Projective objects in categories of locally convex spaces
type: Item
instance of: scholarly article
title: Projective objects in categories of locally convex spaces
zbMATH Open document ID: 0628.46069
DOI: 10.1007/BF01647634
author: S. M. Berger
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: The aim of this paper is to determine the projective objects in the prevariety of locally convex spaces (i.e. a subcategory of the category of locally convex spaces, closed under passage to subspaces and products). Let the categories of (1) Tychonoff spaces (with continuous mappings as morphisms) and (2) Hausdorff locally convex spaces (with linear continuous maps as morphisms) be denoted by (1) HW and (2) HLC. PR(K) denotes the class of projective objects. The subcategory K will be assumed to be complete and saturated. Then the main result of this note is the following: Theorem (a) Let K be a prevariety in HLC. Then \(X\in PR(K)\) iff \(X=F(K,Z)\), where Z is a discrete space and F(K,Z) is the free locally convex space with Hamel basis Z such that injection of Z into F(K,Z) is a homeomorphism. (b) Let \(W_ 1\) be a subcategory of HW such that \(Obj(HLC)\subseteq Obj(W_ j)\). Then \(X\in PR(W_ 1)\) iff X is discrete.
Mathematics Subject Classification ID: 46M10
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 4021943
zbMATH Keywords: reflective subcategories
zbMATH Keywords: projective objects in the prevariety of locally convex spaces
reviewed by: T. K. Mukherjee
MaRDI profile type: Publication
cites work: Homological methods in the theory of locally convex spaces
cites work: Varieties of topological groups a survey
cites work: Varieties of Linear Topological Spaces
cites work: Q5650839
full work available at URL: https://doi.org/10.1007/bf01647634
OpenAlex ID: W1978956282
description: scientific article; zbMATH DE number 4021943