The category of completely regular T-spaces with its reflexive Hausdorff separation

We use the ultrafilter-convergence axiomatics for topological spaces to motivate in detail a gentle categorical introduction, first to barr s set-basedrelational t-algebras, and then to a somewhat hidden result by burroni, of the category of completely regular topological spaces with its reflexive subcategory of compact compact hausdorff topological spaces, under modest assumptions on the monad on set and on the monad on the monad on the monad on the set.