From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1815 Path: news.gmane.org!not-for-mail From: S.J.Vickers@open.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: RE: Why binary products are ordered Date: Tue, 30 Jan 2001 16:43:32 -0000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" X-Trace: ger.gmane.org 1241018122 609 80.91.229.2 (29 Apr 2009 15:15:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:15:22 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Jan 30 20:42:04 2001 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f0UNpZZ12425 for categories-list; Tue, 30 Jan 2001 19:51:35 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Internet Mail Service (5.5.2650.21) Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 46 Original-Lines: 66 Xref: news.gmane.org gmane.science.mathematics.categories:1815 Archived-At: > One might say that the ordering of binary products, with a first projection > and a second projection, is spurious but inevitable. > > The two components of a binary product must be distinguished, as Colin > McLarty explained, but they must be allowed to be isomorphic. The usual > way we handle a situation like this in mathematics is to index them, in > this case by a two-element set. ... Two points. First, even in computer programming, one can lose the need for an ordering by using what are often called "record types", so that denotes the same record as I don't think there's anything mysterious about this. A pair of sets can be described as a function f: X -> 2, and we can quite happily replace 2 by any isomorphic set (but we don't have to choose the isomorphism) such as M = {"your height in inches", "your age in years"} The product of f: X -> M is then a universal solution to the problem of finding g: YxM -> X over M and this is equivalent to the usual characterization once you have chosen the isomorphism between M and 2. A second, and deeper, point is that constructively there are unorderable 2-element sets, so there is a kind of binary product in which the ordering first vs. second projection is impossible. It uses the same "record type" construction. An example in sheaves over the circle O is the twisted double cover M (edge of a Mobius band). It is finite decidable set with cardinality 2. It is isomorphic to 2 (i.e. 2xO) locally but not globally. It has no global elements and no global total ordering. If you have a sheaf X with a map f: X -> M, then locally it falls into two parts whose product you can take. It can be expressed as Pi f = {(i,x,j,y) in MxXxX | f(x) = i and f(y) = j and j = s(i)} / ~ where s: M -> M swaps the two elements and ~ is the equivalence relation generated by (i,x,j,y) ~ (j,y,i,x). Globally, Pi f is the equalizer of two maps from X^M to M^M, namely f^M and the constant identity map: so set theoretically it is the set of sections, {g: M -> X | g;f = Id_M} The universal property is that for any YxM -> X over M you get a unique corresponding Y -> Pi f. The second description with X^M probably looks more natural to a topos theorist, but the first one has the advantage of being geometric. Steve.