From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6761 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: size_question_encore Date: Sun, 10 Jul 2011 19:47:18 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1310398231 14122 80.91.229.12 (11 Jul 2011 15:30:31 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 11 Jul 2011 15:30:31 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Mon Jul 11 17:30:26 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 1QgIRE-00086p-AB for gsmc-categories@m.gmane.org; Mon, 11 Jul 2011 17:30:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42001) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QgIPe-0006Gl-CP; Mon, 11 Jul 2011 12:28:46 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QgIPd-00070c-GM for categories-list@mlist.mta.ca; Mon, 11 Jul 2011 12:28:45 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6761 Archived-At: Even in a category of sets, I don't see why choice is necessary in order to complete a small subcategory under finite limits and obtain a small subcategory. It seems to me that what is needed is rather the axiom of collection, which implies that we can find some *set* of objects containing *at least one* limit for every finite diagram in the original small subcategory; and then we can iterate countably many times to obtain a small category which contains at least one limit for any finite diagram therein. There is of course no canonical result, and the various results obtained will not necessarily be strongly equivalent, but it seems to me that they should all be weakly equivalent. And it also seems to me that the same approach should work internal to any topos. Collection is true internally to any topos (essentially by the internal definition of "indexed family"), so it should still be possible to enlarge a small internal site of definition to one that has finite limits. Unless there is some other subtlety that I'm not seeing. Mike On Tue, Jul 5, 2011 at 4:29 PM, Eduardo Dubuc wrote: > I have now clarified (to myself at least) that there is no canonical > small category of finite sets, but a plethora of them. The canonical one > is large. With choice, they are all equivalent, without choice not. > > When you work with an arbitrary base topos (assume grothendieck) "as if > it were Sets" this may arise problems as they are beautifully > illustrated in Steven Vickers mail. > > In Joyal-Tierney galois theory (memoirs AMS 309) page 60, they say S_f > to be the topos of (cardinal) finite sets, which is an "internal > category" since then they take the exponential S^S_f. Now, in between > parenthesis you see the word "cardinal", which seems to indicate to > which category of finite sets (among all the NON equivalent ones) they > are referring to. > > Now, it is well known the meaning of "cardinal" of a topos ?. > I imagine there are precise definitions, but I need a reference. > > Now, it is often assumed that any small set of generators determine a > small set of generators with finite limits. As before, there is no > canonical small finite limit closure, thus without choice (you have to > choose one limit cone for each finite limit diagram), there is no such a > thing as "the" small finite limit closure. > > Working with an arbitrary base topos, small means internal, thus without > choice it is not clear that a set of generators can be enlarged to have > a set of generators with finite limits (not even with a terminal > object). Unless you add to the topos structure (say in the hypothesis of > Giraud's Theorem) the data of canonical finite limits. > > For example, in Johnstone book (the first, not the elephant) in page 18 > Corollary 0.46 when he proves that there exists a site of definition > with finite limits, in the proof, it appears (between parenthesis) the > word "canonical" with no reference to its meaning. Without that word, > the corollary is false, unless you use choice. With that word, the > corollary is ambiguous, since there is no explanation for the technical > meaning of "canonical". For example, in theorem 0.45 (of which 0.46 is a > corollary), the word does not appear. A topos, is not supposed to have > canonical (whatever this means) finite limits. > > e.d. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]