Topology Atlas Document # ppae-27


Pushout stability of embeddings, injectivity and categories of algebras

Lurdes Sousa

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 295-308

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.

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Copyright © 2002 Charles University and Topology Atlas. Published April 2002.