Raney extensions of frames: algebraic aspects

We draw from Raney duality and generalize the notion of canonical extension for distributive lattices to the context of frames. This is a generalization because for distributive lattices we are in the context of coherent spaces, whereas frames represent arbitrary spaces. We introduce Raney extensions of frames, pairs $(L,C)$ where C is a coframe, L⊆ C is a frame that meet-generates it, and the inclusion L⊆ C preserves the frame operations as well as the strongly exact meets. We think of these as being algebraic versions of the embedding Ω(X)⊆ U(X) of a frame of opens into a lattice of saturated sets (upper sets in the specialization order). We show that a frame may have several Raney extensions, and all satisfy a generalization of the properties of density and compactness from the theory of canonical extensions. We show that every frame L has the largest and the smallest Raney extension, and that these are, respectively, the coframe of fitted sublocales Sₒ(L) and the opposite of the frame Sc(L) of joins of closed sublocales. We thus show that these structures have universal properties which are dual of one another. For Raney extensions $(L,C)$ and $(M,D)$, we characterize the frame morphisms f:L→ M which can be extended to a morphism of Raney extensions. We apply this result to obtain a characterization of morphisms of frames f:L→ M which can be lifted to frame morphisms Sc(L)→ Sc(M). We also show that the canonical extension of a locally compact frame is the free Raney extension over it such that it is algebraic (generated by its compact elements). We give a characterization of sobriety and of strict sobriety based on a variation of the compactness property of the extension (Ω(X),U(X)).