Extending conceptual completeness via virtual ultracategories

TL;DR

This paper introduces virtual ultracategories, a new categorical concept that generalizes ultracategories and relates to topological and logical structures, extending the framework of conceptual completeness for toposes.

Contribution

It defines virtual ultracategories, explores their properties, and extends Makkai--Lurie's conceptual completeness theorem to toposes with enough points.

Findings

Virtual ultracategories generalize ultracategories and multicategories.

Toposes' points form virtual ultracategories, not just ultracategories.

Topos reconstruction from virtual ultracategories extends existing completeness results.

Abstract

We introduce the notion of virtual ultracategory. From a topological point of view, this notion can be seen as a categorification of relational $\beta$-algebras. From a categorical point of view, virtual ultracategories generalize ultracategories in the same way that multicategories generalize monoidal categories. From a logical point of view, whereas the points of a coherent topos form an ultracategory, the points of an arbitrary topos form a virtual ultracategory. We then extend Makkai--Lurie's conceptual completeness: a topos with enough points can be reconstructed from its virtual ultracategory of points.