On deductive varieties of locally convex spaces. (English) Zbl 0673.46051

Berger, S. M.

Commentat. Math. Univ. Carol. 29, No. 3, 465-475 (1988).

The variety of locally convex spaces (LCS’s) is said to be deductive if every of its subprevarieties is a subvariety. The paper contains the following category description: the deductive variety V is generated by the spaces \(R_ P(E)\), \(E\in E+\), where P is a suitable prevariety, \(R_ P\) is the reflector of P from the category of all LCS’s and \(E+\) is the class of LCS’s with the strongest l.c. topology in the prevariety P or (just the same) a subclass of the class of projective objects in the prevariety P.
Some consequences are presented.
Two questions are mentioned: are there suitable characterizations of the structure of the deductive varieties for the category of topological vector spaces and for the category of topological groups? Apparently, the second question isn’t substantial in view of the absence of natural examples of deductive varieties of topological groups.
Also, it seems that the coincidence of the following two classes may be comprehended in topos theory only: the class of the generators for the deductive variety and the class of projective objects in some prevariety (the paper doesn’t contain such a consideration).

Reviewer: S.M.Berger

MSC:

46M10	Projective and injective objects in functional analysis
18A40	Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
54B10	Product spaces in general topology

Keywords:

variety of locally convex spaces; subprevarieties; subvariety; deductive variety; prevariety; reflector; projective objects; topos theory