Subobject classifier

Remarks

Equivalently, when $\mathcal{C}$ is locally small, there is a natural bijection between the subobjects of $X$ and $\Hom(X, \Omega)$, and $\Omega$ represents the subobject functor. Example: in $\mathbf{Set}$, take $\Omega = \set{0, 1}$ and $\top(*) = 1$; for $S \subseteq X$, the unique $\chi$ is the characteristic function, since the pullback of $\set{1}$ along $\chi$ is $\chi^{-1}(1) = S$. Non-example: the category of groups has no subobject classifier. Its terminal object is the trivial group, so every pullback of $\top \colon 1 \to \Omega$ is the kernel of a homomorphism, which is a normal subgroup. A subgroup of order $2$ in $S_3$ is not normal, so it is not classified. In a presheaf category $\mathbf{Set}^{\mathcal{C}^{\mathrm{op}}}$, the object $\Omega(c)$ is the set of sieves on $c$, and $\top$ picks the maximal sieves.