arXiv subjects

Gabriel Saadia

Publications and source records attributed to Gabriel Saadia.

3 recordsLinked to original sources

arXiv2026

Categorified convergence: dropping the 'ultra' from 'virtual ultracategories'

Virtual ultracategories are a recent categorification of topological spaces seen as sets of points together with a convergence relation between points and ultrafilters. However, a topology can also be described in terms of convergence of filters: in this paper, we thus introduce an analogue of virtual ultracategories based on filters rather than ultrafilters. At the topological level, convergence of filters is completely determined by its restriction to ultrafilters. Our main result shows that, under the assumption of tautness, the same phenomenon holds at the categorified level, thereby offering a non-ultra-biased perspective on virtual ultracategories.

math.CT↗

arXiv2025

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.

math.CT↗

arXiv2022

Rigidification of cubical quasi-categories

We construct a cubical analogue of the rigidification functor from quasi-categories to simplicial categories present in the work of Joyal and Lurie. We define a functor from the category of cubical sets of Doherty-Kapulkin-Lindsey-Sattler to the category of (small) simplicial categories. We show that this rigidification functor establishes a Quillen equivalence between the Joyal model structure on cubical sets (as it is called by the four authors) and Bergner's model structure on simplicial categories. We follow the approach to rigidification of Dugger and Spivak, adapting their framework of necklaces to the cubical setting.

math.AT↗