Canonical extensions via fitted sublocales

TL;DR

This paper explores various subcollections of filters in frames, demonstrating their structure as sublocales and subcolocales, and introduces new characterizations of subfitness using polarities and universal properties.

Contribution

It identifies and characterizes several sublocales of the frame of filters, providing concise descriptions and new equivalent definitions of subfitness through polarities.

Findings

All considered subcollections are sublocales of the frame of strongly exact filters.

These sublocales correspond to subcolocales with concise descriptions.

New equivalent definitions of subfitness are provided using polarities.

Abstract

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