Could not include topos theory - contents
basic constructions:
strong axioms
further
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)