nLab sigma-complete Boolean algebra

Idea

A $\sigma$-complete Boolean algebra is a Boolean algebra which is also a $\sigma$-complete lattice; that is, it is a poset with countable limits and colimits that is also cartesian closed and satisfies the law of excluded middle.

 Examples

• Assuming excluded middle, the set of truth values is a $\sigma$-complete Boolean algebra.

• Assuming the limited principle of omniscience, the boolean domain is a $\sigma$-complete Boolean algebra.

• Assuming excluded middle, any $\sigma$-algebra on a set $X$ is a $\sigma$-complete Boolean algebra.

 Related concepts

• sigma-complete lattice, sigma-frame, sigma-complete Heyting algebra

• Boolean algebra, complete Boolean algebra

• sigma-algebra, measurable space, measure space

• point-free measure theory?, Loomis-Sikorski duality

References

• Thierry Coquand, Jonas Höfer?, Hugo Moeneclaey, Model for Synthetic Stone Duality (draft pdf)

• Roman Sikorski, Boolean Algebras, Springer, 1969.

• Ruiyuan Chen, A universal characterization of standard Borel spaces. The Journal of Symbolic Logic, 88(2), 2023. (arXiv)

• Tobias Fritz and Antonio Lorenzin, Categories of abstract and noncommutative measurable spaces, 2025. (arXiv)