Bitopological spaces, d-frames, and Jung-Moshier duality

Stone duality is an adjunction between the category Top of topological spaces and the category Loc of locales, namely the opposite of the category Frm of frames. Is there a similar-looking adjunction between the category biTop of bitopological spaces—namely, sets with two topologies instead of just one—and some category of order-theoretic structures? Well, yes, of course. My preferred one is due to Achim Jung and M. Andrew Moshier, later studied and refined by Tomáš Jakl in his PhD thesis. Read the full post.