From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5639 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: equality is beautiful Date: Mon, 15 Mar 2010 04:25:41 -0700 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1268684178 19935 80.91.229.12 (15 Mar 2010 20:16:18 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 15 Mar 2010 20:16:18 +0000 (UTC) Cc: Richard Garner , David Leduc To: categories@mta.ca Original-X-From: categories@mta.ca Mon Mar 15 21:16:13 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1NrGhx-0004c4-1v for gsmc-categories@m.gmane.org; Mon, 15 Mar 2010 21:16:13 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NrG92-0006gv-Pi for categories-list@mta.ca; Mon, 15 Mar 2010 16:40:08 -0300 Content-Disposition: inline In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5639 Archived-At: David Leduc wrote: >What about the characterization of limits in terms of products and >equalizers? It states that the limit of a functor F:J->C is >constructed by products indexed by the set(oid) of objects and the >set(oid) of arrows of J. But if you don't allow equality on objects in >J, you only have a preset of objects, not a set(oid). Consider the analogy between small and strict categories. (A category is strict if it is equipped with a notion of equality of objects. Logically, this is a structure rather than a property like smallness.) Often when speaking of small categories, one speaks relative to a universe which is a collection of set(oid)s or a collection of set(oid) cardinalities, so every small preset automatically comes equipped with a notion of equality. In this case, every small category is strict. Conversely, every strict category is small relative to some universe, if you accept an axiom such as Grothendieck's axiom of universes. So these are very closely related concepts. As is well known, we are most interested in the limit of F: J -> C for small J. Less well known, but also true, we are most interested in it when J is strict. In that case, there is no problem; the arrows of J form a set(oid), and we can consider products indexed by that set(oid). In principle, J doesn't have to be strict, any more than it has to be small, but if you have some reason to believe that the limit exists, then you can examine that reason to see what product is relevant. Most of the time, you can just assume that J is small and strict. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]