Canonical extensions via fitted sublocales
Abstract
We build on a recent result stating that the frame of strongly exact filters for a frame is anti-isomorphic to the coframe of fitted sublocales. The collection of exact filters of is known to be a sublocale of this frame. We consider several other subcollections of : the collections and of intersections of completely prime and Scott-open filters, respectively, and the collection of regular elements of the frame of filters. We show that all of these are sublocales of , and as such they correspond to subcolocales of , which all turn out to have a concise description. By using the theory of polarities of Birkhoff, one can show that all of the structures mentioned above enjoy universal properties which are variations of that of the canonical extension. We also show how some of these subcollections can be described as polarities and give three new equivalent definitions of subfitness in terms of the lattice of filters.
Keywords
Cite
@article{arxiv.2404.18325,
title = {Canonical extensions via fitted sublocales},
author = {Tomáš Jakl and Anna Laura Suarez},
journal= {arXiv preprint arXiv:2404.18325},
year = {2024}
}