Subobject Classifier

Let $\mathcal{E}$ be a topos (elementary or Grothendieck) with terminal object $1$.

Definition. A subobject classifier in $\mathcal{E}$ is an object $\Omega$ together with a morphism $\mathrm{true}: 1 \to \Omega$ such that for every monomorphism $m: S \hookrightarrow X$, there exists a unique morphism $\chi_m: X \to \Omega$ (the classifying map or characteristic morphism) making the following square a pullback:

Equivalently, $\Omega$ represents the functor $\mathrm{Sub}: \mathcal{E}^{\mathrm{op}} \to \mathbf{Set}$ that sends each object $X$ to its set of subobjects: there is a natural isomorphism $\mathrm{Sub}(X) \cong \mathrm{Hom}_{\mathcal{E}}(X, \Omega)$.

Proposition. In any elementary topos, the subobject classifier $\Omega$ carries the structure of an internal Heyting algebra. The operations $\wedge, \vee, \Rightarrow: \Omega \times \Omega \to \Omega$ and $\neg: \Omega \to \Omega$ are determined by requiring that the classifying maps respect the corresponding operations on subobjects (intersection, union, Heyting implication, complement). In particular, the internal logic of the topos is intuitionistic: the law of excluded middle $p \vee \neg p = \top$ holds for all $p$ iff $\Omega$ is a Boolean algebra.

Proof sketch. Subobjects of any object $X$ form a Heyting algebra under intersection, union, and Heyting implication of subobjects. Since $\mathrm{Sub}(X) \cong \mathrm{Hom}(X, \Omega)$, the Yoneda lemma transfers these operations to morphisms $\Omega \times \Omega \to \Omega$.

Proposition. The subobject classifier is unique up to unique isomorphism (when it exists), being a representing object of the subfunctor $\mathrm{Sub}$.

Proposition. A Lawvere-Tierney topology $j: \Omega \to \Omega$ is a closure operator on $\Omega$ that preserves meets. The image $\Omega_j = \{p \in \Omega : j(p) = p\}$ is the subobject classifier of the subtopos $\mathrm{Sh}_j(\mathcal{E})$.

Examples.

• $\mathbf{Set}$: The subobject classifier is $\Omega = \{0, 1\}$ with $\mathrm{true}: \{*\} \to \{0, 1\}$ picking out $1$. The classifying map $\chi_m$ is the usual characteristic function. The internal logic is classical (Boolean algebra).
• $\mathrm{Sh}(X)$: For a topological space $X$, the subobject classifier is the sheaf $\Omega(U) = \{V \subseteq U : V \text{ open}\}$, the sheaf of open subsets. The global elements $\mathrm{Hom}(1, \Omega) \cong \mathcal{O}(X)$ are the open sets of $X$, so truth values are “open truth values” — a proposition can be “true on $U$” for various opens $U$.
• $\mathbf{Set}^{G}$ (presheaves on a group): For a group $G$, the subobject classifier in the topos of $G$-sets is $\Omega = \{H \leq G : H \text{ is a subgroup}\}$ with $G$-action by conjugation, and $\mathrm{true}$ picks out $G$ itself.