The category of finitary biframes as the category of pointfree bispaces

Abstract: The theory of finitary biframes as order-theoretical duals of bitopological spaces is explored. The category of finitary biframes is a coreflective subcategory of that of biframes. Some of the advantages of adopting finitary biframes as a pointfree notion of bispaces are studied. In particular, it is shown that for every finitary biframe there is a biframe which plays a role analogue to that of the assembly in the theory of frames: for every finitary biframe $\mathcal{L}$ there is a finitary biframe $\mathsf{A}(\mathcal{L})$ with a universal property analogous to that of the assembly of a frame; and such that its main component is isomorphic to the ordered collection of finitary quotients of $\mathcal{L}$ (i.e. its pointfree bisubspaces). Furthermore, in the finitary biframe duality the bispace associated with $\mathsf{A}(\mathcal{L})$ is a natural bitopological analogue of the Skula space of the bispace associated with $\mathcal{L}$. The finitary biframe duality gives us a notion of bisobriety which is weaker than pairwise Hausdorffness, incomparable with the pairwise $T_1$ axiom, and stronger than the pairwise $T_0$ axiom. The notion of pairwise $T_D$ bispaces is introduced, as a natural point-set generalization of the classical $T_D$ axiom. It is shown that in the finitary biframe duality this axiom plays a role analogous to that of the classical $T_D$ axiom for the frame duality.