Lawvere–Tierney topology

In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary elementary 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 ( $j\circ {\mbox{true}}={\mbox{true}}$ ), preserves intersections ( $j\circ \wedge =\wedge \circ (j\times j)$ ), and is idempotent ( $j\circ j=j$ ).

j-closure

Given a subobject $s:S\rightarrowtail A$ of an object A with classifier $\chi _{s}:A\rightarrow \Omega$ , then the composition $j\circ \chi _{s}$ defines another subobject ${\bar {s}}:{\bar {S}}\rightarrowtail A$ 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.
Lawvere–Tierney topologies on the effective topos generalize the notion of an oracle in computability theory.^[1]

Internal point of view

The statement that a morphism $j:\Omega \to \Omega$ is a Lawvere–Tierney topology can be expressed purely in the internal language of the elementary topos. Indeed, a rephrasing of the definition is that $j$ is a Lawvere–Tierney topology when the following three conditions are satisfied internally:

$j(\top )=\top$
$\forall \,p:\Omega ,j(j(p))\Leftrightarrow j(p)$
$\forall \,p\,q:\Omega ,j(p\land q)\Leftrightarrow j(p)\land j(q)$

An equivalent definition uses the following three conditions instead:^[2]^[a]

$\forall \,p\,q:\Omega ,(p\Rightarrow q)\Rightarrow (j(p)\Rightarrow j(q))$
$\forall \,p:\Omega ,p\Rightarrow j(p)$
$\forall \,p:\Omega ,j(j(p))\Rightarrow j(p)$

This reveals that a Lawvere–Tierney topology is the same as a monad on the set of truth values $\Omega$ partially ordered by implication, viewed as a posetal category.

Notes

^ The most interesting part of the equivalence is the fact that if $j$ is a monad on $\Omega$ and $p,q:\Omega$ are such that $j(p)$ and $j(q)$ , then $j(p\land q)$ . For this, observe that $p\Rightarrow p\land j(q)$ trivially since $j(q)$ holds, so $j(p)\Rightarrow j(p\land j(q))$ , but we assumed $j(p)$ , so $j(p\land j(q))$ holds. Assuming $p$ and $j(q)$ , we have $q\Rightarrow p\land q$ , hence $j(q)\Rightarrow j(p\land q)$ , hence $j(p\land q)$ ; this shows that $p\land j(q)\Rightarrow j(p\land q)$ . We deduce that $j(p\land j(q))\Rightarrow j(j(p\land q))$ , and we know that $j(p\land j(q))$ , so $j(j(p\land q))$ holds, so $j(p\land q)$ holds.