A subobject classifier in a category with finite limits is an object Ω together with a morphism true: 1 → Ω such that for every monomorphism m: S ↪ X, there is a unique morphism χ_m: X → Ω (the classifying map) making the following a pullback square:
S → 1 ↓ ↓ true X → Ω χ_m
The classifying map χ_m sends each element of X to “true” if it belongs to S and to some other truth value otherwise. In Set, Ω is the two-element set {0, 1} and χ_m is the characteristic function. In a topos of sheaves over a space, Ω is the sheaf of open sets and truth values are “how openly true” a proposition is — not just true or false, but true on which opens.
The subobject classifier is what gives a topos its internal logic. The elements of Ω are the truth values of the topos, and the operations on Ω (meet, join, implication, negation) define the internal propositional calculus. In an elementary topos, this logic is intuitionistic: Ω forms a Heyting algebra, not necessarily a Boolean algebra. The law of excluded middle (p ∨ ¬p = true) may fail.
A Lawvere-Tierney topology j: Ω → Ω acts on the subobject classifier to redefine which truth values count as “closed” or “stable,” producing a subtopos with a coarser notion of truth.
In the semiotic universe, the Heyting algebra H plays the role of a generalized subobject classifier: its elements are the truth values of the semiotic logic, and the closure operator j determines which semiotic distinctions are stable under interpretation.