From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6722 Path: news.gmane.org!not-for-mail From: Eduardo Dubuc Newsgroups: gmane.science.mathematics.categories Subject: size_question_bisbis Date: Tue, 28 Jun 2011 20:10:58 -0300 Message-ID: Reply-To: Eduardo Dubuc 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: dough.gmane.org 1309353703 24595 80.91.229.12 (29 Jun 2011 13:21:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 29 Jun 2011 13:21:43 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Wed Jun 29 15:21:39 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 1Qbui3-0000ZJ-8Q for gsmc-categories@m.gmane.org; Wed, 29 Jun 2011 15:21:39 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46407) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Qbufz-0002VU-IT; Wed, 29 Jun 2011 10:19:31 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qbufy-0002Zy-P7 for categories-list@mlist.mta.ca; Wed, 29 Jun 2011 10:19:30 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6722 Archived-At: I got a private answer that made me realize I was not clear in the way I put my questions. I send now this mail with my answers, which I hope will clarify the situation. e.d. ****************** Perhaps I was not clear in my question ? It is not belived that finite sets is a small topos with finite limits and all the rest exept the axiom of infinity ? Which category of finite sets ? Of course I know they are all equivalent, but there is not a canonical small one. The answer is not, but people behave as if it were true. More comments below: > I'm replying "in private" to minimize embarrassment. > >> Consider the inclusion S_f C S of finite sets in sets. >> >> Is the category S_f closed under finite limits > > If you mean to understand, as do I, that S_f stands for the > full subcategory of the category S of sets whose objects > are the sets that happen to be finite, then: YES, of course; ... > >> ... and at the same time small ? > > ... and NO, obviously not. > >> For example, there are a proper class of singletons, all finite. Thus a >> proper class of empty limits. > > Quite so. Do you find that objectionable? > >> Question, which is the small category of finite sets ?, which are its >> objects ?. > > I don't understand either the question or its preassumptions. > Why should there be a unique ("the") category of finite sets? > Well, there is a unique category of finite sets, but it is not small, but people works as it it were. Take Joyal's theory of species. > The category S_f of finite sets discussed above isn't small. > But it has lots of equivalent, small subcategories, including > skeletal ones, the best known of which is the full subcategory > of S whose objects are the finite cardinals. Which of these > (if any) is closed under finite limits is an imponderable. Yes, what happens then when people works with a small category of finite sets and with finite limits ? >> A small site with finite limits for a topos would not be closed under >> finite limits ? > > Closed in relation to what ambient setting? The topos. > >> etc etc > > >> But, more basic is the question above: How do you define the small >> category of finite sets ? > > I *don't* define "the" small category of finite sets. > If I need *a* skeletal version of the category of finite sets, > though, I settle for that full subcategory of finite ordinals. What if I need a small category of finite sets with finite limits. Well, using choice I can produce one starting with the finite ordinals (or cardinals). Choosse a limit for each finite diagram, and keep doing this a dennumerable amount of times. >> Or only there are many small categories of finite sets ? > > So I would think :-) . Yes, me too, so this is presisely why I ask the question. > As to what follows here, I either have no answer for it, > or do not even understand the content or relevance of it: Well, I did not explain myself clearly. >> 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 !! > > So if you'd like to amplify a bit, or to explain somewhat, > I'd be grateful. > > PS: I'm suddenly aware that June means WINTER in Argentina, so I hope your > questions are not merely symptomatic of a winter "brain-freeze" Well, it is not "brain freeze" yet ... or it is ? ******************************* [For admin and other information see: http://www.mta.ca/~cat-dist/ ]