From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5595 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: abstraction of notation from sets. Date: Thu, 25 Feb 2010 12:26:26 -0600 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1267208861 32334 80.91.229.12 (26 Feb 2010 18:27:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 26 Feb 2010 18:27:41 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Fri Feb 26 19:27:37 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 1Nl4uW-0006nd-EA for gsmc-categories@m.gmane.org; Fri, 26 Feb 2010 19:27:36 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Nl4KC-0001ih-Fm for categories-list@mta.ca; Fri, 26 Feb 2010 13:50:04 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5595 Archived-At: shaw.ca> writes: > When S is a set, the notation "a \epsilon S" is familiar. > Is this ever extended to CT? All the texts I recall use > natural language such as "A is an object of C". What if > a more symbolic notation is required? As has been pointed out, if category theory is formalized within set theory, so that a category has a set of objects and a set of arrows, then 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. I regard this as precisely analogous to writing "g \in G" when G is a group, since after all when formalized within set theory, a group is not just the set of its elements, but a triple (G,m,e) of a set, a multiplication, and an identity (or some other equivalent encoding). 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. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]