j-sheaf

An object $X$ is a $j$-sheaf when every map $A^{\prime }\to X$ defined on the domain of a j-dense monomorphism $A^{\prime }\hookrightarrow A$ extends uniquely to a map $A\to X$.

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 $i:{\Omega }_j\hookrightarrow \Omega$.

The full subcategory of $j$-sheaves is reflective. Its reflector $L$ sends every $j$-dense monomorphism to an isomorphism and preserves finite limits.

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 $A^{\prime }\hookrightarrow A$ factors as a $j$-closed mono $A^{\prime }\hookrightarrow A^{\prime \prime }$ followed by a $j$-dense mono $A^{\prime \prime }\hookrightarrow A$.