From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5599 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: abstraction of notation from sets. Date: Fri, 26 Feb 2010 18:53:21 +0000 (GMT) Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1267303611 19137 80.91.229.12 (27 Feb 2010 20:46:51 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 27 Feb 2010 20:46:51 +0000 (UTC) To: Michael Shulman , categories@mta.ca Original-X-From: categories@mta.ca Sat Feb 27 21:46:38 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 1NlTYX-0007kL-3L for gsmc-categories@m.gmane.org; Sat, 27 Feb 2010 21:46:33 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NlSzO-0004nQ-A6 for categories-list@mta.ca; Sat, 27 Feb 2010 16:10:14 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5599 Archived-At: > One cannot write "a \in C" to mean that a is an object of > the category C, at least as long as \in is restricted to > its precise set-theoretic meaning. However, in my > experience it is fairly common to write "a \in C" with this > meaning, although perhaps not so common in formal > mathematical writing. > One can refer to this sort of thing perjoratively as an "abuse of > notation," but one can also regard it as a perfectly legitimate part of > informal mathematical language which is not captured by the > set-theoretic encoding. One could also formalize it by regarding the > symbol "\in" as "overloaded" in a precise sense analogous to programming > languages. 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; in practice, this category is free on a graph and we hope to identify a shortest path between two vertices! Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]