---
tag: E6H1CTT439
kind: concept
name: Lawvere–Tierney topology
msc: [18B25, 18F10, 03G30]
sources:
  - url: https://en.wikipedia.org/wiki/Lawvere%E2%80%93Tierney_topology
same_as:
  wikidata: Q30704544
  wikipedia: Lawvere–Tierney topology
---

## Summary

A Lawvere–Tierney topology on a [[topos|6837RS7N82]] is a morphism $j \colon \Omega \to \Omega$ from the [[subobject classifier|NKP5TPQ79X]] to itself.
It behaves like a modality "it is locally true that".
It is the elementary counterpart of a [[Grothendieck topology|GDRFSJ8E2B]]: on a presheaf topos, Lawvere–Tierney topologies correspond to Grothendieck topologies on the base category.
Each such $j$ determines a full subcategory of $j$-sheaves, which is again a topos.
The double negation topology $\neg\neg$ is a standard example and is used in forcing arguments.

## Explanations

### L1 human

In a topos, truth values may be partial, like "true on this part of a space".
A Lawvere–Tierney topology is a rule that makes some statements count as true once they hold "locally enough".
Applying the rule twice gives nothing new, true statements stay true, and the rule respects "and".
For example, take presheaves on the open sets of a space, with the topology given by open covers.
Then a property counts as true on $U$ as soon as it holds on the pieces of a cover of $U$.