From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1941 Path: news.gmane.org!not-for-mail From: Peter Freyd Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits Date: Wed, 2 May 2001 13:02:36 -0400 (EDT) Message-ID: <200105021702.f42H2aT18744@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241018220 1306 80.91.229.2 (29 Apr 2009 15:17:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:17:00 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu May 3 09:36:02 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f43BgwE31822 for categories-list; Thu, 3 May 2001 08:42:58 -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: 9 Original-Lines: 38 Xref: news.gmane.org gmane.science.mathematics.categories:1941 Archived-At: Tobias Schroeder asks: - 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)? A good question. I have no answer, only a similar (and ancient) question: is there a setting in which adjoint operators on Hilbert spaces can be seen to be examples of adjoint functors between categories? As for his second question: - Why have people chosen the term "limit" in category theory? (And, by the way, who has defined it first?) In the beginning, the only diagrams that had limits were "nets", that is, diagrams based on directed posets. I believe it was Norman Steenrod in his dissertation who first used the term. Before his dissertation the Cech cohomology of a space was defined only as the numberical invarients that arose as a limit of a directed set of such invariants. It was Steenrod who perceived that Cech cohomology could be defined as an abelian group. For that he needed to invent the notion of a limit of a directed diagram of groups. In the 50s the fact that one didn't need the diagram to be directed was considered startling. At least two of us tried to avoid the word "limit" in this more general setting. Jim Lambek was pushing "inf" and "sup", a suggestion I wish I had heard. Not having heard it, I was pushing "left root" and "right root" (one was, after all, supplying a root to a generalized tree. sort of). All to no avail. So now we have "finite limits" and "finitely continuous".