Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Gabriel, Saadia, Errol, Yuksel
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214670964