---
tag: C9XM3X631W
kind: definition
name: Lawvere–Tierney topology
msc: [18B25, 18F10, 03G30]
defines: E6H1CTT439
uses: [6837RS7N82, NKP5TPQ79X]
sources:
  - url: https://en.wikipedia.org/wiki/Lawvere%E2%80%93Tierney_topology
---

## Definition

Let $\mathcal{E}$ be a [[topos|6837RS7N82]] with [[subobject classifier|NKP5TPQ79X]] $\top \colon 1 \to \Omega$.
Let $\wedge \colon \Omega \times \Omega \to \Omega$ be the characteristic morphism of the subobject $(\top, \top) \colon 1 \to \Omega \times \Omega$.
A *Lawvere–Tierney topology* on $\mathcal{E}$ is a morphism $j \colon \Omega \to \Omega$ such that:

1. $j \circ \top = \top$;
2. $j \circ j = j$;
3. $j \circ \wedge = \wedge \circ (j \times j)$.

## Remarks

Example: the identity $\id_\Omega$ satisfies all three equations.
Example: the double negation $\neg\neg \colon \Omega \to \Omega$ is a Lawvere–Tierney topology in every topos.
Non-example: in $\mathbf{Set}$, with $\Omega = \set{0, 1}$ and $\top = 1$, the negation $\neg$ sends $1$ to $0$, so $\neg \circ \top \ne \top$ and clause 1 fails.
An object $F$ is a $j$-sheaf when, for every monomorphism $m \colon A \to B$ whose characteristic morphism satisfies $j \circ \chi_m = \top \circ {!}$, where ${!} \colon B \to 1$ is the unique morphism, every morphism $A \to F$ extends uniquely along $m$.
The $j$-sheaves form a topos $\mathrm{Sh}_j(\mathcal{E})$, and the inclusion has a left adjoint that preserves finite limits.
Another name for a Lawvere–Tierney topology is a local operator.