From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3348 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: ramifications of Goldblatt's notion of a skeleton of a category Date: Sat, 01 Jul 2006 19:14:57 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019243 8354 80.91.229.2 (29 Apr 2009 15:34:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:03 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sun Jul 2 20:03:08 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 02 Jul 2006 20:03:08 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1FxAoH-00078m-B7 for categories-list@mta.ca; Sun, 02 Jul 2006 19:53:01 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 122 Xref: news.gmane.org gmane.science.mathematics.categories:3348 Archived-At: Vasili "Bill" [G|H]alchin, in > ramifications of Goldblatt's notion of a skeleton of a category asks, in connection with > 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-obje= ct is > isomorphic > to one and only one C-sub-zero object" , the following: > 1) what "maximal" means in this case? ...some kind of order on > all the > skeletons of category C? > = > 2) would the "operator" skel be a functor? A category "is skeletal" if for each isomorphism A --> B the objects A and B are the same object (A =3D B). With this in mind, the phrase "and only one" in Golblatt's quoted definition is superfluous, being a consequence of (rather than a condition required for) the definition. Once one has a _choice_ of isomorphism from each object of C to = the (unique) object of a skeleton of C that it's isomorphic to (but it may take the axiom of choice to be assured of such an iso), = any two skeleta of C become isomorphic to each other. There's = no inherent "order" among the various possible skeleta of C. Data making the inclusion into C of any full subcategory that = is a skeleton of C an equivalence of categories IS precisely such a "choice of isomorphism from each object of C to the (unique) object of a skeleton of C that it's isomorphic to" mentioned above. There's little hope of skel becoming a functor without data = of the sort just mentioned. Probably the strong temptation = to "wish" that the full inclusion {SKEL} --> {CAT} (of the full subcategory of skeletal categories among all categories) were an equivalence of categories, or at least the inclusion = of a full reflexive subcategory (with skel as inverse, or at = least, reflection back down), is one to be steer clear of, = if at all possible, in general. Cheers, -- Fred (Linton, and as of today, Emeritus from Wesleyan U. :-) ) [original post follows] > 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-obje= ct is > isomorphic > to one and only one C-sub-zero object". This statement seems to imply t= hat > 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)) =3D skel (C) > = > 2) skel (C) =3D a "maximal" skeleton of C. > = > I am struggling with > = > 1) what "maximal" means in this case? E.g. is there some kind of ord= er on > all the > skeletons of category C? > = > 2) would the "operator" skel be a functor? > = > = > Kind regards, Bill Halchin > = > = > =