nLab
Barr's theorem

Contents

Idea

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 𝒮/K\mathcal{S}/K\to\mathcal{E} for a coherent topos \mathcal{E} with set of points KK. 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.

Statement

Theorem

If \mathcal{E} is a Grothendieck topos, then there is a surjective geometric morphism

\mathcal{F} \to \mathcal{E}

where \mathcal{F} satisfies the axiom of choice.

Remark

Deligne's completeness theorem says that a coherent Grothendieck topos has enough points in SetSet and this corresponds to the Gödel-Henkin completeness theorem for first-order theories. Similarly, Barr’s theorem can interpreted as saying that a Grothendieck topos has sufficient Boolean-valued points and is in turn closely related to Mansfield’s Boolean-valued completeness theorem for infinitary first-order theories.

References

Extensive discussion of the context of Barr’s theorem is in chapter 7 of:

  • P. T. Johnstone, Topos Theory , Academic Press New York 1977 (Dover reprint 2014).

For a discussion of the importance of this theorem in constructive algebra see

  • Gavin Wraith, Intuitionistic algebra: some recent developments in topos theory In Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pages 331–337, Helsinki, 1980. Acad. Sci. Fennica. (pdf)

For proof-theoretic approaches to Barr’s theorem see

  • Sara Negri, Contraction-free sequent calculi for geometric theories with an application to Barr’s theorem , Archive for Mathematical Logic 42 (2003) pp.389–401.

For the connection with the Boolean-valued completeness theorem see

  • R. Goldblatt, Topoi - The Categorical Analysis of Logic , North-Holland 1982². (Dover reprint)

  • R. Mansfield, The Completeness Theorem for Infinitary Logic , JSL 37 no.1 (1972) pp.31-34.

  • Michael Makkai, Gonzalo E. Reyes, First-order Categorical Logic , LNM 611 Springer Heidelberg 1977.

See also the following MO discussion: (link)

Revised on July 2, 2015 16:12:30 by Thomas Holder (89.204.155.81)