From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5648 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: equality is beautiful Date: Sat, 20 Mar 2010 21:17:54 -0500 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1269201029 18544 80.91.229.12 (21 Mar 2010 19:50:29 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 21 Mar 2010 19:50:29 +0000 (UTC) Cc: categories@mta.ca To: Richard Garner Original-X-From: categories@mta.ca Sun Mar 21 20:50:24 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 1NtRAF-0005rJ-LM for gsmc-categories@m.gmane.org; Sun, 21 Mar 2010 20:50:23 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NtQXy-0002ab-Ea for categories-list@mta.ca; Sun, 21 Mar 2010 16:10:50 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5648 Archived-At: That's a good point. However, if C is a non-strict category, then while you can define products over its preset of objects, such a product is no longer necessarily a particular case of a limit, since the preset may not have any "discrete" category structure. So while you can construct limits over arbitrary (non-strict) categories via "products" and equalizers if you generalize the notion of "product" in this way, the converse now fails -- having all limits doesn't seem to guarantee that you have all "products" in this generalized sense. Mike On Tue, Mar 16, 2010 at 3:03 AM, Richard Garner wrote: >> To rephrase what Toby said: the construction of limits via products and >> equalizers only works for limits over a domain category which has a >> set(oid) of objects (what Toby calls a "strict category"), whether that >> set is large or small. > > Is this really the case? Given any type (=preset) A and any > term A --> ob C (for C a non-strict category), one can define > what it means to be a product of this family of objects in C. > Now given a non-strict category J and a functor F:J->C, one > may construct the limit of F as an equaliser of two morphisms > between products in the usual way. I don't see where equality > on objects is necessary, or even useful. > > Richard > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]