From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3354 Path: news.gmane.org!not-for-mail From: Bruce Bartlett Newsgroups: gmane.science.mathematics.categories Subject: Re: ramifications of Goldblatt's notion of a skeleton of a category Date: Tue, 04 Jul 2006 17:14:12 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019249 8386 80.91.229.2 (29 Apr 2009 15:34:09 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:09 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Jul 5 08:28:13 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Jul 2006 08:28:13 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Fy5R9-0000u9-9Q for categories-list@mta.ca; Wed, 05 Jul 2006 08:20:55 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 11 Original-Lines: 131 Xref: news.gmane.org gmane.science.mathematics.categories:3354 Archived-At: Fred E.J. Linton wrote: > 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. The situation is actually quite neat. Suppose that, for each category C in Cat, one chooses a skeleton C_0, and isomorphisms from each object to the choice of the skeletal object, as you say. Then, indeed, the operation Skel : Cat -> Cat which sends each category to its chosen skeleton can be made into a (necessarily weak, but thats the interesting part) 2-functor, and can can be extended to an equivalence of 2-categories. Its a special case of a more general notion : if you have a 2-category X, such that every object A in X has an associated object A' and an adjoint equivalence between A and A', then the operation T : X -> X, which (on objects) sends A to A', is in fact a weak 2-functor, and indeed a 2-equivalence. Its all rather nice if you draw it out in string diagrams. Regards, Bruce Bartlett 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-object > 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 = 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-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 > >