I wrote in part:
I had some references (monographs) that I liked too, but I can't find them now; perhaps I can find them tomorrow.
I've (re-)found this one: Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989. This has many pretty diagrams of various categories of spaces, all included in one another by bireflections and the like. The whole subject seems messy, as point-set topology can be, (there are too many categories around to keep track of), and probably needs to be cleaned up somewhat before outsiders (including me) can get a grasp on it; in particular, perhaps locales would be a better approach. OTOH, my reference is nearly 20 years old, and perhaps the situation has improved. Indeed, perhaps this is what Paul Ekland's references are doing; I haven't looked at them carefully yet. --Toby