Extending conceptual completeness via virtual ultracategories

We introduce the notion of virtual ultracategory. From a topological point of view, this notion can be seen as a categorification of relational $β$-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.