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On the injectivity of Boolean algebras

Bernhard Banaschewski

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Abstract

summary:The functor taking global elements of Boolean algebras in the topos $\text{$\bold{Sh}\frak B$}$ of sheaves on a complete Boolean algebra $\frak B$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\frak B$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.

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summary:The functor taking global elements of Boolean algebras in the topos $\text{$\bold{Sh}\frak B$}$ of sheaves on a complete Boolean algebra $\frak B$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\frak B$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.

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Available abstract

summary:The functor taking global elements of Boolean algebras in the topos $\text{$\bold{Sh}\frak B$}$ of sheaves on a complete Boolean algebra $\frak B$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\frak B$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.

Key concepts: Stone's representation theorem for Boolean algebras, Complete Boolean algebra, Free Boolean algebra, Ultrafilter, Mathematics, Boolean algebras canonically defined, Two-element Boolean algebra, Boolean algebra

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