Pushout stability of embeddings, injectivity and categories of algebras

Abstract

In several familiar subcategories of the category $\mathbb{T}$ of topological spaces and continuous maps, embeddings are not pushout-stable. But, an interesting feature, capturable in many categories, namely in categories $\mathcal{B}$ of topological spaces, is the following: For $\mathcal{M}$ the class of all embeddings, the subclass of all pushout-stable $\mathcal{M}$-morphisms (that is, of those $\mathcal{M}$-morphisms whose pushout along an arbitrary morphism always belongs to $\mathcal{M}$) is of the form $A^{Inj}$ for some space $A$, where $A^{Inj}$ consists of all morphisms $m:X\to Y$ such that the map $Hom(m,A):Hom(Y,A)\to Hom(X,A)$ is surjective. We study this phenomenon. We show that, under mild assumptions, the reflective hull of such a space $A$ is the smallest $\mathcal{M}$-reflective subcategory of $\mathcal{B}$; furthermore, the opposite category of this reflective hull is equivalent to a reflective subcategory of the Eilenberg-Moore category $Se{t}^{\mathbb{T}},where${\mathbb T}$isthemonadinducedbytherightadjoint$Hom(-,A): {\mathbb T}^{op} \to Set$.Wealsofindconditionsonacategory$\mathcal{B}$underwhichthepushout-stable$\mathcal{M}$-morphismsareoftheform$\mathcal{A}^{Inj}$forsomecategory$\mathcal{A}$.

14 pages. This article will be revised and submitted for publication elsewhere