A Lawvere-Tierney topology on a topos defines naturally a certain closure operation on subobjects. A subobject inclusion is called dense (a dense monomorphism) if its closure is an isomorphism. In other words, a dense subobject of an object is a subobject whose closure is all of .
\bar A \hookrightarrow B
be the subobject classified by .
The monomorphism is called a dense monomorphism if , that is if is an isomorphism.
Recall that when is a presheaf Grothendieck topos then Lawvere-Tierney topologies on are in bijection with Grothendieck topologies on (making a site). In this case there is the notion of local epimorphism and local isomorphism in with respect to this topology.
We have in this case:
Dense monomorphisms appear around p. 223 of