Lawvere–Tierney topology explained

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 (

j \circ true = true ), preserves intersections (j \circ \wedge = \wedge \circ (j x j) ), and is idempotent (j \circ j = j ).

j-closure

Given a subobject

s:S → tail A of an object A with classifier \chi _{s:A} → \Omega , then the composition j \circ \chi _{s} defines another subobject \bar s:\bar S → tail A of A such that s is a subobject of \bar s , and \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 \subseteq \bar s

• idempotence:

\bar s \equiv \bar \bar s

• preservation of intersections:

\overline{s \cap w} \equiv \bar s \cap \bar w

• preservation of order:

s \subseteq w \Longrightarrow \bar s \subseteq \bar w

• stability under pullback:

\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.