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 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 (), preserves intersections (), and is idempotent ().
j-closure
Given a subobject of an object A with classifier , then the composition defines another subobject of A such that s is a subobject of , and 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: idempotence: preservation of intersections: preservation of order: stability under pullback: .
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.
Internal point of view
The statement that a morphism 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 is a Lawvere–Tierney topology when the following three conditions are satisfied internally:
An equivalent definition uses the following three conditions instead:
This reveals that a Lawvere–Tierney topology is the same as a monad on the set of truth values partially ordered by implication, viewed as a posetal category. In particular, another equivalent definition is by the following conditions, which follow the standard convention of defining monads in functional programming by the ret and bind operators:
The second condition can be informally stated as: to prove a proposition of the form , one can without loss of generality replace any hypothesis with just .
References