nLab
Boolean locale

Boolean locale

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Boolean locale

Definition

A locale LL is Boolean if all of its opens (i.e. elements of the corresponding frame) are complemented, i.e., for any aO(L)a\in O(L) we have a¬a=1a\vee\neg a=1. Equivalently, we could say that for all aAa\in A we have a=¬¬aa=\neg\neg a.

Properties

The underlying frames of Boolean locales are precisely complete Boolean algebras.

Maps of Boolean locales are automatically open. Their underlying morphisms of frames? are precisely complete Boolean homomorphisms?, i.e., suprema-preserving homomorphisms of Boolean algebras.

Thus, the opposite category of Boolean locales is precisely the category of complete Boolean algebras and complete homomorphisms thereof.

By Stonean duality? the category of Boolean locales is equivalent to the category of Stonean locales and open maps thereof. (Not every map of locales? between Stonean locales is open, unlike for Boolean locales.)

Any locale LL has a maximal dense sublocale, whose opens are precisely the regular elements of O(L)O(L), i.e., elements aO(L)a\in O(L) such that a=¬¬aa=\neg\neg a. This sublocale is Boolean and is also known as the double negation sublocale.

Last revised on July 23, 2019 at 20:50:18. See the history of this page for a list of all contributions to it.