A description of HSP-like classes, and applications

Anthony Wood Hager

Vol. 125 (1986), No. 1, 93–102

DOI: 10.2140/pjm.1986.125.93

Abstract

First, in a concrete category, an HS’P-cIass (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 ∈ inj E means ∀e ∈ E, ∀φ ∈ Hom(domain(e),A), ∃φ ∈ Hom(codomain(e),A) with φe = φ. 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 ∈ 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.

Mathematical Subject Classification 2000

Primary: 18A32

Secondary: 06F15, 54D30

Milestones

Received: 9 July 1984

Revised: 10 January 1985

Published: 1 November 1986