From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5598 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: abstraction of notation from sets. Date: Thu, 25 Feb 2010 11:23:06 -0800 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1267208905 32513 80.91.229.12 (26 Feb 2010 18:28:25 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 26 Feb 2010 18:28:25 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Fri Feb 26 19:28:20 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 1Nl4v6-0007A5-O4 for gsmc-categories@m.gmane.org; Fri, 26 Feb 2010 19:28:12 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Nl4L4-0001lY-DP for categories-list@mta.ca; Fri, 26 Feb 2010 13:50:58 -0400 Content-Disposition: inline In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5598 Archived-At: In a context where "C" is a term for a category and "a" is a new variable, I would interpret "a \in C" to declare that a is an object of C, but I wouldn't be surprised if the author really meant to declare that a is a map in C (with notation for source and target derived). Using scripts can make things more clear, as peasthope@shaw.ca wrote: >"$a \in C_0$" = "a is an object in C" >"$f \in C_1$" = "f is a map in C" This generalises immediately to higher categories. If "a" is not a new variable but instead a term with an established meaning, then "a \in C" might mean either that a is an object or a map of C, but the meaning should be clear from context, particularly if "C" is a term for a subcategory of a category D in which a is an object or map. Even in set theory, there can be a logical difference between declaring that a new variable stands for an element of a given set and stating that a given element belongs to a given subset, and people using structural or type-theoretic approaches to set theory sometimes even use different notation to distinguish these. One possibility (not the only one) is to use a colon for the former; then if "a" is a new variable and "S" is a term for a set, then "a: S" would declare that a is an element of S; while if "a" is a term for an element of some set T and "S" is a term for a subset of T, then "a \in S" would denote the proposition that a belongs to S. The former ("a: S") only makes sense as an assertion or hypothesis, but the latter ("a \in S") makes sense in any logical context. (In particular, it makes sense to say "If a \notin S, then ..." but not "If a: S is false, then ...".) This can be extended to category theory as follows: If "a" is a new variable and "C" is a term for a category, then let "a: C" declare that a is an object of C. If "f" is a new variable and "a" and "b" are terms for objects of C, then let "f: a \to b" declare that f is a map in C from a to b, which after all is already a very widely used notation. (We can also write "f: a \to b: C" to declare everything at once.) This generalises to higher categories, as long as you follow the plan of naming sources and targets before you name whatever goes between them. (If you follow the philosophy of avoiding "evil", then this is very natural.) On the other hand, if D is a category and "C" is a term for a subcategory, then "a \in C" has an unambiguous meaning as long as "a" is a term for an element at some level (object or map) in D; this also generalises to higher categories. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]