From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1933 Path: news.gmane.org!not-for-mail From: Bill Rowan Newsgroups: gmane.science.mathematics.categories Subject: Abelian Topological Groups Date: Mon, 30 Apr 2001 21:37:18 -0700 (PDT) Message-ID: <200105010437.f414bIi14109@transbay.net> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241018213 1272 80.91.229.2 (29 Apr 2009 15:16:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:16:53 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue May 1 19:19:52 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f41LdKl11963 for categories-list; Tue, 1 May 2001 18:39:20 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 1 Original-Lines: 22 Xref: news.gmane.org gmane.science.mathematics.categories:1933 Archived-At: I am attempting to construct the ideal abelian category within which live complete, hausdorff abelian topological groups. The idea is that the quotients of such a group, in the abelian category, would be completions of the group with respect to topologies coarser than the given one. The subobjects would be those topologies. Of course, having a topology as an object in the abelian category means we have to have objects in the category other than abelian groups. Of course I want to know if this has been done before. Also, what other ideas are there about the ideal abelian category containing these groups? Mac Lane felt that compactly-generated spaces formed the ideal base category for topological algebra. I seem to be using the category of complete, hausdorff uniform spaces as a base category. I wrote a paper on (universal) algebras with a compatible uniformity, and got some nice results about the congruence (actually, uniformity) lattices. But, admittedly, algebras with compatible uniformities have drawbacks as a foundation for topological algebra because even something like the complex numbers cannot be formalized as such, the multiplication not being uniformly continuous. Bill Rowan