Boolean locale


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In higher category theory



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

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topological homotopy theory

Boolean locale


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.


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.