From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6724 Path: news.gmane.org!not-for-mail From: James Lipton Newsgroups: gmane.science.mathematics.categories Subject: Re: size_question Date: Tue, 28 Jun 2011 22:43:53 -0400 Message-ID: References: Reply-To: James Lipton NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1309353778 25092 80.91.229.12 (29 Jun 2011 13:22:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 29 Jun 2011 13:22:58 +0000 (UTC) Cc: Categories To: Eduardo Dubuc Original-X-From: majordomo@mlist.mta.ca Wed Jun 29 15:22:53 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QbujE-00019E-83 for gsmc-categories@m.gmane.org; Wed, 29 Jun 2011 15:22:52 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46419) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Qbuhf-0002g2-GM; Wed, 29 Jun 2011 10:21:15 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qbuhe-0002c4-OV for categories-list@mlist.mta.ca; Wed, 29 Jun 2011 10:21:14 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6724 Archived-At: Eduardo: For a small cat of finite sets: Why not use the Von Neumann hierarchy (up to omega) for objects, and all set functions as arrows. These are the "hereditarily finite sets" V(0) = empty set V(n+1) = P(V(n)) V(omega) = U{V(n): n in omega} I would not call it "the" cat of finite sets, since there uncountably many countable models of ZF. Obviously the choice of ZF, rather than say Zermelo set theory or some other foundation is also pretty arbitrary. Best, J. Lipton On Tue, Jun 28, 2011 at 1:39 PM, Eduardo Dubuc wrote: > This is a naive question on non naive foundations. > > Consider the inclusion S_f C S of finite sets in sets. > > Is the category S_f closed under finite limits and at the same time small ? > > For example, there are a proper class of singletons, all finite. Thus a > proper class of empty limits. > > Question, which is the small category of finite sets ?, which are its > objects ?. > > A small site with finite limits for a topos would not be closed under > finite limits ? > > etc etc > > But, more basic is the question above: How do you define the small > category of finite sets ? > > Or only there are many small categories of finite sets ? > > You can not define a finite limit as being any universal cone because > then you get a large category. > > Then how do you determine a small category with finite limits without > choosing (vade retro !!) some of them. And if you choose, which ones ? > > The esqueleton is small but a different question !! > > e.d. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]