From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5498 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Small is beautiful Date: Fri, 08 Jan 2010 13:29:09 +0000 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1263048318 9830 80.91.229.12 (9 Jan 2010 14:45:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 9 Jan 2010 14:45:18 +0000 (UTC) To: Robert Pare , categories@mta.ca Original-X-From: categories@mta.ca Sat Jan 09 15:45:11 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 1NTcYv-0006Kv-R8 for gsmc-categories@m.gmane.org; Sat, 09 Jan 2010 15:45:10 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NTc0U-0003Uc-Rt for categories-list@mta.ca; Sat, 09 Jan 2010 10:09:34 -0400 User-Agent: Thunderbird 2.0.0.23 (Windows/20090812) In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5498 Archived-At: Dear Bob, This reminds me of a distinction that arises topologically. In a discrete space, equality is "OK" in the the sense that the diagonal is an open subspace of the square. This works fine also for point-free spaces, by a result in the Joyal--Tierney monograph. (A space X is discrete iff all finite diagonals X -> X^n are open maps.) That suggests working more generally with (point-free) topological categories: the collections of objects and morphisms are spaces. Then the ones with object equality OK are the small ones, where the spaces of objects and morphisms are discrete, i.e. sets. At first sight this doesn't help us with large categories. But actually we go a long way if we generalize spaces a la Grothendieck. For example, the "topologized version of the class of sets" is then the object classifier S[U], a Grothendieck topos whose points are sets. That may look like a clumsy way of replacing something unbeautiful (the large category of sets) by something even worse. But in fact S[U] can be presented in a way that doesn't presuppose knowledge of all of Set, by using a site on a small category of finite sets, whose objects are the natural numbers and whose morphisms correspond to functions between the finite cardinals. Many large categories, including categories of structures such as Group, Ring etc., can be replaced in this way by topical categories, whose collections of objects and morphisms are toposes and whose domain, codomain etc. functors are geometric morphisms. Topical functors, again, are made from geometric morphisms, which imposes continuity conditions on the functors (e.g. preservation of filtered colimits, analogous to Scott continuity). In effect this revises the notion of "class", replacing formulae in set theory by theories in geometric logic. Example: In the topical category of groups, the (generalized) space of objects is the group classifier S[Gp] while the space of morphisms is S[GpHom], the classifier for pairs of groups with homomorphism between them. Similarly for rings we have S[Rg] and S[RgHom]. The group ring construction is then given by geometric morphisms S[Gp] -> S[Rg] and S[GpHom] -> S[RgHom], satisfying the functoriality conditions. (Actually, in this example the morphism parts are given canonically once we have S[Gp] -> S[Rg].) Best wishes, Steve Vickers. Robert Pare wrote: > ... > 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 > Small is beautiful, not evil. > > Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]