Reflexive cum coreflexive subcategories in topology

Abstract

The notions of reflexive and coreflexive subcategories in topology have received much attention in the recent past. (See e.g. Kennison [5], Herrlich [2], Herrlich and Strecker [4], Kannan [6-8].) In this paper we are concerned with the following question and its analogues: Let ~-be the category of all topological spaces with continuous maps as morphisms. Can a proper subcategory of ojbe both reflexive and coreflexive in 9-? The answer turns out to be in the negative. We show further that almost all nice subcategories of 3" have the property that they do not have any proper reflexive cum coreflexive subcategory. Anyhow, examples of subcategories of Y are given which have proper reflexive cum coreflexive subcategories on their own right. The validity of the analogous theorem is discussed in some supercategories of ~-also. An interesting corollary to the proof of Theorem 1 states that a productive intersective divisible topological property (such as e.g., compactness) must fail to be additive. The proof of the main result is based heavily on topological concepts; it is not known how far the result can be extended to arbitrary categories. We start with some preliminary definitions. Let d be a category and be a full subcategory of d . Let X be an object in ~¢. Then Mor(X, ~) denotes the family of all morphisms in d whose domain is X and whose codomain belongs to .~. Similarly we have the symbol Mor(~, X) with an obvious dual meaning. If for each object X of d , there exists an ex e Mor(X, M) through which every element of Mor(X, ~) factors uniquely, then ~ is called a reflexive subcategory of d . In this case e x is called a reflection morphism of X (see Diagram 1). In case this ex is both a monomorphism and an epimorphism for each object X of d , we say that ~ is a simple reflexive subcategory of d . Dually, ~' is said to be a coreflexive subcategory of d if for each object X of d , there exists fx e Mor(~, X) through which every element of Mor (~, X) factors uniquely, f x is called a coreflection morphism of X (see Diagram 2).