PreprintCC BY 4.0arXiv
Categorified convergence: dropping the 'ultra' from 'virtual ultracategories'
Abstract
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.
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BibTeX
@misc{saadia2026categorified,
title = {{Categorified convergence: dropping the 'ultra' from 'virtual ultracategories'}},
author = {Gabriel Saadia and Umberto Tarantino},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08124},
eprint = {2611.08124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08124},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Categorified convergence: dropping the 'ultra' from 'virtual ultracategories'
AU - Gabriel Saadia
AU - Umberto Tarantino
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08124
UR - https://arxiv.org/abs/2611.08124
AB - 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.
ER -