on this category, so it's not even an artifact of Heyting pseudocomplementation. In general, I don't think there's a lot you can say about when this kind of factorization happens.

Answer 2

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.