From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3349 Path: news.gmane.org!not-for-mail From: "Ronnie Brown" Newsgroups: gmane.science.mathematics.categories Subject: Re: ramifications of Goldblatt's notion of a skeleton of a category Date: Mon, 3 Jul 2006 12:07:08 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=response Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019244 8361 80.91.229.2 (29 Apr 2009 15:34:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:04 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Mon Jul 3 08:53:36 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 03 Jul 2006 08:53:36 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1FxMvp-0002xx-7f for categories-list@mta.ca; Mon, 03 Jul 2006 08:49:37 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 79 Xref: news.gmane.org gmane.science.mathematics.categories:3349 Archived-At: Following on from Fred Linton's comment, an easy example is groupoids. For a connected groupoid G, any vertex (object) group G(x)= G(x,x) is skeletal in G. See my book: www.bangor.ac.uk/r.brown/topgpds.html (since I do all the publicity, I have to take every opportunity....!) For many mathematicians, this meant that `groupoids reduced to groups'. But this reduction involves choices, and so cannot be made natural, which takes us back to the first paper on categories by E-M! Heller commented to me in the 1980s that on this reductionist basis, vector spaces reduce to a cardinality. But, as he said, the classification of vector spaces with n endomorphisms is interesting for n=1, hard for n=2, and unknown for n=3. I have not seen a classification of groupoids with one endomorphism! Ronnie ----- Original Message ----- From: "Galchin Vasili" To: Sent: Thursday, June 29, 2006 6:29 AM Subject: categories: ramifications of Goldblatt's notion of a skeleton of a category > Hello, > > Rob Goldblatt in section 9.2 of his book "Topoi: The Categorical > Analysis of > Logic" introduces the notion of a "skeleton of a category C" which he > defines as a "full > subcategory C-sub-zero of C that is skeletal, and such that each C-object > is > isomorphic > to one and only one C-sub-zero object". This statement seems to imply that > we can have an "operator": > > skel: CAT -> CAT where CAT is the categories of (small) categories > > such that > > 1) skel is idempotent on any member of C of CAT, i.e. > ] > skel (skel (C)) = skel (C) > > 2) skel (C) = a "maximal" skeleton of C. > > I am struggling with > > 1) what "maximal" means in this case? E.g. is there some kind of order > on > all the > skeletons of category C? > > 2) would the "operator" skel be a functor? > > > Kind regards, Bill Halchin > > > > > > -- > Internal Virus Database is out-of-date. > Checked by AVG Free Edition. > Version: 7.1.392 / Virus Database: 268.9.0/368 - Release Date: 16/06/2006 >