From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5458 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Small is beautiful Date: Sat, 02 Jan 2010 23:57:14 -0800 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1262531166 11482 80.91.229.12 (3 Jan 2010 15:06:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 3 Jan 2010 15:06:06 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sun Jan 03 16:05:59 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 4.50) id 1NRS1j-0000ik-Pc for gsmc-categories@m.gmane.org; Sun, 03 Jan 2010 16:05:55 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NRRls-0000K2-69 for categories-list@mta.ca; Sun, 03 Jan 2010 10:49:32 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5458 Archived-At: Robert Pare wrote: > Then > there are the small categories which are used to study the large ones. > These are syntactic in nature. Don't get me started. Oops, too late. > For these, one can't expect the kinds of > universal constructions that large categories have, Not following. FinSet is an essentially small category, what do you mean that it doesn't enjoy universal constructions? It's even a topos. Then there are the categories enriched in small categories, again subject to cardinality restrictions, which too are perfectly capable of enjoying universal constructions. > but now it's okay, > even necessary, to consider equality between objects. For small as opposed to essentially small categories, yes in some cases. But consider the category of ordinals truncated at say beth_2, certainly a small category when the morphisms are the inequalities. Are you comfortable defining equality on the objects of this category? (PTJ would correctly accuse me of being inconsistent on this point.) > Well, after these ramblings, perhaps my message is lost. So here it is: > Small categories -> equality of objects okay > Large categories -> equality of objects not okay I hate to seem argumentative, Bob, but this can't possibly be the difference between small and large. > Small is beautiful, not evil. Agreed, so long as this is not at the expense of large. Nice to be able to close on a note of consensus. :) Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]