HEREDITARY TOPOLOGICAL CATEGORIES AND TOPOLOGICAL UNIVERSES

Abstract

This paper contains an investigation of hereditary topological categories. Useful and illustrative descriptions and pleasant properties of their one-point-extensions are developed and used for a characterization of heredity. The results are applied to the problem to find necessary and sufficient conditions for a bi(co)reflective subcategory of a hereditary topological category to be hereditary. In the reflective case, preservation of certain subobjects by the reflector is sufficient; under the additional assumption of final density it is also necessary. Since a topological category is a topological universe (or quasitopos) iff it is Cartesian closed and hereditary. combination with results on Cartesian closedness yields results on topological universes similar to the ones on heredity.