Path: news.gmane.org!not-for-mail From: Aleks Kissinger Newsgroups: gmane.science.mathematics.categories Subject: Re: Functors and limits Date: Tue, 26 Apr 2011 22:12:36 +0100 Lines: 38 Approved: news@gmane.org Message-ID: References: Reply-To: Aleks Kissinger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1303862067 12218 80.91.229.12 (26 Apr 2011 23:54:27 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 26 Apr 2011 23:54:27 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Wed Apr 27 01:54:22 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QEs5F-0006ug-QR for gsmc-categories@m.gmane.org; Wed, 27 Apr 2011 01:54:21 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58660) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QEs3L-0001Pp-AI; Tue, 26 Apr 2011 20:52:23 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QEs3H-0006jm-Me for categories-list@mlist.mta.ca; Tue, 26 Apr 2011 20:52:20 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6644 Archived-At: Correction: 1. reflects limits if F*(k) contains only limiting cones On 26 April 2011 16:15, Aleks Kissinger wrote: > Dear categorists, > > The "standard" definitions for functors doing nice things with limits > have always seemed a bit clumsy to me. Here's what I think is quite a > natural way to unroll the quantifiers: > > For a functor F : C --> D and a cone k, let F*(k) be the class of all > cones k' in C s.t. F(k') =3D k. > > For all limiting cones k in D, F.... > =A01. reflects limits if F*(k) !=3D {} implies F*(k) contains a limiting = cone > =A02. lifts limits if F*(k) contains a limiting cone > =A03. lifts limits uniquely if F*(k) contains exactly 1 limiting cone, > but possibly other cones > =A04. creates limits if F*(k) =3D {k'}, for k' a limiting cone > > This seems to read much more cleanly than the usual, quantifier-laden > version that seems to be in most standard texts. Of course, they're > all still there in the def, but there is no ambiguity in how they > nest. For example, the difference in 3 in 4 ranges from subtle to > all-but-invisible in most of the places I've seen them defined. Does > this definition, or some close relative exist somewhere? If not, is it > problematic somehow? For example, do you get into trouble when F*(k) > is a proper class? > > Thanks! > Aleks > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]