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

TL;DR

This paper explores the relationship between proximity homomorphisms and frame maps through a comonad, connecting to the concept of stable compactification and revealing new structures in proximity frames.

Contribution

It establishes a correspondence between proximity homomorphisms and frame maps using a comonad, providing a novel categorical perspective on proximity frames.

Findings

Proximity frames form a Kleisli category of a specific comonad.

The frame of round ideals carries two natural proximities inducing two comonads.

The construction relates to the known concept of stable compactification.

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.