nLab Deligne completeness theorem

Idea

The Deligne completeness theorem, or Deligne’s theorem, as it is also called, was initially proved by Pierre Deligne in an appendix of SGA4¹ in the context of algebraic geometry as a theorem concerning the abundance of points for a coherent topos.

When in the early 70s the connection between topos theory and logic became manifest William Lawvere (1975) pointed out that the theorem may be viewed as a variant of the classical Gödel-Henkin completeness theorem for first-order logic: since points of a topos $\mathcal{E}$ correspond to set-theoretic models of the theory classified by $\mathcal{E}$ it amounts not only to saying that a finitary geometric theory $\mathbb{T}$ has models in $Set$ but that the provability of a sequent in coherent logic relative to $\mathbb{T}$ is equivalent to its validity in all set-theoretical models of $\mathbb{T}$.

Statement

Recall that a point of a topos $\mathcal{E}$ is simply a geometric morphism $p:Set\to\mathcal{E}$ and that $\mathcal{E}$ is said to have enough points when for any two distinct parallel $f,g:A\to B$ in $\mathcal{E}$ there is a point $p$ that separates $f$ and $g$: $p*(f)\neq p*(g)$.

Theorem

A coherent topos has enough points.

As a corollary of the Deligne-Lurie completeness theorem this appears as (Lurie SpecSchm, corollary 4.2).

Remarks

Since a Grothendieck topos $\mathcal{E}$ has enough points iff it has a sufficient set of points in the sense that there is a surjection $Set/K\to \mathcal{E}$ with $K$ a set (cf. Johnstone 1977, pp.224-229) and, furthermore, $Set/K\cong Sh(2^K)$, one sees that Deligne’s theorem yields a special form $Sh(2^K)\to\mathcal{E}$ of Barr's theorem.

A general Grothendieck topos may fail to have enough points or even fail to have points at all, but it nevertheless has ‘enough Boolean-valued points’ (cf. Barr's theorem).

Related entries

• Deligne-Lurie completeness theorem

• Barr's theorem

• coherent logic

• geometric theory

• coherent topos

• compactness theorem

References

• M. Artin, A. Grothendieck, J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4. II , LNM 270 Springer Heidelberg 1972.

• B. Frot, Gödel’s Completeness Theorem and Deligne’s Theorem , arXiv:1309.0389 (2013). (pdf)

• P. T. Johnstone, Topos Theory , Academic Press New York 1977 (Dover reprint 2014). (ch. 7)

• F. W. Lawvere, Continuously Variable Sets: Algebraic Geometry= Geometric Logic , pp.135-156 in Proc. Logic Colloquium Bristol 1973, North-Holland Amsterdam 1975.

• S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (sec. IX.11, pp.521f)

• G. E. Reyes, Sheaves and concepts: A model-theoretic interpretation of Grothendieck topoi , Cah. Top. Diff. Géo. Cat. XVIII no.2 (1977) pp.405-437. (numdam)

An historical analysis and a collection of improvements of Deligne can be found in

• Ivan Di Liberti, Morgan Rogers?, Topoi with enough points [arXiv:2403.15338]

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1. SGA 4, vol. II, exposé VI, p.336. ↩