Lemma 2.51.
The subobject classifier is \(0\)-truncated. That is, for every object \(X \in T\), the anima \(\Hom_T(X,\Omega)\) is 0-truncated.
Proof
By definition, \(\Hom_T(X,\Omega)\) is the full subanima of \((T_{/X})^{\simeq}\) spanned by the monomorphisms \(U \hookrightarrow X\). If \(U \hookrightarrow X\) is a monomorphism, then its anima of automorphisms in \(T_{/X}\) is contractible. Thus this full subanima has no nontrivial self-identifications and is therefore \(0\)-truncated.