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 $Set^{\mathbb T}, where ${\mathbb T}$ is the monad induced by the right adjoint $Hom(-,A): {\mathbb T}^{op} \to Set$. We also find conditions on a category $\mathcal{B}$ under which the pushout-stable $\mathcal{M}$-morphisms are of the form $\mathcal{A}^{Inj}$ for some category $\mathcal{A}$.