Generalised ultracategories and conceptual completeness of geometric logic

TL;DR

This paper develops the theory of generalised ultracategories, extending ultracategories to include topological spaces and points of toposes, and proves a reconstruction theorem for toposes with enough points.

Contribution

It introduces generalised ultracategories and establishes a conceptual completeness theorem linking toposes with enough points to their ultracategories.

Findings

Topological spaces are examples of generalised ultracategories.

Any topos with enough points can be reconstructed from its ultracategory of points.

The reconstruction uses left ultrafunctors and parallels known results in topos theory.

Abstract

We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in the theory of generalised ultracategories. Another example of these generalised ultracategories is points of toposes. In this paper, we show a conceptual completeness theorem for toposes with enough points, stating that any such topos can be reconstructed from its generalised ultracategory of points. This is done by considering left ultrafunctors from topological spaces to the category of points and paralleling this construction with another known fundamental result in topos theory, namely that any topos with enough points is a colimit of a topological groupoid.