Deligne completeness theorem



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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 𝕋\mathbb{T} has models in SetSet but that the provability of a sequent in coherent logic relative to 𝕋\mathbb{T} is equivalent to its validity in all set-theoretical models of 𝕋\mathbb{T}.


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


A coherent topos has enough points.

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


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/KSet/K\to \mathcal{E} with KK a set (cf. Johnstone 1977, pp.224-229) and, furthermore, Set/KSh(2 K)Set/K\cong Sh(2^K), one sees that Deligne’s theorem yields a special form Sh(2 K)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.

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