From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5170 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: question Date: Wed, 23 Sep 2009 11:00:29 +0100 (BST) Message-ID: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1253796679 15286 80.91.229.12 (24 Sep 2009 12:51:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 24 Sep 2009 12:51:19 +0000 (UTC) To: John Kennison , categories@mta.ca Original-X-From: categories@mta.ca Thu Sep 24 14:51:08 2009 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 1Mqnmr-0006CZ-0C for gsmc-categories@m.gmane.org; Thu, 24 Sep 2009 14:51:05 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MqnBi-0003WS-Fv for categories-list@mta.ca; Thu, 24 Sep 2009 09:12:42 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5170 Archived-At: I agree with John about the accepted meaning of "replete", but my understanding of "skeletal" (supported by Mac Lane's "Categories for the Working Mathematician") is that it means a category in which every isomorphism class of objects has exactly one member. If I had to find a word for a subcategory which meets every isomorphism class of objects of the ambient category (but possibly in more than one member) I'd call it "representative", or something like that. A skeleton would then be a subcategory which is full, representative and skeletal. Regarding the question of whether such a subcategory is equivalent to the ambient category, I recall that Peter Freyd once showed that each of the following statements is equivalent to the axiom of choice: (a) Every small category has a skeleton. (b) A small category is equivalent to any of its skeletons. (c) Any two skeletons of a given small category are isomorphic. The first equivalence is trivial, but the other two require a bit of ingenuity. I don't think he ever published this. Peter Johnstone On Tue, 22 Sep 2009, John Kennison wrote: > > My understanding of this ancient terminology ids that a replete subcategory= > is one that is closed under the forming of isomorphic copies.A subcategory= > which contains an isomorphic copy of every object in the containing catego= > ry is called skeletal > A subcategory ois both replete and skeletal if and only if it contains all = > objects of the larger category. > > ---John > > On 9/22/09 3:04 AM, "Fred Linton" wrote: > > Jim Stasheff asked, > >> What do you call it when you have one (small) category being a (full) >> subcategory of another, and every object in the big category is >> isomorphic to one in the small category ? ... > > One adjective that *had* been used for such a subcategory > (whether small, or full, or not) was "replete". I'll defer > to others on the question of whether that terminology is > still in use today, or is ... um ... *deprecated* :-) . > > Cheers, -- Fred > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]