Lawvere–Tierney topology

Analog of Grothendieck 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.

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 {\displaystyle j\circ {\mbox{true}}={\mbox{true}}} ), preserves intersections ( j ∘ ∧ = ∧ ∘ ( j × j ) {\displaystyle j\circ \wedge =\wedge \circ (j\times j)} ), and is idempotent ( j ∘ j = j {\displaystyle j\circ j=j} ).

j-closure

Commutative diagrams showing how j-closure operates. Ω and t are the subobject classifier. χs is the characteristic morphism of s as a subobject of A and j ∘ χ s {\displaystyle j\circ \chi _{s}} is the characteristic morphism of s ¯ {\displaystyle {\bar {s}}} 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.

Given a subobject s : S ↣ A {\displaystyle s:S\rightarrowtail A} of an object A with classifier χ s : A → Ω {\displaystyle \chi _{s}:A\rightarrow \Omega } , then the composition j ∘ χ s {\displaystyle j\circ \chi _{s}} defines another subobject s ¯ : S ¯ ↣ A {\displaystyle {\bar {s}}:{\bar {S}}\rightarrowtail A} of A such that s is a subobject of s ¯ {\displaystyle {\bar {s}}} , and s ¯ {\displaystyle {\bar {s}}} is said to be the j-closure of s.

Some theorems related to j-closure are (for some subobjects s and w of A):

  • inflationary property: s ⊆ s ¯ {\displaystyle s\subseteq {\bar {s}}}
  • idempotence: s ¯ ≡ s ¯ ¯ {\displaystyle {\bar {s}}\equiv {\bar {\bar {s}}}}
  • preservation of intersections: s ∩ w ¯ ≡ s ¯ ∩ w ¯ {\displaystyle {\overline {s\cap w}}\equiv {\bar {s}}\cap {\bar {w}}}
  • preservation of order: s ⊆ w ⟹ s ¯ ⊆ w ¯ {\displaystyle s\subseteq w\Longrightarrow {\bar {s}}\subseteq {\bar {w}}}
  • stability under pullback: f − 1 ( s ) ¯ ≡ f − 1 ( s ¯ ) {\displaystyle {\overline {f^{-1}(s)}}\equiv f^{-1}({\bar {s}})} .

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" (PDF), Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 1, Paris: Gauthier-Villars, pp. 329–334, MR 0430021, S2CID 2337874, archived from the original (PDF) on 2018-03-17
  • Mac Lane, Saunders; Moerdijk, Ieke (2012) [1994], Sheaves in geometry and logic. A first introduction to topos theory, Universitext, Springer, ISBN 978-1-4612-0927-0
  • McLarty, Colin (1995) [1992], Elementary Categories, Elementary Toposes, Oxford Logic Guides, vol. 21, Oxford University Press, p. 196, ISBN 978-0-19-158949-2