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

TL;DR

This work develops a pointfree analogue of Raney duality by introducing Raney extensions (pairs $(L,C)$ with $C$ a coframe and $L\subseteq C$ a meet-generating subframe that preserves strongly exact meets). It establishes a dual adjunction between $\mathbf{Raney}$ and $\mathbf{Top}$ whose fixpoints are precisely the $T_0$ spaces, via the functors $X\mapsto (\\Omega(X),\\mathcal{U}(X))$ and the corresponding spectrum construction. The authors then analyze the spectra of Raney extensions, showing a symmetry between sublocales and fitted sublocales, with the spectra ranging between $\mathsf{pt}(L)$ and its $T_D$-spectrum $\mathsf{pt}_D(L)$. They characterize sobriety, the $T_D$ and $T_1$ axioms, and scatteredness in terms of density and compactness within Raney extensions, and develop free and cofree constructions that extend frame maps to Raney morphisms. The canonical extension of frames emerges as a canonical example: for pre-spatial frames, it coincides with the free algebraic Raney extension, tying canonical extension to a robust, pointfree Raney-theoretic framework and providing new insights into $T_D$-duality and sobriety in a uniform, algebraic setting.

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 $(Ω(X),\mathcal{U}(X))$, 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 $(Ω(X),\mathcal{U}(X))$. 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.

Theorems & Definitions (164)

• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Lemma 2.4
• Lemma 2.5
• Proposition 2.6
• Theorem 2.7
• Theorem 2.8
• Proposition 2.9
• Theorem 2.10: see ball14, Theorem 5.2.3