From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8207 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: functors defined by well-founded induction Date: Wed, 9 Jul 2014 10:39:31 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1404994218 16548 80.91.229.3 (10 Jul 2014 12:10:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 10 Jul 2014 12:10:18 +0000 (UTC) Cc: categories To: Paul Taylor Original-X-From: majordomo@mlist.mta.ca Thu Jul 10 14:10:12 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 1X5DAv-0004AB-W3 for gsmc-categories@m.gmane.org; Thu, 10 Jul 2014 14:10:10 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45868) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X5DAW-0008P5-5v; Thu, 10 Jul 2014 09:09:44 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X5DAU-0006rm-Gf for categories-list@mlist.mta.ca; Thu, 10 Jul 2014 09:09:42 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8207 Archived-At: 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/ ]