INITIAL COMPLETIONS OF MONOTOPOLOGICAL CATEGORIES,AND CARTESIAN CLOSEDNESS

Abstract

Herrlich and Strecker 9] give examples of monotopological categories for which the MacNeille completion coincides with the universal initial completion. It is shown here that this situation always holds for monotopological categories. If the category is a proper monotopological c-category, then the MacNeille completion also coincides with the largest epi-reflective initial completion. During the course of the proof a lemma is given which characterizes monotopological categories (not necessarily c-categories) which are already topological. (Schwarz 14] gave such a characterization for the case of c-categories.)

It is also shown that a monotopological c-category is Cartesian closed if and only if its largest epi-reflective initial completion is Cartesian closed. A similar result holds for the case of a topological category which is not necessarily a c-category.