Raney extensions: a pointfree theory of T_0 spaces based on canonical extension

Abstract

We introduce a pointfree version of Raney duality. Our objects are \emph{Raney extensions} of frames, pairs $(L,C)$ where $C$ is a coframe and $L\subseteq C$ is a subframe that meet-generates it and whose embedding preserves strongly exact meets. We show that there is a dual adjunction between $\mathbf{Raney}$ and $\mathbf{Top}$, with all $T_0$ spaces as fixpoints, assigning to a space $X$ the pair $(\Omega(X),\mathcal{U}(X))$, with $\mathcal{U}(X)$ are the intersections of open sets. We show that for every Raney extension $(L,C)$ there are subcolocale inclusions $\mathcal{S}_c(L)^{op}\subseteq C\subseteq \mathcal{S}_o(L)$ where these are the opposite of the frame of joins of closed sublocales and the coframe of intersections of open sublocales. We thus exhibit a symmetry between these two well-studied structures in pointfree topology. The spectra of these are, respectively, the classical spectrum $\mathsf{pt}(L)$ of the underlying frame and its $T_D$ spectrum $\mathsf{pt}_D(L)$. This confirms the view advanced in \cite{banaschewskitd} that sobriety and the $T_D$ property are mirror images of each other, and suggests that the symmetry above is a pointfree view of it. All Raney extensions satisfy some variation of the properties \emph{density} and \emph{compactness} from the theory of canonical extensions. We characterize sobriety, the $T_1$, and the $T_D$ axioms in terms of density and compactness of $(\Omega(X),\mathcal{U}(X))$. We characterize frame morphisms $f:L\to M$ that extend to Raney morphisms $\overline{f}:(L,C)\to (M,D)$. We use this result to exhibit the existence of various free and cofree constructions. We use Raney extensions to give a new perspective on canonical extension generalized to frames as well as $T_D$ duality.