Sam van Gool: Toposes with enough points as categories of étale spaces
Time: Wed 2025-11-19 10.00 - 12.00
Location: Albano house 1, floor 3, Room U (Kovalevsky)
Participating: Sam van Gool (ENS Paris-Saclay)
Barr (1970) showed that topological spaces correspond to relational modules for the ultrafilter monad. The aim of this talk is to discuss a recent lifting of this result to Grothendieck toposes with enough points. More precisely, we will discuss a duality between such toposes and "categories equipped with relational ultrastructure", a new concept, for which we propose the name "ultraconvergence space".
Our duality extends Makkai's duality between coherent toposes and ultracategories, and our proof generalizes and simplifies Makkai's original proof. In logical terms, we obtain a strong conceptual completeness theorem for geometric theories with enough Set-models.
In order to achieve this, we identify a novel notion of étale space over an ultraconvergence space, and show as our main theorem that any topos with enough points is equivalent to the category of étale spaces over its ultraconvergence space of points.
This talk is based on our recent preprint arXiv:2508.09604, joint with Jérémie Marquès and Umberto Tarantino. The same result has recently been obtained independently by G. Saadia (arXiv:2506.23935) and by A. Hamad (arXiv:2507.07922), via a rather different route.
Our duality extends Makkai's duality between coherent toposes and ultracategories, and our proof generalizes and simplifies Makkai's original proof. In logical terms, we obtain a strong conceptual completeness theorem for geometric theories with enough Set-models.
In order to achieve this, we identify a novel notion of étale space over an ultraconvergence space, and show as our main theorem that any topos with enough points is equivalent to the category of étale spaces over its ultraconvergence space of points.
This talk is based on our recent preprint arXiv:2508.09604, joint with Jérémie Marquès and Umberto Tarantino. The same result has recently been obtained independently by G. Saadia (arXiv:2506.23935) and by A. Hamad (arXiv:2507.07922), via a rather different route.