From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8211 Path: news.gmane.org!not-for-mail From: "Oosten, J. van" Newsgroups: gmane.science.mathematics.categories Subject: Re: functors defined by well-founded induction Date: Thu, 10 Jul 2014 14:40:18 +0200 Message-ID: References: Reply-To: "Oosten, J. van" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8"; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1405086026 25245 80.91.229.3 (11 Jul 2014 13:40:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 11 Jul 2014 13:40:26 +0000 (UTC) Cc: categories To: Michael Shulman Original-X-From: majordomo@mlist.mta.ca Fri Jul 11 15:40:19 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1X5b3i-00080b-Cg for gsmc-categories@m.gmane.org; Fri, 11 Jul 2014 15:40:18 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46042) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X5b2u-0004er-1o; Fri, 11 Jul 2014 10:39:28 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X5b2s-0004k2-Rp for categories-list@mlist.mta.ca; Fri, 11 Jul 2014 10:39:26 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8211 Archived-At: Dear Mike, is the following too simple-minded? Given a well-founded poset (X,<), a category C and a function F which, to every functor G from an initial segment of X to C, assigns a cocone for G. Then there is a unique functor H:X-->C with the property that for every x\in X, H(x) is the vertex of the cocone which is F applied to the restriction of H to {y|y Actually, my question is much more basic. > > On Wed, Jul 9, 2014 at 2:39 AM, Paul Taylor wrote: >> The simple answer is that the recursion has to define the functor, >> ie the morphisms corresponding to instances of the order relation, >> and not just the values at individual ordinals, in order to make sense >> of defining the values at limit ordinals as colimits. > That's exactly what I said: > >>> since we have to define the value of the functor on morphisms too, >>> and its value at a given object may depend on its value at morphisms >>> between previous objects. > All I'm looking for is a general theorem of the form "given a > well-founded relation < on a set X, and a category C, and > such-and-such data, there is an induced functor X -> C." I don't care > about set-theoretic issues right now, I'm just looking for a place > where someone has written out exactly how to construct such a functor > using the well-foundedness of <. It seems like it should be a > well-known thing, so that I can just cite it rather than having to > write out my own proof. > > Mike > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]