Lawvere–Tierney topology

Summary

A Lawvere–Tierney topology on a topos is a morphism $j \colon \Omega \to \Omega$ from the subobject classifier to itself. It behaves like a modality "it is locally true that". It is the elementary counterpart of a Grothendieck topology: on a presheaf topos, Lawvere–Tierney topologies correspond to Grothendieck topologies on the base category. Each such $j$ determines a full subcategory of $j$-sheaves, which is again a topos. The double negation topology $\neg\neg$ is a standard example and is used in forcing arguments.