From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1944 Path: news.gmane.org!not-for-mail From: Dusko Pavlovic Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits Date: Thu, 03 May 2001 16:15:02 -0700 Message-ID: <3AF1E676.918BC691@kestrel.edu> References: <15089.14134.867261.350412@henry.cs.bham.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018222 1329 80.91.229.2 (29 Apr 2009 15:17:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:17:02 +0000 (UTC) To: Category Mailing List Original-X-From: rrosebru@mta.ca Fri May 4 09:11:11 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f44BWnH32431 for categories-list; Fri, 4 May 2001 08:32:49 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 4.72 [en] (X11; U; SunOS 5.5.1 sun4u) X-Accept-Language: en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 12 Original-Lines: 23 Xref: news.gmane.org gmane.science.mathematics.categories:1944 Archived-At: > Tobias Schroeder writes: > > - Can the limit of a sequence of real numbers be expressed > > as a categorical limit (of course it can if the sequence is > > monotone, but what if it is not)? > > I think I have an answer to this question (without cheating). It may > be well known or wrong (I haven't carefully checked the details, but I > believe that they are correct). the view of (quasi)metric spaces as R+-categories with the hom-objects d(x,y) goes back to lawvere's "metric spaces, generalized logic and closed categories" from 1973. the cauchy completeness of a space was identified with what came to be known as the cauchy completeness of the corresponding category (see kelly's book on enriched categories, or francis borceux handbook). and cauchy completeness of a category amounts to the existence of certain absolute (co)limits: eg over Set, Ab, Cat... to splitting idempotents (which can be done by taking (co)equalizers with id). -- dusko