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Keywords:
sheaves on a complete Boolean algebra; injective Boolean algebra; complete Boolean algebra; injective complete Boolean algebra; absolute frame retract
sheaves on a complete Boolean algebra; injective Boolean algebra; complete Boolean algebra; injective complete Boolean algebra; absolute frame retract
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.
References:
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