# nLab Deligne completeness theorem

Contents

topos theory

model theory

## Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

# Contents

## 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

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).

• 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)

1. SGA 4, vol. II, exposé VI, p.336.