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arXiv · 2506.23935

Extending conceptual completeness via virtual ultracategories

Gabriel Saadia
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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.

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Gabriel Saadia. 2025-06-30. Extending conceptual completeness via virtual ultracategories. https://arxiv.org/abs/2506.23935

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