From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1940 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits Date: 02 May 2001 19:10:56 +0200 Message-ID: References: Reply-To: Andrej.Bauer@andrej.com NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018219 1300 80.91.229.2 (29 Apr 2009 15:16:59 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:16:59 +0000 (UTC) To: Category Mailing List Original-X-From: rrosebru@mta.ca Thu May 3 09:33:27 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f43BhpX18409 for categories-list; Thu, 3 May 2001 08:43:51 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: Tobias Schroeder's message of "Wed, 2 May 2001 15:04:00 +0200 (CEST)" User-Agent: Gnus/5.0807 (Gnus v5.8.7) XEmacs/21.1 (Capitol Reef) Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 8 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:1940 Archived-At: Tobias Schroeder writes: > So I'd be very grateful for answers to one of the following: > - 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)? With a little bit of cheating, you can use domain theory to express the limit as a sequence as a _colimit_ in a partially ordered set. Let D be the partial order consisting of all the closed intervals, including singletons [a,a], ordered by reverse inclusion. We can embed R into D by mapping it to the maximal elements a |---> [a,a], and under a suitable topology on D (the Scott topology), this is a topological embedding--purists may want to throw in R as the smallest element to obtain an honest continuous domain. Let x_i be a Cauchy sequence of real numbers. To say that x_i is a Cauchy sequence is to say that there exist numbers d_i such that (1) For j >= i, the interval [x_i - d_i, x_i + d_i] contains [x_j + d_j, x_j + d_j]. (2) The numbers d_i become arbitrarily small: for every k there is i such that for all j >= i, d_i < 1/k. (Exercise for your students: show that this is equivalent to the usual definition of Cauchy sequence.) In terms of the partial order D, (1) says that the intervals [x_i - d_i, x_i + d_i] form an increasing sequence. Every increasing sequence in D has a supremum, because an intersection of a nested sequence of closed intervals is a closed interval, so let [u,v] = sup_i [x_i - d_i, x_i + d_i] By (2), we get that u = v, and we have obtained the limit of the sequence (x_i) as a supremum. Supremums are the _colimits_ in a partial order. If you prefer limits, you can stand on your head. I do not see how to get by without using the _evidence_ that (x_i) is a Cauchy sequence, i.e., the numbers d_i. This is intuitionistic mathematics creeping in, which is just as well. > - Why have people chosen the term "limit" in category theory? > (And, by the way, who has defined it first?) I am way too young to know the answer to this. Andrej