Raney extensions of frames: topological aspects

Abstract

We explore a pointfree approach to spaces which extends the category of $T_0$ spaces. Our pointfree objects are Raney extensions, pairs $(L,C)$ where $C$ is a coframe, $L\subseteq C$ is a frame which meet-generates it, and the inclusion $L\subseteq C$ preserves the frame operations as well as the strongly exact meets. We show that the category $\mathbf{Raney}$ extends that of $T_0$ spaces, by showing the existence of an adjunction which extends that between frames and spaces. We map a space $X$ to the pair $(\Omega(X),\mathcal{U}(X))$, where $\Omega(X)$ are its opens and $\mathcal{U}(X)$ its saturated sets. The spectrum functor $\mathsf{pt}_R$ maps a Raney extension $(L,C)$ to the collection of completely join-prime elements of $C$, suitably topologized. For a frame $L$ the spectra of the largest and the smallest Raney extensions over it are, respectively, the classical spectrum $\mathsf{pt}(L)$ and the $T_D$ spectrum $\mathsf{pt}_D(L)$. We characterize sobriety as well as the $T_D$ and the $T_1$ axioms for spaces in terms of algebraic properties of their Raney duals. We use this to define sobriety for general Raney extensions, as well as the $T_D$ and $T_1$ properties, and show that a sober coreflection always exists, whereas a $T_D$ reflection exists when we restrict morphisms to exact maps. We show that a frame is subfit if and only if it admits a $T_1$ Raney extension, and that a subfit frame is scattered if and only if it admits a unique Raney extension. We show that the dual adjunction between frames and spaces restricts to a dual adjunction between the category of $T_D$ spaces and the category of $\mathbf{Frm}_{\mathcal{E}}$ of frames and exact maps, and that exact sublocales (sublocales whose surjection is exact) form a subcolocale of the coframe of all sublocales.