Canonical extensions via fitted sublocales

Abstract

We study restrictions of the correspondence between the lattice \(\textsf{SE}(L)\) of strongly exact filters, of a frame L, and the coframe \(\mathcal {S}_o(L)\) of fitted sublocales. In particular, we consider the classes of exact filters \(\textsf{E}(L)\), regular filters \(\textsf{R}(L)\), and the intersections \(\mathcal {J}(\textsf{CP}(L))\) and \(\mathcal {J}(\textsf{SO}(L))\) of completely prime and Scott-open filters, respectively. We show that all these classes of filters are sublocales of \(\textsf{SE}(L)\) and as such correspond to subcolocales of \(\mathcal {S}_o(L)\) with a concise description. The theory of polarities of Birkhoff is central to our investigations. We automatically derive universal properties for the said classes of filters by giving their descriptions in terms of polarities. The obtained universal properties strongly resemble that of the canonical extensions of lattices. We also give new equivalent definitions of subfitness in terms of the lattice of filters.