Revisiting Stone duality for bitopological spaces

Exactly one year ago, I presented a form of Stone duality for bitopological spaces due to Jung and Moshier, and refined by Jakl. This relied on a notion called d-frames, which are a pair of frames linked by a totality and a consistency relation. There are natural relations we would like to be able to express that apparently cannot be stated in the setting of d-frames, and I will show that replacing totality and consistency by a unique relation ⊢ cures this problem. This will give rise to a notion that I have decided to call 2-frames (d-frames and biframes already exist), and we will obtain an expected adjunction between the category of bitopological spaces and the opposite of the category of 2-frames, which is idempotent, and leads to a notion of 2-sobrification of bitopological spaces, and of 2-sober bitopological spaces. I will end this by giving a description of the free 2-frame on a pair of sets. Read the full post.