In several familiar subcategories of the category T of topological spaces and continuous maps, embeddings are not pushout-stable. But, an interesting feature, capturable in many categories, namely in categories B of topological spaces, is the following: For M the class of all embeddings, the subclass of all pushout-stable M-morphisms (that is, of those M-morphisms whose pushout along an arbitrary morphism always belongs to M) is of the form AInj for some space A, where AInj consists of all morphisms m:X --> Y such that the map Hom(m, A): Hom(Y, A) --> Hom(X, A) is surjective. We study this phenomenon. We show that, under mild assumptions, the reflective hull of such a space A is the smallest M-reflective subcategory of B; furthermore, the opposite category of this reflective hull is equivalent to a reflective subcategory of the Eilenberg-Moore category SetT, where T is the monad induced by the right adjointHom(-, A): T^op et.We also find conditions on a category under which thepushout-stable -morphisms are of the form^Inj for some category
Mathematics Subject Classification. 18A20 18A40 18B30 18G05 54B30 54C10 54C25.
Keywords. embeddings, injectivity, pushout-stability,(epi)reflective
subcategories of ${\mathbb T}$, closure operator, Eilenberg-Moore
categories.
Comments. This article will be revised and submitted for publication elsewhere.