Lawvere–Tierney topology


In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by and Myles Tierney.

Definition

If E is a topos, then a topology on E is a morphism j from the subobject classifier Ω to Ω such that j preserves truth, preserves intersections, and is idempotent.

''j''-closure

Given a subobject of an object A with classifier, then the composition defines another subobject of A such that s is a subobject of, and is said to be the j-closure of s.
Some theorems related to j-closure are :
Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets over C.