Subsets $𝐴$ of a set $𝑋$ may classically be identified with a characteristic map ${𝜒}_{𝐴}:𝑋\to \text{𝖻𝗈𝗈𝗅}$. Intuitively, for every $𝑥:𝑋$, ${𝜒}_{𝐴}$ gives a truth value to the statement “$𝑥$ is in the subset $𝐴$”. In this manner, the domain of the characteristic map, $\text{𝖻𝗈𝗈𝗅}$, classifies the subsets of $𝑋$.

Generalizing over this principle, in a category $\mathrm{𝒞︀}$ an object $\mathrm{\Omega }$ is a subobject classifier if maps into it from some object $𝑐$ likewise uniquely identify a subobject of $𝑐$.

We can understand $\mathrm{\Omega }$ to behave like an object of truth values that are not necessarily boolean valued. A morphism $𝑝:𝑐\to \mathrm{\Omega }$ can be thought of like a predicate on $𝑐$. If $𝑐$ were a set, this would precisely be the characteristic function on it. However, this idea can generalize beyond sets. For instance in the category of graphs, $\mathrm{\Omega }$ is a cleverly constructed graph such that any graph homomorphism $𝑔\to \mathrm{\Omega }$ into it picks out a unique subgraph of $𝑔$.

There are always two (suggestively named) disjoint “points” of $\mathrm{\Omega }$, thought of as morphisms out of the terminal object $\text{𝗍𝗋𝗎𝖾},\text{𝖿𝖺𝗅𝗌𝖾}:\top \to \mathrm{\Omega }$. I think if we’re being careful, $\text{𝗍𝗋𝗎𝖾}$ may properly be the “subobject classifier” but I always use the term to refer to $\mathrm{\Omega }$ itself.

A subobject $𝜙:𝑑\hookrightarrow 𝑐$ induces a unique characteristic morphism ${𝑝}_{𝜙}:𝑐\to \mathrm{\Omega }$ such that

Moreover, the appropriate square must be a pullback.