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 Grothendieck topology or topology or geometric modality. They were introduced by William Lawvere (1971) 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 (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, F. W. (1971), "Quantifiers and sheaves", Actes du Congrès International des Mathématiciens (Nice, 1970), 1, Paris: Gauthier-Villars, pp. 329–334, MR 0430021
• Mac Lane, Saunders; Moerdijk, Ieke (1994), Sheaves in geometry and logic. A first introduction to topos theory, Universitext, New York: Springer-Verlag. Corrected reprint of the 1992 edition.