Could not include topos theory - contents
basic constructions:
strong axioms
Barr’s theorem was originally conjectured by William Lawvere as an infinitary generalization of the Deligne completeness theorem for coherent toposes which can be expressed as the existence of a surjection $\mathcal{S}/K\to\mathcal{E}$ for a coherent topos $\mathcal{E}$ with set of points $K$. General toposes $\mathcal{E}$ may fail to have enough points but Michael Barr showed that a surjection from a suitable Boolean topos still exists.
As surjections permit the transfer of logical properties, Barr’s theorem has the following important consequence:
If a statement in geometric logic is deducible from a geometric theory using classical logic and the axiom of choice, then it is also deducible from it in constructive mathematics.
The proof of Barr’s theorem itself, however, is highly non-constructive.
If $\mathcal{E}$ is a Grothendieck topos, then there is a surjective geometric morphism
where $\mathcal{F}$ satisfies the axiom of choice.
Extensive discussion of the context of Barr’s theorem is in chapter 7 of:
For a discussion of the importance of this theorem in constructive algebra see
See also the following MO discussion: (link)