We introduce the concept of ultraconvergence space: a category equipped with additional structure that captures "how objects converge along ultrafilter-indexed families of objects". The main instance of interest is the category of points of any Grothendieck topos, which can be endowed with such an ultraconvergence structure in a canonical way. When viewed through a logical lens, the ultraconvergence space canonically associated to a geometric theory is its category of models, equipped with additional structure that captures "how models map to ultraproducts of models". Our main theorem is a representation result: if a topos has enough points, then it is equivalent to a category of étale spaces over the canonical ultraconvergence space of its points. This theorem simultaneously generalizes both M. Barr's 1970 "Relational algebras" representing topological spaces via the ultrafilter monad, and M. Makkai's 1987 "Stone duality for first-order logic" between coherent toposes and ultracategories.

This is joint work with Jérémie Marquès and Umberto Tarantino, recently appeared in Journal of Pure and Applied Algebra 2026 (https://doi.org/10.1016/j.jpaa.2026.108269), also see https://arxiv.org/abs/2508.09604 .