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Toposes with enough points as categories of étale spaces

Sam van Gool, Jérémie Marquès, Umberto Tarantino

Abstract

We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be seen as an extension to ionads of Barr's equivalence between topological spaces and relational modules for the ultrafilter monad. In view of the correspondence between toposes and geometric theories, we obtain a strong conceptual completeness theorem, in the sense of Makkai, for geometric theories with enough Set-models. The same result has recently been obtained independently by Saadia (arXiv:2506.23935) and by Hamad (arXiv:2507.07922). Both of their proofs rely on groupoid representations of toposes, which our proof here does not assume.

Toposes with enough points as categories of étale spaces

Abstract

We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be seen as an extension to ionads of Barr's equivalence between topological spaces and relational modules for the ultrafilter monad. In view of the correspondence between toposes and geometric theories, we obtain a strong conceptual completeness theorem, in the sense of Makkai, for geometric theories with enough Set-models. The same result has recently been obtained independently by Saadia (arXiv:2506.23935) and by Hamad (arXiv:2507.07922). Both of their proofs rely on groupoid representations of toposes, which our proof here does not assume.

Paper Structure

This paper contains 26 sections, 20 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. Outline.
  4. Acknowledgements.
  5. Category-theoretic preliminaries
  6. Ultraconvergence spaces
  7. Ultrafilters and ultrafamilies
  8. Ultrafilters.
  9. Ultrafamilies and ultraproducts.
  10. Categories of ultrafamilies.
  11. Ultraconvergence spaces
  12. Definition of ultraconvergence spaces.
  13. Examples of ultraconvergence spaces.
  14. The 2-category of ultraconvergence spaces
  15. Subspaces.
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\mathcal{E}$ be a topos and let $X$ be a separating set of points of $\mathcal{E}$, equipped with the canonical ultraconvergence structure. Then ${\llbracket}{-}{\rrbracket}$ is an equivalence between $\mathcal{E}$ and the category of continuous maps from $X$ to $\mathop{\mathrm{\mathsf{Set}}}\

Theorems & Definitions (71)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 61 more