English

Canonical extensions via fitted sublocales

Category Theory 2024-04-30 v1 General Topology

Abstract

We build on a recent result stating that the frame SE(L)\mathsf{SE}(L) of strongly exact filters for a frame LL is anti-isomorphic to the coframe So(L)\mathsf{S}_o(L) of fitted sublocales. The collection E(L)\mathsf{E}(L) of exact filters of LL is known to be a sublocale of this frame. We consider several other subcollections of SE(L)\mathsf{SE}(L): the collections J(CP(L))\mathcal{J}(\mathsf{CP}(L)) and J(SO(L))\mathcal{J}(\mathsf{SO}(L)) of intersections of completely prime and Scott-open filters, respectively, and the collection R(L)\mathsf{R}(L) of regular elements of the frame of filters. We show that all of these are sublocales of SE(L)\mathsf{SE}(L), and as such they correspond to subcolocales of So(L)\mathsf{S}_o(L), 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}
}
R2 v1 2026-06-28T16:09:09.162Z