From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2458 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Categories of elements Date: Fri, 3 Oct 2003 14:20:05 +1000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Trace: ger.gmane.org 1241018674 4244 80.91.229.2 (29 Apr 2009 15:24:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:34 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Oct 3 15:33:33 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Oct 2003 15:33:33 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A5Ueu-00013M-00 for categories-list@mta.ca; Fri, 03 Oct 2003 15:28:08 -0300 X-Sender: street@icsmail.ics.mq.edu.au Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 8 Original-Lines: 26 Xref: news.gmane.org gmane.science.mathematics.categories:2458 Archived-At: > "An element of a functor is an attaching functor into the category of > elements of the functor," is unacceptably confusing due to the fact that > the category of elements of a functor does not in any sense consist of > the elements of the functor (as you would describe them). I'm not sure where the above quote is taken from but I agree it is confusing. Here is my argument in favour of the traditional name. As Bill says, an element of an object F in a category is generally any morphism A --> F into F. It just happens that in many categories F is determined by elements with a restricted class of domains A. In Set, we can restrict A to be terminal. In a presheaf category, we can restrict A to be representable. The objects of the category elF of elements of F are (up to isomorphism) elements A --> F with A representable. It is also conventional to name categories after their objects (although the Ehresmann convention of naming them after their morphisms is more precise). Hence elF is the category of elements of F. Regards, Ross