B. Banaschewski
On the injectivity of Boolean algebras

Comment.Math.Univ.Carolinae 34,3 (1993) 501-511.

Abstract: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.

Keywords: sheaves on a complete Boolean algebra, injective Boolean algebra, complete Boolean algebra, injective complete Boolean algebra, absolute frame retract
AMS Subject Classification: 03E25, 03E40, 03G05, 06A23, 06E99