Regular and effective regular categories of locales

Abstract

We examine the analogues for the respective categories of locales of two well-known results about regularity and effectiveness of some categories of spaces. We show that the category of compact regular locales is effective regular (=Barr-exact). We also show that the category of compactly generated Hausdorff locales is regular, provided that it is coreflective within Hausdorff locales. We do not appeal to the existence of points (which would render the two results trivial) but rely on the treatment of the subject by methods that are valid in the internal logic of a topos. On the course to the result about compactly generated locales we arrive at a generalization of a result of B. Day and R. Street, deriving regularity for a cocomplete category containing a dense regular subcategory closed under finite limits and colimits and satisfying a certain compatibility condition of pullbacks with appropriate colimits.