From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5483 Path: news.gmane.org!not-for-mail From: John Power Newsgroups: gmane.science.mathematics.categories Subject: re: Small is beautiful Date: Wed, 06 Jan 2010 06:53:47 +0000 Message-ID: References: Reply-To: John Power NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262833895 32017 80.91.229.12 (7 Jan 2010 03:11:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 7 Jan 2010 03:11:35 +0000 (UTC) Cc: categories@mta.ca To: Robert Pare Original-X-From: categories@mta.ca Thu Jan 07 04:11:28 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 1NSimV-00061B-8t for gsmc-categories@m.gmane.org; Thu, 07 Jan 2010 04:11:27 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NSiCe-00002S-Px for categories-list@mta.ca; Wed, 06 Jan 2010 22:34:24 -0400 In-Reply-To: Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5483 Archived-At: Dear Colleagues, I have not quite absorbed all the email on this yet, so may be =20 repeating something already said. But perhaps it would be helpful to =20 mention that, in regard to questions like this, I have found enriched =20 categories helpful: consider either 1 the functor category [->,Set] (an object is a pair of sets X and Y =20 and a function from X to Y) or 2 the category Sub(Set) (an object is a set X together with a subset =20 X', and a map from (X,X') to (Y,Y') is a function from X to Y for =20 which the image of X' lies in Y' These categories, especially the first, both have the properties one =20 typically seeks for a V in studying V-categories. Spelling out what a V-category is in the second case yields a category =20 C with a subcategory for which the inclusion is the identity on objects. Happy New Year to all, John. Quoting Robert Pare : > > I would like to add a few thoughts to the "evil" discussion. > > My 30+ years involvement with indexed categories have led me > to the following understanding. There are two kinds of categories, > small and large (surprise!). But the difference is not mainly one > of size. Rather it's how well we can pin down the objects. The > distinction between sets and classes is often thought of in terms > of size but Russell's problem with the set of all sets was not one of > size but rather of the nature of sets. Once you think you have the set > of all sets, you can construct another set which you had missed. > I.e. the notion is changing, slippery. There are set theories where > you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.) > Smallness is more a question of representability: a functor may fail to > be representable because it's too big (no solution set) or, more often, > because it's badly behaved (doesn't preserve products, say). Subfunctors > of representables are not usually representable. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]