Subobject classifier ($\top :1\to \Omega$)

A subobject classifier is a monomorphism $\top :1\to \Omega$ such that every monomorphism $A^{\prime }\hookrightarrow A$ is, uniquely, a pullback of $\top$ along a characteristic map ${\chi }_{A^{\prime }}:A\to \Omega$.

In the covariant functor category $[\mathbb{N},\mathbf{Set}]$, the classifier at $n$ consists of thresholds $k\ge n$ and the empty threshold $\infty$. Its transition to $n+1$ sends $k$ to $\max (k,n+1)$. The displayed maps commute with transitions and are all surjective, so define an epimorphism. They identify $n$ and $n+1$, so the endomorphism is not an isomorphism.

In a category with finite limits and a subobject classifier, every monic endomorphism $f$ of $\Omega$ is an involution. The subobject $g:U\hookrightarrow \Omega$ classified by $f$ is subterminal, since $fg=\top {!}_U$ is monic. If $V\hookrightarrow U$ is classified by $g$, pullback uniqueness gives $f\top {!}_U=g$, hence $f^{2}g=g$. If $u$ classifies $U\hookrightarrow 1$, the pullback of $g$ along $f$ is $\top {!}_U$, so $f^2(p)=u\wedge p$. Monicity implies $u=\top$, because $u\wedge u=u\wedge \top$, proving the displayed identity.

The subobject classifier $\Omega$ of a topos carries internal truth, falsity, meet, join and implication. Its generalized elements over $X$ are subobjects of $X$; implication satisfies $R\le (P\Rightarrow Q)$ exactly when $R\cap P\le Q$. Pullback-compatible operations make this an internal Heyting algebra, not merely a structure on global truth values.