Why the category of sheaves of abelian groups does not have a subobject classifier?

Question

Let $X$ be a topological space. Why the category of sheaves of abelian groups over $X$ does not have a subobject classifier?

Answer 1

Let $\mathcal{C}$ be a (locally small and wellpowered) category with finite limits. We have a functor $\textrm{Sub} : \mathcal{C}^\textrm{op} \to \textbf{Set}$ sending each object $X$ in $\mathcal{C}$ to the set $\textrm{Sub} (X)$ of isomorphism classes of subobjects of $X$, with morphisms in $\mathcal{C}$ acting by pullback. Since $\mathcal{C}$ has pullbacks, each $\textrm{Sub} (X)$ is a meet semilattice. It is clear that, for every morphism $f : X \to Y$, $f^* : \textrm{Sub} (Y) \to \textrm{Sub} (X)$ preserves the top element. The pullback pasting lemma implies it also preserves binary meets. Thus the meet semilattice structure on $\textrm{Sub} (X)$ is natural (in the technical sense!), so if $\textrm{Sub}$ is a representable by some object $\Omega$ in $\mathcal{C}$, $\Omega$ would have the structure of an internal meet semilattice in $\mathcal{C}$, i.e. we would have morphisms $\top : 1 \to \Omega$ and ${\wedge} : \Omega \times \Omega \to \Omega$ satisfying certain equations (which I omit).

Now suppose $\mathcal{C}$ is also additive. If $\textrm{Sub}$ is representable then every $\textrm{Sub} (X)$ would also have the structure of an abelian group, and furthermore the abelian group structure and the meet semilattice structure would be compatible in the following sense: $$0 = \top$$ $$(x + y) \land (z + w) = (x \land z) + (y \land w)$$ But then the Eckmann–Hilton argument applies, so $\land$ and $+$ would be the same operation. In particular $\land$ would be cancellative – but $x \land x = x$ so we would have $x = 0$ for all $x$. Thus we have proved:

Proposition. An additive category with pullbacks has a subobject classifier if and only if every object is zero.

In particular, no non-trivial abelian category has a subobject classifier.