Subobject-classifier preservation criterion for a sheaf reflector
= Subobject-classifier preservation criterion for a sheaf reflector
For a <local operator> $j$, the following are equivalent: $L$ preserves the subobject classifier; $L(q):L\Omega\to\Omega_j$ is an isomorphism; the canonical inclusion $i:\Omega_j\hookrightarrow\Omega$ is $j$-dense; and every monomorphism $A'\hookrightarrow A$ factors as a $j$-closed mono $A'\hookrightarrow A''$ followed by a $j$-dense mono $A''\hookrightarrow A$.