nLab complete Boolean algebra

Definition

A complete Boolean algebra is a complete lattice that is also a Boolean algebra. Since lattice homomorphisms of Boolean algebras automatically preserves the Boolean structure, the complete Boolean algebras form a full subcategory CompBoolAlg of CompLat.

Category of complete Boolean algebras

A natural notion of morphism for complete Boolean algebras is that of a continuous homomorphism of Boolean algebras, also known as complete Boolean homomorphisms. These can be defined as homomorphisms of Boolean algebras that preserve suprema, or, equivalently, infima. It suffices to require preservation of suprema of directed subsets.

With this notion of morphisms, complete Boolean algebras form a category.

Relation to Stonean locales and Boolean locales

The category of complete Boolean algebras is equivalent to the category of Boolean locales and Stonean locales. The latter fact is also known as the (localic) Stonean duality.

In the presence of the axiom of choice, the category of Stonean locales is equivalent to the category of Stonean spaces, so the latter is contravariantly equivalent to the category of complete Boolean algebras. The latter fact is also known as the (traditional) Stonean duality.

Stonean duality

The Stone duality establishes a contravariant equivalence? of categories between the category of Boolean algebras and the category of Stone spaces. The latter is a full subcategory of the category Top of topological spaces and continuous maps on compact totally disconnected Hausdorff topological spaces.

Recall that a Stonean space is a compact extremally disconnected Hausdorff topological space. Morphisms of Stonean spaces are defined to be open continuous maps. Restricting the Stone duality produces a contravariant equivalence? between the category of complete Boolean algebras and the category of Stonean spaces. See Corollary 6.10(2) in Bezhanishvili.

CABAs

Assuming excluded middle, complete atomic Boolean algebras are (up to isomorphism) precisely power sets. In fact, taking power sets defines a fully faithful functor from the opposite category of Set to Comp Bool Alg whose essential image consists of the complete atomic boolean algebras. See at Set – Properties – Opposite category. These abstract representations of power sets are important enough to have their own abbreviation: ‘CABA’.

This property of CABAs is not applicable in constructive mathematics, where power sets are rarely boolean algebras. However, we can use discrete locales instead (or rather, their corresponding frames). That is, define a CABA to be (not a complete atomic boolean algebra but) a frame $X$ such that the locale maps $X \to 1$ and $X \to X \times X$ (which in the category of frames are maps $0 \to X$ and $X + X \to X$) are open (as locale maps). (Of course, that $X \to 1$ is open is the condition that $X$ is overt.) Then it should be (I will check) a classical theorem that CABAs and complete atomic boolean algebras are the same, and a constructive theorem that CABAs and power sets are the same (in the same functorial manner as above).

Realtedly, CABAs are the same as completely distributive complete Boolean algebras.

Another approach is via overlap algebras. An overlap algebra is a frame with two extra conditions (one of which is overtness of the corresponding locale). Classically, overlap algebras are the same thing as complete Boolean algebras; constructively, atomic overlap algebras are the same thing as powersets. See Ciraulo 2010.

Algebraicity

Complete Boolean algebras are the models of an algebraic theory (in which the operations, notably $j$-indexed suprema and infima, have arities $j$ unbounded by any cardinal). It follows from general principles that the underlying-set functor $U: CompBoolAlg \to Set$ preserves and reflects limits and isomorphisms.

However, this functor $U$ is not monadic; in fact, it does not even possess a left adjoint. Indeed, while the free complete Boolean algebra on a finite set $X$ exists and coincides with the free Boolean algebra on $X$ (it is finite, being isomorphic to the double power set $P(P X)$), we have

Theorem (Gaifman-Hales; Solovay)

There is no free complete Boolean algebra on countably many generators.

As a consequence, $CompBoolAlg$ is not cocomplete (otherwise there would exist a countable coproduct of copies of $P(P 1)$, which is ruled out by the previous theorem).

Related concepts

• Boolean locale
• Stonean locale
• completely distributive lattice
• $\sigma$-complete Boolean algebra

References

For instance around theorem 2.4 of

• Jaap van Oosten, Basic category theory (pdf)

and

• PlanetMath, representing a complete atomic Boolean algebra by power set

The Stone duality for complete Boolean algebras is explained in

• Guram Bezhanishvili?,

  Stone duality and Gleason covers through de Vries duality. Topology and its Applications 157 (2010), 1064–1080.

For overlap algebras, see

• Francesco Ciraulo?, 2010. The Role of the Overlap Relation in Constructive Mathematics. Slides.