International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3387-3404
doi:10.1155/IJMMS.2005.3387

The envelope of a subcategory in topology and group theory

Ahmed Ayache1 and Othman Echi2

1Department of Mathematics, College of Sciences, University of Bahrain, P.O. Box 32038, Isa Town, Bahrain
 2Department of Mathematics, Faculty of Sciences, Tunis-El Manar University, Campus Universitaire, Tunis 2092, Tunisia

Abstract

A collection of results are presented which are loosely centered around the notion of reflective subcategory. For example, it is shown that reflective subcategories are orthogonality classes, that the morphisms orthogonal to a reflective subcategory are precisely the morphisms inverted under the reflector, and that each subcategory has a largest “envelope” in the ambient category in which it is reflective. Moreover, known results concerning the envelopes of the category of sober spaces, spectral spaces, and jacspectral spaces, respectively, are summarized and reproved. Finally, attention is focused on the envelopes of one-object subcategories, and examples are considered in the category of groups.