From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5602 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: abstraction of notation from sets. Date: Sat, 27 Feb 2010 23:20:44 +0000 Message-ID: References: Reply-To: Paul Levy NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1267368320 22566 80.91.229.12 (28 Feb 2010 14:45:20 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 28 Feb 2010 14:45:20 +0000 (UTC) To: Richard Garner Original-X-From: categories@mta.ca Sun Feb 28 15:45:14 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1NlkOO-00008D-E8 for gsmc-categories@m.gmane.org; Sun, 28 Feb 2010 15:45:12 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NljiO-0001iN-BN for categories-list@mta.ca; Sun, 28 Feb 2010 10:01:48 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5602 Archived-At: On 26 Feb 2010, at 18:53, Richard Garner wrote: > > I think it is more perspicuous to treat this, not as an > overloading of \in, but as an "implicit conversion" > associated to the notion of category; that is, we allow the > forgetful functor Cat->Set to be applied silently in contexts > which otherwise would not type-check. In fact the vast > majority of "abuses of notation" are of this character, when > applied to, for example, any forgetful functor from an > Eilenberg-Moore category; the discrete category functor > Set->Cat; the Yoneda embedding C -> [C^op, Set]; the > forgetful functor from universal cones to their vertex, etc, > etc. In principle this becomes problematic as soon as the > category generated by all such implicit conversions has > non-identity idempotents; Why restrict this to idempotents? Surely the category needs to be a preorder for the usage to be unambiguous? Paul > in practice, this category is free > on a graph and we hope to identify a shortest path between > two vertices! > > Richard > -- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]