From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5906 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Equality again Date: Tue, 1 Jun 2010 10:38:04 -0400 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1275485512 9530 80.91.229.12 (2 Jun 2010 13:31:52 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 2 Jun 2010 13:31:52 +0000 (UTC) To: "Marco Grandis" Original-X-From: categories@mta.ca Wed Jun 02 15:31:51 2010 connect(): No such file or directory 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 1OJo2w-0006Ch-NB for gsmc-categories@m.gmane.org; Wed, 02 Jun 2010 15:31:50 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OJnW1-0005rK-5q for categories-list@mta.ca; Wed, 02 Jun 2010 09:57:49 -0300 Thread-Index: AcsBh3AxPlrLNto4TCyuV4IspYXj4wAC+FsZ Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5906 Archived-At: Dear Marco, We could use of the dotted-equality symbol only when the=20 canonical isomorphism under consideration is part of a contractible network of isomorphisms. The network does not need to be explicitly identified if the context is clear enough.=20 For example, the dotted equality (A times B)times C =3D. A times (B times C) is refering to the associativity constraint. The dotted equality A times B =3D. B times A is refering to the symmetry constraint. But the dotted equality A times A =3D. A times A is ambiguous and should be excluded (actually, it is not ambiguous, since the identity of A times A is denoted A times A =3D A times A ). I am proposing a rule of thumb, not a new formalism. Mathematics is as much an art as it is an exact science. Best, Andr=E9 -------- Message d'origine-------- De: categories@mta.ca de la part de Marco Grandis Date: mar. 01/06/2010 02:36 =C0: Prof. Peter Johnstone; categories@mta.ca Objet : categories: Re: Equality again =20 On 27 May 2010, at 13:30, Prof. Peter Johnstone wrote: > > TeX provides a command \doteq for an equality sign with a dot over it; > this is used in other areas of mathematics to mean "is approximately > equal to", but as far as I know it hasn't yet been used by category- > theorists. Perhaps we could use it to mean "is canonically > isomorphic to"? > > I'd also like to use it (or something like it) between pairs of > morphisms, meaning that (they are not equal but) they become equal > when composed with the appropriate canonical isomorphisms (to which > I can't be bothered to give names) in order to match up their domains > and codomains. (Of course, this is simply saying that they are > canonically isomorphic as objects of the functor category [2,C], > where C is the category in which they live.) > > Peter Johnstone Dear Peter, Isn't this very dangerous? 1. First, I think you are referring to some (specified) *coherent* (contractible) system of isomorphisms, otherwise you can easily prove that 1 =3D - 1 (see an example below). 2. Even in that case, we know that coherence can be a delicate thing. Let us take the cartesian product in Set (or the tensor product in a symmetric monoidal category). Would you write XxY =3D. YxX for the symmetry isomorphism s? Then by XxX =3D. XxX do you mean s or the identity? For XxXxX =3D. XxXxX we have six permutations of variables, generated by sxX and Xxs; and so on. I hope nobody will suggest some complicated trick to account for this; transpositions and permutations are already there, known to everybody; but we have to name them. 3. Coming back to point 1, "canonical" isomorphisms need not be coherent. There are a lot of such situations; I like to refer to the induced isomorphisms in homological algebra, because much of my early work was linked with that. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]