nLab Boolean locale

Definition

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

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

Related concepts

• complete Boolean algebra
• Stonean locale
• Stonean space