Closed-subobject classifier (${\Omega }_j$)

The closed-subobject classifier is the equalizerIt classifies $j$-closed subobjects and is itself a j-sheaf. The local operator factors as a split epimorphism $q:\Omega \twoheadrightarrow {\Omega }_j$ followed by the canonical inclusion $i:{\Omega }_j\hookrightarrow \Omega$.