Strictly zero-dimensional biframes and Raney extensions

Abstract: Raney extensions and strictly zero-dimensional biframes both faithfully extend the dual of the category of $T_0$ spaces. We use tools from pointfree topology to look at the connection between the two. Raney extensions may be equivalently described as pairs $(L,\mathcal{F})$ where $L$ is a frame and $\mathcal{F}\subseteq \mathcal{S}_{o}(L)$ a subcolocale containing all open sublocales. Here, $\mathcal{S}_o(L)$ is the collection of all intersections of open sublocales of $L$. Similarly, a strictly zero-dimensional biframe is a pair $(L,\mathcal{D})$ where $\mathcal{D}\subseteq \mathcal{S}(L)$ is a codense subcolocale. We show that there is an adjunction between certain subcolocales of $\mathcal{S}_o(L)$ and codense subcolocales of $\mathcal{S}(L)$. We show that the adjunction maximally restricts to an order-isomorphism between the subcolocales of $\mathcal{S}_o(L)$ where the joins of open sublocales distribute over binary meets, which we call the proper subcolocales, and what we call the essential codense subcolocales. As an application of our main result, we establish a bijection between proper Raney extensions and the strictly zero-dimensional biframes $(L_1,L_2,L)$ such that $L$ is an essential extension of $L_2$ in the category of frames. We show that this correspondence cannot be made functorial in the obvious way, as a frame morphism $f:L\to M$ may lift to a map $f:(L,\mathcal{F})\to (L,\mathcal{G})$ of Raney extensions without lifting to a map between the associated strictly zero-dimensional biframes.