From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2451 Path: news.gmane.org!not-for-mail From: Jpdonaly@aol.com Newsgroups: gmane.science.mathematics.categories Subject: Re: Categories of elements Date: Wed, 1 Oct 2003 01:33:39 EDT Message-ID: <2d.348a2564.2cabc133@aol.com> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018670 4217 80.91.229.2 (29 Apr 2009 15:24:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:30 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Oct 2 11:20:12 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 11:20:12 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A54HU-0001Iy-00 for categories-list@mta.ca; Thu, 02 Oct 2003 11:18:12 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 1 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:2451 Archived-At: Dear Professor Lawvere, Thanks for your clarifications and views in response to my latest note. Coming from an applications-oriented environment, I do assume a set of Zermelo-Fraenkel axioms with a universe of small sets (as prescribed in CWM) in order to ensure access to a fully viable arithmetic of natural transformations. This seems to allow for more than enough categories for my purposes, but it certainly does give the category of small functions a prominence which can feel artificially restrictive at times. Thus I would be especially attentive to any comments which you might make specifically on the functorial isomorphism (I presume to call it a "Lawvere isomorphism" ) which, in converting the Yoneda picture (function-valued natural transformations) of categorical duality into the Lawvere picture (cocompatible functors), represses the category of small functions and, as I do realize, moves things into the context of the general existence theory of adjunctions and Kan extensions, possibly providing a functorial interpretation of your explanation of the origin of comma categories. By now this isomorphism seems to me to be more of a perspicuous relabelling than a redefiner of concepts, so that I have to plead innocent to your apparent conviction that I agonize over the definition of elements. I am in full accord with the doctrine of elements as you have described it, and the Lawvere isomorphism actually relieves some conceptual agony in this regard by smoothly ensuring that, to within a label, the elements of a function-valued functor constitute a (limit) object which is in the functor's codomain category. But I have to restate my belief that the otherwise perfectly redeemable sentence, "An element of a functor is an attaching functor into the category of elements of the functor," is unacceptably confusing due to the fact that the category of elements of a functor does not in any sense consist of the elements of the functor (as you would describe them). So I would rename it. Pat Donaly