Toposes with enough points as categories of étale spaces

Barr [1] showed that topological spaces correspond to relational modules for the ultrafilter monad $\beta$ on $\mathrm{𝖲𝖾𝗍}$. The aim of this talk is to discuss a recent lifting of this result to Grothendieck toposes with enough points. More precisely, when $ℰ$ is such a topos, we equip the category $\mathrm{𝗉𝗍}ℰ$ of its points with a relational generalization of Makkai’s ultrastructure [8]. We propose to call the resulting new concept of a ‘category equipped with relational ultrastructure’ an ultraconvergence space. We argue that ultraconvergence spaces are an appropriate categorical generalization of Barr’s relational modules for $\beta$. Once the topological notions of continuous and étale maps are also naturally extended to this setting, our main theorem reads as follows:

Theorem. Let $ℰ$ be a Grothendieck topos with enough points. Then

$$ℰ\simeq \mathrm{𝐂𝐨𝐧𝐭}(\mathrm{𝗉𝗍}ℰ,\mathrm{𝖲𝖾𝗍})\simeq \mathrm{𝐄𝐭𝐚𝐥𝐞}(\mathrm{𝗉𝗍}ℰ).$$

That is, $ℰ$ is equivalent to the topos of continuous maps from $\mathrm{𝗉𝗍}ℰ$ to the ultraconvergence space $\mathrm{𝖲𝖾𝗍}$, and to the topos of étale ultraconvergence spaces over $\mathrm{𝗉𝗍}ℰ$.

Our work extends Makkai’s duality [8] between coherent toposes and ultracategories, and our proof simultaneously generalizes and simplifies Makkai’s original proof. In logical terms, our theorem is a strong conceptual completeness theorem for geometric theories with enough models in $\mathrm{𝖲𝖾𝗍}$.

This talk is based on our recent preprint [5]. The same result has recently been obtained independently by both G. Saadia [9] and A. Hamad [6], albeit via a rather different route. Notably, the proofs in [9, 6] both make crucial use of a classical representation theorem for toposes via topological groupoids [7, 2]. Our proof of the theorem, on the other hand, is obtained without relying on any such result. Time permitting, we will also touch on connections to ionad theory [3, 4], and describe some future directions to explore.