A description of HSP-like classes, and applications (Q1065920)

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.