Closure operation of a local operator

If a subobject $A^{\prime }\hookrightarrow A$ has characteristic map $\chi :A\to \Omega$, its $j$-closure is classified by $j\chi$. This closure operation is inflationary, idempotent, and stable under pullback.

A mono is j-closed if it equals its closure under the local operator $j$. Its characteristic map factors through the closed-subobject classifier ${\Omega }_j$. A j-closed subobject of a j-sheaf is a j-sheaf; a mono that is both closed and j-dense is an isomorphism.

A monomorphism is $j$-dense when its $j$-closure is its whole codomain. It is $j$-closed when it equals its closure. Dense monomorphisms are stable under pullback, and a monomorphism that is both dense and closed is an isomorphism.