From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5510 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: A challenge to all Date: Tue, 12 Jan 2010 11:24:39 -0500 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: ger.gmane.org 1263341117 16260 80.91.229.12 (13 Jan 2010 00:05:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 13 Jan 2010 00:05:17 +0000 (UTC) To: Original-X-From: categories@mta.ca Wed Jan 13 01:05:09 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 4.50) id 1NUqhw-0001I5-9C for gsmc-categories@m.gmane.org; Wed, 13 Jan 2010 01:03:32 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NUq5m-0004Mh-UB for categories-list@mta.ca; Tue, 12 Jan 2010 19:24:07 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5510 Archived-At: Dear All, I cannot imagine a category without an equality relation between the = objects of this category. Ok, I may have been brainwashed by my training in mathematics at an = early age. But more seriously, I think that the equality relation is inseparable=20 from the idea of a set. I do not understand what a preset is: http://ncatlab.org/nlab/show/preset Two things are equal if they are the same, if they coincide (whatever = that mean!). Without this notion, an element of a set has no identity, no = individuality. Of course, a set is often constructed from other sets,=20 as in arithmetic with congruence classes.=20 I am fully aware that the equality relation between the objects of a=20 category is not preserved by equivalences in general. But the art of category theory consists partly in knowing which construction on the objects and arrows of a category is invariant under equivalences.=20 I would like to propose a test for verifying if the=20 notion of category can be freed from the equality relation on its set of objects. The equality relation on an ordinary=20 set S is defined by the diagonal S-->S times S. The objects of a symmetric monoidal category have no diagonal in = general, ie no coalgebra structure. The test: Can we define a notion of category internal to a symmetric monoidal category without using a coalgebra structure on the object of objects? Best, Andr=E9=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]