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 Grothendieck topology or topology or geometric modality. They were introduced by William Lawvere (1971) and Myles Tierney.
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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 (j(true) = true), commutes with intersections (j∧ = ∧(j×j)), and is idempotent (jj = j).
Examples
Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets over C.
References
Lawvere–Tierney topology Wikipedia(Text) CC BY-SA