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 William Lawvere (1971) and Myles Tierney.

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 (

of an object A with classifier

defines another subobject

is said to be the j-closure of s. Some theorems related to j-closure are (for some subobjects s and w of A): Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets over C.

Commutative diagrams showing how j -closure operates. Ω and t are the subobject classifier . χ s is the characteristic morphism of s as a subobject of A and is the characteristic morphism of which is the j -closure of s . The bottom two squares are pullback squares and they are contained in the top diagram as well: the first one as a trapezoid and the second one as a two-square rectangle.