A categorical review of complete regularity

TL;DR

This paper provides a categorical framework for understanding complete regularity in topological spaces using ultrafilter convergence, T-algebras, and T-spaces, connecting classical notions with modern category theory.

Contribution

It introduces a categorical approach to complete regularity via T-spaces and establishes a fibrational characterization of completely regular T-spaces and compact Hausdorff T-spaces.

Findings

Categorical framework for complete regularity using T-spaces.

Fibrational characterization of completely regular T-spaces.

Connection between classical topology and category theory.

Abstract

We use the ultrafilter-convergence axiomatics for topological spaces to motivate in detail a gentle categorical introduction, first to Barr's Set-based relational T-algebras, and then to Burroni's T-preorders internal to a category C, here called T-spaces in C, for a monad T on C that substitutes the ultrafilter monad on Set. Within these settings one finds not only the notions of compactness and Hausdorff separation, originally due to Manes, but also that of complete regularity. Based on a somewhat hidden result by Burroni, the main theorem of this paper establishes an external fibrational characterization of the category of completely regular T-spaces with its reflexive subcategory of compact Hausdorff T-spaces, under modest assumptions on C and T.