Pushout Stability of Embeddings, Injectivity and Categories of Algebras

In several familiar subcategories of the category Top of topological spaces and continuous maps, embeddings are not pushoutstable. But, an interesting feature, capturable in many categories, namely in categories B of topological spaces, is the following: ForM 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 smallestM-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 adjoint Hom(−, A) : Topop → Set. We also find conditions on a category B under which the pushout-stable M-morphisms are of the form AInj for some category A.