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
A^{Inj} for some space A, where A^{Inj}
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 *S*et^{T}, 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.

- Document formats
- AtlasImage (for online previewing)
- LaTeX
`43.4 Kb` requires diagrams.tex
- DVI
`62.8 Kb`
- PostScript
`223.2 Kb`
- gzipped PostScript
`87.8 Kb`
- PDF
`255.1 Kb`
- arXiv
- math.CT/0204140
- Metadata
- Citation
- Reference list in BibTeX

Comments. This article will be revised and submitted for publication elsewhere.

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