A general framework for the faithful pointfree representation of $T_0$-spaces

Abstract

We introduce a general framework for studying natural contravariant adjunctions that refine the adjunction between frames and spaces so that the fixpoints are $T_0$-spaces. Our objects of study are \textit{spatializable $\mathbf{Frm}$-concrete categories}, or \textit{SFC-categories}. These consist of a faithful functor $\mathcal O:\mathcal C\to \mathbf{Frm}$ equipped with an object $2_{\mathcal C} \in \mathcal C$, satisfying compatibility conditions that ensure that $(2_{\mathcal C},\mathbb{S})$ forms a dualizing object in the sense of Porst and Tholen, where $\mathbb{S}$ denotes the Sierpiński space. Three important instances of pointfree $T_0$ spaces present in the literature fit into this framework: strictly zero-dimensional biframes, MT-algebras, and Raney extensions. We show SFC-categories are assembled in an ordered category -- a category enriched in preordered sets -- whose morphisms are suitable functors which preserve certain initial liftings. SFC-categories induce natural dual adjunctions, and morphisms between them will respectively induce suitable morphisms between these adjunctions. Motivated by the characterization of sober spaces as maximal objects in the fibers of $Ω:\mathbf{Top}\to \mathbf{Frm}^{\mathsf{op}}$, and of $T_D$-spaces as the minimal ones, due to Banaschewski and Pultr, we study initial and terminal objects of fibers for an arbitrary SFC-category. We prove that the natural adjunction for fiber-initials has exactly the sober spaces as fixpoints, while for fiber-terminals contains at most $T_D$-spaces, recovering their results of in a much more general setting.