Could not include topos theory - contents
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 for a coherent topos with set of points . General toposes 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:
The proof of Barr’s theorem itself, however, is highly non-constructive.
where 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
For proof-theoretic approaches to Barr’s theorem see
See also the following MO discussion: (link)