 Subobject classifier

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 category theory, a **subobject classifier** is a fundamental concept that generalizes the notion of characteristic functions and subobjects in set theory. It plays an important role in topos theory and categorical logic.