PreprintCC BY 4.0arXiv
Geometric morphisms of virtual ultracategories
Abstract
We build on the recent development that toposes with enough points can be represented as categories of points equipped with ultraconvergence data (a.k.a. virtual ultracategories), characterizing usual classes of geometric morphisms --- surjections, embeddings, hyperconnected maps, and localic maps --- in terms of their points. In particular, the characterisation of geometric surjections relies on a notion of "virtual ultraretractions", which generalises usual retractions and filtered colimits, categorifies Johnstone's notion of subclosure for topological spaces, and provides a new characterisation of separating classes of points of a topos.
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BibTeX
@misc{gabriel2026geometric,
title = {{Geometric morphisms of virtual ultracategories}},
author = {Saadia Gabriel and Yuksel Errol},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31445},
eprint = {2609.31445},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31445},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric morphisms of virtual ultracategories
AU - Saadia Gabriel
AU - Yuksel Errol
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31445
UR - https://arxiv.org/abs/2609.31445
AB - We build on the recent development that toposes with enough points can be represented as categories of points equipped with ultraconvergence data (a.k.a. virtual ultracategories), characterizing usual classes of geometric morphisms --- surjections, embeddings, hyperconnected maps, and localic maps --- in terms of their points. In particular, the characterisation of geometric surjections relies on a notion of "virtual ultraretractions", which generalises usual retractions and filtered colimits, categorifies Johnstone's notion of subclosure for topological spaces, and provides a new characterisation of separating classes of points of a topos.
ER -