A correspondence between proximity homomorphisms and certain frame maps via a comonad

Abstract: We exhibit the proximity frames and proximity homomorphisms as a Kleisli category of a comonad whose underlying functor takes a proximity frame to its frame of round ideals. This construction is known in the literature as {\em stable compactification} (\cite{BezHar2}). We show that the frame of round ideals naturally carries with it two proximities of interest from which two comonads are induced.