Subobject Classifier

Definition

Let $\mathcal{C}$ be a category with a terminal object.
Then a subobject classifer for $\mathcal{C}$ is:

• a $\mathcal{C}$-object $\Omega$
• a morphism $\mathrm{true}:1\to \Omega$
  Such that the $\Omega$-axiom holds:
  For all monomorphisms $f:a\rightarrowtail b$ there is a unique morphism ${\chi }_f:b\to \Omega$ such that the following is a pullback:

${\chi }_f$ is the characteristic morphism for $f$.

Remarks

This is a cruicial requirement for a category to be a topos.