A description of HSP-like classes, and applications
type: Item
instance of: scholarly article
title: A description of HSP-like classes, and applications
zbMATH Open document ID: 0577.18002
author: Anthony W. Hager
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: First, in a concrete category, an HS'P-class (of objects) is one closed under P : products, S' : some kind of subobjects, H : surjective images. Next, given a class E of morphisms, the object class of ''injectives for E'' is defined: \(A\in inj E\) means \(\forall e\in E\), \(\forall \phi \in Hom(domain(e),A),\exists {\bar \phi}\in Hom(codomain(e),A)\) with \({\bar \phi}\)e\(=\phi\). Then, the ''description'' of the title is, in a concrete category with enough free objects, and well-behaved in other ways: the HS'P-classes are exactly the classes of the form inj E, for just those E which have domain(e) free for each \(e\in E\) (with the meaning of S' and the nature of the maps in E depending on each other). This includes a version of Birkhoff's variety theorem, but more to the present point, is interpreted easily in various specific settings from topology, algebra, and abstract analysis to provide quite concrete descriptions of HSP-like classes.
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08B20
Mathematics Subject Classification ID: 08B30
zbMATH DE Number: 3922925
zbMATH Keywords: injectives
zbMATH Keywords: Birkhoff's variety theorem
zbMATH Keywords: HSP-like classes
MaRDI profile type: Publication
full work available at URL: pjm.1986.125.93
OpenAlex ID: W2077470279
DOI: 10.2140/PJM.1986.125.93
description: scientific article; zbMATH DE number 3922925