From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6643 Path: news.gmane.org!not-for-mail From: Aleks Kissinger Newsgroups: gmane.science.mathematics.categories Subject: Functors and limits Date: Tue, 26 Apr 2011 16:15:54 +0100 Message-ID: Reply-To: Aleks Kissinger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1303862053 12152 80.91.229.12 (26 Apr 2011 23:54:13 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 26 Apr 2011 23:54:13 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Wed Apr 27 01:54:07 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 1QEs51-0006mY-25 for gsmc-categories@m.gmane.org; Wed, 27 Apr 2011 01:54:07 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58655) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QEs2j-0001Hl-4s; Tue, 26 Apr 2011 20:51:45 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QEs2h-0006ig-0T for categories-list@mlist.mta.ca; Tue, 26 Apr 2011 20:51:43 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6643 Archived-At: 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') = k. For all limiting cones k in D, F.... 1. reflects limits if F*(k) != {} implies F*(k) contains a limiting cone 2. lifts limits if F*(k) contains a limiting cone 3. lifts limits uniquely if F*(k) contains exactly 1 limiting cone, but possibly other cones 4. creates limits if F*(k) = {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/ ]