Toposes with enough points as categories of \'etale spaces

TL;DR

This paper generalizes Makkai duality to toposes with enough points, linking them to ultraconvergence spaces and extending classical topological dualities, resulting in a new conceptual completeness theorem for geometric theories.

Contribution

It extends Makkai duality to a broader class of toposes and simplifies the proof, also connecting to Barr's equivalence and providing a new proof for a conceptual completeness theorem.

Findings

Established duality between toposes with enough points and ultraconvergence spaces.

Simplified proof of Makkai duality.

Proved a conceptual completeness theorem for geometric theories.

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.